Subgroups — #10950823

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	#78569	C₂	1	1	yes	maximal, characteristic
3	#71354	C₂	1	1	yes	maximal, characteristic, complemented
4	#71352	C₂	1	1	yes	maximal, characteristic, complemented
5	#71313	C₂	1	1	yes	maximal, characteristic, complemented
6	#71311	C₂	1	1	yes	maximal, characteristic, complemented
7	#71084	C₂	1	2	yes	maximal, complemented
8	#71082	C₂	1	2	yes	maximal, complemented
9	#70913	C₂	1	2	yes	maximal, complemented
10	C₄×#828	C₂	1	2	yes	maximal, complemented
11	#69881	C₂	1	1	yes	maximal, characteristic, complemented
12	#33743	—	3	1	no	maximal, Sylow
13	C₃×#541	—	2	1	no	—
14	#5535	—	2	1	no	—
15	C₃×#499	C₂²	1	1	yes	characteristic
16	C₃×#494	C₂²	1	1	yes	characteristic
17	C₃×#491	—	2	1	no	—
18	C₃×#470	—	2	1	no	—
19	C₃×#467	C₂²	1	2	yes	—
20	C₃×#441	—	2	1	no	—
21	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
22	C₃×#405	C₂²	1	1	yes	characteristic
23	C₂×#828	C₂²	1	2	yes	—
24	C₂×#828	C₂²	1	2	yes	—
25	#5135	—	2	1	no	—
26	#5115	C₂²	1	2	yes	—
27	C₂×#812	C₂²	1	1	yes	characteristic
28	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
29	#4713	—	2	1	no	—
30	#4711	—	2	1	no	—
31	#4695	C₂²	1	1	yes	characteristic
32	C₄×D₂₄	C₂²	1	2	yes	—
33	#4683	C₂²	1	1	yes	characteristic
34	#4564	C₂²	1	2	yes	—
35	#4564	C₂²	1	2	yes	—
36	#4491	C₂²	1	2	yes	—
37	#4491	C₂²	1	2	yes	—
38	#4490	C₂²	1	2	yes	—
39	#4489	C₂²	1	2	yes	—
40	#4461	C₂²	1	1	yes	characteristic
41	#4461	C₂²	1	1	yes	characteristic
42	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
43	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
44	#3177	D₃	1	1	yes	characteristic, complemented
45	#3176	—	3	1	no	—
46	#2253	—	3	1	no	—
47	#2248	—	3	1	no	—
48	#2245	—	3	1	no	—
49	#2244	—	3	1	no	—
50	#2220	—	3	2	no	—
51	#2217	—	3	2	no	—
52	#2175	—	3	2	no	—
53	C₄×SD₃₂	—	3	2	no	—
54	#2108	—	3	1	no	—
55	C₂×C₃×Q₁₆	—	2	1	no	—
56	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
57	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
59	C₃×C₄⋊Q₈	—	2	1	no	—
60	C₃×#126	—	4	2	no	—
61	C₃×#125	—	2	1	no	—
62	C₃×#125	—	4	2	no	—
63	C₃×C₂²⋊Q₈	—	2	2	no	—
64	C₃×C₄×Q₈	—	2	2	no	—
65	C₃×#117	D₄	1	1	yes	characteristic
66	#930	—	2	2	no	—
67	C₂×D₂₄	—	2	1	no	—
68	C₂×D₂₄	C₂³	1	2	yes	—
69	#894	—	4	2	no	—
70	#884	—	4	2	no	—
71	#874	—	2	1	no	—
72	C₄×D₁₂	—	2	2	no	—
73	#870	—	2	1	no	—
74	C₃×#107	C₂³	1	2	yes	—
75	C₃×#106	—	2	1	no	—
76	C₃×#103	—	2	2	no	—
77	C₃×#103	—	2	2	no	—
78	C₃×#97	D₄	1	1	yes	characteristic
79	C₃×#97	D₄	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
82	#828	—	2	4	no	—
83	#828	—	2	4	no	—
84	#821	—	2	2	no	—
85	#821	—	2	2	no	—
86	#818	C₂³	1	2	yes	—
87	#817	—	2	1	no	—
88	#812	—	2	2	no	—
89	C₂×C₃⋊C₁₆	—	2	2	no	—
90	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
91	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
92	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
93	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
94	#541	—	2	1	no	—
95	#540	—	6	1	no	—
96	C₂×SD₃₂	—	3	2	no	—
97	C₂×SD₃₂	—	3	2	no	—
