Subgroups — C₃×#33585 #10652192

Order 768.

748 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3155	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2210	C₂	1	4	yes	maximal, complemented
6	C₃×#2206	C₂	1	4	yes	maximal, complemented
7	C₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	#33585	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×AGL(1,ℤ/8)	—	2	1	no	—
12	C₂×C₃×#135	C₂²	1	1	yes	characteristic
13	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
15	C₃×#535	—	2	1	no	—
16	C₃×#518	—	2	1	no	—
17	C₃×#509	—	2	4	no	—
18	C₂×C₃×SD₃₂	C₂²	1	2	yes	complemented
19	C₂×C₃×D₁₆	—	2	4	no	—
20	C₂×C₃×D₁₆	C₂²	1	4	yes	—
21	C₂×C₃×D₁₆	C₂²	1	2	yes	complemented
22	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
23	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
24	C₃×#496	—	2	2	no	—
25	C₃×#495	C₂²	1	2	yes	complemented
26	C₃×#493	C₂²	1	2	yes	complemented
27	C₃×#442	—	2	2	no	—
28	C₃×C₄×D₈	C₂²	1	4	yes	complemented
29	C₃×#427	C₂²	1	1	yes	characteristic
30	C₃×#418	—	2	1	no	—
31	C₃×#405	C₂²	1	1	yes	characteristic
32	C₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
33	C₃×#358	C₂²	1	2	yes	complemented
34	C₃×#357	C₂²	1	2	yes	complemented
35	C₃×#357	C₂²	1	4	yes	complemented
36	C₂×#509	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂²×D₁₆	C₂×C₃	1	1	yes	characteristic, complemented
38	#3155	C₂×C₃	1	1	yes	characteristic, complemented
39	#2948	C₂×C₃	1	1	yes	characteristic, complemented
40	#2210	C₂×C₃	1	4	yes	complemented
41	#2206	C₂×C₃	1	4	yes	complemented
42	#2151	C₂×C₃	1	1	yes	characteristic, complemented
43	#2140	C₂×C₃	1	1	yes	characteristic, complemented
44	#2138	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×#41	—	2	1	no	—
46	C₂²×C₃×D₄	—	2	1	no	—
47	C₂²×C₃×D₄	—	2	1	no	—
48	C₃×AGL(1,ℤ/8)	—	4	4	no	—
49	C₃×#135	—	2	2	no	—
50	C₃×#135	—	2	2	no	—
51	C₂×C₃×Q₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₃×SD₁₆	—	2	1	no	—
53	C₂×C₃×SD₁₆	—	2	2	no	—
54	C₂×C₃×D₈	—	2	1	no	—
55	C₂×C₃×D₈	—	2	2	no	—
56	C₂×C₃×D₈	—	2	1	no	—
57	C₂×C₃×D₈	—	2	2	no	—
58	C₂×C₃×D₈	—	2	2	no	—
59	C₂×C₃×D₈	—	2	2	no	—
60	C₂×C₃×D₈	D₄	1	4	yes	complemented
61	C₂×C₃×D₈	C₂³	1	2	yes	—
62	C₂×C₃×D₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×D₈	C₂³	1	2	yes	—
64	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
65	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₃×#127	—	2	2	no	—
67	C₃×#124	—	2	2	no	—
68	C₃×#121	—	4	2	no	—
69	C₃×#120	—	4	2	no	—
70	C₃×C₄×D₄	—	2	4	no	—
71	C₃×C₄×D₄	—	4	2	no	—
72	C₃×#117	—	2	1	no	—
73	C₃×#117	D₄	1	1	yes	characteristic
74	C₂×C₃×C₂²⋊C₄	—	4	2	no	—
75	C₃×SD₃₂	—	2	4	no	—
76	C₃×D₁₆	—	2	8	no	—
77	C₃×D₁₆	—	2	4	no	—
78	C₃×M₃₂	—	2	2	no	—
79	C₂×C₃×C₁₆	—	2	2	no	—
80	C₂×C₃×C₁₆	C₂³	1	2	yes	—
81	C₂×C₃×C₁₆	C₂³	1	2	yes	—
82	C₃×#107	C₂³	1	2	yes	—
83	C₃×#106	—	2	1	no	—
84	C₃×#102	—	2	4	no	—
85	C₃×#97	D₄	1	2	yes	—
86	C₃×C₄×C₈	C₂³	1	2	yes	—
