Subgroups — C₃×#33629 #10652236

Order 768.

574 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×#135	C₂	1	1	yes	maximal, characteristic
3	C₃×#2227	C₂	1	2	yes	maximal, complemented
4	C₃×#2226	C₂	1	2	yes	maximal, complemented
5	C₃×#2225	C₂	1	2	yes	maximal, complemented
6	C₃×#2224	C₂	1	2	yes	maximal, complemented
7	C₃×#2159	C₂	1	2	yes	maximal, complemented
8	C₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2139	C₂	1	2	yes	maximal
10	#33629	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#135	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
13	C₃×C₄×#41	—	2	1	no	—
14	C₂×C₃×M₃₂	C₂²	1	2	yes	—
15	C₃×#500	—	2	1	no	—
16	C₃×#500	—	2	1	no	—
17	C₃×#500	C₂²	1	2	yes	—
18	C₃×#500	C₂²	1	2	yes	—
19	C₃×C₄×Q₁₆	C₂²	1	1	yes	characteristic
20	C₃×C₄×Q₁₆	C₂²	1	1	yes	characteristic
21	C₃×C₄×SD₁₆	—	2	1	no	—
22	C₃×C₄×SD₁₆	—	2	1	no	—
23	C₃×C₄×D₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×D₈	C₂²	1	1	yes	characteristic
25	C₃×#427	C₂²	1	1	yes	characteristic
26	C₂×C₃×#107	—	2	1	no	—
27	C₂×C₃×#107	C₂²	1	2	yes	—
28	C₃×#416	—	2	1	no	—
29	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₃×#367	C₂²	1	4	yes	—
31	C₃×#366	C₂²	1	4	yes	—
32	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₃×#358	C₂²	1	4	yes	—
35	C₃×#357	C₂²	1	4	yes	—
36	C₂×#500	C₂×C₃	1	1	yes	characteristic, complemented
37	C₄×#135	C₂×C₃	1	1	yes	characteristic
38	#2227	C₂×C₃	1	2	yes	complemented
39	#2226	C₂×C₃	1	2	yes	complemented
40	#2225	C₂×C₃	1	2	yes	complemented
41	#2224	C₂×C₃	1	2	yes	complemented
42	#2159	C₂×C₃	1	2	yes	complemented
43	#2150	C₂×C₃	1	1	yes	characteristic, complemented
44	#2139	C₂×C₃	1	2	yes	—
45	C₂×C₃×#41	—	2	1	no	—
46	C₂²×C₃×Q₈	—	4	1	no	—
47	C₃×#135	—	2	2	no	—
48	C₃×#135	Q₈	1	4	yes	complemented
49	C₂×C₃×Q₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₃×SD₁₆	—	2	1	no	—
51	C₂×C₃×D₈	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
54	C₃×C₄⋊Q₈	—	2	2	no	—
55	C₃×C₄⋊Q₈	—	2	2	no	—
56	C₃×C₄⋊Q₈	—	4	1	no	—
57	C₃×C₄⋊Q₈	—	4	1	no	—
58	C₃×C₄×Q₈	—	2	1	no	—
59	C₃×C₄×Q₈	—	2	1	no	—
60	C₃×C₄×D₄	—	2	1	no	—
61	C₃×C₄×D₄	—	2	1	no	—
62	C₃×C₄×D₄	—	4	1	no	—
63	C₃×C₄×D₄	—	4	1	no	—
64	C₃×#117	—	2	1	no	—
65	C₃×#117	—	4	1	no	—
66	C₂×C₃×C₄⋊C₄	—	2	2	no	—
67	C₂×C₃×C₄⋊C₄	—	4	1	no	—
68	C₂×C₃×C₄²	—	4	1	no	—
69	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
70	C₃×M₃₂	—	2	4	no	—
71	C₂×C₃×C₁₆	C₂³	1	4	yes	—
72	C₃×#107	—	2	2	no	—
73	C₃×#107	—	2	2	no	—
74	C₃×#107	—	2	2	no	—
75	C₃×#107	C₂³	1	1	yes	characteristic
76	C₃×#107	C₂³	1	4	yes	—
77	C₃×#107	C₂³	1	1	yes	characteristic
78	C₃×#106	—	2	1	no	—
79	C₃×#103	—	2	1	no	—
80	C₃×#103	—	2	1	no	—
81	C₃×#102	—	2	1	no	—
82	C₃×#102	—	2	1	no	—
83	C₃×C₄×C₈	D₄	1	1	yes	characteristic
84	C₃×C₄×C₈	D₄	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
87	C₂×#135	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₄⋊Q₈	—	2	1	no	—
89	C₄×#41	—	2	1	no	—
90	C₂×M₃₂	C₂²×C₃	1	2	yes	—
91	#500	—	2	1	no	—
92	#500	—	2	1	no	—
93	#500	C₂²×C₃	1	2	yes	—
94	#500	C₂²×C₃	1	2	yes	—
95	C₄×Q₁₆	C₂²×C₃	1	1	yes	characteristic
96	C₄×Q₁₆	C₂²×C₃	1	1	yes	characteristic
97	C₄×SD₁₆	—	2	1	no	—
98	C₄×SD₁₆	—	2	1	no	—
99	C₄×D₈	C₂²×C₃	1	1	yes	characteristic
100	C₄×D₈	C₂²×C₃	1	1	yes	characteristic
101	#427	C₂²×C₃	1	1	yes	characteristic
102	C₂×#107	—	2	1	no	—
103	C₂×#107	C₂²×C₃	1	2	yes	—
104	#416	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
106	#367	C₂²×C₃	1	4	yes	—
107	#366	C₂²×C₃	1	4	yes	—
