Subgroups — C₃²×C₄⋊Q₈ #64177

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	2	yes	maximal
2	C₃²×C₄⋊C₄	C₂	1	4	yes	maximal
3	C₃²×C₄²	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄⋊Q₈	C₃	1	4	yes	maximal, direct product factor
5	C₃²×Q₈	—	2	4	no	—
6	C₂×C₃²×C₄	C₂²	1	4	yes	—
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄	C₂²	1	2	yes	—
9	C₂×C₃×Q₈	C₂×C₃	1	8	yes	—
10	C₃×C₄⋊C₄	C₂×C₃	1	16	yes	—
11	C₃×C₄²	C₂×C₃	1	4	yes	—
12	C₂²×C₃²	C₂³	1	1	yes	characteristic
13	C₃²×C₄	—	2	4	no	—
14	C₃²×C₄	D₄	1	2	yes	—
15	C₃²×C₄	Q₈	1	4	yes	complemented
16	C₄⋊Q₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
17	C₃×Q₈	—	2	16	no	—
18	C₂×C₃×C₄	C₂²×C₃	1	16	yes	—
19	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
20	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
21	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
22	C₂×C₃²	C₂×Q₈	1	2	yes	—
23	C₂×Q₈	C₂×C₃²	1	2	yes	—
24	C₄⋊C₄	C₂×C₃²	1	4	yes	—
25	C₄²	C₂×C₃²	1	1	yes	characteristic
26	C₂²×C₃	C₂³×C₃	1	4	yes	—
27	C₃×C₄	—	2	16	no	—
28	C₃×C₄	C₃×D₄	1	8	yes	—
29	C₃×C₄	C₃×Q₈	1	16	yes	complemented
30	C₃²	C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
31	Q₈	—	2	4	no	—
32	C₂×C₄	C₂²×C₃²	1	4	yes	—
33	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
34	C₂×C₄	C₂²×C₃²	1	2	yes	—
35	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
36	C₂×C₃	C₂×C₃×Q₈	1	8	yes	—
37	C₂²	C₂³×C₃²	1	1	yes	characteristic
38	C₄	—	2	4	no	—
39	C₄	C₃²×D₄	1	2	yes	—
40	C₄	C₃²×Q₈	1	4	yes	complemented
41	C₃	C₃×C₄⋊Q₈	1	4	yes	direct product factor
42	C₂	C₂×C₃²×D₄	1	1	yes	characteristic
43	C₂	C₂×C₃²×Q₈	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₄⋊Q₈	C₃	direct (N × H)
15. C₃²×C₄	Q₈	semidirect
16. C₄⋊Q₈	C₃²	direct (N × H)
29. C₃×C₄	C₃×Q₈	semidirect
30. C₃²	C₄⋊Q₈	direct (N × H)
40. C₄	C₃²×Q₈	semidirect
41. C₃	C₃×C₄⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×Q₈	2
C₃²×C₄⋊C₄	2
C₃²×C₄²	2
C₃×C₄⋊Q₈	3

Subgroups — C32×C4⋊Q8 #64177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄⋊Q₈ #64177