Subgroups — C₂²×C₃×Q₃₂ #85274

Order 384.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₁₆	C₂	1	2	yes	maximal
2	C₂×C₃×Q₃₂	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×Q₈	—	2	2	no	—
6	C₂×C₃×Q₁₆	—	2	6	no	—
7	C₂×C₃×Q₁₆	C₂²	1	12	yes	—
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₃×Q₃₂	C₂²	1	16	yes	direct product factor
10	C₂×C₃×C₁₆	C₂²	1	6	yes	—
11	C₂²×Q₁₆	C₂×C₃	1	2	yes	—
12	C₂×Q₃₂	C₂×C₃	1	12	yes	direct product factor
13	C₂²×C₁₆	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×Q₈	—	2	12	no	—
15	C₂×C₃×Q₈	—	4	6	no	—
16	C₂²×C₃×C₄	—	4	2	no	—
17	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
18	C₃×Q₁₆	—	2	12	no	—
19	C₃×Q₁₆	C₂³	1	8	yes	—
20	C₂×C₃×C₈	D₄	1	3	yes	—
21	C₂×C₃×C₈	C₂³	1	3	yes	—
22	C₃×C₁₆	C₂³	1	4	yes	—
23	C₂²×Q₈	—	2	2	no	—
24	C₂×Q₁₆	—	2	6	no	—
25	C₂×Q₁₆	C₂²×C₃	1	12	yes	—
26	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
27	Q₃₂	C₂²×C₃	1	16	yes	direct product factor
28	C₂×C₁₆	C₂²×C₃	1	6	yes	—
29	C₂³×C₃	D₈	1	1	yes	characteristic
30	C₃×Q₈	—	2	8	no	—
31	C₃×Q₈	—	4	12	no	—
32	C₂×C₃×C₄	—	4	12	no	—
33	C₂×C₃×C₄	D₈	1	3	yes	—
34	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
35	C₃×C₈	C₂×D₄	1	3	yes	—
36	C₃×C₈	C₂⁴	1	1	yes	characteristic
37	C₂×Q₈	—	2	12	no	—
38	C₂×Q₈	—	4	6	no	—
39	C₂²×C₄	—	4	2	no	—
40	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
41	Q₁₆	—	2	12	no	—
42	Q₁₆	C₂³×C₃	1	8	yes	—
43	C₂×C₈	C₃×D₄	1	3	yes	—
44	C₂×C₈	C₂³×C₃	1	3	yes	—
45	C₁₆	C₂³×C₃	1	4	yes	—
46	C₂²×C₃	Q₃₂	1	4	yes	direct product factor
47	C₂²×C₃	C₂×D₈	1	3	yes	—
48	C₃×C₄	—	4	8	no	—
49	C₃×C₄	C₂×D₈	1	3	yes	—
50	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
51	C₂³	C₃×D₈	1	1	yes	characteristic
52	Q₈	—	2	8	no	—
53	Q₈	—	4	12	no	—
54	C₂×C₄	—	4	12	no	—
55	C₂×C₄	C₃×D₈	1	3	yes	—
56	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
57	C₈	C₂×C₃×D₄	1	3	yes	—
58	C₈	C₂⁴×C₃	1	1	yes	characteristic
59	C₂×C₃	C₂×Q₃₂	1	6	yes	direct product factor
60	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
61	C₂²	C₃×Q₃₂	1	4	yes	direct product factor
62	C₂²	C₂×C₃×D₈	1	3	yes	—
63	C₄	—	4	8	no	—
64	C₄	C₂×C₃×D₈	1	3	yes	—
65	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
66	C₃	C₂²×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
67	C₂	C₂×C₃×Q₃₂	1	6	yes	direct product factor
68	C₂	C₂²×C₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×Q₃₂	C₂	direct (N × H)
4. C₂²×Q₃₂	C₃	direct (N × H)
9. C₃×Q₃₂	C₂²	direct (N × H)
12. C₂×Q₃₂	C₂×C₃	direct (N × H)
27. Q₃₂	C₂²×C₃	direct (N × H)
46. C₂²×C₃	Q₃₂	direct (N × H)
59. C₂×C₃	C₂×Q₃₂	direct (N × H)
61. C₂²	C₃×Q₃₂	direct (N × H)
66. C₃	C₂²×Q₃₂	direct (N × H)
67. C₂	C₂×C₃×Q₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₁₆	2
C₂×C₃×Q₃₂	2
C₂²×C₃×C₁₆	2
C₂²×Q₃₂	3

Subgroups — C22×C3×Q32 #85274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×Q₃₂ #85274