Subgroups — C₃×#26400 #10645007

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3083	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3081	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2946	C₂	1	1	yes	maximal, characteristic
4	C₃×#2913	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2093	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2063	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2060	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#1922	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#1907	C₂	1	1	yes	maximal, characteristic
12	C₃×#1883	C₂	1	1	yes	maximal, characteristic
13	C₃×#1866	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃×#1859	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃×#1856	C₂	1	1	yes	maximal, characteristic, complemented
16	#26400	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₂×C₃×#131	C₂²	1	1	yes	characteristic
19	C₃×#544	C₂²	1	1	yes	characteristic
20	C₃×#520	C₂²	1	1	yes	characteristic
21	C₃×#520	C₂²	1	1	yes	characteristic
22	C₃×#518	C₂²	1	1	yes	characteristic
23	C₃×#479	—	2	1	no	—
24	C₃×#478	—	2	1	no	—
25	C₃×#478	—	2	1	no	—
26	C₃×#478	—	2	1	no	—
27	C₃×#477	—	2	1	no	—
28	C₃×#476	—	2	1	no	—
29	C₃×#474	—	2	1	no	—
30	C₃×#474	—	2	1	no	—
31	C₃×#442	—	2	1	no	—
32	C₃×#440	—	2	1	no	—
33	C₃×C₄×SD₁₆	—	2	1	no	—
34	C₃×C₄×D₈	—	2	1	no	—
35	C₂×C₃×#108	C₂²	1	1	yes	characteristic
36	C₃×#428	—	2	1	no	—
37	C₃×#428	—	2	1	no	—
38	C₃×#428	C₂²	1	1	yes	characteristic
39	C₃×#428	C₂²	1	1	yes	characteristic
40	C₃×#428	C₂²	1	1	yes	characteristic
41	C₃×#428	C₂²	1	1	yes	characteristic
42	C₃×#428	C₂²	1	1	yes	characteristic
43	C₃×#427	—	2	1	no	—
44	C₃×#427	C₂²	1	1	yes	characteristic
45	C₂×C₃×#107	—	2	1	no	—
46	C₂×C₃×#107	C₂²	1	1	yes	characteristic
47	C₂×C₃×#107	C₂²	1	1	yes	characteristic
48	C₃×#424	C₂²	1	1	yes	characteristic
49	C₃×#424	C₂²	1	1	yes	characteristic
50	C₂×C₃×#104	C₂²	1	1	yes	characteristic
51	C₃×#418	C₂²	1	1	yes	characteristic
52	C₃×#417	C₂²	1	1	yes	characteristic
53	C₃×#416	C₂²	1	1	yes	characteristic
54	C₂×C₃×#102	C₂²	1	1	yes	characteristic
55	C₂×C₃×#102	C₂²	1	1	yes	characteristic
56	C₂×C₃×#100	C₂²	1	1	yes	characteristic
57	C₃×#405	C₂²	1	1	yes	characteristic
58	C₃×#390	C₂²	1	1	yes	characteristic
59	C₃×#384	C₂²	1	1	yes	characteristic
60	C₃×#343	C₂²	1	1	yes	characteristic
61	C₃×#343	C₂²	1	1	yes	characteristic
62	C₃×#343	C₂²	1	1	yes	characteristic
63	C₃×#341	C₂²	1	1	yes	characteristic
64	C₃×#339	C₂²	1	1	yes	characteristic
