Subgroups — #10694089

Order 768.

482 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#322	C₂	1	1	yes	maximal, characteristic
3	#69788	C₂	1	2	yes	maximal
4	C₂×#4456	C₂	1	2	yes	maximal
5	C₂×#4457	C₂	1	1	yes	maximal, characteristic
6	#9664	—	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×#322	—	2	2	no	—
18	#4550	—	2	4	no	—
19	#4457	—	2	2	no	—
20	#4456	—	2	4	no	—
21	C₂×C₄×Q₁₆	—	3	1	no	—
22	#1984	—	3	2	no	—
23	C₂×#334	—	3	1	no	—
24	C₂×Q₈⋊C₈	—	3	2	no	—
25	C₂×#322	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	4	no	—
37	C₂×Dic₁₂	—	2	2	no	—
38	C₄×Dic₆	—	2	2	no	—
39	C₄×Dic₆	—	2	2	no	—
40	C₄×Dic₆	—	2	2	no	—
41	C₄×Dic₆	—	2	2	no	—
42	C₃×C₄×C₈	—	2	1	no	—
43	C₃×C₄×C₈	—	2	1	no	—
44	C₃×C₄×C₈	—	2	1	no	—
45	C₃×C₄×C₈	D₄	1	2	yes	—
46	C₃×C₄×C₈	D₄	1	2	yes	—
47	C₃×C₄×C₈	D₄	1	2	yes	—
48	#818	—	2	2	no	—
49	#816	—	2	4	no	—
50	C₄⋊(C₃⋊C₈)	—	2	2	no	—
51	C₄⋊(C₃⋊C₈)	—	2	2	no	—
52	C₂²×Q₁₆	—	3	1	no	—
53	C₂×C₄×Q₈	—	3	2	no	—
54	C₄×Q₁₆	—	6	2	no	—
55	C₄×Q₁₆	—	6	2	no	—
56	C₂×#107	—	3	1	no	—
57	C₂×C₄⋊C₈	—	3	2	no	—
58	C₂×#103	—	3	2	no	—
59	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
62	#336	—	6	4	no	—
63	#334	—	6	2	no	—
64	Q₈⋊C₈	—	6	4	no	—
65	#322	—	2	2	no	—
66	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	#41	1	1	yes	characteristic
69	C₂²×Dic₃	—	2	2	no	—
70	C₂²×Dic₃	—	2	2	no	—
71	C₂×Dic₆	—	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	M₁₆	1	4	yes	—
77	C₂×C₃×C₈	—	2	4	no	—
78	C₂×C₃×C₈	—	2	2	no	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	4	no	—
83	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
85	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
86	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
87	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
88	C₃×C₄²	#41	1	1	yes	characteristic
89	C₄⋊Dic₃	—	2	2	no	—
90	C₄⋊Dic₃	—	2	2	no	—
91	Dic₃⋊C₄	—	4	2	no	—
92	Dic₃⋊C₄	—	4	2	no	—
93	C₄×Dic₃	—	4	2	no	—
94	C₄×Dic₃	—	4	2	no	—
95	C₂×C₃⋊C₈	—	4	2	no	—
96	C₂×C₃⋊C₈	—	4	4	no	—
97	Dic₁₂	—	2	8	no	—
98	C₂²×Q₈	—	3	2	no	—
99	C₂×Q₁₆	—	6	4	no	—
100	C₂×Q₁₆	—	6	2	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	6	2	no	—
103	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
105	C₂²×C₈	D₁₂	1	2	yes	—
106	C₄×Q₈	—	6	2	no	—
107	C₄×Q₈	—	6	2	no	—
108	C₄×Q₈	—	6	2	no	—
109	C₄×Q₈	—	6	2	no	—
110	C₂×C₄⋊C₄	—	3	2	no	—
111	C₂×C₄⋊C₄	—	6	2	no	—
112	C₂×C₄²	—	6	2	no	—
113	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
114	#107	—	6	2	no	—
115	C₄⋊C₈	—	6	2	no	—
116	C₄⋊C₈	—	6	2	no	—
117	#103	—	6	4	no	—
118	C₄×C₈	—	2	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	D₁₂	1	2	yes	—
122	C₄×C₈	D₁₂	1	2	yes	—
123	C₄×C₈	#67	1	2	yes	—
124	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	#121	1	1	yes	characteristic
129	C₂×C₃×C₄	#121	1	1	yes	characteristic
130	C₂×C₃×C₄	#124	1	1	yes	characteristic
131	C₂×C₃×C₄	#127	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
133	C₂×C₃×C₄	#131	1	2	yes	—
134	C₂×C₃×C₄	#137	1	4	yes	—
135	C₂×Dic₃	—	2	4	no	—
136	C₂×Dic₃	—	4	2	no	—
137	C₂×Dic₃	—	4	2	no	—
