Subgroups — #69369

Order 384.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4904	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₄₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#4449	C₂	1	2	yes	maximal, complemented
5	D₉₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#2262	—	3	1	no	maximal, Sylow
7	#904	—	2	2	no	—
8	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	Dic₂₄	C₂²	1	2	yes	—
10	#800	—	2	2	no	—
11	D₄₈	C₂²	1	2	yes	—
12	C₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₃×C₃₂	C₂²	1	1	yes	characteristic
14	#508	—	3	2	no	—
15	Q₆₄	—	3	1	no	—
16	SD₆₄	—	3	2	no	—
17	D₃₂	—	3	1	no	—
18	C₂×C₃₂	D₃	1	1	yes	characteristic, complemented
19	#235	—	4	2	no	—
20	C₂×C₃×C₈	D₄	1	1	yes	characteristic
21	Dic₁₂	—	2	2	no	—
22	D₂₄	—	2	2	no	—
23	#204	—	4	2	no	—
24	C₃×C₁₆	D₄	1	1	yes	characteristic
25	C₃×C₁₆	C₂³	1	1	yes	characteristic
26	#135	—	6	2	no	—
27	Q₃₂	—	3	2	no	—
28	SD₃₂	—	6	2	no	—
29	D₁₆	—	3	2	no	—
30	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
31	C₃₂	C₂×D₃	1	1	yes	characteristic
32	C₃₂	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃×C₄	D₈	1	1	yes	characteristic
34	#67	—	8	2	no	—
35	D₁₂	—	4	2	no	—
36	C₄×D₃	—	8	2	no	—
37	Dic₆	—	4	2	no	—
38	C₃×C₈	D₈	1	1	yes	characteristic
39	C₃×C₈	C₂×D₄	1	1	yes	characteristic
40	#41	—	12	2	no	—
41	Q₁₆	—	6	2	no	—
42	SD₁₆	—	12	2	no	—
43	D₈	—	6	2	no	—
44	C₂×C₈	D₁₂	1	1	yes	characteristic
45	C₁₆	D₁₂	1	1	yes	characteristic
46	C₁₆	C₂²×D₃	1	1	yes	characteristic
47	C₂²×C₃	D₁₆	1	1	yes	characteristic
48	C₂×D₃	—	8	2	no	—
49	C₃×C₄	D₁₆	1	1	yes	characteristic
50	C₃×C₄	C₂×D₈	1	1	yes	characteristic
51	Dic₃	—	8	2	no	—
52	Q₈	—	12	2	no	—
53	D₄	—	12	2	no	—
54	D₄	—	24	2	no	—
55	C₂×C₄	—	24	2	no	—
56	C₂×C₄	D₂₄	1	1	yes	characteristic
57	C₈	D₂₄	1	1	yes	characteristic
58	C₈	C₂×D₁₂	1	1	yes	characteristic
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
61	D₃	—	16	2	no	—
62	C₂²	—	24	2	no	—
63	C₂²	D₄₈	1	1	yes	characteristic
64	C₄	—	24	2	no	—
65	C₄	D₄₈	1	1	yes	characteristic
66	C₄	C₂×D₂₄	1	1	yes	characteristic
67	C₃	#2262	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	1	no	—
69	C₂	—	48	2	no	—
70	C₂	C₂×D₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₈
commutator, cosocle	C₃×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₃
Fitting	C₂×C₃×C₃₂	C₂
socle	C₂×C₃	C₂×D₁₆
hypercenter	C₂×C₃₂	D₃
hypocenter	C₃	#2262
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #4904	C₂	semidirect
2. C₂×C₃×C₃₂	C₂	semidirect
3. Dic₄₈	C₂	semidirect
4. #4449	C₂	semidirect
5. D₉₆	C₂	semidirect
18. C₂×C₃₂	D₃	semidirect
67. C₃	#2262	semidirect

Maximal subgroups

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

subgroup	index
#4904	2
C₂×C₃×C₃₂	2
Dic₄₈	2
#4449	2
D₉₆	2
#2262	3