98	C₂×M₃₂	—	3	1	no	—
99	C₂²×C₁₆	—	3	1	no	—
100	#499	—	3	1	no	—
101	#499	C₂×D₃	1	1	yes	characteristic
102	#494	C₂×D₃	1	1	yes	characteristic
103	#493	—	3	1	no	—
104	#491	—	2	1	no	—
105	#490	—	6	1	no	—
106	#470	—	2	1	no	—
107	#469	—	6	1	no	—
108	#467	C₂×D₃	1	2	yes	—
109	#466	—	3	2	no	—
110	#442	—	6	1	no	—
111	#441	—	2	1	no	—
112	C₄×Q₁₆	C₂×D₃	1	2	yes	—
113	C₄×D₈	—	3	2	no	—
114	#405	C₂×D₃	1	1	yes	characteristic
115	#367	—	3	2	no	—
116	#366	—	3	2	no	—
117	#358	—	3	2	no	—
118	#358	—	3	2	no	—
119	#357	—	3	2	no	—
120	#357	—	3	2	no	—
121	#346	—	3	1	no	—
122	#346	—	3	1	no	—
123	C₄×C₁₆	—	3	1	no	—
124	C₄×C₁₆	—	3	1	no	—
125	C₂×C₃×Q₈	—	2	2	no	—
126	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
127	C₂×#67	—	4	2	no	—
128	C₂×D₁₂	—	2	2	no	—
129	C₂×D₁₂	—	4	2	no	—
130	C₂×C₄×D₃	—	4	2	no	—
131	C₃×Q₁₆	—	2	2	no	—
132	C₃×Q₁₆	—	2	2	no	—
133	C₃×Q₁₆	—	4	2	no	—
134	C₃×M₁₆	—	2	2	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
140	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
141	C₃×C₄⋊C₄	—	2	2	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	2	no	—
146	C₃×C₄⋊C₄	D₈	1	2	yes	—
147	C₃×C₂²⋊C₄	—	4	2	no	—
148	C₃×C₂²⋊C₄	D₈	1	2	yes	—
149	C₃×C₄²	—	4	2	no	—
150	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
151	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
152	(C₂×D₃)⋊C₄	—	4	2	no	—
153	(C₂×D₃)⋊C₄	—	4	2	no	—
154	C₄⋊Dic₃	—	2	2	no	—
155	Dic₃⋊C₄	—	4	2	no	—
156	D₂₄	—	2	2	no	—
157	D₂₄	—	2	2	no	—
158	D₂₄	—	4	2	no	—
159	C₃⋊C₁₆	—	2	1	no	—
160	C₃⋊C₁₆	—	2	1	no	—
161	C₃⋊C₁₆	#41	1	2	yes	—
162	C₃⋊C₁₆	#41	1	2	yes	—
163	C₂×Q₁₆	—	2	1	no	—
164	C₂×Q₁₆	C₂²×D₃	1	2	yes	—
165	C₂×D₈	—	3	2	no	—
166	C₂×D₈	—	6	1	no	—
167	C₂×M₁₆	#67	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	1	no	—
170	#127	—	6	1	no	—
171	#126	—	4	2	no	—
172	#125	—	2	1	no	—
173	#125	—	4	2	no	—
174	#125	—	6	1	no	—
175	#123	—	12	2	no	—
176	C₂²⋊Q₈	—	2	2	no	—
177	#121	—	6	2	no	—
178	#121	—	12	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×D₄	—	6	2	no	—
181	#117	#67	1	1	yes	characteristic
182	SD₃₂	—	6	4	no	—
183	SD₃₂	—	6	4	no	—
184	M₃₂	—	6	2	no	—
185	C₂×C₁₆	—	3	1	no	—
186	C₂×C₁₆	—	3	1	no	—
187	C₂×C₁₆	—	3	1	no	—
188	C₂×C₁₆	—	3	1	no	—
189	C₂×C₁₆	—	6	2	no	—
190	#107	—	3	2	no	—
191	#107	C₂²×D₃	1	2	yes	—
192	#106	—	2	1	no	—
193	#106	—	6	1	no	—
194	#103	—	2	2	no	—
195	#103	—	2	2	no	—
196	#102	—	6	2	no	—
197	#102	—	6	2	no	—
198	#97	#67	1	1	yes	characteristic
199	#97	#67	1	1	yes	characteristic
200	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
202	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
203	C₂²×D₃	—	4	2	no	—
204	C₃×Q₈	—	4	2	no	—
205	C₃×Q₈	—	4	2	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	4	2	no	—
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
211	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
213	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
214	#67	—	16	2	no	—
215	C₂×Dic₃	—	4	2	no	—
216	D₁₂	—	4	2	no	—
217	D₁₂	—	4	2	no	—
218	D₁₂	—	16	2	no	—
219	C₄×D₃	—	16	2	no	—
220	C₃×C₈	—	2	1	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	#135	1	2	yes	—
