Subgroups — #10950822

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	#78577	C₂	1	1	yes	maximal, characteristic, complemented
2	#78569	C₂	1	1	yes	maximal, characteristic
3	C₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
4	#71444	C₂	1	2	yes	maximal, complemented
5	#71414	C₂	1	2	yes	maximal, complemented
6	#69083	C₂	1	1	yes	maximal, characteristic, complemented
7	#69080	C₂	1	1	yes	maximal, characteristic, complemented
8	#69077	C₂	1	2	yes	maximal, complemented
9	C₄×#800	C₂	1	2	yes	maximal, complemented
10	#69061	C₂	1	1	yes	maximal, characteristic, complemented
11	#69045	C₂	1	1	yes	maximal, characteristic, complemented
12	#33743	—	3	1	no	maximal, Sylow
13	#5535	—	2	1	no	—
14	#5518	—	2	1	no	—
15	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₃×#405	C₂²	1	1	yes	characteristic
18	#5137	—	2	1	no	—
19	#5135	—	2	1	no	—
20	#5116	C₂²	1	2	yes	—
21	#5115	C₂²	1	2	yes	—
22	C₂×#800	C₂²	1	2	yes	—
23	C₂×#800	C₂²	1	2	yes	—
24	#4716	—	2	1	no	—
25	#4713	—	2	1	no	—
26	#4711	—	2	1	no	—
27	#4703	—	2	1	no	—
28	#4699	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₂	C₂²	1	2	yes	—
30	#4695	C₂²	1	1	yes	characteristic
31	C₄×D₂₄	C₂²	1	2	yes	—
32	#4683	C₂²	1	1	yes	characteristic
33	#4683	C₂²	1	1	yes	characteristic
34	C₃×#346	C₂²	1	1	yes	characteristic
35	C₃×#346	C₂²	1	1	yes	characteristic
36	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
37	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
38	#4509	C₂²	1	2	yes	—
39	#4509	C₂²	1	2	yes	—
40	#4505	C₂²	1	2	yes	—
41	#4504	C₂²	1	2	yes	—
42	#4503	C₂²	1	2	yes	—
43	#4503	C₂²	1	2	yes	—
44	#3177	—	3	1	no	—
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	D₃	1	1	yes	characteristic, complemented
55	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₃×#117	D₄	1	1	yes	characteristic
58	#930	—	2	2	no	—
59	#924	—	2	2	no	—
60	C₂×Dic₁₂	—	2	1	no	—
61	C₂×Dic₁₂	C₂³	1	2	yes	—
62	C₂×D₂₄	—	2	1	no	—
63	C₂×D₂₄	C₂³	1	2	yes	—
64	#894	—	4	2	no	—
65	#889	—	4	2	no	—
66	#884	—	4	2	no	—
67	#879	—	4	2	no	—
68	#874	—	2	1	no	—
69	C₄×D₁₂	—	2	2	no	—
70	#870	—	2	1	no	—
71	#870	—	2	1	no	—
72	#869	—	2	1	no	—
73	C₄×Dic₆	—	2	2	no	—
74	C₃×M₃₂	—	2	2	no	—
75	C₂×C₃×C₁₆	—	2	2	no	—
76	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
80	C₃×#97	D₄	1	1	yes	characteristic
81	C₃×#97	D₄	1	1	yes	characteristic
82	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
83	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
84	#821	—	2	2	no	—
85	#821	—	2	2	no	—
86	#818	C₂³	1	2	yes	—
87	#818	C₂³	1	2	yes	—
88	#817	—	2	1	no	—
89	#817	—	2	1	no	—
90	#816	—	2	2	no	—
91	#816	—	2	2	no	—
92	#800	—	2	4	no	—
93	#800	—	2	4	no	—
94	#541	—	6	1	no	—
95	#540	—	6	1	no	—
96	C₂×SD₃₂	—	3	2	no	—
97	C₂×SD₃₂	—	3	2	no	—
98	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
99	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
100	#499	—	3	1	no	—
101	#499	—	3	1	no	—
102	#494	—	3	1	no	—
103	#493	—	3	1	no	—
104	#491	—	6	1	no	—
105	#490	—	6	1	no	—
106	#470	—	6	1	no	—
107	#469	—	6	1	no	—
108	#467	—	3	2	no	—
109	#466	—	3	2	no	—
110	#442	—	6	1	no	—
111	#441	—	6	1	no	—
112	C₄×Q₁₆	—	3	2	no	—
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	C₂×D₃	1	1	yes	characteristic
122	#346	C₂×D₃	1	1	yes	characteristic
123	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
124	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
126	C₂×#67	—	4	2	no	—
127	C₂×D₁₂	—	2	2	no	—
128	C₂×D₁₂	—	4	2	no	—
129	C₂×C₄×D₃	—	4	2	no	—
130	C₂×Dic₆	—	2	2	no	—
131	C₃×M₁₆	—	2	2	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	1	no	—
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×D₄	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
138	C₃×C₄⋊C₄	D₈	1	2	yes	—
139	C₃×C₂²⋊C₄	D₈	1	2	yes	—
140	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
141	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
142	#217	—	4	2	no	—
143	(C₂×D₃)⋊C₄	—	4	2	no	—
144	(C₂×D₃)⋊C₄	—	4	2	no	—
145	C₄⋊Dic₃	—	2	2	no	—
146	C₄⋊Dic₃	—	2	2	no	—
147	C₄⋊Dic₃	—	4	2	no	—
148	Dic₃⋊C₄	—	4	2	no	—
149	Dic₃⋊C₄	—	4	2	no	—
