Subgroups — C₃²×Q₈² #10596359

Order 576.

636 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊Q₈	C₂	1	9	yes	maximal
2	C₃²×C₄×Q₈	C₂	1	6	yes	maximal
3	C₃×Q₈²	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×Q₈	C₂²	1	6	yes	—
5	C₂×C₃²×Q₈	C₂²	1	2	yes	—
6	C₃²×C₄⋊C₄	C₂²	1	18	yes	—
7	C₃²×C₄²	C₂²	1	9	yes	—
8	C₃×C₄⋊Q₈	C₂×C₃	1	36	yes	—
9	C₃×C₄×Q₈	C₂×C₃	1	24	yes	—
10	C₃²×Q₈	—	4	6	no	—
11	C₃²×Q₈	Q₈	1	8	yes	direct product factor
12	C₂×C₃²×C₄	C₂³	1	9	yes	—
13	C₂×C₃²×C₄	C₂³	1	6	yes	—
14	Q₈²	C₃²	1	1	yes	characteristic, direct product factor, Sylow
15	C₂×C₃×Q₈	C₂²×C₃	1	24	yes	—
16	C₂×C₃×Q₈	C₂²×C₃	1	8	yes	—
17	C₃×C₄⋊C₄	C₂²×C₃	1	72	yes	—
18	C₃×C₄²	C₂²×C₃	1	36	yes	—
19	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
20	C₃²×C₄	—	2	9	no	—
21	C₃²×C₄	C₂×Q₈	1	12	yes	—
22	C₄⋊Q₈	C₂×C₃²	1	9	yes	—
23	C₄×Q₈	C₂×C₃²	1	6	yes	—
24	C₃×Q₈	—	4	24	no	—
25	C₃×Q₈	C₃×Q₈	1	32	yes	direct product factor
26	C₂×C₃×C₄	C₂³×C₃	1	36	yes	—
27	C₂×C₃×C₄	C₂³×C₃	1	24	yes	—
28	C₂×C₃²	C₂²×Q₈	1	2	yes	—
29	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
30	C₂×Q₈	C₂²×C₃²	1	6	yes	—
31	C₂×Q₈	C₂²×C₃²	1	2	yes	—
32	C₄⋊C₄	C₂²×C₃²	1	18	yes	—
33	C₄²	C₂²×C₃²	1	9	yes	—
34	C₂²×C₃	C₂⁴×C₃	1	4	yes	—
35	C₃×C₄	—	2	36	no	—
36	C₃×C₄	C₂×C₃×Q₈	1	48	yes	—
37	C₃²	Q₈²	1	1	yes	characteristic, direct product factor, Sylow
38	Q₈	—	4	6	no	—
39	Q₈	C₃²×Q₈	1	8	yes	direct product factor
40	C₂×C₄	C₂³×C₃²	1	9	yes	—
41	C₂×C₄	C₂³×C₃²	1	6	yes	—
42	C₂×C₃	C₂²×C₃×Q₈	1	8	yes	—
43	C₂×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
44	C₂²	C₂⁴×C₃²	1	1	yes	characteristic
45	C₄	—	2	9	no	—
46	C₄	C₂×C₃²×Q₈	1	12	yes	—
47	C₃	C₃×Q₈²	1	4	yes	direct product factor
48	C₂	C₂²×C₃²×Q₈	1	2	yes	—
49	C₂	C₃²×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×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
3. C₃×Q₈²	C₃	direct (N × H)
11. C₃²×Q₈	Q₈	direct (N × H)
14. Q₈²	C₃²	direct (N × H)
25. C₃×Q₈	C₃×Q₈	direct (N × H)
37. C₃²	Q₈²	direct (N × H)
39. Q₈	C₃²×Q₈	direct (N × H)
47. C₃	C₃×Q₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄⋊Q₈	2
C₃²×C₄×Q₈	2
C₃×Q₈²	3

Subgroups — C32×Q82 #10596359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×Q₈² #10596359