Subgroups — C₂×C₃×Q₈² #87361

Order 384.

1112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈²	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×C₄⋊Q₈	C₂	1	9	yes	maximal
3	C₂×C₃×C₄×Q₈	C₂	1	6	yes	maximal
4	C₂×Q₈²	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×Q₈	C₂²	1	6	yes	—
6	C₂²×C₃×Q₈	C₂²	1	2	yes	—
7	C₃×C₄⋊Q₈	C₂²	1	72	yes	—
8	C₃×C₄×Q₈	C₂²	1	48	yes	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	18	yes	—
10	C₂×C₃×C₄²	C₂²	1	9	yes	—
11	Q₈²	C₂×C₃	1	16	yes	direct product factor
12	C₂×C₄⋊Q₈	C₂×C₃	1	9	yes	—
13	C₂×C₄×Q₈	C₂×C₃	1	6	yes	—
14	C₂×C₃×Q₈	—	4	12	no	—
15	C₂×C₃×Q₈	Q₈	1	16	yes	direct product factor
16	C₂×C₃×Q₈	C₂³	1	24	yes	—
17	C₂×C₃×Q₈	C₂³	1	8	yes	—
18	C₂²×C₃×C₄	C₂³	1	9	yes	—
19	C₂²×C₃×C₄	C₂³	1	6	yes	—
20	C₃×C₄⋊C₄	C₂³	1	72	yes	—
21	C₃×C₄²	C₂³	1	36	yes	—
22	C₂²×Q₈	C₂²×C₃	1	6	yes	—
23	C₂²×Q₈	C₂²×C₃	1	2	yes	—
24	C₄⋊Q₈	C₂²×C₃	1	72	yes	—
25	C₄×Q₈	C₂²×C₃	1	48	yes	—
26	C₂×C₄⋊C₄	C₂²×C₃	1	18	yes	—
27	C₂×C₄²	C₂²×C₃	1	9	yes	—
28	C₂³×C₃	C₂⁴	1	1	yes	characteristic
29	C₃×Q₈	—	4	24	no	—
30	C₃×Q₈	C₂×Q₈	1	32	yes	direct product factor
31	C₂×C₃×C₄	—	2	18	no	—
32	C₂×C₃×C₄	C₂×Q₈	1	24	yes	—
33	C₂×C₃×C₄	C₂⁴	1	18	yes	—
34	C₂×C₃×C₄	C₂⁴	1	12	yes	—
35	C₂×Q₈	—	4	12	no	—
36	C₂×Q₈	C₃×Q₈	1	16	yes	direct product factor
37	C₂×Q₈	C₂³×C₃	1	24	yes	—
38	C₂×Q₈	C₂³×C₃	1	8	yes	—
39	C₂²×C₄	C₂³×C₃	1	9	yes	—
40	C₂²×C₄	C₂³×C₃	1	6	yes	—
41	C₄⋊C₄	C₂³×C₃	1	72	yes	—
42	C₄²	C₂³×C₃	1	36	yes	—
43	C₂²×C₃	C₂²×Q₈	1	4	yes	—
44	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
45	C₂²×C₃	C₂⁵	1	1	yes	characteristic
46	C₃×C₄	—	2	18	no	—
47	C₃×C₄	C₂²×Q₈	1	24	yes	—
48	C₂³	C₂⁴×C₃	1	1	yes	characteristic
49	Q₈	—	4	24	no	—
50	Q₈	C₂×C₃×Q₈	1	32	yes	direct product factor
51	C₂×C₄	—	2	18	no	—
52	C₂×C₄	C₂×C₃×Q₈	1	24	yes	—
53	C₂×C₄	C₂⁴×C₃	1	18	yes	—
54	C₂×C₄	C₂⁴×C₃	1	12	yes	—
55	C₂×C₃	Q₈²	1	4	yes	direct product factor
56	C₂×C₃	C₂³×Q₈	1	2	yes	—
57	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
58	C₂²	C₂²×C₃×Q₈	1	4	yes	—
59	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
60	C₂²	C₂⁵×C₃	1	1	yes	characteristic
61	C₄	—	2	18	no	—
62	C₄	C₂²×C₃×Q₈	1	24	yes	—
63	C₃	C₂×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
64	C₂	C₃×Q₈²	1	4	yes	direct product factor
65	C₂	C₂³×C₃×Q₈	1	2	yes	—
66	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈²	C₂	direct (N × H)
4. C₂×Q₈²	C₃	direct (N × H)
11. Q₈²	C₂×C₃	direct (N × H)
15. C₂×C₃×Q₈	Q₈	direct (N × H)
30. C₃×Q₈	C₂×Q₈	direct (N × H)
36. C₂×Q₈	C₃×Q₈	direct (N × H)
50. Q₈	C₂×C₃×Q₈	direct (N × H)
55. C₂×C₃	Q₈²	direct (N × H)
63. C₃	C₂×Q₈²	direct (N × H)
64. C₂	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
C₃×Q₈²	2
C₂×C₃×C₄⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₂×Q₈²	3

Subgroups — C2×C3×Q82 #87361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Q₈² #87361