Subgroups — C₂²⋊(Q₈⋊C₉) #63702

Order 288.

73 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₃	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₉	—	4	1	no	maximal
3	Q₈⋊C₉	—	4	4	no	maximal
4	C₂×C₃×Q₈	—	3	4	no	—
5	C₂²×C₃×C₄	—	3	1	no	—
6	C₂²⋊C₉	—	4	1	no	—
7	C₂²×Q₈	C₉	1	1	yes	characteristic, complemented, Sylow
8	C₂³×C₃	A₄	1	1	yes	characteristic
9	C₃×Q₈	—	3	4	no	—
10	C₃×Q₈	A₄	1	4	yes	—
11	C₂×C₃×C₄	—	3	6	no	—
12	C₂×C₉	—	16	1	no	—
13	C₂×Q₈	—	3	4	no	—
14	C₂²×C₄	—	3	1	no	—
15	C₂²×C₃	—	3	1	no	—
16	C₂²×C₃	—	3	1	no	—
17	C₂²×C₃	SL(2,𝔽₃)	1	1	yes	characteristic
18	C₃×C₄	—	3	4	no	—
19	C₉	—	16	1	no	Sylow
20	C₂³	C₂²⋊C₉	1	1	yes	characteristic
21	Q₈	—	3	4	no	—
22	Q₈	C₂²⋊C₉	1	4	yes	complemented
23	C₂×C₄	—	3	6	no	—
24	C₂×C₃	—	3	1	no	—
25	C₂×C₃	—	3	1	no	—
26	C₂×C₃	C₂²⋊A₄	1	1	yes	characteristic
27	C₂²	—	3	1	no	—
28	C₂²	—	3	1	no	—
29	C₂²	Q₈⋊C₉	1	1	yes	characteristic, complemented
30	C₄	—	3	4	no	—
31	C₃	C₂²⋊SL(2,𝔽₃)	1	1	yes	characteristic
32	C₂	—	3	1	no	—
33	C₂	—	3	1	no	—
34	C₂	C₂²⋊(C₂²⋊C₉)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	C₂²⋊A₄
commutator, hypocenter	C₂²×Q₈	C₉
Fitting, cosocle	C₂²×C₃×Q₈	C₃
socle, prime radical	C₂³×C₃	A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂×C₃

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

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