Subgroups — #10712927

Order 768.

326 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4515	C₂	1	1	yes	maximal, characteristic
3	#71432	C₂	1	1	yes	maximal, characteristic
4	#71419	C₂	1	1	yes	maximal, characteristic, complemented
5	#69789	C₂	1	1	yes	maximal, characteristic
6	#67815	C₂	1	1	yes	maximal, characteristic, complemented
7	#67657	C₂	1	1	yes	maximal, characteristic, complemented
8	#13122	—	3	1	no	maximal, Sylow
9	C₂×#901	C₂²	1	1	yes	characteristic
10	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#5113	C₄	1	1	yes	characteristic
14	#5101	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×#805	C₄	1	1	yes	characteristic
18	C₃×#364	—	2	2	no	—
19	#4552	C₄	1	1	yes	characteristic
20	#4523	C₄	1	1	yes	characteristic
21	#4515	—	2	2	no	—
22	(C₂×D₃)⋊C₁₆	—	2	1	no	—
23	(C₂×D₃)⋊C₁₆	—	2	1	no	—
24	#4502	—	2	1	no	—
25	Dic₃×C₁₆	—	2	1	no	—
26	C₂×#364	D₃	1	1	yes	characteristic, complemented
27	C₂×#350	—	3	1	no	—
28	#2113	—	3	1	no	—
29	#2108	—	3	1	no	—
30	#1764	—	3	1	no	—
31	#1379	—	3	1	no	—
32	#1374	—	3	1	no	—
33	C₂×#235	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	#926	C₂×C₄	1	1	yes	characteristic
38	C₂×#208	C₂×C₄	1	1	yes	characteristic
39	C₂×#208	C₂×C₄	1	1	yes	characteristic
40	#901	—	2	1	no	—
41	#901	—	2	1	no	—
42	#901	—	2	1	no	—
43	#901	—	2	1	no	—
44	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
45	C₂×D₃×C₈	—	2	1	no	—
46	C₃×M₃₂	—	2	1	no	—
47	C₃×M₃₂	—	2	1	no	—
48	C₃×M₃₂	—	2	1	no	—
49	C₃×M₃₂	D₄	1	2	yes	—
50	C₂×C₃×C₁₆	—	2	1	no	—
51	C₂×C₃×C₁₆	—	2	1	no	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	#821	C₈	1	1	yes	characteristic
55	#821	C₈	1	1	yes	characteristic
56	#816	C₈	1	1	yes	characteristic
57	#816	C₈	1	1	yes	characteristic
58	#815	—	2	1	no	—
59	C₈×Dic₃	—	2	1	no	—
60	C₂×C₃⋊C₁₆	—	2	1	no	—
61	C₂×C₃⋊C₁₆	—	2	1	no	—
62	C₂×C₃⋊C₁₆	—	2	1	no	—
63	C₂×C₃⋊C₁₆	—	2	1	no	—
64	C₂×C₃⋊C₁₆	—	2	1	no	—
65	C₂×C₃⋊C₁₆	D₄	1	1	yes	characteristic
66	C₂×C₃⋊C₁₆	D₄	1	1	yes	characteristic
67	#805	—	2	2	no	—
68	C₂×#131	—	3	1	no	—
69	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
70	C₂×M₃₂	C₂×D₃	1	1	yes	characteristic
71	C₂²×C₁₆	—	3	1	no	—
72	C₂²×C₁₆	—	3	1	no	—
73	C₂×#104	—	3	1	no	—
74	#416	—	3	1	no	—
75	#405	—	3	1	no	—
76	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
77	#364	—	2	2	no	—
78	#350	—	6	2	no	—
79	C₂²⋊C₁₆	—	6	1	no	—
80	C₂²⋊C₁₆	—	6	1	no	—
81	#346	—	6	1	no	—
82	C₄×C₁₆	—	6	1	no	—
83	#341	—	3	1	no	—
84	#339	—	3	1	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×#67	—	2	1	no	—
88	#235	—	2	1	no	—
89	#235	—	2	1	no	—
90	#235	—	2	1	no	—
91	#235	—	2	1	no	—
92	C₂×D₁₂	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄×D₃	—	2	1	no	—
94	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₃×C₈	#41	1	1	yes	characteristic
102	C₂×C₃×C₈	#41	1	1	yes	characteristic
103	C₂×C₃×C₈	#41	1	1	yes	characteristic
