Subgroups — C₂³×C₃×SD₁₆ #87459

Order 384.

1720 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, complemented
2	C₂³×C₃×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×SD₁₆	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×SD₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂⁵×C₃	—	2	1	no	—
7	C₂²×C₃×Q₈	—	2	7	no	—
8	C₂²×C₃×Q₈	C₂²	1	14	yes	complemented
9	C₂²×C₃×D₄	—	2	7	no	—
10	C₂²×C₃×D₄	C₂²	1	14	yes	—
11	C₂³×C₃×C₄	—	2	1	no	—
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×SD₁₆	C₂²	1	112	yes	direct product factor
14	C₂²×C₃×C₈	C₂²	1	14	yes	complemented
15	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
16	C₂³×D₄	C₂×C₃	1	1	yes	characteristic
17	C₂²×SD₁₆	C₂×C₃	1	28	yes	direct product factor
18	C₂³×C₈	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂⁴×C₃	—	2	14	no	—
20	C₂⁴×C₃	—	4	8	no	—
21	C₂⁴×C₃	D₄	1	1	yes	characteristic
22	C₂×C₃×Q₈	—	2	42	no	—
23	C₂×C₃×Q₈	C₂³	1	28	yes	complemented
24	C₂×C₃×D₄	—	2	42	no	—
25	C₂×C₃×D₄	C₂³	1	28	yes	—
26	C₂²×C₃×C₄	—	2	14	no	—
27	C₂²×C₃×C₄	D₄	1	7	yes	—
28	C₂²×C₃×C₄	C₂³	1	7	yes	—
29	C₃×SD₁₆	C₂³	1	64	yes	direct product factor
30	C₂×C₃×C₈	C₂³	1	28	yes	complemented
31	C₂⁵	—	2	1	no	—
32	C₂²×Q₈	—	2	7	no	—
33	C₂²×Q₈	C₂²×C₃	1	14	yes	complemented
34	C₂²×D₄	—	2	7	no	—
35	C₂²×D₄	C₂²×C₃	1	14	yes	—
36	C₂³×C₄	—	2	1	no	—
37	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
38	C₂×SD₁₆	C₂²×C₃	1	112	yes	direct product factor
39	C₂²×C₈	C₂²×C₃	1	14	yes	complemented
40	C₂³×C₃	—	2	28	no	—
41	C₂³×C₃	—	4	56	no	—
42	C₂³×C₃	SD₁₆	1	8	yes	direct product factor
43	C₂³×C₃	C₂×D₄	1	7	yes	—
44	C₃×Q₈	—	2	28	no	—
45	C₃×Q₈	C₂⁴	1	8	yes	complemented
46	C₃×D₄	—	2	28	no	—
47	C₃×D₄	C₂⁴	1	8	yes	—
48	C₂×C₃×C₄	—	2	28	no	—
49	C₂×C₃×C₄	C₂×D₄	1	21	yes	—
50	C₂×C₃×C₄	C₂⁴	1	7	yes	—
51	C₃×C₈	C₂⁴	1	8	yes	complemented
52	C₂⁴	—	2	14	no	—
53	C₂⁴	—	4	8	no	—
54	C₂⁴	C₃×D₄	1	1	yes	characteristic
55	C₂×Q₈	—	2	42	no	—
56	C₂×Q₈	C₂³×C₃	1	28	yes	complemented
57	C₂×D₄	—	2	42	no	—
58	C₂×D₄	C₂³×C₃	1	28	yes	—
59	C₂²×C₄	—	2	14	no	—
60	C₂²×C₄	C₃×D₄	1	7	yes	—
61	C₂²×C₄	C₂³×C₃	1	7	yes	—
62	SD₁₆	C₂³×C₃	1	64	yes	direct product factor
63	C₂×C₈	C₂³×C₃	1	28	yes	complemented
64	C₂²×C₃	—	2	8	no	—
65	C₂²×C₃	—	4	56	no	—