108	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
109	C₄⋊C₁₆	C₂²×C₃	1	2	yes	—
110	#358	C₂²×C₃	1	4	yes	—
111	#357	C₂²×C₃	1	4	yes	—
112	C₃×#41	—	2	4	no	—
113	C₃×#41	—	4	2	no	—
114	C₂×C₃×Q₈	—	2	1	no	—
115	C₂×C₃×Q₈	—	4	2	no	—
116	C₂×C₃×Q₈	—	4	2	no	—
117	C₂×C₃×Q₈	—	8	4	no	—
118	C₂×C₃×D₄	—	2	1	no	—
119	C₂×C₃×D₄	—	4	1	no	—
120	C₂²×C₃×C₄	—	4	1	no	—
121	C₂²×C₃×C₄	—	4	2	no	—
122	C₂²×C₃×C₄	—	4	1	no	—
123	C₂²×C₃×C₄	—	4	1	no	—
124	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
125	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
127	C₃×Q₁₆	—	2	1	no	—
128	C₃×Q₁₆	C₂×Q₈	1	2	yes	complemented
129	C₃×SD₁₆	—	2	4	no	—
130	C₃×D₈	—	2	1	no	—
131	C₃×D₈	C₂×Q₈	1	2	yes	complemented
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
137	C₂×C₃×C₈	#41	1	2	yes	—
138	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	2	1	no	—
141	C₃×C₄⋊C₄	—	2	4	no	—
142	C₃×C₄⋊C₄	—	4	2	no	—
143	C₃×C₄⋊C₄	—	4	2	no	—
144	C₃×C₄⋊C₄	—	4	2	no	—
145	C₃×C₄⋊C₄	—	4	1	no	—
146	C₃×C₄⋊C₄	—	4	1	no	—
147	C₃×C₂²⋊C₄	—	4	1	no	—
148	C₃×C₂²⋊C₄	—	4	1	no	—
149	C₃×C₂²⋊C₄	—	4	1	no	—
150	C₃×C₄²	—	4	1	no	—
151	C₃×C₄²	—	4	1	no	—
152	C₃×C₄²	—	4	1	no	—
153	C₃×C₄²	D₈	1	1	yes	characteristic
154	C₃×C₄²	D₈	1	1	yes	characteristic
155	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
156	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
157	C₃×C₁₆	—	2	4	no	—
158	C₂×#41	—	2	1	no	—
159	C₂²×Q₈	—	4	1	no	—
160	#135	—	2	2	no	—
161	#135	C₃×Q₈	1	4	yes	complemented
162	C₂×Q₁₆	C₂³×C₃	1	1	yes	characteristic
163	C₂×SD₁₆	—	2	1	no	—
164	C₂×D₈	C₂³×C₃	1	1	yes	characteristic
165	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	4	1	no	—
170	C₄⋊Q₈	—	4	1	no	—
171	C₄×Q₈	—	2	1	no	—
172	C₄×Q₈	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	—	4	1	no	—
176	C₄×D₄	—	4	1	no	—
177	#117	—	2	1	no	—
178	#117	—	4	1	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	4	1	no	—
181	C₂×C₄²	—	4	1	no	—
182	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
183	M₃₂	—	2	4	no	—
184	C₂×C₁₆	C₂³×C₃	1	4	yes	—
185	#107	—	2	2	no	—
186	#107	—	2	2	no	—
187	#107	—	2	2	no	—
188	#107	C₂³×C₃	1	4	yes	—
189	#107	C₂³×C₃	1	1	yes	characteristic
190	#107	C₂³×C₃	1	1	yes	characteristic
191	#106	—	2	1	no	—
192	#103	—	2	1	no	—
193	#103	—	2	1	no	—
194	#102	—	2	1	no	—
195	#102	—	2	1	no	—
196	C₄×C₈	C₃×D₄	1	1	yes	characteristic
197	C₄×C₈	C₃×D₄	1	1	yes	characteristic
198	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
199	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
200	C₂³×C₃	—	4	1	no	—
201	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
202	C₃×Q₈	—	2	2	no	—
203	C₃×Q₈	—	4	1	no	—
204	C₃×Q₈	—	8	4	no	—
205	C₃×Q₈	—	8	4	no	—
206	C₃×D₄	—	2	2	no	—
207	C₃×D₄	—	4	4	no	—
208	C₃×D₄	—	4	1	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	4	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	4	4	no	—
216	C₂×C₃×C₄	—	8	4	no	—
217	C₂×C₃×C₄	—	8	2	no	—
218	C₂×C₃×C₄	—	8	2	no	—
219	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
220	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
225	C₂×C₃×C₄	#137	1	2	yes	—
226	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
230	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
231	C₃×C₈	C₂×#41	1	1	yes	characteristic
232	#41	—	2	4	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	8	4	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
245	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
