Subgroups — #10694045

Order 768.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈²	C₂	1	1	yes	maximal, characteristic
3	#69788	C₂	1	2	yes	maximal
4	C₂×#4456	C₂	1	2	yes	maximal
5	C₂×#4458	C₂	1	1	yes	maximal, characteristic
6	#9656	—	3	1	no	maximal, Sylow
7	C₂²×Dic₁₂	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#818	C₄	1	1	yes	characteristic
13	C₂×#816	C₄	1	2	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₄×Dic₁₂	—	2	2	no	—
16	C₄×Dic₁₂	—	2	2	no	—
17	C₃×C₈²	—	2	2	no	—
18	#4550	—	2	4	no	—
19	#4458	—	2	2	no	—
20	#4456	—	2	4	no	—
21	C₂×C₄×Q₁₆	—	3	1	no	—
22	#1984	—	3	2	no	—
23	C₂×#335	—	3	1	no	—
24	C₂×Q₈⋊C₈	—	3	2	no	—
25	C₂×C₈²	D₃	1	1	yes	characteristic
26	C₂²×Dic₆	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₈	—	2	1	no	—
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
33	C₂×Dic₃⋊C₄	—	2	2	no	—
34	C₂×C₄×Dic₃	—	2	2	no	—
35	C₂²×C₃⋊C₈	—	2	2	no	—
36	C₂×Dic₁₂	—	2	2	no	—
37	C₂×Dic₁₂	—	2	2	no	—
38	C₂×Dic₁₂	C₈	1	4	yes	complemented
39	C₄×Dic₆	—	2	2	no	—
40	C₄×Dic₆	—	2	2	no	—
41	C₄×Dic₆	—	2	2	no	—
42	C₄×Dic₆	—	2	2	no	—
43	C₃×C₄×C₈	—	2	1	no	—
44	C₃×C₄×C₈	—	2	1	no	—
45	C₃×C₄×C₈	—	2	1	no	—
46	C₃×C₄×C₈	D₄	1	2	yes	—
47	C₃×C₄×C₈	D₄	1	2	yes	—
48	C₃×C₄×C₈	D₄	1	2	yes	—
49	#818	—	2	2	no	—
50	#816	—	2	4	no	—
51	C₄⋊(C₃⋊C₈)	—	2	2	no	—
52	C₄⋊(C₃⋊C₈)	—	2	2	no	—
53	C₂²×Q₁₆	—	3	1	no	—
54	C₂×C₄×Q₈	—	3	2	no	—
55	C₄×Q₁₆	—	6	2	no	—
56	C₄×Q₁₆	—	6	2	no	—
57	C₂×#107	—	3	1	no	—
58	C₂×C₄⋊C₈	—	3	2	no	—
59	C₂×#103	—	3	2	no	—
60	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
63	#336	—	6	4	no	—
64	#335	—	6	2	no	—
65	Q₈⋊C₈	—	6	4	no	—
66	C₈²	—	2	2	no	—
67	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	#41	1	1	yes	characteristic
70	C₂²×Dic₃	—	2	2	no	—
71	C₂²×Dic₃	—	2	2	no	—
72	C₂×Dic₆	—	2	2	no	—
73	C₂×Dic₆	—	2	2	no	—
74	C₂×Dic₆	—	2	2	no	—
75	C₂×Dic₆	—	2	2	no	—
76	C₂×Dic₆	—	2	2	no	—
77	C₂×Dic₆	C₂×C₈	1	4	yes	—
78	C₂×C₃×C₈	—	2	4	no	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	1	no	—
82	C₂×C₃×C₈	—	2	1	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
87	C₂×C₃×C₈	M₁₆	1	2	yes	—
88	C₂×C₃×C₈	Q₁₆	1	4	yes	complemented
89	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
91	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₃×C₄²	#41	1	1	yes	characteristic
93	C₄⋊Dic₃	—	2	2	no	—
94	C₄⋊Dic₃	—	2	2	no	—
95	Dic₃⋊C₄	—	4	2	no	—
96	Dic₃⋊C₄	—	4	2	no	—
97	C₄×Dic₃	—	4	2	no	—
98	C₄×Dic₃	—	4	2	no	—
99	C₂×C₃⋊C₈	—	4	2	no	—
100	C₂×C₃⋊C₈	—	4	4	no	—
101	Dic₁₂	—	2	8	no	—
102	C₂²×Q₈	—	3	2	no	—
103	C₂×Q₁₆	—	3	4	no	—
104	C₂×Q₁₆	—	6	2	no	—
105	C₂×Q₁₆	—	6	2	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	—	6	2	no	—
108	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₈	D₁₂	1	2	yes	—
111	C₄×Q₈	—	6	2	no	—
112	C₄×Q₈	—	6	2	no	—
113	C₄×Q₈	—	6	2	no	—
114	C₄×Q₈	—	6	2	no	—
115	C₂×C₄⋊C₄	—	3	2	no	—
116	C₂×C₄⋊C₄	—	6	2	no	—
117	C₂×C₄²	—	6	2	no	—
118	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
119	#107	—	6	2	no	—
120	C₄⋊C₈	—	6	2	no	—
121	C₄⋊C₈	—	6	2	no	—
122	#103	—	6	4	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	D₁₂	1	2	yes	—
127	C₄×C₈	D₁₂	1	2	yes	—
128	C₄×C₈	#67	1	2	yes	—
129	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	#121	1	1	yes	characteristic
134	C₂×C₃×C₄	#121	1	1	yes	characteristic
135	C₂×C₃×C₄	#124	1	1	yes	characteristic
136	C₂×C₃×C₄	#127	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₃×C₄	#131	1	1	yes	characteristic
140	C₂×C₃×C₄	#131	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
142	C₂×C₃×C₄	#135	1	2	yes	—
143	C₂×Dic₃	—	2	4	no	—
144	C₂×Dic₃	—	4	2	no	—
145	C₂×Dic₃	—	4	2	no	—
146	C₂×Dic₃	—	4	2	no	—
147	C₂×Dic₃	—	4	4	no	—
148	Dic₆	—	2	4	no	—
149	Dic₆	—	2	4	no	—
150	Dic₆	—	4	2	no	—
151	Dic₆	—	4	2	no	—
152	C₃×C₈	—	2	4	no	—
