Subgroups — C₂³×#215 #87240

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×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂²×#215	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×SD₁₆	—	3	1	no	maximal, Sylow
6	C₂⁴×D₃	—	2	1	no	—
7	C₂²×C₃×Q₈	—	2	7	no	—
8	C₂²×C₃×Q₈	C₂²	1	14	yes	complemented
9	C₂³×C₃×C₄	—	2	1	no	—
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂²×D₁₂	—	2	7	no	—
12	C₂²×D₁₂	C₂²	1	14	yes	—
13	C₂×#215	C₂²	1	112	yes	direct product factor
14	C₂²×C₃⋊C₈	C₂²	1	14	yes	complemented
15	C₂³×Q₈	D₃	1	1	yes	characteristic, complemented
16	C₂³×D₄	—	3	1	no	—
17	C₂²×SD₁₆	—	3	28	no	—
18	C₂³×C₈	—	3	1	no	—
19	C₂⁴×C₃	D₄	1	1	yes	characteristic
20	C₂³×D₃	—	2	14	no	—
21	C₂³×D₃	—	4	8	no	—
22	C₂×C₃×Q₈	—	2	42	no	—
23	C₂×C₃×Q₈	C₂³	1	28	yes	complemented
24	C₂²×C₃×C₄	—	2	14	no	—
25	C₂²×C₃×C₄	D₄	1	7	yes	—
26	C₂²×C₃×C₄	C₂³	1	7	yes	—
27	C₂×D₁₂	—	2	42	no	—
28	C₂×D₁₂	C₂³	1	28	yes	—
29	#215	C₂³	1	64	yes	direct product factor
30	C₂×C₃⋊C₈	C₂³	1	28	yes	complemented
31	C₂⁵	—	6	1	no	—
32	C₂²×Q₈	—	2	7	no	—
33	C₂²×Q₈	C₂×D₃	1	14	yes	complemented
34	C₂²×D₄	—	3	14	no	—
35	C₂²×D₄	—	6	7	no	—
36	C₂³×C₄	—	2	1	no	—
37	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂×SD₁₆	—	3	112	no	—
39	C₂²×C₈	—	3	14	no	—
40	C₂³×C₃	SD₁₆	1	8	yes	complemented
41	C₂³×C₃	C₂×D₄	1	7	yes	—
42	C₂²×D₃	—	2	28	no	—
43	C₂²×D₃	—	4	56	no	—
44	C₃×Q₈	—	2	28	no	—
45	C₃×Q₈	C₂⁴	1	8	yes	complemented
46	C₂×C₃×C₄	—	2	28	no	—
47	C₂×C₃×C₄	C₂×D₄	1	21	yes	—
48	C₂×C₃×C₄	C₂⁴	1	7	yes	—
49	D₁₂	—	2	28	no	—
50	D₁₂	C₂⁴	1	8	yes	—
51	C₃⋊C₈	C₂⁴	1	8	yes	complemented
52	C₂⁴	—	6	14	no	—
53	C₂⁴	—	12	8	no	—
54	C₂⁴	#67	1	1	yes	characteristic
55	C₂×Q₈	—	2	42	no	—
56	C₂×Q₈	C₂²×D₃	1	28	yes	complemented
57	C₂×D₄	—	3	28	no	—
58	C₂×D₄	—	6	42	no	—
59	C₂²×C₄	—	2	14	no	—
60	C₂²×C₄	#67	1	7	yes	—
61	C₂²×C₄	C₂²×D₃	1	7	yes	—
62	SD₁₆	—	3	64	no	—
63	C₂×C₈	—	3	28	no	—
64	C₂²×C₃	C₂×SD₁₆	1	28	yes	complemented
65	C₂²×C₃	C₂²×D₄	1	7	yes	—
66	C₂×D₃	—	2	8	no	—
67	C₂×D₃	—	4	56	no	—
68	C₃×C₄	—	2	8	no	—
69	C₃×C₄	C₂²×D₄	1	7	yes	—
70	C₃×C₄	C₂⁵	1	1	yes	characteristic
71	C₂³	—	6	28	no	—
72	C₂³	—	12	56	no	—
73	C₂³	#215	1	8	yes	direct product factor
74	C₂³	C₂×#67	1	7	yes	—
75	Q₈	—	2	28	no	—
76	Q₈	C₂³×D₃	1	8	yes	complemented
77	D₄	—	3	8	no	—
78	D₄	—	6	28	no	—
79	C₂×C₄	—	2	28	no	—
80	C₂×C₄	C₂×#67	1	21	yes	—
81	C₂×C₄	C₂³×D₃	1	7	yes	—
82	C₈	—	3	8	no	—
83	C₂×C₃	C₂²×SD₁₆	1	14	yes	complemented
84	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
85	D₃	—	4	8	no	—
86	C₂²	—	6	8	no	—
87	C₂²	—	12	56	no	—
88	C₂²	C₂×#215	1	28	yes	direct product factor
89	C₂²	C₂²×#67	1	7	yes	—
90	C₄	—	2	8	no	—
91	C₄	C₂²×#67	1	7	yes	—
92	C₄	C₂⁴×D₃	1	1	yes	characteristic
93	C₃	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Sylow
94	C₂	—	12	8	no	—
95	C₂	C₂²×#215	1	14	yes	direct product factor
96	C₂	C₂³×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#67
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂³×C₃×Q₈	C₂
socle	C₂⁴×C₃	D₄
prime radical	C₂³×D₁₂	C₂
hypercenter	C₂³×Q₈	D₃
hypocenter	C₃	C₂³×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×Q₈	C₂	semidirect
3. C₂²×#215	C₂	direct (N × H)
4. C₂³×C₃⋊C₈	C₂	semidirect
8. C₂²×C₃×Q₈	C₂²	semidirect
13. C₂×#215	C₂²	direct (N × H)
14. C₂²×C₃⋊C₈	C₂²	semidirect
15. C₂³×Q₈	D₃	semidirect
23. C₂×C₃×Q₈	C₂³	semidirect
29. #215	C₂³	direct (N × H)
30. C₂×C₃⋊C₈	C₂³	semidirect
33. C₂²×Q₈	C₂×D₃	semidirect
40. C₂³×C₃	SD₁₆	semidirect
45. C₃×Q₈	C₂⁴	semidirect
51. C₃⋊C₈	C₂⁴	semidirect
56. C₂×Q₈	C₂²×D₃	semidirect
64. C₂²×C₃	C₂×SD₁₆	semidirect
73. C₂³	#215	direct (N × H)
76. Q₈	C₂³×D₃	semidirect
83. C₂×C₃	C₂²×SD₁₆	semidirect
88. C₂²	C₂×#215	direct (N × H)
93. C₃	C₂³×SD₁₆	semidirect
95. C₂	C₂²×#215	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×D₁₂	2
C₂²×#215	2
C₂³×C₃⋊C₈	2
C₂³×SD₁₆	3

Subgroups — C23×#215 #87240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#215 #87240