Subgroups — C₃×C₄×Q₃₂ #72889

Order 384.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Q₁₆	C₂	1	2	yes	maximal
3	C₃×#366	C₂	1	1	yes	maximal, characteristic
4	C₃×#358	C₂	1	2	yes	maximal
5	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
8	C₃×C₄×Q₈	—	2	2	no	—
9	C₃×Q₃₂	C₄	1	4	yes	direct product factor
10	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×#107	C₂²	1	2	yes	—
13	C₃×#103	—	2	2	no	—
14	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂	C₂×C₃	1	1	yes	characteristic
16	C₄×Q₁₆	C₂×C₃	1	2	yes	—
17	#366	C₂×C₃	1	1	yes	characteristic
18	#358	C₂×C₃	1	2	yes	—
19	C₄×C₁₆	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×Q₈	—	2	2	no	—
21	C₃×Q₁₆	—	2	2	no	—
22	C₃×Q₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₃×C₄⋊C₄	—	2	2	no	—
26	C₃×C₄⋊C₄	—	4	2	no	—
27	C₃×C₄²	—	4	2	no	—
28	C₃×C₄²	D₄	1	1	yes	characteristic
29	C₃×C₁₆	—	2	1	no	—
30	C₃×C₁₆	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
32	C₄×Q₈	—	2	2	no	—
33	Q₃₂	C₃×C₄	1	4	yes	direct product factor
34	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
36	#107	C₂²×C₃	1	2	yes	—
37	#103	—	2	2	no	—
38	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
39	C₃×Q₈	—	2	4	no	—
40	C₃×Q₈	—	4	2	no	—
41	C₂×C₃×C₄	—	4	2	no	—
42	C₂×C₃×C₄	—	4	2	no	—
43	C₂×C₃×C₄	D₈	1	1	yes	characteristic
44	C₂×C₃×C₄	D₈	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₃×C₈	—	2	1	no	—
47	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₈	#41	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	Q₁₆	—	2	2	no	—
51	Q₁₆	C₂×C₃×C₄	1	4	yes	—
52	C₂×C₈	C₃×D₄	1	1	yes	characteristic
53	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	4	2	no	—
56	C₄²	—	4	2	no	—
57	C₄²	C₃×D₄	1	1	yes	characteristic
58	C₁₆	—	2	1	no	—
59	C₁₆	C₂×C₃×C₄	1	2	yes	—
60	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	4	4	no	—
63	C₃×C₄	—	8	2	no	—
64	C₃×C₄	Q₃₂	1	2	yes	direct product factor
65	C₃×C₄	C₄×D₄	1	1	yes	characteristic
66	C₃×C₄	#135	1	1	yes	characteristic
67	Q₈	—	2	4	no	—
68	Q₈	—	4	2	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	C₃×D₈	1	1	yes	characteristic
72	C₂×C₄	C₃×D₈	1	1	yes	characteristic
73	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
76	C₈	C₃×#41	1	1	yes	characteristic
77	C₂×C₃	C₄×D₈	1	1	yes	characteristic
78	C₂×C₃	C₂×Q₃₂	1	1	yes	characteristic
79	C₂×C₃	#508	1	1	yes	characteristic
80	C₂²	C₂×C₃×D₈	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	4	4	no	—
83	C₄	—	8	2	no	—
84	C₄	C₃×Q₃₂	1	2	yes	direct product factor
85	C₄	C₃×C₄×D₄	1	1	yes	characteristic
86	C₄	C₃×#135	1	1	yes	characteristic
87	C₃	C₄×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
88	C₂	C₃×C₄×D₈	1	1	yes	characteristic
89	C₂	C₂×C₃×Q₃₂	1	1	yes	characteristic
90	C₂	C₃×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₈
commutator	C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃
socle, prime radical	C₂²×C₃	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄×Q₃₂	C₃	direct (N × H)
9. C₃×Q₃₂	C₄	direct (N × H)
33. Q₃₂	C₃×C₄	direct (N × H)
64. C₃×C₄	Q₃₂	direct (N × H)
84. C₄	C₃×Q₃₂	direct (N × H)
87. C₃	C₄×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
C₂×C₃×Q₃₂	2
C₃×C₄×Q₁₆	2
C₃×#366	2
C₃×#358	2
C₃×C₄×C₁₆	2
C₄×Q₃₂	3

Subgroups — C3×C4×Q32 #72889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×Q₃₂ #72889