Subgroups — #67992

Order 384.

47 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#1335	—	3	1	no	maximal, Sylow
3	#857	—	4	1	no	maximal
4	#443	—	3	1	no	—
5	#364	—	3	1	no	—
6	C₈²	D₃	1	1	yes	characteristic, complemented
7	C₄²⋊C₃	—	4	1	no	—
8	#135	—	3	1	no	—
9	#131	—	6	1	no	—
10	M₃₂	—	6	1	no	—
11	#108	—	3	1	no	—
12	#104	—	6	1	no	—
13	C₄×C₈	—	3	1	no	—
14	S₄	—	16	1	no	—
15	#41	—	6	1	no	—
16	Q₁₆	—	3	1	no	—
17	SD₁₆	—	6	1	no	—
18	D₈	—	3	1	no	—
19	M₁₆	—	6	1	no	—
20	M₁₆	—	12	1	no	—
21	C₂×C₈	—	3	1	no	—
22	C₂×C₈	—	3	1	no	—
23	C₂×C₈	—	12	1	no	—
24	C₄²	S₄	1	1	yes	characteristic
25	C₁₆	—	12	1	no	—
26	A₄	—	16	1	no	—
27	Q₈	—	6	1	no	—
28	D₄	—	6	1	no	—
29	D₄	—	12	1	no	—
30	C₂×C₄	—	3	1	no	—
31	C₂×C₄	—	12	1	no	—
32	C₈	—	3	1	no	—
33	C₈	—	3	1	no	—
34	C₈	—	3	1	no	—
35	C₈	—	3	1	no	—
36	C₈	—	12	1	no	—
37	D₃	—	64	1	no	—
38	C₂²	—	12	1	no	—
39	C₂²	#857	1	1	yes	characteristic
40	C₄	—	3	1	no	—
41	C₄	—	3	1	no	—
42	C₄	—	12	1	no	—
43	C₃	—	64	1	no	Sylow
44	C₂	—	3	1	no	—
45	C₂	—	24	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₈²⋊C₃	C₂
Frattini	C₄²	S₄
Fitting	C₈²	D₃
socle	C₂²	#857
LCS	G ▷ C₈²⋊C₃
UCS	C₁

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

N	H ≅ G/N	type
1. C₈²⋊C₃	C₂	semidirect
6. C₈²	D₃	semidirect

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