65	C₃×#339	C₂²	1	1	yes	characteristic
66	C₃×#339	C₂²	1	1	yes	characteristic
67	C₃×#336	C₂²	1	1	yes	characteristic
68	#3083	C₂×C₃	1	1	yes	characteristic, complemented
69	#3081	C₂×C₃	1	1	yes	characteristic, complemented
70	#2946	C₂×C₃	1	1	yes	characteristic
71	#2913	C₂×C₃	1	1	yes	characteristic, complemented
72	#2093	C₂×C₃	1	1	yes	characteristic, complemented
73	#2089	C₂×C₃	1	1	yes	characteristic, complemented
74	#2063	C₂×C₃	1	1	yes	characteristic, complemented
75	#2060	C₂×C₃	1	1	yes	characteristic, complemented
76	#1952	C₂×C₃	1	1	yes	characteristic, complemented
77	#1922	C₂×C₃	1	1	yes	characteristic, complemented
78	#1907	C₂×C₃	1	1	yes	characteristic
79	#1883	C₂×C₃	1	1	yes	characteristic
80	#1866	C₂×C₃	1	1	yes	characteristic, complemented
81	#1859	C₂×C₃	1	1	yes	characteristic, complemented
82	#1856	C₂×C₃	1	1	yes	characteristic, complemented
83	C₂×C₃×#41	C₂³	1	1	yes	characteristic
84	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
85	C₃×AGL(1,ℤ/8)	—	2	2	no	—
86	C₃×AGL(1,ℤ/8)	Q₈	1	4	yes	complemented
87	C₂×C₃×SD₁₆	—	2	1	no	—
88	C₂×C₃×D₈	—	2	1	no	—
89	C₃×#131	—	2	2	no	—
90	C₃×#131	—	2	2	no	—
91	C₂×C₃×M₁₆	—	2	1	no	—
92	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
93	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
94	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
95	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
96	C₂²×C₃×C₈	—	2	1	no	—
97	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
98	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
99	C₃×C₄⋊Q₈	—	2	1	no	—
100	C₃×C₄⋊Q₈	—	2	1	no	—
101	C₃×#125	—	2	1	no	—
102	C₃×#125	—	2	1	no	—
103	C₃×C₂²⋊Q₈	—	4	1	no	—
104	C₃×C₄×Q₈	—	2	1	no	—
105	C₃×C₄×Q₈	—	2	1	no	—
106	C₃×C₄×D₄	—	2	1	no	—
107	C₃×C₄×D₄	—	2	1	no	—
108	C₃×C₄×D₄	—	2	1	no	—
109	C₃×C₄×D₄	—	2	1	no	—
110	C₃×C₄×D₄	—	2	1	no	—
111	C₃×C₄×D₄	—	2	1	no	—
112	C₃×C₄×D₄	—	4	1	no	—
113	C₃×C₄×D₄	—	4	1	no	—
114	C₃×#117	—	2	1	no	—
115	C₃×#117	—	2	1	no	—
116	C₃×#117	C₂³	1	1	yes	characteristic
117	C₃×#117	C₂³	1	1	yes	characteristic
118	C₃×#117	C₂³	1	1	yes	characteristic
119	C₂×C₃×C₄⋊C₄	—	2	1	no	—
120	C₂×C₃×C₄⋊C₄	—	2	1	no	—
121	C₂×C₃×C₄⋊C₄	—	4	1	no	—
122	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
123	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
124	C₂×C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
125	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
126	C₂×C₃×C₂²⋊C₄	—	4	1	no	—
127	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