153	C₃×C₈	—	2	2	no	—
154	C₃×C₈	—	2	4	no	—
155	C₃×C₈	C₂²⋊C₈	1	4	yes	—
156	C₃⋊C₈	—	8	2	no	—
157	C₂×Q₈	—	3	4	no	—
158	C₂×Q₈	—	6	2	no	—
159	C₂×Q₈	—	6	2	no	—
160	C₂×Q₈	—	6	2	no	—
161	C₂×Q₈	—	6	2	no	—
162	C₂×Q₈	—	6	2	no	—
163	C₂²×C₄	—	6	2	no	—
164	C₂²×C₄	—	6	2	no	—
165	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
167	C₂²×C₄	#235	1	1	yes	characteristic
168	Q₁₆	—	6	8	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	12	2	no	—
176	C₂×C₈	—	12	4	no	—
177	C₂×C₈	D₃×C₈	1	2	yes	—
178	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
179	C₂×C₈	Dic₁₂	1	4	yes	complemented
180	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
181	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
182	C₄⋊C₄	—	6	2	no	—
183	C₄⋊C₄	—	6	2	no	—
184	C₄⋊C₄	—	12	2	no	—
185	C₄⋊C₄	—	12	2	no	—
186	C₄²	—	12	2	no	—
187	C₄²	—	12	2	no	—
188	C₄²	C₂×D₁₂	1	1	yes	characteristic
189	C₄²	C₂×D₁₂	1	1	yes	characteristic
190	C₄²	#235	1	1	yes	characteristic
191	C₄²	C₂×#67	1	1	yes	characteristic
192	C₂²×C₃	#390	1	1	yes	characteristic
193	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
194	C₂²×C₃	#436	1	1	yes	characteristic
195	C₂²×C₃	C₄×Q₁₆	1	2	yes	—
196	C₂²×C₃	#443	1	2	yes	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
200	C₃×C₄	#408	1	1	yes	characteristic
201	C₃×C₄	C₈×D₄	1	1	yes	characteristic
202	C₃×C₄	#436	1	1	yes	characteristic
203	C₃×C₄	#467	1	2	yes	—
204	C₃×C₄	#494	1	1	yes	characteristic
205	C₃×C₄	#495	1	1	yes	characteristic
206	Dic₃	—	4	4	no	—
207	Dic₃	—	8	2	no	—
208	C₂³	C₄×D₁₂	1	1	yes	characteristic
209	Q₈	—	6	4	no	—
210	Q₈	—	6	4	no	—
211	Q₈	—	12	2	no	—
212	Q₈	—	12	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	6	4	no	—
215	C₂×C₄	—	12	2	no	—
216	C₂×C₄	—	12	2	no	—
217	C₂×C₄	—	12	2	no	—
218	C₂×C₄	—	12	4	no	—
219	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
220	C₂×C₄	#874	1	1	yes	characteristic
221	C₂×C₄	#875	1	1	yes	characteristic
222	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
223	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
224	C₂×C₄	#901	1	1	yes	characteristic
225	C₂×C₄	#901	1	1	yes	characteristic
226	C₂×C₄	#904	1	2	yes	—
227	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
228	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	#930	1	1	yes	characteristic
230	C₂×C₄	#930	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	2	2	no	—
233	C₈	—	2	4	no	—
234	C₈	—	24	2	no	—
235	C₈	(C₂×D₃)⋊C₈	1	4	yes	—
236	C₂×C₃	C₈×Q₁₆	1	2	yes	—
237	C₂×C₃	#1591	1	2	yes	—
238	C₂×C₃	#1954	1	1	yes	characteristic
239	C₂×C₃	#1969	1	1	yes	characteristic
240	C₂×C₃	#1974	1	1	yes	characteristic
241	C₂²	C₈×D₁₂	1	1	yes	characteristic
242	C₂²	#4688	1	1	yes	characteristic
243	C₂²	C₄×Dic₁₂	1	2	yes	—
244	C₂²	#4700	1	2	yes	—
245	C₂²	#4939	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	12	4	no	—
249	C₄	—	24	2	no	—
250	C₄	C₈×D₁₂	1	1	yes	characteristic
251	C₄	#4688	1	1	yes	characteristic
252	C₄	#4696	1	1	yes	characteristic
253	C₄	#4699	1	1	yes	characteristic
254	C₄	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
255	C₄	#5110	1	1	yes	characteristic
256	C₄	#5116	1	2	yes	—
257	C₃	#9656	1	1	yes	characteristic, complemented, Sylow
258	C₂	C₈×Dic₁₂	1	2	yes	—
259	C₂	#68056	1	2	yes	—
260	C₂	#69775	1	1	yes	characteristic
261	C₂	#69787	1	1	yes	characteristic
262	C₂	#69795	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×C₈²	C₂
socle, prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×C₈²	D₃
hypocenter	C₃	#9656
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
38. C₂×Dic₁₂	C₈	semidirect
88. C₂×C₃×C₈	Q₁₆	semidirect
179. C₂×C₈	Dic₁₂	semidirect
257. C₃	#9656	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₂	2
C₂×C₃×C₈²	2
#69788	2
C₂×#4456	2
C₂×#4458	2
#9656	3