Subgroups — #69383

Order 384.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4919	C₂	1	2	yes	maximal, complemented
2	C₃×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#4521	C₂	1	2	yes	maximal, complemented
4	D₉₆	C₂	1	1	yes	maximal, characteristic, complemented
5	D₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#2262	—	3	1	no	maximal, Sylow
7	#919	—	2	2	no	—
8	C₃×Q₃₂	C₂²	1	2	yes	—
9	#828	—	2	2	no	—
10	D₄₈	C₂²	1	2	yes	—
11	D₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₃⋊C₃₂	C₂²	1	1	yes	characteristic
14	#508	—	3	2	no	—
15	Q₆₄	D₃	1	1	yes	characteristic, complemented
16	SD₆₄	—	3	2	no	—
17	D₃₂	—	3	1	no	—
18	C₂×C₃₂	—	3	1	no	—
19	#239	—	4	2	no	—
20	C₃×Q₁₆	—	2	2	no	—
21	#215	—	4	2	no	—
22	D₂₄	—	2	2	no	—
23	D₃×C₈	D₄	1	1	yes	characteristic
24	C₃×C₁₆	C₂³	1	1	yes	characteristic
25	C₃⋊C₁₆	D₄	1	1	yes	characteristic
26	#135	—	6	2	no	—
27	Q₃₂	C₂×D₃	1	2	yes	—
28	SD₃₂	—	6	2	no	—
29	D₁₆	—	3	2	no	—
30	C₂×C₁₆	—	3	1	no	—
31	C₃₂	—	3	1	no	—
32	C₃₂	C₂×D₃	1	1	yes	characteristic
33	C₃×Q₈	—	4	2	no	—
34	D₁₂	—	4	2	no	—
35	D₁₂	—	8	2	no	—
36	C₄×D₃	—	8	2	no	—
37	C₄×D₃	D₈	1	1	yes	characteristic
38	C₃×C₈	C₂×D₄	1	1	yes	characteristic
39	C₃⋊C₈	D₈	1	1	yes	characteristic
40	#41	—	12	2	no	—
41	Q₁₆	—	2	2	no	—
42	SD₁₆	—	12	2	no	—
43	D₈	—	6	2	no	—
44	C₂×C₈	—	3	1	no	—
45	C₁₆	—	3	1	no	—
46	C₁₆	C₂²×D₃	1	1	yes	characteristic
47	C₂×D₃	—	8	2	no	—
48	C₂×D₃	D₁₆	1	1	yes	characteristic
49	C₃×C₄	—	8	2	no	—
50	C₃×C₄	C₂×D₈	1	1	yes	characteristic
51	Dic₃	D₁₆	1	1	yes	characteristic
52	Q₈	—	4	2	no	—
53	D₄	—	12	2	no	—
54	D₄	—	24	2	no	—
55	C₂×C₄	—	3	1	no	—
56	C₂×C₄	—	24	2	no	—
57	C₈	—	3	1	no	—
58	C₈	D₃×D₄	1	1	yes	characteristic
59	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
60	D₃	—	2	1	no	—
61	D₃	—	16	2	no	—
62	C₂²	—	3	1	no	—
63	C₂²	—	24	2	no	—
64	C₄	—	3	1	no	—
65	C₄	—	8	2	no	—
66	C₄	D₃×D₈	1	1	yes	characteristic
67	C₃	#2262	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	6	1	no	—
69	C₂	—	48	2	no	—
70	C₂	D₃×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×D₁₆
commutator, cosocle	C₃×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₃
Fitting	C₃×Q₆₄	C₂
socle	C₂×C₃	C₂×D₁₆
hypercenter	Q₆₄	D₃
hypocenter	C₃	#2262
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #4919	C₂	semidirect
2. C₃×Q₆₄	C₂	semidirect
3. #4521	C₂	semidirect
4. D₉₆	C₂	semidirect
5. D₃×C₃₂	C₂	semidirect
15. Q₆₄	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
#4919	2
C₃×Q₆₄	2
#4521	2
D₉₆	2
D₃×C₃₂	2
#2262	3