104	C₂×C₃×C₈	#41	1	1	yes	characteristic
105	C₃×C₄²	—	2	1	no	—
106	C₃×C₄²	—	2	1	no	—
107	#217	—	2	1	no	—
108	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
109	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
110	C₄×Dic₃	—	2	1	no	—
111	#208	—	2	1	no	—
112	#208	—	2	1	no	—
113	#208	—	2	1	no	—
114	#208	M₁₆	1	2	yes	—
115	C₂×C₃⋊C₈	—	2	1	no	—
116	C₂×C₃⋊C₈	—	2	1	no	—
117	ΓL(1,𝔽₂₅)	—	4	1	no	—
118	ΓL(1,𝔽₂₅)	—	4	1	no	—
119	D₃×C₈	—	4	1	no	—
120	D₃×C₈	—	4	1	no	—
121	C₃×C₁₆	—	2	2	no	—
122	C₃×C₁₆	—	4	1	no	—
123	C₃⋊C₁₆	—	2	1	no	—
124	C₃⋊C₁₆	—	2	1	no	—
125	C₃⋊C₁₆	—	4	1	no	—
126	C₂×#41	—	3	1	no	—
127	#131	—	6	1	no	—
128	#131	—	6	1	no	—
129	#131	—	6	1	no	—
130	#131	—	6	1	no	—
131	C₂×M₁₆	—	3	1	no	—
132	C₂×M₁₆	—	3	1	no	—
133	C₂×M₁₆	—	6	1	no	—
134	C₂²×C₈	—	6	1	no	—
135	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
137	#117	—	3	1	no	—
138	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
139	M₃₂	—	2	1	no	—
140	M₃₂	—	2	1	no	—
141	M₃₂	—	2	1	no	—
142	M₃₂	#67	1	2	yes	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	3	1	no	—
146	C₂×C₁₆	—	3	1	no	—
147	C₂×C₁₆	—	6	1	no	—
148	C₂×C₁₆	—	6	1	no	—
149	C₂×C₁₆	—	6	1	no	—
150	C₂×C₁₆	—	6	1	no	—
151	C₂×C₁₆	—	6	1	no	—
152	#104	—	6	2	no	—
153	#103	—	3	1	no	—
154	#103	—	3	1	no	—
155	#102	—	3	1	no	—
156	#102	—	3	1	no	—
157	#97	—	6	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	6	1	no	—
161	C₂³×C₃	#131	1	1	yes	characteristic
162	C₂²×D₃	—	2	1	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	#117	1	1	yes	characteristic
167	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
171	C₂×C₃×C₄	#131	1	1	yes	characteristic
172	#67	—	4	1	no	—
173	#67	—	4	1	no	—
174	C₂×Dic₃	—	2	1	no	—
175	C₂×Dic₃	—	2	1	no	—
176	C₂×Dic₃	—	2	1	no	—
177	D₁₂	—	2	1	no	—
178	D₁₂	—	2	1	no	—
179	C₄×D₃	—	4	1	no	—
180	C₄×D₃	—	4	1	no	—
181	Dic₆	—	2	1	no	—
182	Dic₆	—	2	1	no	—
183	C₃×C₈	—	2	1	no	—
184	C₃×C₈	—	2	1	no	—
185	C₃×C₈	#121	1	1	yes	characteristic
186	C₃×C₈	#123	1	1	yes	characteristic
187	C₃×C₈	#124	1	1	yes	characteristic
188	C₃×C₈	#126	1	1	yes	characteristic
189	C₃⋊C₈	—	2	2	no	—
190	C₃⋊C₈	—	4	1	no	—
191	#41	—	6	1	no	—
192	#41	—	6	1	no	—
193	#41	—	6	1	no	—
194	#41	—	6	1	no	—
195	C₂×Q₈	—	3	1	no	—
196	C₂×D₄	—	3	1	no	—
197	C₂×D₄	—	6	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	6	1	no	—
200	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
201	M₁₆	—	3	2	no	—
202	M₁₆	—	6	1	no	—
203	M₁₆	—	6	1	no	—
204	M₁₆	—	6	1	no	—
205	M₁₆	—	12	1	no	—
206	M₁₆	—	12	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	6	1	no	—
210	C₂×C₈	—	6	1	no	—
211	C₂×C₈	—	12	1	no	—
212	C₂×C₈	—	12	1	no	—
213	C₂×C₈	D₃×C₈	1	1	yes	characteristic
214	C₂×C₈	D₃×C₈	1	1	yes	characteristic
215	C₂×C₈	#235	1	1	yes	characteristic
216	C₂×C₈	D₃×D₄	1	1	yes	characteristic
217	C₂×C₈	#237	1	1	yes	characteristic
218	C₂×C₈	#239	1	1	yes	characteristic
219	C₂×C₈	#239	1	1	yes	characteristic