128	C₃×#108	—	2	2	no	—
129	C₃×#107	—	2	1	no	—
130	C₃×#107	—	2	1	no	—
131	C₃×#107	—	2	1	no	—
132	C₃×#107	—	2	1	no	—
133	C₃×#107	—	2	1	no	—
134	C₃×#107	—	2	1	no	—
135	C₃×#107	—	2	1	no	—
136	C₃×#107	—	2	1	no	—
137	C₃×#107	—	2	1	no	—
138	C₃×#107	D₄	1	1	yes	characteristic
139	C₃×#107	D₄	1	1	yes	characteristic
140	C₃×#106	—	2	1	no	—
141	C₃×#106	—	2	1	no	—
142	C₃×#106	—	2	1	no	—
143	C₃×#106	—	2	1	no	—
144	C₃×#106	—	2	1	no	—
145	C₃×#106	D₄	1	1	yes	characteristic
146	C₃×#106	D₄	1	1	yes	characteristic
147	C₃×C₄⋊C₈	—	2	1	no	—
148	C₃×C₄⋊C₈	—	2	1	no	—
149	C₃×C₄⋊C₈	—	2	1	no	—
150	C₃×C₄⋊C₈	—	2	1	no	—
151	C₃×#104	—	2	2	no	—
152	C₃×#103	—	2	1	no	—
153	C₃×#103	D₄	1	1	yes	characteristic
154	C₃×#103	D₄	1	1	yes	characteristic
155	C₃×#102	—	2	1	no	—
156	C₃×#102	—	2	1	no	—
157	C₃×#102	—	2	1	no	—
158	C₃×#102	—	2	1	no	—
159	C₃×#102	—	2	1	no	—
160	C₃×#102	—	2	1	no	—
161	C₃×#102	—	2	1	no	—
162	C₃×#102	D₄	1	1	yes	characteristic
163	C₃×#102	D₄	1	1	yes	characteristic
164	C₃×#100	—	2	2	no	—
165	C₃×#97	—	2	1	no	—
166	C₃×C₄×C₈	—	2	1	no	—
167	C₂×AGL(1,ℤ/8)	C₂²×C₃	1	1	yes	characteristic
168	C₂×#131	C₂²×C₃	1	1	yes	characteristic
169	#544	C₂²×C₃	1	1	yes	characteristic
170	#520	C₂²×C₃	1	1	yes	characteristic
171	#520	C₂²×C₃	1	1	yes	characteristic
172	#518	C₂²×C₃	1	1	yes	characteristic
173	#479	—	2	1	no	—
174	#478	—	2	1	no	—
175	#478	—	2	1	no	—
176	#478	—	2	1	no	—
177	#477	—	2	1	no	—
178	#476	—	2	1	no	—
179	#474	—	2	1	no	—
180	#474	—	2	1	no	—
181	#442	—	2	1	no	—
182	#440	—	2	1	no	—
183	C₄×SD₁₆	—	2	1	no	—
184	C₄×D₈	—	2	1	no	—
185	C₂×#108	C₂²×C₃	1	1	yes	characteristic
186	#428	—	2	1	no	—
187	#428	—	2	1	no	—
188	#428	C₂²×C₃	1	1	yes	characteristic
189	#428	C₂²×C₃	1	1	yes	characteristic
190	#428	C₂²×C₃	1	1	yes	characteristic
191	#428	C₂²×C₃	1	1	yes	characteristic
192	#428	C₂²×C₃	1	1	yes	characteristic
193	#427	—	2	1	no	—
194	#427	C₂²×C₃	1	1	yes	characteristic
195	C₂×#107	—	2	1	no	—
196	C₂×#107	C₂²×C₃	1	1	yes	characteristic
197	C₂×#107	C₂²×C₃	1	1	yes	characteristic
198	#424	C₂²×C₃	1	1	yes	characteristic
199	#424	C₂²×C₃	1	1	yes	characteristic
200	C₂×#104	C₂²×C₃	1	1	yes	characteristic
201	#418	C₂²×C₃	1	1	yes	characteristic
202	#417	C₂²×C₃	1	1	yes	characteristic
203	#416	C₂²×C₃	1	1	yes	characteristic
204	C₂×#102	C₂²×C₃	1	1	yes	characteristic
205	C₂×#102	C₂²×C₃	1	1	yes	characteristic
206	C₂×#100	C₂²×C₃	1	1	yes	characteristic
207	#405	C₂²×C₃	1	1	yes	characteristic
208	#390	C₂²×C₃	1	1	yes	characteristic
209	#384	C₂²×C₃	1	1	yes	characteristic
