Subgroups — #69366

Order 384.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4901	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#4499	C₂	1	1	yes	maximal, characteristic, complemented
4	#4447	C₂	1	2	yes	maximal, complemented
5	D₃×C₃₂	C₂	1	2	yes	maximal, complemented
6	#2258	—	3	1	no	maximal, Sylow
7	#901	C₄	1	1	yes	characteristic
8	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	#812	C₄	1	1	yes	characteristic
10	#798	C₄	1	2	yes	—
11	D₃×C₁₆	C₂²	1	2	yes	—
12	C₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₃×C₃₂	C₂²	1	1	yes	characteristic
14	C₃⋊C₃₂	C₂²	1	2	yes	—
15	#504	—	3	1	no	—
16	M₆₄	—	3	1	no	—
17	M₆₄	—	3	2	no	—
18	C₂×C₃₂	—	3	2	no	—
19	C₂×C₃₂	D₃	1	1	yes	characteristic, complemented
20	#235	C₈	1	1	yes	characteristic
21	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	#208	C₈	1	1	yes	characteristic
23	ΓL(1,𝔽₂₅)	C₈	1	2	yes	—
24	D₃×C₈	C₂×C₄	1	2	yes	—
25	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₃×C₁₆	C₂³	1	1	yes	characteristic
27	C₃⋊C₁₆	C₂×C₄	1	2	yes	—
28	#131	—	3	1	no	—
29	M₃₂	—	3	1	no	—
30	M₃₂	—	3	2	no	—
31	C₂×C₁₆	—	3	2	no	—
32	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
33	C₃₂	—	3	2	no	—
34	C₃₂	C₂×D₃	1	1	yes	characteristic
35	C₃₂	C₂×D₃	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
37	#67	C₁₆	1	2	yes	—
38	D₁₂	C₁₆	1	1	yes	characteristic
39	C₄×D₃	C₂×C₈	1	2	yes	—
40	Dic₆	C₁₆	1	1	yes	characteristic
41	C₃×C₈	C₂×C₈	1	1	yes	characteristic
42	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₃⋊C₈	C₂×C₈	1	2	yes	—
44	#41	—	3	1	no	—
45	M₁₆	—	3	1	no	—
46	M₁₆	—	3	2	no	—
47	C₂×C₈	—	3	2	no	—
48	C₂×C₈	C₄×D₃	1	1	yes	characteristic
49	C₁₆	—	3	2	no	—
50	C₁₆	C₄×D₃	1	1	yes	characteristic
51	C₁₆	C₂²×D₃	1	1	yes	characteristic
52	C₂²×C₃	C₂×C₁₆	1	1	yes	characteristic
53	C₂×D₃	C₂×C₁₆	1	2	yes	—
54	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
55	C₃×C₄	C₂²×C₈	1	1	yes	characteristic
56	Dic₃	C₂×C₁₆	1	2	yes	—
57	Q₈	—	3	1	no	—
58	D₄	—	3	2	no	—
59	D₄	—	3	1	no	—
60	C₂×C₄	—	3	2	no	—
61	C₂×C₄	D₃×C₈	1	1	yes	characteristic
62	C₈	—	3	2	no	—
63	C₈	D₃×C₈	1	1	yes	characteristic
64	C₈	C₂×C₄×D₃	1	1	yes	characteristic
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	C₂²×C₁₆	1	1	yes	characteristic
67	D₃	—	2	2	no	—
68	C₂²	—	3	2	no	—
69	C₂²	D₃×C₁₆	1	1	yes	characteristic
70	C₄	—	3	2	no	—
71	C₄	D₃×C₁₆	1	1	yes	characteristic
72	C₄	C₂×D₃×C₈	1	1	yes	characteristic
73	C₃	#2258	1	1	yes	characteristic, complemented, Sylow
74	C₂	—	2	1	no	—
75	C₂	—	6	2	no	—
76	C₂	C₂×D₃×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂×D₃
commutator, socle	C₂×C₃	C₂²×C₁₆
Frattini	C₁₆	C₂²×D₃
Fitting	C₂×C₃×C₃₂	C₂
prime radical	#235	C₈
cosocle	C₃×C₁₆	C₂³
hypercenter	C₂×C₃₂	D₃
hypocenter	C₃	#2258
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃₂	C₂	semidirect
3. #4499	C₂	semidirect
4. #4447	C₂	semidirect
5. D₃×C₃₂	C₂	semidirect
19. C₂×C₃₂	D₃	semidirect
73. C₃	#2258	semidirect

Maximal subgroups

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

subgroup	index
#4901	2
C₂×C₃×C₃₂	2
#4499	2
#4447	2
D₃×C₃₂	2
#2258	3