66	C₂²×C₃	C₂×SD₁₆	1	28	yes	direct product factor
67	C₂²×C₃	C₂²×D₄	1	7	yes	—
68	C₃×C₄	—	2	8	no	—
69	C₃×C₄	C₂²×D₄	1	7	yes	—
70	C₃×C₄	C₂⁵	1	1	yes	characteristic
71	C₂³	—	2	28	no	—
72	C₂³	—	4	56	no	—
73	C₂³	C₃×SD₁₆	1	8	yes	direct product factor
74	C₂³	C₂×C₃×D₄	1	7	yes	—
75	Q₈	—	2	28	no	—
76	Q₈	C₂⁴×C₃	1	8	yes	complemented
77	D₄	—	2	28	no	—
78	D₄	C₂⁴×C₃	1	8	yes	—
79	C₂×C₄	—	2	28	no	—
80	C₂×C₄	C₂×C₃×D₄	1	21	yes	—
81	C₂×C₄	C₂⁴×C₃	1	7	yes	—
82	C₈	C₂⁴×C₃	1	8	yes	complemented
83	C₂×C₃	—	4	8	no	—
84	C₂×C₃	C₂²×SD₁₆	1	14	yes	direct product factor
85	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
86	C₂²	—	2	8	no	—
87	C₂²	—	4	56	no	—
88	C₂²	C₂×C₃×SD₁₆	1	28	yes	direct product factor
89	C₂²	C₂²×C₃×D₄	1	7	yes	—
90	C₄	—	2	8	no	—
91	C₄	C₂²×C₃×D₄	1	7	yes	—
92	C₄	C₂⁵×C₃	1	1	yes	characteristic
93	C₃	C₂³×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
94	C₂	—	4	8	no	—
95	C₂	C₂²×C₃×SD₁₆	1	14	yes	direct product factor
96	C₂	C₂³×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₃
prime radical	C₂³×C₃×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×Q₈	C₂	semidirect
3. C₂²×C₃×SD₁₆	C₂	direct (N × H)
4. C₂³×C₃×C₈	C₂	semidirect
5. C₂³×SD₁₆	C₃	direct (N × H)
8. C₂²×C₃×Q₈	C₂²	semidirect
13. C₂×C₃×SD₁₆	C₂²	direct (N × H)
14. C₂²×C₃×C₈	C₂²	semidirect
15. C₂³×Q₈	C₂×C₃	semidirect
17. C₂²×SD₁₆	C₂×C₃	direct (N × H)
18. C₂³×C₈	C₂×C₃	semidirect
23. C₂×C₃×Q₈	C₂³	semidirect
29. C₃×SD₁₆	C₂³	direct (N × H)
30. C₂×C₃×C₈	C₂³	semidirect
33. C₂²×Q₈	C₂²×C₃	semidirect
38. C₂×SD₁₆	C₂²×C₃	direct (N × H)
39. C₂²×C₈	C₂²×C₃	semidirect
42. C₂³×C₃	SD₁₆	direct (N × H)
45. C₃×Q₈	C₂⁴	semidirect
51. C₃×C₈	C₂⁴	semidirect
56. C₂×Q₈	C₂³×C₃	semidirect
62. SD₁₆	C₂³×C₃	direct (N × H)
63. C₂×C₈	C₂³×C₃	semidirect
66. C₂²×C₃	C₂×SD₁₆	direct (N × H)
73. C₂³	C₃×SD₁₆	direct (N × H)
76. Q₈	C₂⁴×C₃	semidirect
82. C₈	C₂⁴×C₃	semidirect
84. C₂×C₃	C₂²×SD₁₆	direct (N × H)
88. C₂²	C₂×C₃×SD₁₆	direct (N × H)
93. C₃	C₂³×SD₁₆	direct (N × H)
95. C₂	C₂²×C₃×SD₁₆	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
C₂³×C₃×D₄	2
C₂²×C₃×SD₁₆	2
C₂³×C₃×C₈	2
C₂³×SD₁₆	3

Subgroups — C23×C3×SD16 #87459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×SD₁₆ #87459