210	#343	C₂²×C₃	1	1	yes	characteristic
211	#343	C₂²×C₃	1	1	yes	characteristic
212	#343	C₂²×C₃	1	1	yes	characteristic
213	#341	C₂²×C₃	1	1	yes	characteristic
214	#339	C₂²×C₃	1	1	yes	characteristic
215	#339	C₂²×C₃	1	1	yes	characteristic
216	#339	C₂²×C₃	1	1	yes	characteristic
217	#336	C₂²×C₃	1	1	yes	characteristic
218	C₂⁴×C₃	—	2	1	no	—
219	C₃×#41	—	2	1	no	—
220	C₃×#41	—	2	1	no	—
221	C₃×#41	C₂×Q₈	1	2	yes	—
222	C₃×#41	#41	1	2	yes	—
223	C₂×C₃×Q₈	—	4	1	no	—
224	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
225	C₂×C₃×D₄	—	2	1	no	—
226	C₂×C₃×D₄	—	2	1	no	—
227	C₂×C₃×D₄	—	2	1	no	—
228	C₂×C₃×D₄	—	2	1	no	—
229	C₂×C₃×D₄	—	2	1	no	—
230	C₂×C₃×D₄	—	2	1	no	—
231	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
232	C₂×C₃×D₄	C₂×Q₈	1	2	yes	—
233	C₂²×C₃×C₄	—	2	1	no	—
234	C₂²×C₃×C₄	—	2	1	no	—
235	C₂²×C₃×C₄	—	2	1	no	—
236	C₂²×C₃×C₄	—	2	1	no	—
237	C₂²×C₃×C₄	—	2	1	no	—
238	C₂²×C₃×C₄	—	4	1	no	—
239	C₂²×C₃×C₄	—	4	1	no	—
240	C₂²×C₃×C₄	—	4	1	no	—
241	C₂²×C₃×C₄	—	4	1	no	—
242	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
243	C₃×SD₁₆	—	2	4	no	—
244	C₃×D₈	—	2	4	no	—
245	C₃×M₁₆	—	2	4	no	—
246	C₃×M₁₆	—	2	1	no	—
247	C₃×M₁₆	—	2	1	no	—
248	C₃×M₁₆	—	2	1	no	—
249	C₃×M₁₆	—	2	1	no	—
250	C₃×M₁₆	—	2	1	no	—
251	C₃×M₁₆	—	2	1	no	—
252	C₃×M₁₆	C₂×Q₈	1	2	yes	—
253	C₃×M₁₆	#41	1	2	yes	—
254	C₂×C₃×C₈	—	2	4	no	—
255	C₂×C₃×C₈	—	2	1	no	—
256	C₂×C₃×C₈	—	2	1	no	—
257	C₂×C₃×C₈	—	2	1	no	—
258	C₂×C₃×C₈	—	2	1	no	—
259	C₂×C₃×C₈	—	2	1	no	—
260	C₂×C₃×C₈	—	2	1	no	—
261	C₂×C₃×C₈	—	2	1	no	—
262	C₂×C₃×C₈	—	2	1	no	—
263	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
264	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
265	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
266	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
267	C₃×C₄⋊C₄	—	2	1	no	—
268	C₃×C₄⋊C₄	—	2	1	no	—
269	C₃×C₄⋊C₄	—	2	1	no	—
270	C₃×C₄⋊C₄	—	2	1	no	—
271	C₃×C₄⋊C₄	—	2	1	no	—
272	C₃×C₄⋊C₄	—	2	1	no	—
273	C₃×C₄⋊C₄	—	4	1	no	—
274	C₃×C₄⋊C₄	—	4	1	no	—
275	C₃×C₄⋊C₄	—	4	1	no	—
276	C₃×C₄⋊C₄	—	4	1	no	—
277	C₃×C₄⋊C₄	—	4	1	no	—
278	C₃×C₄⋊C₄	—	8	2	no	—
279	C₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
280	C₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
281	C₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
282	C₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
283	C₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
284	C₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
285	C₃×C₂²⋊C₄	—	2	1	no	—
286	C₃×C₂²⋊C₄	—	2	1	no	—