87	C₂×AGL(1,ℤ/8)	—	2	1	no	—
88	C₂×#135	C₂²×C₃	1	1	yes	characteristic
89	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
90	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
91	#535	—	2	1	no	—
92	#518	—	2	1	no	—
93	#509	—	2	4	no	—
94	C₂×SD₃₂	C₂²×C₃	1	2	yes	complemented
95	C₂×D₁₆	—	2	4	no	—
96	C₂×D₁₆	C₂²×C₃	1	4	yes	—
97	C₂×D₁₆	C₂²×C₃	1	2	yes	complemented
98	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
100	#496	—	2	2	no	—
101	#495	C₂²×C₃	1	2	yes	complemented
102	#493	C₂²×C₃	1	2	yes	complemented
103	#442	—	2	2	no	—
104	C₄×D₈	C₂²×C₃	1	4	yes	complemented
105	#427	C₂²×C₃	1	1	yes	characteristic
106	#418	—	2	1	no	—
107	#405	C₂²×C₃	1	1	yes	characteristic
108	C₄⋊C₁₆	C₂²×C₃	1	4	yes	complemented
109	#358	C₂²×C₃	1	2	yes	complemented
110	#357	C₂²×C₃	1	2	yes	complemented
111	#357	C₂²×C₃	1	4	yes	complemented
112	C₂⁴×C₃	—	4	1	no	—
113	C₂⁴×C₃	—	4	1	no	—
114	C₃×#41	—	4	2	no	—
115	C₃×#41	—	4	2	no	—
116	C₂×C₃×Q₈	—	2	1	no	—
117	C₂×C₃×D₄	—	2	4	no	—
118	C₂×C₃×D₄	—	2	2	no	—
119	C₂×C₃×D₄	—	2	2	no	—
120	C₂×C₃×D₄	—	2	1	no	—
121	C₂×C₃×D₄	—	4	1	no	—
122	C₂×C₃×D₄	—	4	2	no	—
123	C₂×C₃×D₄	—	4	1	no	—
124	C₂×C₃×D₄	—	4	1	no	—
125	C₂×C₃×D₄	—	4	4	no	—
126	C₂×C₃×D₄	—	4	2	no	—
127	C₂×C₃×D₄	—	4	2	no	—
128	C₂²×C₃×C₄	—	4	1	no	—
129	C₂²×C₃×C₄	—	4	4	no	—
130	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
131	C₃×Q₁₆	—	2	1	no	—
132	C₃×Q₁₆	—	2	1	no	—
133	C₃×SD₁₆	—	4	4	no	—
134	C₃×SD₁₆	—	4	2	no	—
135	C₃×D₈	—	2	4	no	—
136	C₃×D₈	—	2	2	no	—
137	C₃×D₈	—	2	1	no	—
138	C₃×D₈	—	2	2	no	—
139	C₃×D₈	—	2	2	no	—
140	C₃×D₈	—	2	1	no	—
141	C₃×D₈	—	4	4	no	—
142	C₃×D₈	—	4	2	no	—
143	C₃×D₈	C₂×D₄	1	4	yes	—
144	C₃×M₁₆	—	2	2	no	—
145	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
146	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
148	C₂×C₃×C₈	#41	1	2	yes	—
149	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	D₈	1	2	yes	—
152	C₃×C₂²⋊C₄	—	4	1	no	—
153	C₃×C₂²⋊C₄	—	4	4	no	—
154	C₃×C₂²⋊C₄	—	4	4	no	—
155	C₃×C₂²⋊C₄	D₈	1	2	yes	—
156	C₃×C₄²	—	4	1	no	—
157	C₃×C₄²	C₂×D₄	1	2	yes	—
158	C₃×C₁₆	—	2	2	no	—
159	C₃×C₁₆	—	2	2	no	—
160	C₂×#41	—	2	1	no	—
161	C₂²×D₄	—	2	1	no	—
162	C₂²×D₄	—	2	1	no	—
163	AGL(1,ℤ/8)	—	4	4	no	—
164	#135	—	2	2	no	—
165	#135	—	2	2	no	—
166	C₂×Q₁₆	C₂³×C₃	1	1	yes	characteristic
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	2	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	C₃×D₄	1	4	yes	complemented
176	C₂×D₈	C₂³×C₃	1	2	yes	—
177	C₂×D₈	C₂³×C₃	1	1	yes	characteristic
178	C₂×D₈	C₂³×C₃	1	2	yes	—
179	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
181	#127	—	2	2	no	—
182	#124	—	2	2	no	—
183	#121	—	4	2	no	—
184	#120	—	4	2	no	—
185	C₄×D₄	—	2	4	no	—
186	C₄×D₄	—	4	2	no	—
187	#117	—	2	1	no	—
188	#117	C₃×D₄	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	—	4	2	no	—
190	SD₃₂	—	2	4	no	—
191	D₁₆	—	2	8	no	—
192	D₁₆	—	2	4	no	—