150	Dic₃⋊C₄	—	4	2	no	—
151	Dic₃⋊C₄	—	4	2	no	—
152	C₄×Dic₃	—	4	2	no	—
153	Dic₁₂	—	2	2	no	—
154	Dic₁₂	—	2	2	no	—
155	Dic₁₂	—	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	—	3	2	no	—
164	C₂×Q₁₆	—	6	1	no	—
165	C₂×D₈	—	3	2	no	—
166	C₂×D₈	—	6	1	no	—
167	C₂×M₁₆	D₁₂	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	6	1	no	—
170	#127	—	6	1	no	—
171	#126	—	12	2	no	—
172	#125	—	6	1	no	—
173	#125	—	6	1	no	—
174	#125	—	12	2	no	—
175	#123	—	12	2	no	—
176	C₂²⋊Q₈	—	6	2	no	—
177	#121	—	6	2	no	—
178	#121	—	12	2	no	—
179	C₄×Q₈	—	6	2	no	—
180	C₄×D₄	—	6	2	no	—
181	#117	D₁₂	1	1	yes	characteristic
182	SD₃₂	—	6	4	no	—
183	SD₃₂	—	6	4	no	—
184	M₃₂	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
187	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
188	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
189	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
190	#107	—	3	2	no	—
191	#107	—	3	2	no	—
192	#106	—	6	1	no	—
193	#106	—	6	1	no	—
194	#103	—	6	2	no	—
195	#103	—	6	2	no	—
196	#102	—	6	2	no	—
197	#102	—	6	2	no	—
198	#97	D₁₂	1	1	yes	characteristic
199	#97	D₁₂	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	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
207	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
208	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
210	#67	—	16	2	no	—
211	C₂×Dic₃	—	4	2	no	—
212	C₂×Dic₃	—	4	2	no	—
213	C₂×Dic₃	—	4	2	no	—
214	D₁₂	—	4	2	no	—
215	D₁₂	—	4	2	no	—
216	D₁₂	—	16	2	no	—
217	C₄×D₃	—	16	2	no	—
218	Dic₆	—	4	2	no	—
219	Dic₆	—	4	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	—	6	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×D₁₂	1	1	yes	characteristic
231	Q₁₆	—	6	2	no	—
232	Q₁₆	—	6	2	no	—
233	Q₁₆	—	12	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×D₁₂	1	1	yes	characteristic
241	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
242	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
243	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
244	C₄⋊C₄	—	6	2	no	—
245	C₄⋊C₄	—	6	2	no	—
246	C₄⋊C₄	—	12	2	no	—
247	C₄⋊C₄	—	12	2	no	—
248	C₄⋊C₄	—	12	2	no	—
249	C₄⋊C₄	—	12	2	no	—
250	C₄⋊C₄	—	12	2	no	—
251	C₄⋊C₄	D₂₄	1	2	yes	—
252	C₂²⋊C₄	—	12	2	no	—
253	C₂²⋊C₄	—	12	2	no	—
254	C₂²⋊C₄	—	12	2	no	—
255	C₂²⋊C₄	D₂₄	1	2	yes	—
256	C₄²	—	12	2	no	—
257	C₄²	C₂×D₁₂	1	1	yes	characteristic
258	C₄²	C₂×D₁₂	1	1	yes	characteristic
259	C₁₆	—	2	1	no	—
260	C₁₆	—	2	1	no	—
261	C₁₆	#235	1	2	yes	—
262	C₁₆	#235	1	2	yes	—
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	#532	1	1	yes	characteristic
273	C₃×C₄	C₂×#135	1	1	yes	characteristic
274	Dic₃	—	8	2	no	—
275	Dic₃	—	8	2	no	—
276	Dic₃	—	8	2	no	—
277	C₂³	—	12	2	no	—
278	C₂³	C₂×D₂₄	1	1	yes	characteristic
279	Q₈	—	12	2	no	—
280	Q₈	—	12	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	—	12	2	no	—
288	C₂×C₄	—	12	2	no	—
289	C₂×C₄	—	12	2	no	—
290	C₂×C₄	—	48	2	no	—
291	C₂×C₄	C₂×D₂₄	1	2	yes	—
292	C₂×C₄	C₂×D₂₄	1	2	yes	—
293	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	#904	1	2	yes	—
298	C₈	C₂×#235	1	1	yes	characteristic
299	C₈	C₂×#235	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×D₂₄	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	24	2	no	—
314	C₄	—	24	2	no	—
315	C₄	—	24	2	no	—
316	C₄	#5477	1	1	yes	characteristic
317	C₄	C₂×#904	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	#78470	1	1	yes	characteristic
322	C₂	#82014	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₃×#2108	C₂
socle	C₂²×C₃	C₂²×D₈
prime radical	#78569	C₂
hypercenter	#2108	D₃
hypocenter	C₃	#33743
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
1. #78577	C₂	semidirect
3. C₃×#2108	C₂	semidirect
4. #71444	C₂	semidirect
5. #71414	C₂	semidirect
6. #69083	C₂	semidirect
7. #69080	C₂	semidirect
8. #69077	C₂	semidirect
9. C₄×#800	C₂	semidirect
10. #69061	C₂	semidirect
11. #69045	C₂	semidirect
54. #2108	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
#78577	2
#78569	2
C₃×#2108	2
#71444	2
#71414	2
#69083	2
#69080	2
#69077	2
C₄×#800	2
#69061	2
#69045	2
#33743	3