Subgroups — C₃²×C₂²⋊Q₈ #64171

Order 288.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃²×C₂²⋊C₄	C₂	1	2	yes	maximal
6	C₃×C₂²⋊Q₈	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²	C₂²	1	1	yes	characteristic
8	C₃²×Q₈	—	2	2	no	—
9	C₂×C₃²×C₄	—	2	2	no	—
10	C₂×C₃²×C₄	C₂²	1	2	yes	—
11	C₂×C₃²×C₄	C₂²	1	2	yes	—
12	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×Q₈	C₂×C₃	1	4	yes	complemented
15	C₂²×C₃×C₄	C₂×C₃	1	4	yes	—
16	C₃×C₄⋊C₄	C₂×C₃	1	4	yes	complemented
17	C₃×C₄⋊C₄	C₂×C₃	1	8	yes	complemented
18	C₃×C₂²⋊C₄	C₂×C₃	1	8	yes	—
19	C₂²×C₃²	—	2	1	no	—
20	C₂²×C₃²	—	2	1	no	—
21	C₂²×C₃²	Q₈	1	2	yes	complemented
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₃²×C₄	—	2	1	no	—
24	C₃²×C₄	—	2	2	no	—
25	C₃²×C₄	—	2	2	no	—
26	C₃²×C₄	D₄	1	2	yes	—
27	C₂²⋊Q₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
28	C₂³×C₃	C₂²×C₃	1	4	yes	—
29	C₃×Q₈	—	2	8	no	—
30	C₂×C₃×C₄	—	2	8	no	—
31	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
32	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
33	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
34	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
35	C₂×C₃²	—	2	2	no	—
36	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
37	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₃²	#41	1	1	yes	characteristic
39	C₂×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
40	C₂²×C₄	C₂×C₃²	1	1	yes	characteristic
41	C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic, complemented
42	C₄⋊C₄	C₂×C₃²	1	2	yes	complemented
43	C₂²⋊C₄	C₂×C₃²	1	2	yes	—
44	C₂²×C₃	—	2	4	no	—
45	C₂²×C₃	—	2	4	no	—
46	C₂²×C₃	C₃×Q₈	1	8	yes	complemented
47	C₂²×C₃	C₂³×C₃	1	4	yes	—
48	C₃×C₄	—	2	4	no	—
49	C₃×C₄	—	2	8	no	—
50	C₃×C₄	—	2	8	no	—
51	C₃×C₄	C₃×D₄	1	8	yes	—
52	C₃²	C₂²⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³	C₂²×C₃²	1	1	yes	characteristic
54	Q₈	—	2	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	C₂²×C₃²	1	2	yes	—
57	C₂×C₄	C₂²×C₃²	1	2	yes	—
58	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
59	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
60	C₂×C₃	—	2	8	no	—
61	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
62	C₂×C₃	C₂×C₃×Q₈	1	4	yes	—
63	C₂×C₃	C₃×#41	1	4	yes	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₃²×Q₈	1	2	yes	complemented
67	C₂²	C₂³×C₃²	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	2	no	—
71	C₄	C₃²×D₄	1	2	yes	—
72	C₃	C₃×C₂²⋊Q₈	1	4	yes	direct product factor
73	C₂	—	2	2	no	—
74	C₂	C₂×C₃²×D₄	1	1	yes	characteristic
75	C₂	C₂×C₃²×Q₈	1	1	yes	characteristic
76	C₂	C₃²×#41	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²
prime radical	C₂³×C₃²	C₂²
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
1. C₂×C₃²×Q₈	C₂	semidirect
3. C₃²×C₄⋊C₄	C₂	semidirect
4. C₃²×C₄⋊C₄	C₂	semidirect
6. C₃×C₂²⋊Q₈	C₃	direct (N × H)
14. C₂×C₃×Q₈	C₂×C₃	semidirect
16. C₃×C₄⋊C₄	C₂×C₃	semidirect
17. C₃×C₄⋊C₄	C₂×C₃	semidirect
21. C₂²×C₃²	Q₈	semidirect
27. C₂²⋊Q₈	C₃²	direct (N × H)
39. C₂×Q₈	C₂×C₃²	semidirect
41. C₄⋊C₄	C₂×C₃²	semidirect
42. C₄⋊C₄	C₂×C₃²	semidirect
46. C₂²×C₃	C₃×Q₈	semidirect
52. C₃²	C₂²⋊Q₈	direct (N × H)
66. C₂²	C₃²×Q₈	semidirect
72. C₃	C₃×C₂²⋊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
C₂²×C₃²×C₄	2
C₃²×C₄⋊C₄	2
C₃²×C₂²⋊C₄	2
C₃×C₂²⋊Q₈	3

Subgroups — C32×C22⋊Q8 #64171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₂²⋊Q₈ #64171