Subgroups — C₂²×C₃×M₃₂ #85269

Order 384.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×M₃₂	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂²×M₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₄	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₄	1	6	yes	—
8	C₂²×C₃×C₈	C₂²	1	6	yes	—
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×M₃₂	C₂²	1	16	yes	direct product factor
11	C₂×C₃×C₁₆	C₂²	1	12	yes	complemented
12	C₂³×C₈	C₂×C₃	1	1	yes	characteristic
13	C₂×M₃₂	C₂×C₃	1	12	yes	direct product factor
14	C₂²×C₁₆	C₂×C₃	1	2	yes	complemented
15	C₂⁴×C₃	C₈	1	1	yes	characteristic
16	C₂²×C₃×C₄	C₈	1	6	yes	—
17	C₂²×C₃×C₄	C₈	1	1	yes	characteristic
18	C₂²×C₃×C₄	C₂×C₄	1	6	yes	—
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₈	C₂×C₄	1	6	yes	—
21	C₂×C₃×C₈	C₂×C₄	1	12	yes	—
22	C₂×C₃×C₈	C₂×C₄	1	3	yes	—
23	C₂×C₃×C₈	C₂³	1	3	yes	—
24	C₂×C₃×C₈	C₂³	1	4	yes	—
25	C₃×C₁₆	C₂³	1	8	yes	complemented
26	C₂³×C₄	C₃×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₃×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₃×C₄	1	6	yes	—
29	C₂²×C₈	C₂²×C₃	1	6	yes	—
30	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
31	M₃₂	C₂²×C₃	1	16	yes	direct product factor
32	C₂×C₁₆	C₂²×C₃	1	12	yes	complemented
33	C₂³×C₃	—	2	4	no	—
34	C₂³×C₃	C₂×C₈	1	6	yes	—
35	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂×C₈	1	12	yes	—
37	C₂×C₃×C₄	C₂×C₈	1	6	yes	—
38	C₂×C₃×C₄	C₂×C₈	1	3	yes	—
39	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
40	C₂×C₃×C₄	C₂²×C₄	1	3	yes	—
41	C₃×C₈	C₂²×C₄	1	4	yes	—
42	C₃×C₈	C₂²×C₄	1	3	yes	—
43	C₃×C₈	C₂⁴	1	1	yes	characteristic
44	C₂⁴	C₃×C₈	1	1	yes	characteristic
45	C₂²×C₄	C₃×C₈	1	6	yes	—
46	C₂²×C₄	C₃×C₈	1	1	yes	characteristic
47	C₂²×C₄	C₂×C₃×C₄	1	6	yes	—
48	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
49	C₂×C₈	C₂×C₃×C₄	1	12	yes	—
50	C₂×C₈	C₂×C₃×C₄	1	6	yes	—
51	C₂×C₈	C₂×C₃×C₄	1	3	yes	—
52	C₂×C₈	C₂³×C₃	1	3	yes	—
53	C₂×C₈	C₂³×C₃	1	4	yes	—
54	C₁₆	C₂³×C₃	1	8	yes	complemented
55	C₂²×C₃	—	2	12	no	—
56	C₂²×C₃	M₃₂	1	4	yes	direct product factor
57	C₂²×C₃	C₂²×C₈	1	4	yes	—
58	C₂²×C₃	C₂²×C₈	1	3	yes	—
59	C₃×C₄	C₂²×C₈	1	3	yes	—
60	C₃×C₄	C₂²×C₈	1	4	yes	—
61	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
62	C₂³	—	2	4	no	—
63	C₂³	C₂×C₃×C₈	1	6	yes	—
64	C₂³	C₂×C₃×C₈	1	1	yes	characteristic
65	C₂×C₄	C₂×C₃×C₈	1	12	yes	—
66	C₂×C₄	C₂×C₃×C₈	1	3	yes	—
67	C₂×C₄	C₂×C₃×C₈	1	6	yes	—
68	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
69	C₂×C₄	C₂²×C₃×C₄	1	3	yes	—
70	C₈	C₂²×C₃×C₄	1	3	yes	—
71	C₈	C₂²×C₃×C₄	1	4	yes	—
72	C₈	C₂⁴×C₃	1	1	yes	characteristic
73	C₂×C₃	—	2	4	no	—
74	C₂×C₃	C₂×M₃₂	1	6	yes	direct product factor
75	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
76	C₂²	—	2	12	no	—
77	C₂²	C₃×M₃₂	1	4	yes	direct product factor
78	C₂²	C₂²×C₃×C₈	1	4	yes	—
79	C₂²	C₂²×C₃×C₈	1	3	yes	—
80	C₄	C₂²×C₃×C₈	1	3	yes	—
81	C₄	C₂²×C₃×C₈	1	4	yes	—
82	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
83	C₃	C₂²×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	—	2	4	no	—
85	C₂	C₂×C₃×M₃₂	1	6	yes	direct product factor
86	C₂	C₂³×C₃×C₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×M₃₂	C₂	direct (N × H)
3. C₂²×C₃×C₁₆	C₂	semidirect
4. C₂²×M₃₂	C₃	direct (N × H)
10. C₃×M₃₂	C₂²	direct (N × H)
11. C₂×C₃×C₁₆	C₂²	semidirect
13. C₂×M₃₂	C₂×C₃	direct (N × H)
14. C₂²×C₁₆	C₂×C₃	semidirect
25. C₃×C₁₆	C₂³	semidirect
31. M₃₂	C₂²×C₃	direct (N × H)
32. C₂×C₁₆	C₂²×C₃	semidirect
54. C₁₆	C₂³×C₃	semidirect
56. C₂²×C₃	M₃₂	direct (N × H)
74. C₂×C₃	C₂×M₃₂	direct (N × H)
77. C₂²	C₃×M₃₂	direct (N × H)
83. C₃	C₂²×M₃₂	direct (N × H)
85. C₂	C₂×C₃×M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₈	2
C₂×C₃×M₃₂	2
C₂²×C₃×C₁₆	2
C₂²×M₃₂	3

Subgroups — C22×C3×M32 #85269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×M₃₂ #85269