287	C₃×C₂²⋊C₄	—	2	1	no	—
288	C₃×C₂²⋊C₄	—	4	1	no	—
289	C₃×C₂²⋊C₄	—	4	1	no	—
290	C₃×C₂²⋊C₄	—	4	1	no	—
291	C₃×C₂²⋊C₄	—	4	1	no	—
292	C₃×C₂²⋊C₄	—	8	2	no	—
293	C₃×C₄²	—	2	1	no	—
294	C₃×C₄²	—	2	1	no	—
295	C₃×C₄²	—	2	1	no	—
296	C₃×C₄²	—	2	1	no	—
297	C₃×C₄²	—	2	1	no	—
298	C₃×C₄²	—	4	1	no	—
299	C₃×C₄²	—	4	1	no	—
300	C₂×#41	C₂³×C₃	1	1	yes	characteristic
301	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
302	AGL(1,ℤ/8)	—	2	2	no	—
303	AGL(1,ℤ/8)	C₃×Q₈	1	4	yes	complemented
304	C₂×SD₁₆	—	2	1	no	—
305	C₂×D₈	—	2	1	no	—
306	#131	—	2	2	no	—
307	#131	—	2	2	no	—
308	C₂×M₁₆	—	2	1	no	—
309	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
310	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
311	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
312	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
313	C₂²×C₈	—	2	1	no	—
314	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
316	C₄⋊Q₈	—	2	1	no	—
317	C₄⋊Q₈	—	2	1	no	—
318	#125	—	2	1	no	—
319	#125	—	2	1	no	—
320	C₂²⋊Q₈	—	4	1	no	—
321	C₄×Q₈	—	2	1	no	—
322	C₄×Q₈	—	2	1	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	—	2	1	no	—
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	—	2	1	no	—
328	C₄×D₄	—	2	1	no	—
329	C₄×D₄	—	4	1	no	—
330	C₄×D₄	—	4	1	no	—
331	#117	—	2	1	no	—
332	#117	—	2	1	no	—
333	#117	C₂³×C₃	1	1	yes	characteristic
334	#117	C₂³×C₃	1	1	yes	characteristic
335	#117	C₂³×C₃	1	1	yes	characteristic
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	4	1	no	—
339	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
342	C₂×C₂²⋊C₄	—	4	1	no	—
343	C₂×C₂²⋊C₄	—	4	1	no	—
344	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
345	#108	—	2	2	no	—
346	#107	—	2	1	no	—
347	#107	—	2	1	no	—
348	#107	—	2	1	no	—
349	#107	—	2	1	no	—
350	#107	—	2	1	no	—
351	#107	—	2	1	no	—
352	#107	—	2	1	no	—
353	#107	—	2	1	no	—
354	#107	—	2	1	no	—
355	#107	C₃×D₄	1	1	yes	characteristic
356	#107	C₃×D₄	1	1	yes	characteristic
357	#106	—	2	1	no	—
358	#106	—	2	1	no	—
359	#106	—	2	1	no	—
360	#106	—	2	1	no	—
361	#106	—	2	1	no	—
362	#106	C₃×D₄	1	1	yes	characteristic
363	#106	C₃×D₄	1	1	yes	characteristic
364	C₄⋊C₈	—	2	1	no	—
365	C₄⋊C₈	—	2	1	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	#104	—	2	2	no	—
369	#103	—	2	1	no	—
370	#103	C₃×D₄	1	1	yes	characteristic
371	#103	C₃×D₄	1	1	yes	characteristic
372	#102	—	2	1	no	—
373	#102	—	2	1	no	—
374	#102	—	2	1	no	—
375	#102	—	2	1	no	—
376	#102	—	2	1	no	—
377	#102	—	2	1	no	—
378	#102	—	2	1	no	—
379	#102	C₃×D₄	1	1	yes	characteristic