193	M₃₂	—	2	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	C₂³×C₃	1	2	yes	—
196	C₂×C₁₆	C₂³×C₃	1	2	yes	—
197	#107	C₂³×C₃	1	2	yes	—
198	#106	—	2	1	no	—
199	#102	—	2	4	no	—
200	#97	C₃×D₄	1	2	yes	—
201	C₄×C₈	C₂³×C₃	1	2	yes	—
202	C₂³×C₃	—	4	2	no	—
203	C₂³×C₃	—	4	4	no	—
204	C₂³×C₃	—	4	2	no	—
205	C₂³×C₃	—	4	1	no	—
206	C₂³×C₃	—	8	2	no	—
207	C₂³×C₃	—	16	2	no	—
208	C₂³×C₃	—	16	2	no	—
209	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
210	C₃×Q₈	—	4	1	no	—
211	C₃×Q₈	—	4	1	no	—
212	C₃×D₄	—	2	4	no	—
213	C₃×D₄	—	4	2	no	—
214	C₃×D₄	—	4	1	no	—
215	C₃×D₄	—	4	2	no	—
216	C₃×D₄	—	4	1	no	—
217	C₃×D₄	—	4	2	no	—
218	C₃×D₄	—	4	4	no	—
219	C₃×D₄	—	8	2	no	—
220	C₃×D₄	—	8	2	no	—
221	C₃×D₄	—	16	4	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	—	4	1	no	—
224	C₂×C₃×C₄	—	8	2	no	—
225	C₂×C₃×C₄	—	16	4	no	—
226	C₂×C₃×C₄	#121	1	2	yes	—
227	C₂×C₃×C₄	C₂×D₈	1	4	yes	—
228	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
229	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
230	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
231	C₃×C₈	—	2	2	no	—
232	C₃×C₈	#121	1	2	yes	—
233	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
234	C₃×C₈	C₂×#41	1	1	yes	characteristic
235	C₂⁴	—	4	1	no	—
236	C₂⁴	—	4	1	no	—
237	#41	—	4	2	no	—
238	#41	—	4	2	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	4	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	4	no	—
253	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	1	no	—
256	SD₁₆	—	4	4	no	—
257	SD₁₆	—	4	2	no	—
258	D₈	—	2	4	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	2	no	—
262	D₈	—	2	2	no	—
263	D₈	—	2	1	no	—
264	D₈	—	4	4	no	—
265	D₈	—	4	2	no	—
266	D₈	C₂×C₃×D₄	1	4	yes	—
267	M₁₆	—	2	2	no	—
268	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
269	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
270	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
271	C₂×C₈	C₃×#41	1	2	yes	—
272	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	C₃×D₈	1	2	yes	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	4	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	C₃×D₈	1	2	yes	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×C₃×D₄	1	2	yes	—
281	C₁₆	—	2	2	no	—
282	C₁₆	—	2	2	no	—
283	C₂²×C₃	—	2	2	no	—
284	C₂²×C₃	—	4	4	no	—
285	C₂²×C₃	—	8	4	no	—
286	C₂²×C₃	—	8	2	no	—
287	C₂²×C₃	—	8	4	no	—
288	C₂²×C₃	—	8	1	no	—
289	C₂²×C₃	—	8	2	no	—
290	C₂²×C₃	—	16	4	no	—
291	C₂²×C₃	—	16	4	no	—
292	C₂²×C₃	#459	1	2	yes	—
293	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
294	C₃×C₄	—	2	4	no	—
295	C₃×C₄	—	8	2	no	—
296	C₃×C₄	—	8	1	no	—
297	C₃×C₄	#459	1	2	yes	—
298	C₃×C₄	C₂×#121	1	1	yes	characteristic
299	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂³	—	4	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	—	8	2	no	—
305	C₂³	—	16	2	no	—
306	C₂³	—	16	2	no	—
307	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