220	C₂×C₈	C₂×#67	1	1	yes	characteristic
221	C₄⋊C₄	—	3	1	no	—
222	C₄⋊C₄	—	3	1	no	—
223	C₂²⋊C₄	—	6	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	6	1	no	—
227	C₁₆	—	2	2	no	—
228	C₁₆	—	4	1	no	—
229	C₁₆	—	6	1	no	—
230	C₁₆	—	6	1	no	—
231	C₁₆	—	12	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	#433	1	1	yes	characteristic
235	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
236	C₂²×C₃	#436	1	1	yes	characteristic
237	C₂×D₃	—	4	1	no	—
238	C₂×D₃	—	4	1	no	—
239	C₂×D₃	—	4	1	no	—
240	C₃×C₄	—	4	2	no	—
241	C₃×C₄	#388	1	1	yes	characteristic
242	C₃×C₄	#431	1	1	yes	characteristic
243	C₃×C₄	C₈×D₄	1	1	yes	characteristic
244	C₃×C₄	#435	1	1	yes	characteristic
245	Dic₃	—	4	1	no	—
246	Dic₃	—	4	1	no	—
247	Dic₃	—	4	1	no	—
248	C₂³	—	6	1	no	—
249	C₂³	#907	1	1	yes	characteristic
250	Q₈	—	6	1	no	—
251	Q₈	—	6	1	no	—
252	D₄	—	6	1	no	—
253	D₄	—	6	1	no	—
254	D₄	—	12	1	no	—
255	D₄	—	12	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	6	1	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	—	12	1	no	—
263	C₂×C₄	—	12	1	no	—
264	C₂×C₄	#892	1	1	yes	characteristic
265	C₂×C₄	#893	1	1	yes	characteristic
266	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
267	C₂×C₄	#901	1	1	yes	characteristic
268	C₂×C₄	D₃×M₁₆	1	1	yes	characteristic
269	C₂×C₄	C₄×#67	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	6	2	no	—
273	C₈	—	12	1	no	—
274	C₈	#894	1	1	yes	characteristic
275	C₈	#898	1	1	yes	characteristic
276	C₈	#939	1	1	yes	characteristic
277	C₈	#947	1	1	yes	characteristic
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	#1923	1	1	yes	characteristic
281	C₂×C₃	#2170	1	1	yes	characteristic
282	C₂×C₃	#2171	1	1	yes	characteristic
283	D₃	—	8	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	12	1	no	—
287	C₂²	—	12	1	no	—
288	C₂²	—	12	1	no	—
289	C₂²	#4834	1	1	yes	characteristic
290	C₂²	#4840	1	1	yes	characteristic
291	C₂²	#5128	1	1	yes	characteristic
292	C₄	—	4	2	no	—
293	C₄	—	12	1	no	—
294	C₄	—	12	1	no	—
295	C₄	—	12	1	no	—
296	C₄	#4833	1	1	yes	characteristic
297	C₄	#4836	1	1	yes	characteristic
298	C₄	#4990	1	1	yes	characteristic
299	C₄	C₈×#67	1	1	yes	characteristic
300	C₃	#13122	1	1	yes	characteristic, complemented, Sylow
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	24	1	no	—
304	C₂	#69300	1	1	yes	characteristic
305	C₂	#69301	1	1	yes	characteristic
306	C₂	#70449	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂²×C₃	C₈×D₄
prime radical	C₂×#235	C₂×C₄
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#13122
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#364	C₂	semidirect
4. #71419	C₂	semidirect
6. #67815	C₂	semidirect
7. #67657	C₂	semidirect
26. C₂×#364	D₃	semidirect
300. C₃	#13122	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#364	2
C₂×#4515	2
#71432	2
#71419	2
#69789	2
#67815	2
#67657	2
#13122	3