380	#102	C₃×D₄	1	1	yes	characteristic
381	#100	—	2	2	no	—
382	#97	—	2	1	no	—
383	C₄×C₈	—	2	1	no	—
384	C₂³×C₃	—	2	1	no	—
385	C₂³×C₃	—	2	2	no	—
386	C₂³×C₃	—	4	1	no	—
387	C₂³×C₃	—	4	1	no	—
388	C₂³×C₃	—	8	1	no	—
389	C₂³×C₃	—	8	1	no	—
390	C₂³×C₃	H(5,𝔽₂)	1	1	yes	characteristic
391	C₃×Q₈	—	2	1	no	—
392	C₃×Q₈	—	16	1	no	—
393	C₃×Q₈	C₂²⋊Q₈	1	2	yes	—
394	C₃×D₄	—	2	2	no	—
395	C₃×D₄	—	2	2	no	—
396	C₃×D₄	—	2	2	no	—
397	C₃×D₄	—	2	1	no	—
398	C₃×D₄	—	2	2	no	—
399	C₃×D₄	—	4	1	no	—
400	C₃×D₄	—	4	1	no	—
401	C₃×D₄	C₂²⋊Q₈	1	2	yes	—
402	C₂×C₃×C₄	—	2	1	no	—
403	C₂×C₃×C₄	—	2	2	no	—
404	C₂×C₃×C₄	—	2	2	no	—
405	C₂×C₃×C₄	—	2	2	no	—
406	C₂×C₃×C₄	—	2	1	no	—
407	C₂×C₃×C₄	—	2	1	no	—
408	C₂×C₃×C₄	—	2	1	no	—
409	C₂×C₃×C₄	—	2	1	no	—
410	C₂×C₃×C₄	—	2	1	no	—
411	C₂×C₃×C₄	—	2	1	no	—
412	C₂×C₃×C₄	—	4	1	no	—
413	C₂×C₃×C₄	—	4	1	no	—
414	C₂×C₃×C₄	—	4	1	no	—
415	C₂×C₃×C₄	—	4	1	no	—
416	C₂×C₃×C₄	—	4	1	no	—
417	C₂×C₃×C₄	—	8	1	no	—
418	C₂×C₃×C₄	—	8	1	no	—
419	C₂×C₃×C₄	—	8	1	no	—
420	C₂×C₃×C₄	—	16	1	no	—
421	C₂×C₃×C₄	—	16	1	no	—
422	C₂×C₃×C₄	—	16	1	no	—
423	C₂×C₃×C₄	—	16	1	no	—
424	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
425	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
426	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
427	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
428	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
429	C₂×C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
430	C₃×C₈	—	2	2	no	—
431	C₃×C₈	—	2	1	no	—
432	C₃×C₈	—	2	2	no	—
433	C₃×C₈	—	2	1	no	—
434	C₃×C₈	—	2	1	no	—
435	C₃×C₈	—	2	1	no	—
436	C₃×C₈	—	4	1	no	—
437	C₃×C₈	—	4	1	no	—
438	C₂⁴	—	2	1	no	—
439	#41	—	2	1	no	—
440	#41	—	2	1	no	—
441	#41	C₂×C₃×Q₈	1	2	yes	—
442	#41	C₃×#41	1	2	yes	—
443	C₂×Q₈	—	4	1	no	—
444	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
452	C₂×D₄	C₂×C₃×Q₈	1	2	yes	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	4	1	no	—
459	C₂²×C₄	—	4	1	no	—
460	C₂²×C₄	—	4	1	no	—
461	C₂²×C₄	—	4	1	no	—
462	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
463	SD₁₆	—	2	4	no	—
464	D₈	—	2	4	no	—
465	M₁₆	—	2	4	no	—
466	M₁₆	—	2	1	no	—
467	M₁₆	—	2	1	no	—
468	M₁₆	—	2	1	no	—
469	M₁₆	—	2	1	no	—
470	M₁₆	—	2	1	no	—
471	M₁₆	—	2	1	no	—
472	M₁₆	C₂×C₃×Q₈	1	2	yes	—
473	M₁₆	C₃×#41	1	2	yes	—
474	C₂×C₈	—	2	4	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
484	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
485	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