308	Q₈	—	4	1	no	—
309	Q₈	—	4	1	no	—
310	D₄	—	2	4	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	1	no	—
317	D₄	—	8	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	16	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	16	4	no	—
324	C₂×C₄	C₃×#121	1	2	yes	—
325	C₂×C₄	C₂×C₃×D₈	1	4	yes	—
326	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
327	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
328	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	C₃×#121	1	2	yes	—
331	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
332	C₈	C₂×C₃×#41	1	1	yes	characteristic
333	C₂×C₃	—	2	2	no	—
334	C₂×C₃	—	8	4	no	—
335	C₂×C₃	—	16	1	no	—
336	C₂×C₃	—	16	2	no	—
337	C₂×C₃	C₂×#459	1	1	yes	characteristic
338	C₂×C₃	#3415	1	2	yes	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	8	1	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	16	4	no	—
347	C₂²	—	16	4	no	—
348	C₂²	C₃×#459	1	2	yes	—
349	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
350	C₄	—	2	4	no	—
351	C₄	—	8	2	no	—
352	C₄	—	8	1	no	—
353	C₄	C₃×#459	1	2	yes	—
354	C₄	C₂×C₃×#121	1	1	yes	characteristic
355	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
356	C₃	#33585	1	1	yes	characteristic, direct product factor, Sylow
357	C₂	—	2	2	no	—
358	C₂	—	8	4	no	—
359	C₂	—	16	2	no	—
360	C₂	—	16	1	no	—
361	C₂	C₂×C₃×#459	1	1	yes	characteristic
362	C₂	C₃×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ C₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#509	C₂	semidirect
2. C₂²×C₃×D₁₆	C₂	semidirect
3. C₃×#3155	C₂	semidirect
4. C₃×#2948	C₂	semidirect
5. C₃×#2210	C₂	semidirect
6. C₃×#2206	C₂	semidirect
7. C₃×#2151	C₂	semidirect
8. C₃×#2140	C₂	semidirect
9. C₃×#2138	C₂	semidirect
10. #33585	C₃	direct (N × H)
18. C₂×C₃×SD₃₂	C₂²	semidirect
21. C₂×C₃×D₁₆	C₂²	semidirect
25. C₃×#495	C₂²	semidirect
26. C₃×#493	C₂²	semidirect
28. C₃×C₄×D₈	C₂²	semidirect
32. C₃×C₄⋊C₁₆	C₂²	semidirect
33. C₃×#358	C₂²	semidirect
34. C₃×#357	C₂²	semidirect
35. C₃×#357	C₂²	semidirect
36. C₂×#509	C₂×C₃	semidirect
37. C₂²×D₁₆	C₂×C₃	semidirect
38. #3155	C₂×C₃	semidirect
39. #2948	C₂×C₃	semidirect
40. #2210	C₂×C₃	semidirect
41. #2206	C₂×C₃	semidirect
42. #2151	C₂×C₃	semidirect
43. #2140	C₂×C₃	semidirect
44. #2138	C₂×C₃	semidirect
60. C₂×C₃×D₈	D₄	semidirect
94. C₂×SD₃₂	C₂²×C₃	semidirect
97. C₂×D₁₆	C₂²×C₃	semidirect
101. #495	C₂²×C₃	semidirect
102. #493	C₂²×C₃	semidirect
104. C₄×D₈	C₂²×C₃	semidirect
108. C₄⋊C₁₆	C₂²×C₃	semidirect
109. #358	C₂²×C₃	semidirect
110. #357	C₂²×C₃	semidirect
111. #357	C₂²×C₃	semidirect
175. C₂×D₈	C₃×D₄	semidirect
356. C₃	#33585	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#509	2
C₂²×C₃×D₁₆	2
C₃×#3155	2
C₃×#2948	2
C₃×#2210	2
C₃×#2206	2
C₃×#2151	2
C₃×#2140	2
C₃×#2138	2
#33585	3

Subgroups — C3×#33585 #10652192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33585 #10652192