138	C₂×Dic₃	—	4	2	no	—
139	C₂×Dic₃	—	4	4	no	—
140	Dic₆	—	2	4	no	—
141	Dic₆	—	2	4	no	—
142	Dic₆	—	4	2	no	—
143	Dic₆	—	4	2	no	—
144	C₃×C₈	—	2	4	no	—
145	C₃×C₈	—	2	2	no	—
146	C₃×C₈	—	2	2	no	—
147	C₃×C₈	—	4	2	no	—
148	C₃⋊C₈	—	8	2	no	—
149	C₂×Q₈	—	3	4	no	—
150	C₂×Q₈	—	6	2	no	—
151	C₂×Q₈	—	6	2	no	—
152	C₂×Q₈	—	6	2	no	—
153	C₂×Q₈	—	6	2	no	—
154	C₂×Q₈	—	6	2	no	—
155	C₂²×C₄	—	6	2	no	—
156	C₂²×C₄	—	6	2	no	—
157	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
159	C₂²×C₄	#235	1	1	yes	characteristic
160	Q₁₆	—	6	8	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	12	2	no	—
168	C₂×C₈	—	12	4	no	—
169	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
170	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
171	C₄⋊C₄	—	6	2	no	—
172	C₄⋊C₄	—	6	2	no	—
173	C₄⋊C₄	—	12	2	no	—
174	C₄⋊C₄	—	12	2	no	—
175	C₄²	—	12	2	no	—
176	C₄²	—	12	2	no	—
177	C₄²	C₂×D₁₂	1	1	yes	characteristic
178	C₄²	C₂×D₁₂	1	1	yes	characteristic
179	C₄²	#235	1	1	yes	characteristic
180	C₄²	C₂×#67	1	1	yes	characteristic
181	C₂²×C₃	#390	1	1	yes	characteristic
182	C₂²×C₃	#435	1	2	yes	—
183	C₂²×C₃	#441	1	2	yes	—
184	C₂²×C₃	#443	1	2	yes	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	#407	1	1	yes	characteristic
188	C₃×C₄	#408	1	1	yes	characteristic
189	C₃×C₄	#435	1	2	yes	—
190	C₃×C₄	#470	1	2	yes	—
191	C₃×C₄	#497	1	2	yes	—
192	Dic₃	—	4	4	no	—
193	Dic₃	—	8	2	no	—
194	C₂³	C₄×D₁₂	1	1	yes	characteristic
195	Q₈	—	6	4	no	—
196	Q₈	—	6	4	no	—
197	Q₈	—	12	2	no	—
198	Q₈	—	12	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	6	4	no	—
201	C₂×C₄	—	12	2	no	—
202	C₂×C₄	—	12	2	no	—
203	C₂×C₄	—	12	2	no	—
204	C₂×C₄	—	12	4	no	—
205	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
206	C₂×C₄	#874	1	1	yes	characteristic
207	C₂×C₄	#875	1	1	yes	characteristic
208	C₂×C₄	D₃×M₁₆	1	2	yes	—
209	C₂×C₄	#907	1	2	yes	—
210	C₂×C₄	#909	1	4	yes	—
211	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₂×C₄	#930	1	1	yes	characteristic
213	C₂×C₄	#930	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	—	4	2	no	—
218	C₈	—	24	2	no	—
219	C₂×C₃	#1587	1	4	yes	—
220	C₂×C₃	#1955	1	1	yes	characteristic
221	C₂×C₃	#1971	1	1	yes	characteristic
222	C₂×C₃	#1974	1	1	yes	characteristic
223	C₂²	#4700	1	2	yes	—
224	C₂²	#4706	1	2	yes	—
225	C₂²	#4716	1	2	yes	—
226	C₂²	#4939	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	12	4	no	—
230	C₄	—	24	2	no	—
231	C₄	#4706	1	2	yes	—
232	C₄	#4715	1	2	yes	—
233	C₄	#5126	1	1	yes	characteristic
234	C₄	#5129	1	1	yes	characteristic
235	C₄	#5137	1	2	yes	—
236	C₃	#9664	1	1	yes	characteristic, complemented, Sylow
237	C₂	#68087	1	4	yes	—
238	C₂	#69795	1	1	yes	characteristic
239	C₂	#69824	1	1	yes	characteristic
240	C₂	#69837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#322	C₂
socle, prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#322	D₃
hypocenter	C₃	#9664
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
236. C₃	#9664	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×#322	2
#69788	2
C₂×#4456	2
C₂×#4457	2
#9664	3