247	Q₁₆	—	2	1	no	—
248	Q₁₆	C₂×C₃×Q₈	1	2	yes	complemented
249	SD₁₆	—	2	4	no	—
250	D₈	—	2	1	no	—
251	D₈	C₂×C₃×Q₈	1	2	yes	complemented
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
254	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
255	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
257	C₂×C₈	C₃×#41	1	2	yes	—
258	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₃×D₈	1	1	yes	characteristic
274	C₄²	C₃×D₈	1	1	yes	characteristic
275	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
276	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
277	C₁₆	—	2	4	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	2	1	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	8	1	no	—
282	C₂²×C₃	—	8	1	no	—
283	C₂²×C₃	#474	1	2	yes	—
284	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	4	2	no	—
288	C₃×C₄	—	8	1	no	—
289	C₃×C₄	—	8	4	no	—
290	C₃×C₄	—	8	1	no	—
291	C₃×C₄	#474	1	2	yes	—
292	C₃×C₄	#509	1	2	yes	—
293	C₃×C₄	#510	1	2	yes	—
294	C₃×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
295	C₃×C₄	C₂×#137	1	1	yes	characteristic
296	C₂³	—	4	1	no	—
297	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
298	Q₈	—	2	2	no	—
299	Q₈	—	4	1	no	—
300	Q₈	—	8	4	no	—
301	Q₈	—	8	4	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	4	no	—
304	D₄	—	4	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	8	4	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
316	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
320	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
321	C₂×C₄	C₃×#137	1	2	yes	—
322	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
326	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
327	C₈	C₂×C₃×#41	1	1	yes	characteristic
328	C₂×C₃	—	2	2	no	—
329	C₂×C₃	—	8	2	no	—
330	C₂×C₃	C₂×#474	1	1	yes	characteristic
331	C₂×C₃	C₂×#509	1	1	yes	characteristic
332	C₂×C₃	C₂×#510	1	1	yes	characteristic
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	C₃×#474	1	2	yes	—
339	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	2	no	—
343	C₄	—	8	4	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	C₃×#474	1	2	yes	—
347	C₄	C₃×#509	1	2	yes	—
348	C₄	C₃×#510	1	2	yes	—
349	C₄	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
350	C₄	C₂×C₃×#137	1	1	yes	characteristic
351	C₃	#33629	1	1	yes	characteristic, direct product factor, Sylow
352	C₂	—	2	2	no	—
353	C₂	—	8	2	no	—
354	C₂	C₂×C₃×#474	1	1	yes	characteristic
355	C₂	C₂×C₃×#509	1	1	yes	characteristic
356	C₂	C₂×C₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#500	C₂	semidirect
3. C₃×#2227	C₂	semidirect
4. C₃×#2226	C₂	semidirect
5. C₃×#2225	C₂	semidirect
6. C₃×#2224	C₂	semidirect
7. C₃×#2159	C₂	semidirect
8. C₃×#2150	C₂	semidirect
10. #33629	C₃	direct (N × H)
36. C₂×#500	C₂×C₃	semidirect
38. #2227	C₂×C₃	semidirect
39. #2226	C₂×C₃	semidirect
40. #2225	C₂×C₃	semidirect
41. #2224	C₂×C₃	semidirect
42. #2159	C₂×C₃	semidirect
43. #2150	C₂×C₃	semidirect
48. C₃×#135	Q₈	semidirect
128. C₃×Q₁₆	C₂×Q₈	semidirect
131. C₃×D₈	C₂×Q₈	semidirect
161. #135	C₃×Q₈	semidirect
248. Q₁₆	C₂×C₃×Q₈	semidirect
251. D₈	C₂×C₃×Q₈	semidirect
351. C₃	#33629	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×#500	2
C₃×C₄×#135	2
C₃×#2227	2
C₃×#2226	2
C₃×#2225	2
C₃×#2224	2
C₃×#2159	2
C₃×#2150	2
C₃×#2139	2
#33629	3

Subgroups — C3×#33629 #10652236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33629 #10652236