486	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	4	1	no	—
494	C₄⋊C₄	—	4	1	no	—
495	C₄⋊C₄	—	4	1	no	—
496	C₄⋊C₄	—	4	1	no	—
497	C₄⋊C₄	—	4	1	no	—
498	C₄⋊C₄	—	8	2	no	—
499	C₄⋊C₄	C₂×C₃×D₄	1	1	yes	characteristic
500	C₄⋊C₄	C₂×C₃×D₄	1	1	yes	characteristic
501	C₄⋊C₄	C₂×C₃×D₄	1	1	yes	characteristic
502	C₄⋊C₄	C₂×C₃×D₄	1	1	yes	characteristic
503	C₄⋊C₄	C₂×C₃×D₄	1	1	yes	characteristic
504	C₄⋊C₄	C₂×C₃×D₄	1	1	yes	characteristic
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	4	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₂²⋊C₄	—	4	1	no	—
511	C₂²⋊C₄	—	4	1	no	—
512	C₂²⋊C₄	—	8	2	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	2	1	no	—
517	C₄²	—	2	1	no	—
518	C₄²	—	4	1	no	—
519	C₄²	—	4	1	no	—
520	C₂²×C₃	—	2	1	no	—
521	C₂²×C₃	—	2	1	no	—
522	C₂²×C₃	—	2	2	no	—
523	C₂²×C₃	—	4	1	no	—
524	C₂²×C₃	—	4	1	no	—
525	C₂²×C₃	—	4	2	no	—
526	C₂²×C₃	—	8	2	no	—
527	C₂²×C₃	—	8	1	no	—
528	C₂²×C₃	—	8	1	no	—
529	C₂²×C₃	—	16	1	no	—
530	C₂²×C₃	#539	1	1	yes	characteristic
531	C₂²×C₃	D₄²	1	1	yes	characteristic
532	C₂²×C₃	#554	1	1	yes	characteristic
533	C₃×C₄	—	2	2	no	—
534	C₃×C₄	—	4	2	no	—
535	C₃×C₄	—	4	1	no	—
536	C₃×C₄	—	4	1	no	—
537	C₃×C₄	—	4	1	no	—
538	C₃×C₄	—	4	1	no	—
539	C₃×C₄	—	4	1	no	—
540	C₃×C₄	—	4	1	no	—
541	C₃×C₄	—	8	1	no	—
542	C₃×C₄	—	8	1	no	—
543	C₃×C₄	—	8	1	no	—
544	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
545	C₃×C₄	#525	1	1	yes	characteristic
546	C₃×C₄	#547	1	1	yes	characteristic
547	C₃×C₄	D₄×Q₈	1	1	yes	characteristic
548	C₂³	—	2	1	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	4	1	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	8	1	no	—
553	C₂³	—	8	1	no	—
554	C₂³	C₃×H(5,𝔽₂)	1	1	yes	characteristic
555	Q₈	—	2	1	no	—
556	Q₈	—	16	1	no	—
557	Q₈	C₃×C₂²⋊Q₈	1	2	yes	—
558	D₄	—	2	2	no	—
559	D₄	—	2	2	no	—
560	D₄	—	2	2	no	—
561	D₄	—	2	1	no	—
562	D₄	—	2	2	no	—
563	D₄	—	4	1	no	—
564	D₄	—	4	1	no	—
565	D₄	C₃×C₂²⋊Q₈	1	2	yes	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	8	1	no	—
582	C₂×C₄	—	8	1	no	—
583	C₂×C₄	—	8	1	no	—
584	C₂×C₄	—	16	1	no	—
585	C₂×C₄	—	16	1	no	—
586	C₂×C₄	—	16	1	no	—
587	C₂×C₄	—	16	1	no	—
588	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
589	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
590	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
591	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
592	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
593	C₂×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
594	C₈	—	2	2	no	—
595	C₈	—	2	1	no	—
596	C₈	—	2	2	no	—
