Subgroups — #88391

Order 432.

34 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈⋊C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3675	—	3	1	no	maximal
3	C₂×D₂₇	—	4	1	no	maximal
4	Q₈×C₉	D₃	1	1	yes	characteristic
5	D₃₆	—	3	1	no	—
6	C₉⋊C₈	—	3	1	no	—
7	C₂×C₂₇	—	4	1	no	—
8	D₂₇	—	4	2	no	—
9	#215	—	9	1	no	—
10	C₂×D₉	—	6	1	no	—
11	C₄×C₉	—	3	1	no	—
12	C₂₇	—	4	1	no	Sylow
13	C₃×Q₈	D₉	1	1	yes	characteristic
14	D₁₂	—	9	1	no	—
15	C₃⋊C₈	—	9	1	no	—
16	C₂×C₉	S₄	1	1	yes	characteristic
17	D₉	—	12	1	no	—
18	SD₁₆	—	27	1	no	Sylow
19	C₂×D₃	—	18	1	no	—
20	C₃×C₄	—	3	1	no	—
21	C₉	GL(2,𝔽₃)	1	1	yes	characteristic
22	Q₈	D₂₇	1	1	yes	characteristic, complemented
23	D₄	—	27	1	no	—
24	C₈	—	27	1	no	—
25	C₂×C₃	(C₂²⋊C₉)⋊C₂	1	1	yes	characteristic
26	D₃	—	36	1	no	—
27	C₂²	—	54	1	no	—
28	C₄	—	3	1	no	—
29	C₃	#3689	1	1	yes	characteristic
30	C₂	—	108	1	no	—
31	C₂	(C₂²⋊C₂₇)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	(C₂²⋊C₂₇)⋊C₂
commutator, cosocle, hypocenter	Q₈⋊C₂₇	C₂
Frattini	C₂×C₉	S₄
Fitting	Q₈×C₉	D₃
socle	C₂×C₃	(C₂²⋊C₉)⋊C₂
LCS	G ▷ Q₈⋊C₂₇
UCS	C₁ ◁ C₂

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

N	H ≅ G/N	type
1. Q₈⋊C₂₇	C₂	semidirect
22. Q₈	D₂₇	semidirect

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