Subgroups — C₂²×C₃×C₄×Q₈ #87307

Order 384.

1664 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃×C₄⋊C₄	C₂	1	3	yes	maximal
4	C₂²×C₃×C₄²	C₂	1	3	yes	maximal
5	C₂²×C₄×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×Q₈	C₄	1	16	yes	direct product factor
7	C₂²×C₃×Q₈	C₂²	1	12	yes	—
8	C₂³×C₃×C₄	C₂²	1	3	yes	—
9	C₂³×C₃×C₄	C₂²	1	3	yes	—
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₃×C₄×Q₈	C₂²	1	64	yes	direct product factor
12	C₂×C₃×C₄⋊C₄	C₂²	1	36	yes	—
13	C₂×C₃×C₄²	C₂²	1	36	yes	—
14	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic
15	C₂×C₄×Q₈	C₂×C₃	1	24	yes	direct product factor
16	C₂²×C₄⋊C₄	C₂×C₃	1	3	yes	—
17	C₂²×C₄²	C₂×C₃	1	3	yes	—
18	C₂⁴×C₃	C₂³	1	1	yes	characteristic
19	C₂×C₃×Q₈	C₂×C₄	1	96	yes	direct product factor
20	C₂×C₃×Q₈	C₂³	1	16	yes	—
21	C₂²×C₃×C₄	—	2	12	no	—
22	C₂²×C₃×C₄	C₂×C₄	1	24	yes	—
23	C₂²×C₃×C₄	Q₈	1	8	yes	direct product factor
24	C₂²×C₃×C₄	C₂³	1	18	yes	—
25	C₂²×C₃×C₄	C₂³	1	18	yes	—
26	C₂²×C₃×C₄	C₂³	1	6	yes	—
27	C₃×C₄⋊C₄	C₂³	1	48	yes	—
28	C₃×C₄²	C₂³	1	48	yes	—
29	C₂²×Q₈	C₃×C₄	1	16	yes	direct product factor
30	C₂²×Q₈	C₂²×C₃	1	12	yes	—
31	C₂³×C₄	C₂²×C₃	1	3	yes	—
32	C₂³×C₄	C₂²×C₃	1	3	yes	—
33	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₄×Q₈	C₂²×C₃	1	64	yes	direct product factor
35	C₂×C₄⋊C₄	C₂²×C₃	1	36	yes	—
36	C₂×C₄²	C₂²×C₃	1	36	yes	—
37	C₂³×C₃	C₂²×C₄	1	4	yes	—
38	C₂³×C₃	C₂×Q₈	1	4	yes	—
39	C₂³×C₃	#41	1	4	yes	—
40	C₂³×C₃	C₂⁴	1	3	yes	—
41	C₃×Q₈	C₂²×C₄	1	64	yes	direct product factor
42	C₂×C₃×C₄	—	2	36	no	—
43	C₂×C₃×C₄	C₂²×C₄	1	72	yes	—
44	C₂×C₃×C₄	C₂×Q₈	1	24	yes	direct product factor
45	C₂×C₃×C₄	C₂⁴	1	12	yes	—
46	C₂×C₃×C₄	C₂⁴	1	12	yes	—
47	C₂×C₃×C₄	C₂⁴	1	4	yes	—
48	C₂⁴	C₂³×C₃	1	1	yes	characteristic
49	C₂×Q₈	C₂×C₃×C₄	1	96	yes	direct product factor
50	C₂×Q₈	C₂³×C₃	1	16	yes	—
51	C₂²×C₄	—	2	12	no	—
52	C₂²×C₄	C₂×C₃×C₄	1	24	yes	—
53	C₂²×C₄	C₃×Q₈	1	8	yes	direct product factor
54	C₂²×C₄	C₂³×C₃	1	18	yes	—
55	C₂²×C₄	C₂³×C₃	1	18	yes	—
56	C₂²×C₄	C₂³×C₃	1	6	yes	—
57	C₄⋊C₄	C₂³×C₃	1	48	yes	—
58	C₄²	C₂³×C₃	1	48	yes	—
59	C₂²×C₃	C₄×Q₈	1	16	yes	direct product factor
60	C₂²×C₃	C₂³×C₄	1	6	yes	—
61	C₂²×C₃	C₂²×Q₈	1	6	yes	—
62	C₂²×C₃	C₂×#41	1	6	yes	—
63	C₂²×C₃	C₂⁵	1	1	yes	characteristic
64	C₃×C₄	—	2	12	no	—