223	C₃×C₈	C₂×#41	1	1	yes	characteristic
224	C₃×C₈	C₂×#41	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	6	2	no	—
227	C₂×D₄	—	12	2	no	—
228	C₂×D₄	—	12	2	no	—
229	C₂²×C₄	—	12	2	no	—
230	C₂²×C₄	C₂×#67	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	4	2	no	—
234	D₈	—	6	2	no	—
235	D₈	—	6	2	no	—
236	D₈	—	12	2	no	—
237	M₁₆	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	C₂×#67	1	1	yes	characteristic
241	C₂×C₈	C₂×#67	1	1	yes	characteristic
242	C₂×C₈	C₂×#67	1	1	yes	characteristic
243	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	6	2	no	—
250	C₄⋊C₄	—	12	2	no	—
251	C₄⋊C₄	#213	1	2	yes	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	12	2	no	—
254	C₂²⋊C₄	—	12	2	no	—
255	C₂²⋊C₄	#213	1	2	yes	—
256	C₄²	—	4	2	no	—
257	C₄²	C₂×#67	1	1	yes	characteristic
258	C₄²	C₂×#67	1	1	yes	characteristic
259	C₁₆	—	3	2	no	—
260	C₁₆	—	3	2	no	—
261	C₁₆	—	6	1	no	—
262	C₁₆	—	6	1	no	—
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
266	C₂×D₃	—	8	2	no	—
267	C₂×D₃	—	8	2	no	—
268	C₂×D₃	—	8	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	8	2	no	—
273	C₃×C₄	—	8	2	no	—
274	C₃×C₄	#532	1	1	yes	characteristic
275	C₃×C₄	C₂×#135	1	1	yes	characteristic
276	Dic₃	—	8	2	no	—
277	C₂³	—	12	2	no	—
278	C₂³	C₂×#213	1	1	yes	characteristic
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	12	2	no	—
282	D₄	—	12	2	no	—
283	D₄	—	48	2	no	—
284	D₄	—	48	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	12	2	no	—
290	C₂×C₄	—	48	2	no	—
291	C₂×C₄	C₂×#213	1	2	yes	—
292	C₂×C₄	C₂×#213	1	2	yes	—
293	C₂×C₄	C₂×#213	1	1	yes	characteristic
294	C₂×C₄	C₂²×#67	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#950	1	2	yes	—
298	C₈	D₃×#41	1	1	yes	characteristic
299	C₈	D₃×#41	1	1	yes	characteristic
300	C₂×C₃	—	4	1	no	—
301	C₂×C₃	#3169	1	1	yes	characteristic
302	C₂×C₃	#3416	1	2	yes	—
303	D₃	—	16	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	24	2	no	—
307	C₂²	—	24	2	no	—
308	C₂²	—	24	2	no	—
309	C₂²	C₂²×#213	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	24	2	no	—
316	C₄	C₂×#950	1	1	yes	characteristic
317	C₄	#5832	1	1	yes	characteristic
318	C₃	#33743	1	1	yes	characteristic, complemented, Sylow
319	C₂	—	4	1	no	—
320	C₂	—	48	2	no	—
321	C₂	#83291	1	1	yes	characteristic
322	C₂	#84016	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#213
commutator, cosocle	C₂×C₃×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×#3177	C₂
socle	C₂²×C₃	C₂²×D₈
prime radical	#78569	C₂
hypercenter	#3177	D₃
hypocenter	C₃	#33743
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
1. C₃×#3177	C₂	semidirect
3. #71354	C₂	semidirect
4. #71352	C₂	semidirect
5. #71313	C₂	semidirect
6. #71311	C₂	semidirect
7. #71084	C₂	semidirect
8. #71082	C₂	semidirect
9. #70913	C₂	semidirect
10. C₄×#828	C₂	semidirect
11. #69881	C₂	semidirect
44. #3177	D₃	semidirect
318. C₃	#33743	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3177	2
#78569	2
#71354	2
#71352	2
#71313	2
#71311	2
#71084	2
#71082	2
#70913	2
C₄×#828	2
#69881	2
#33743	3