597	C₈	—	2	1	no	—
598	C₈	—	2	1	no	—
599	C₈	—	2	1	no	—
600	C₈	—	4	1	no	—
601	C₈	—	4	1	no	—
602	C₂×C₃	—	2	1	no	—
603	C₂×C₃	—	2	1	no	—
604	C₂×C₃	—	4	2	no	—
605	C₂×C₃	—	8	2	no	—
606	C₂×C₃	#2449	1	1	yes	characteristic
607	C₂×C₃	#3289	1	1	yes	characteristic
608	C₂×C₃	#3292	1	1	yes	characteristic
609	C₂²	—	2	1	no	—
610	C₂²	—	2	1	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	4	1	no	—
613	C₂²	—	4	1	no	—
614	C₂²	—	4	2	no	—
615	C₂²	—	8	2	no	—
616	C₂²	—	8	1	no	—
617	C₂²	—	8	1	no	—
618	C₂²	—	16	1	no	—
619	C₂²	C₃×#539	1	1	yes	characteristic
620	C₂²	C₃×D₄²	1	1	yes	characteristic
621	C₂²	C₃×#554	1	1	yes	characteristic
622	C₄	—	2	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	1	no	—
625	C₄	—	4	1	no	—
626	C₄	—	4	1	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	1	no	—
630	C₄	—	8	1	no	—
631	C₄	—	8	1	no	—
632	C₄	—	8	1	no	—
633	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
634	C₄	C₃×#525	1	1	yes	characteristic
635	C₄	C₃×#547	1	1	yes	characteristic
636	C₄	C₃×D₄×Q₈	1	1	yes	characteristic
637	C₃	#26400	1	1	yes	characteristic, direct product factor, Sylow
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	4	2	no	—
641	C₂	—	8	2	no	—
642	C₂	C₃×#2449	1	1	yes	characteristic
643	C₂	C₃×#3289	1	1	yes	characteristic
644	C₂	C₃×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂×C₃×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×#3083	C₂	semidirect
2. C₃×#3081	C₂	semidirect
4. C₃×#2913	C₂	semidirect
5. C₃×#2093	C₂	semidirect
6. C₃×#2089	C₂	semidirect
7. C₃×#2063	C₂	semidirect
8. C₃×#2060	C₂	semidirect
9. C₃×#1952	C₂	semidirect
10. C₃×#1922	C₂	semidirect
13. C₃×#1866	C₂	semidirect
14. C₃×#1859	C₂	semidirect
15. C₃×#1856	C₂	semidirect
16. #26400	C₃	direct (N × H)
68. #3083	C₂×C₃	semidirect
69. #3081	C₂×C₃	semidirect
71. #2913	C₂×C₃	semidirect
72. #2093	C₂×C₃	semidirect
73. #2089	C₂×C₃	semidirect
74. #2063	C₂×C₃	semidirect
75. #2060	C₂×C₃	semidirect
76. #1952	C₂×C₃	semidirect
77. #1922	C₂×C₃	semidirect
80. #1866	C₂×C₃	semidirect
81. #1859	C₂×C₃	semidirect
82. #1856	C₂×C₃	semidirect
86. C₃×AGL(1,ℤ/8)	Q₈	semidirect
303. AGL(1,ℤ/8)	C₃×Q₈	semidirect
637. C₃	#26400	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#3083	2
C₃×#3081	2
C₃×#2946	2
C₃×#2913	2
C₃×#2093	2
C₃×#2089	2
C₃×#2063	2
C₃×#2060	2
C₃×#1952	2
C₃×#1922	2
C₃×#1907	2
C₃×#1883	2
C₃×#1866	2
C₃×#1859	2
C₃×#1856	2
#26400	3

Subgroups — C3×#26400 #10645007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#26400 #10645007