Subgroups — C₂²⋊(C₂²⋊SL(2,𝔽₃)) #87549

Order 384.

1129 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₂×C₂²⋊A₄	—	4	1	no	maximal
3	C₂²⋊SL(2,𝔽₃)	—	4	20	no	maximal
4	C₂³×Q₈	—	3	20	no	—
5	C₂⁴×C₄	—	3	1	no	—
6	C₂²⋊A₄	—	4	1	no	—
7	C₂⁵	A₄	1	1	yes	characteristic
8	C₂²×Q₈	—	3	160	no	—
9	C₂²×Q₈	—	3	20	no	—
10	C₂²×Q₈	A₄	1	20	yes	complemented
11	C₂³×C₄	—	3	30	no	—
12	C₂×A₄	—	16	5	no	—
13	SL(2,𝔽₃)	—	16	16	no	—
14	C₂⁴	—	3	5	no	—
15	C₂⁴	—	3	5	no	—
16	C₂⁴	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
17	C₂×Q₈	—	3	240	no	—
18	C₂×Q₈	—	3	80	no	—
19	C₂²×C₄	—	3	120	no	—
20	C₂²×C₄	—	3	20	no	—
21	A₄	—	16	5	no	—
22	C₂³	—	3	10	no	—
23	C₂³	—	3	30	no	—
24	C₂³	—	3	5	no	—
25	C₂³	—	3	5	no	—
26	C₂³	C₂²⋊A₄	1	5	yes	—
27	Q₈	—	3	80	no	—
28	Q₈	C₂²⋊A₄	1	16	yes	complemented
29	C₂×C₄	—	3	120	no	—
30	C₂×C₃	—	64	1	no	—
31	C₂²	—	3	5	no	—
32	C₂²	—	3	30	no	—
33	C₂²	—	3	10	no	—
34	C₂²	—	3	5	no	—
35	C₂²	C₂²⋊SL(2,𝔽₃)	1	5	yes	complemented
36	C₄	—	3	16	no	—
37	C₃	—	64	1	no	Sylow
38	C₂	—	3	5	no	—
39	C₂	—	3	5	no	—
40	C₂	C₂²⋊(C₂²⋊A₄)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₂²⋊(C₂²⋊A₄)
commutator, Fitting, cosocle, hypocenter	C₂⁴×Q₈	C₃
socle	C₂⁵	A₄
LCS	G ▷ C₂⁴×Q₈
UCS	C₁ ◁ C₂

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.