Subgroups — #10711545

Order 768.

314 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#4463	C₂	1	2	yes	maximal
3	#67664	C₂	1	2	yes	maximal
4	#67663	C₂	1	2	yes	maximal
5	#12882	—	3	1	no	maximal, Sylow
6	C₂×C₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₃×#97	C₄	1	1	yes	characteristic
8	C₂×C₃×#97	C₂²	1	1	yes	characteristic
9	C₂×C₃×#97	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×C₈	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄³	C₄	1	1	yes	characteristic
13	C₂×#812	C₂²	1	4	yes	—
14	#4463	—	2	4	no	—
15	C₂×#364	—	3	2	no	—
16	C₄×#97	D₃	1	1	yes	characteristic
17	#1377	—	3	2	no	—
18	#1375	—	3	2	no	—
19	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄²	—	2	2	no	—
24	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₃×#97	—	2	2	no	—
28	C₃×#97	—	2	2	no	—
29	C₃×#97	—	2	2	no	—
30	C₃×#97	C₈	1	4	yes	—
31	C₃×C₄×C₈	—	2	2	no	—
32	C₃×C₄×C₈	D₄	1	2	yes	—
33	C₃×C₄×C₈	Q₈	1	2	yes	—
34	#812	—	2	4	no	—
35	#812	—	2	4	no	—
36	C₂×C₃⋊C₁₆	—	2	4	no	—
37	C₂×M₃₂	—	3	4	no	—
38	C₂×#97	Dic₃	1	1	yes	characteristic
39	C₂×#97	Dic₃	1	1	yes	characteristic
40	C₂×#97	C₂×D₃	1	1	yes	characteristic
41	C₂×#97	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄×C₈	Dic₃	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
44	C₄³	Dic₃	1	1	yes	characteristic
45	#364	—	6	4	no	—
46	C₂²×C₃×C₄	—	2	1	no	—
47	C₂²×C₃×C₄	—	2	1	no	—
48	C₂²×C₃×C₄	M₁₆	1	2	yes	—
49	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×C₈	—	2	1	no	—
51	C₂×C₃×C₈	—	2	1	no	—
52	C₂×C₃×C₈	—	2	1	no	—
53	C₂×C₃×C₈	—	2	1	no	—
54	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
56	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
57	C₂×C₃×C₈	M₁₆	1	2	yes	—
58	C₂×C₃×C₈	M₁₆	1	2	yes	—
59	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₈	#41	1	2	yes	—
62	C₂×C₃×C₈	#41	1	2	yes	—
63	C₃×C₄²	—	2	1	no	—
64	C₃×C₄²	—	2	1	no	—
65	C₃×C₄²	—	2	1	no	—
66	C₃×C₄²	—	2	1	no	—
67	C₃×C₄²	—	4	4	no	—
68	C₃×C₄²	C₄⋊C₄	1	2	yes	—
69	C₃×C₄²	C₂×C₈	1	2	yes	—
70	C₃⋊C₁₆	—	4	4	no	—
71	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
73	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
74	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₄²	—	2	2	no	—
76	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
77	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
78	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
79	M₃₂	—	6	4	no	—
80	M₃₂	—	6	4	no	—
81	C₂×C₁₆	—	6	4	no	—
82	#97	—	2	2	no	—
83	#97	—	2	2	no	—
84	#97	—	2	2	no	—
85	#97	C₃⋊C₈	1	4	yes	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	Dic₆	1	2	yes	—
88	C₄×C₈	D₁₂	1	2	yes	—
89	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₃×C₄	—	2	4	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	4	2	no	—
95	C₂×C₃×C₄	—	4	2	no	—
96	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₃×C₄	#117	1	1	yes	characteristic
98	C₂×C₃×C₄	#117	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
102	C₃×C₈	—	2	2	no	—
103	C₃×C₈	—	2	2	no	—
104	C₃×C₈	C₄⋊C₈	1	4	yes	—
105	C₃×C₈	#124	1	2	yes	—
106	C₃×C₈	#125	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	#208	1	2	yes	—
110	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
116	C₂×C₈	C₂×C₃⋊C₈	1	2	yes	—
117	C₂×C₈	#208	1	2	yes	—
118	C₂×C₈	#208	1	2	yes	—
119	C₂×C₈	C₄⋊Dic₃	1	2	yes	—
120	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
121	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
122	C₂×C₈	#235	1	2	yes	—
123	C₂×C₈	#235	1	2	yes	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	4	4	no	—
129	C₄²	C₂×C₃⋊C₈	1	2	yes	—
130	C₄²	C₄⋊Dic₃	1	2	yes	—
131	C₁₆	—	12	4	no	—
132	C₂²×C₃	—	2	2	no	—
133	C₂²×C₃	#423	1	1	yes	characteristic
134	C₂²×C₃	#431	1	1	yes	characteristic
135	C₂²×C₃	#432	1	1	yes	characteristic
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	—	4	2	no	—
138	C₃×C₄	—	4	2	no	—
139	C₃×C₄	#382	1	1	yes	characteristic
140	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₃×C₄	#431	1	1	yes	characteristic
142	C₃×C₄	#432	1	1	yes	characteristic
143	C₂³	C₂×#208	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
151	C₂×C₄	C₂×#208	1	1	yes	characteristic
152	C₂×C₄	C₂×#208	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
154	C₂×C₄	#926	1	1	yes	characteristic
155	C₂×C₄	#926	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	2	2	no	—
158	C₈	C₄⋊(C₃⋊C₈)	1	4	yes	—
159	C₈	#870	1	2	yes	—
160	C₈	#875	1	2	yes	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	1	no	—
163	C₂×C₃	#1831	1	1	yes	characteristic
164	C₂×C₃	#2165	1	2	yes	—
165	C₂²	—	2	2	no	—
166	C₂²	#4924	1	1	yes	characteristic
167	C₂²	#4925	1	1	yes	characteristic
168	C₂²	#4926	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	2	no	—
172	C₄	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
173	C₄	#4925	1	1	yes	characteristic
174	C₄	#4926	1	1	yes	characteristic
175	C₄	#4936	1	1	yes	characteristic
176	C₃	#12882	1	1	yes	characteristic, complemented, Sylow
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#69392	1	1	yes	characteristic
180	C₂	#69864	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#926
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×C₄×#97	C₂
socle	C₂²×C₃	#432
prime radical	C₂³×C₃	C₂×M₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₄×#97	D₃
hypocenter	C₃	#12882
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
176. C₃	#12882	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#97	2
C₂×#4463	2
#67664	2
#67663	2
#12882	3