Subgroups — #63733

Order 288.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3787	C₂	1	1	yes	maximal, characteristic
2	#3712	C₂	1	2	yes	maximal
3	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
4	#622	C₄	1	2	yes	—
5	C₃×C₃⋊C₈	—	2	2	no	—
6	ΓL(1,𝔽₂₅)	—	6	2	no	—
7	C₂×C₃⋊D₃	Q₈	1	1	yes	characteristic
8	C₃²×C₄	D₄	1	1	yes	characteristic
9	C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
10	#108	—	9	1	no	maximal, Sylow
11	C₄×D₃	—	6	2	no	—
12	C₃×C₈	—	6	2	no	—
13	C₃⋊C₈	—	2	2	no	—
14	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
15	C₃⋊D₃	—	2	1	no	—
16	M₁₆	—	9	2	no	—
17	C₂×C₈	—	9	1	no	—
18	C₂×D₃	—	6	2	no	—
19	C₃×C₄	—	2	2	no	—
20	Dic₃	—	6	2	no	—
21	C₃²	#108	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₄	—	9	1	no	—
23	C₈	—	6	2	no	—
24	C₈	—	9	2	no	—
25	C₂×C₃	—	2	2	no	—
26	D₃	—	12	2	no	—
27	C₂²	—	9	1	no	—
28	C₄	—	9	1	no	—
29	C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
30	C₃	—	2	2	no	—
31	C₂	—	18	1	no	—
32	C₂	#3773	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	GO⁺(4,𝔽₂)
commutator	C₃⋊Dic₃	C₂×C₄
Fitting	C₃²×C₄	D₄
socle	C₂×C₃²	C₄⋊C₄
prime radical	C₂×C₃⋊D₃	Q₈
cosocle	C₄×C₃⋊D₃	C₂²
hypocenter	C₃²	#108
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄

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

N	H ≅ G/N	type
21. C₃²	#108	semidirect

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