65	C₃×C₄	C₂³×C₄	1	24	yes	—
66	C₃×C₄	C₂²×Q₈	1	8	yes	direct product factor
67	C₂³	C₂²×C₃×C₄	1	4	yes	—
68	C₂³	C₂×C₃×Q₈	1	4	yes	—
69	C₂³	C₃×#41	1	4	yes	—
70	C₂³	C₂⁴×C₃	1	3	yes	—
71	Q₈	C₂²×C₃×C₄	1	64	yes	direct product factor
72	C₂×C₄	—	2	36	no	—
73	C₂×C₄	C₂²×C₃×C₄	1	72	yes	—
74	C₂×C₄	C₂×C₃×Q₈	1	24	yes	direct product factor
75	C₂×C₄	C₂⁴×C₃	1	12	yes	—
76	C₂×C₄	C₂⁴×C₃	1	12	yes	—
77	C₂×C₄	C₂⁴×C₃	1	4	yes	—
78	C₂×C₃	C₂×C₄×Q₈	1	12	yes	direct product factor
79	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
80	C₂×C₃	C₂³×Q₈	1	1	yes	characteristic
81	C₂×C₃	C₂²×#41	1	1	yes	characteristic
82	C₂²	C₃×C₄×Q₈	1	16	yes	direct product factor
83	C₂²	C₂³×C₃×C₄	1	6	yes	—
84	C₂²	C₂²×C₃×Q₈	1	6	yes	—
85	C₂²	C₂×C₃×#41	1	6	yes	—
86	C₂²	C₂⁵×C₃	1	1	yes	characteristic
87	C₄	—	2	12	no	—
88	C₄	C₂³×C₃×C₄	1	24	yes	—
89	C₄	C₂²×C₃×Q₈	1	8	yes	direct product factor
90	C₃	C₂²×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
91	C₂	C₂×C₃×C₄×Q₈	1	12	yes	direct product factor
92	C₂	C₂⁴×C₃×C₄	1	1	yes	characteristic
93	C₂	C₂³×C₃×Q₈	1	1	yes	characteristic
94	C₂	C₂²×C₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₄	C₂²
commutator	C₂	C₂⁴×C₃×C₄
Frattini, cosocle	C₂²	C₂⁵×C₃
socle, prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×Q₈	C₂	direct (N × H)
5. C₂²×C₄×Q₈	C₃	direct (N × H)
6. C₂²×C₃×Q₈	C₄	direct (N × H)
11. C₃×C₄×Q₈	C₂²	direct (N × H)
15. C₂×C₄×Q₈	C₂×C₃	direct (N × H)
19. C₂×C₃×Q₈	C₂×C₄	direct (N × H)
23. C₂²×C₃×C₄	Q₈	direct (N × H)
29. C₂²×Q₈	C₃×C₄	direct (N × H)
34. C₄×Q₈	C₂²×C₃	direct (N × H)
41. C₃×Q₈	C₂²×C₄	direct (N × H)
44. C₂×C₃×C₄	C₂×Q₈	direct (N × H)
49. C₂×Q₈	C₂×C₃×C₄	direct (N × H)
53. C₂²×C₄	C₃×Q₈	direct (N × H)
59. C₂²×C₃	C₄×Q₈	direct (N × H)
66. C₃×C₄	C₂²×Q₈	direct (N × H)
71. Q₈	C₂²×C₃×C₄	direct (N × H)
74. C₂×C₄	C₂×C₃×Q₈	direct (N × H)
78. C₂×C₃	C₂×C₄×Q₈	direct (N × H)
82. C₂²	C₃×C₄×Q₈	direct (N × H)
89. C₄	C₂²×C₃×Q₈	direct (N × H)
90. C₃	C₂²×C₄×Q₈	direct (N × H)
91. C₂	C₂×C₃×C₄×Q₈	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×Q₈	2
C₂×C₃×C₄×Q₈	2
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₄²	2
C₂²×C₄×Q₈	3

Subgroups — C22×C3×C4×Q8 #87307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×Q₈ #87307