Subgroups — C₂³×C₃×C₄⋊C₄ #87304

Order 384.

2488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄	C₂	1	2	yes	maximal
2	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₄⋊C₄	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁵×C₃	C₂²	1	1	yes	characteristic
6	C₂³×C₃×C₄	—	2	16	no	—
7	C₂³×C₃×C₄	C₄	1	16	yes	complemented
8	C₂³×C₃×C₄	C₂²	1	28	yes	—
9	C₂³×C₃×C₄	C₂²	1	14	yes	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₄	C₂×C₃	1	2	yes	—
12	C₂⁴×C₄	C₂×C₃	1	1	yes	characteristic
13	C₂²×C₄⋊C₄	C₂×C₃	1	28	yes	direct product factor
14	C₂⁴×C₃	C₂×C₄	1	8	yes	—
15	C₂⁴×C₃	D₄	1	8	yes	—
16	C₂⁴×C₃	Q₈	1	8	yes	—
17	C₂⁴×C₃	C₂³	1	7	yes	—
18	C₂²×C₃×C₄	—	2	112	no	—
19	C₂²×C₃×C₄	C₂×C₄	1	112	yes	complemented
20	C₂²×C₃×C₄	C₂³	1	56	yes	—
21	C₂²×C₃×C₄	C₂³	1	28	yes	—
22	C₃×C₄⋊C₄	C₂³	1	64	yes	direct product factor
23	C₂⁵	C₂²×C₃	1	1	yes	characteristic
24	C₂³×C₄	—	2	16	no	—
25	C₂³×C₄	C₃×C₄	1	16	yes	complemented
26	C₂³×C₄	C₂²×C₃	1	28	yes	—
27	C₂³×C₄	C₂²×C₃	1	14	yes	—
28	C₂×C₄⋊C₄	C₂²×C₃	1	112	yes	direct product factor
29	C₂³×C₃	C₄⋊C₄	1	64	yes	direct product factor
30	C₂³×C₃	C₂²×C₄	1	28	yes	—
31	C₂³×C₃	C₂×D₄	1	28	yes	—
32	C₂³×C₃	C₂×Q₈	1	28	yes	—
33	C₂³×C₃	C₂⁴	1	7	yes	—
34	C₂×C₃×C₄	—	2	112	no	—
35	C₂×C₃×C₄	C₂²×C₄	1	112	yes	complemented
36	C₂×C₃×C₄	C₂⁴	1	16	yes	—
37	C₂×C₃×C₄	C₂⁴	1	8	yes	—
38	C₂⁴	C₂×C₃×C₄	1	8	yes	—
39	C₂⁴	C₃×D₄	1	8	yes	—
40	C₂⁴	C₃×Q₈	1	8	yes	—
41	C₂⁴	C₂³×C₃	1	7	yes	—
42	C₂²×C₄	—	2	112	no	—
43	C₂²×C₄	C₂×C₃×C₄	1	112	yes	complemented
44	C₂²×C₄	C₂³×C₃	1	56	yes	—
45	C₂²×C₄	C₂³×C₃	1	28	yes	—
46	C₄⋊C₄	C₂³×C₃	1	64	yes	direct product factor
47	C₂²×C₃	C₂×C₄⋊C₄	1	112	yes	direct product factor
48	C₂²×C₃	C₂³×C₄	1	14	yes	—
49	C₂²×C₃	C₂²×D₄	1	14	yes	—
50	C₂²×C₃	C₂²×Q₈	1	14	yes	—
51	C₂²×C₃	C₂⁵	1	1	yes	characteristic
52	C₃×C₄	—	2	16	no	—
53	C₃×C₄	C₂³×C₄	1	16	yes	complemented
54	C₂³	C₃×C₄⋊C₄	1	64	yes	direct product factor
55	C₂³	C₂²×C₃×C₄	1	28	yes	—
56	C₂³	C₂×C₃×D₄	1	28	yes	—
57	C₂³	C₂×C₃×Q₈	1	28	yes	—
58	C₂³	C₂⁴×C₃	1	7	yes	—
59	C₂×C₄	—	2	112	no	—
60	C₂×C₄	C₂²×C₃×C₄	1	112	yes	complemented
61	C₂×C₄	C₂⁴×C₃	1	16	yes	—
62	C₂×C₄	C₂⁴×C₃	1	8	yes	—
63	C₂×C₃	C₂²×C₄⋊C₄	1	28	yes	direct product factor
64	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
65	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
66	C₂×C₃	C₂³×Q₈	1	1	yes	characteristic
67	C₂²	C₂×C₃×C₄⋊C₄	1	112	yes	direct product factor
68	C₂²	C₂³×C₃×C₄	1	14	yes	—
69	C₂²	C₂²×C₃×D₄	1	14	yes	—
70	C₂²	C₂²×C₃×Q₈	1	14	yes	—
71	C₂²	C₂⁵×C₃	1	1	yes	characteristic
72	C₄	—	2	16	no	—
73	C₄	C₂³×C₃×C₄	1	16	yes	complemented
74	C₃	C₂³×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
75	C₂	C₂²×C₃×C₄⋊C₄	1	28	yes	direct product factor
76	C₂	C₂⁴×C₃×C₄	1	1	yes	characteristic
77	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
78	C₂	C₂³×C₃×Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃	C₂²
commutator	C₂	C₂⁴×C₃×C₄
Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₃×C₄⋊C₄	C₂	direct (N × H)
4. C₂³×C₄⋊C₄	C₃	direct (N × H)
7. C₂³×C₃×C₄	C₄	semidirect
10. C₂×C₃×C₄⋊C₄	C₂²	direct (N × H)
13. C₂²×C₄⋊C₄	C₂×C₃	direct (N × H)
19. C₂²×C₃×C₄	C₂×C₄	semidirect
22. C₃×C₄⋊C₄	C₂³	direct (N × H)
25. C₂³×C₄	C₃×C₄	semidirect
28. C₂×C₄⋊C₄	C₂²×C₃	direct (N × H)
29. C₂³×C₃	C₄⋊C₄	direct (N × H)
35. C₂×C₃×C₄	C₂²×C₄	semidirect
43. C₂²×C₄	C₂×C₃×C₄	semidirect
46. C₄⋊C₄	C₂³×C₃	direct (N × H)
47. C₂²×C₃	C₂×C₄⋊C₄	direct (N × H)
53. C₃×C₄	C₂³×C₄	semidirect
54. C₂³	C₃×C₄⋊C₄	direct (N × H)
60. C₂×C₄	C₂²×C₃×C₄	semidirect
63. C₂×C₃	C₂²×C₄⋊C₄	direct (N × H)
67. C₂²	C₂×C₃×C₄⋊C₄	direct (N × H)
73. C₄	C₂³×C₃×C₄	semidirect
74. C₃	C₂³×C₄⋊C₄	direct (N × H)
75. C₂	C₂²×C₃×C₄⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₄	2
C₂²×C₃×C₄⋊C₄	2
C₂³×C₄⋊C₄	3

Subgroups — C23×C3×C4⋊C4 #87304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₄⋊C₄ #87304