Subgroups — #10716185

Order 768.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₈	C₂	1	4	yes	maximal
2	#69066	C₂	1	2	yes	maximal
3	C₃×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
4	#13726	—	3	1	no	maximal, Sylow
5	C₂×Dic₂₄	—	2	2	no	—
6	C₂×Dic₂₄	C₂²	1	4	yes	—
7	#4699	—	2	2	no	—
8	C₂×C₃×C₃₂	C₂²	1	2	yes	—
9	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	Dic₄₈	—	2	8	no	—
11	C₂×Q₆₄	—	3	4	no	—
12	#2247	—	3	2	no	—
13	C₄×C₃₂	D₃	1	1	yes	characteristic
14	C₂×Dic₁₂	—	2	4	no	—
15	C₂×Dic₁₂	—	4	2	no	—
16	#869	—	4	2	no	—
17	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
18	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₈	D₄	1	1	yes	characteristic
20	Dic₂₄	—	2	8	no	—
21	Dic₂₄	—	4	4	no	—
22	C₃×C₃₂	D₄	1	4	yes	—
23	C₂×Q₃₂	—	3	4	no	—
24	C₂×Q₃₂	—	6	2	no	—
25	#494	—	6	2	no	—
26	Q₆₄	—	6	8	no	—
27	C₂×C₃₂	C₂×D₃	1	2	yes	—
28	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
29	C₂×Dic₆	—	4	4	no	—
30	C₂×C₃×C₈	D₈	1	1	yes	characteristic
31	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
32	C₃×C₄²	D₈	1	1	yes	characteristic
33	C₄⋊Dic₃	—	8	4	no	—
34	Dic₁₂	—	4	4	no	—
35	Dic₁₂	—	4	4	no	—
36	Dic₁₂	—	8	4	no	—
37	C₃×C₁₆	D₈	1	2	yes	—
38	C₃×C₁₆	C₂×D₄	1	2	yes	—
39	C₂×Q₁₆	—	6	4	no	—
40	C₂×Q₁₆	—	12	2	no	—
41	C₄⋊Q₈	—	12	2	no	—
42	Q₃₂	—	6	8	no	—
43	Q₃₂	—	12	4	no	—
44	C₂×C₁₆	D₁₂	1	1	yes	characteristic
45	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
46	C₄×C₈	D₁₂	1	1	yes	characteristic
47	C₃₂	D₁₂	1	4	yes	—
48	C₂×C₃×C₄	D₁₆	1	2	yes	—
49	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₃	—	8	4	no	—
51	Dic₆	—	8	4	no	—
52	Dic₆	—	8	4	no	—
53	C₃×C₈	D₁₆	1	2	yes	—
54	C₃×C₈	#127	1	1	yes	characteristic
55	C₃×C₈	C₂×D₈	1	1	yes	characteristic
56	C₂×Q₈	—	12	4	no	—
57	Q₁₆	—	12	4	no	—
58	Q₁₆	—	12	4	no	—
59	Q₁₆	—	24	4	no	—
60	C₂×C₈	D₂₄	1	1	yes	characteristic
61	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
62	C₄⋊C₄	—	24	4	no	—
63	C₄²	D₂₄	1	1	yes	characteristic
64	C₁₆	D₂₄	1	2	yes	—
65	C₁₆	C₂×D₁₂	1	2	yes	—
66	C₂²×C₃	C₂×D₁₆	1	1	yes	characteristic
67	C₃×C₄	Q₆₄	1	4	yes	complemented
68	C₃×C₄	#493	1	1	yes	characteristic
69	C₃×C₄	C₂×D₁₆	1	1	yes	characteristic
70	Dic₃	—	16	4	no	—
71	Q₈	—	24	4	no	—
72	Q₈	—	24	4	no	—
73	C₂×C₄	—	24	4	no	—
74	C₂×C₄	D₄₈	1	2	yes	—
75	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
76	C₈	D₄₈	1	2	yes	—
77	C₈	#874	1	1	yes	characteristic
78	C₈	C₂×D₂₄	1	1	yes	characteristic
79	C₂×C₃	#2246	1	1	yes	characteristic
80	C₂×C₃	C₂×Q₆₄	1	2	yes	—
81	C₂²	C₂×D₄₈	1	1	yes	characteristic
82	C₄	—	48	4	no	—
83	C₄	Dic₄₈	1	4	yes	complemented
84	C₄	#4695	1	1	yes	characteristic
85	C₄	C₂×D₄₈	1	1	yes	characteristic
86	C₃	#13726	1	1	yes	characteristic, complemented, Sylow
87	C₂	#69060	1	1	yes	characteristic
88	C₂	C₂×Dic₄₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₈
commutator, cosocle	C₂×C₃×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₃
Fitting	C₃×C₄×C₃₂	C₂
socle, prime radical	C₂²×C₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₃
hypocenter	C₃	#13726
LCS	G ▷ C₂×C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
67. C₃×C₄	Q₆₄	semidirect
83. C₄	Dic₄₈	semidirect
86. C₃	#13726	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₄₈	2
#69066	2
C₃×C₄×C₃₂	2
#13726	3