Subgroups — #83231

Order 384.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#5229	C₂	1	4	yes	maximal
3	#4980	C₂	1	8	yes	maximal
4	#4937	C₂	1	2	yes	maximal
5	#2381	—	3	1	no	maximal, Sylow
6	C₂²×C₃×Q₈	C₂²	1	2	yes	—
7	C₃×C₄⋊Q₈	C₄	1	8	yes	complemented
8	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₂×C₄×Dic₃	C₂²	1	4	yes	—
12	C₃⋊#95	C₂²	1	4	yes	—
13	C₃⋊#95	C₂²	1	8	yes	—
14	C₃⋊#95	C₂²	1	8	yes	—
15	C₂×C₄⋊Q₈	D₃	1	1	yes	characteristic
16	#391	—	3	4	no	—
17	#387	—	3	8	no	—
18	#383	—	3	2	no	—
19	C₂×C₃×Q₈	—	4	2	no	—
20	C₂×C₃×Q₈	—	4	2	no	—
21	C₂×C₃×Q₈	C₂×C₄	1	8	yes	—
22	C₂²×C₃×C₄	C₂³	1	2	yes	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂³	1	4	yes	—
25	C₂²×Dic₃	C₂³	1	8	yes	—
26	C₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
27	C₃×C₄²	C₂×C₄	1	4	yes	—
28	C₄⋊Dic₃	—	4	4	no	—
29	C₄×Dic₃	—	4	4	no	—
30	C₂²×Q₈	C₂×D₃	1	2	yes	—
31	C₄⋊Q₈	Dic₃	1	8	yes	complemented
32	C₂×C₄⋊C₄	—	3	4	no	—
33	C₂×C₄⋊C₄	C₂×D₃	1	4	yes	—
34	C₂×C₄²	—	3	4	no	—
35	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
36	#95	—	3	8	no	—
37	#95	—	3	4	no	—
38	#95	—	3	8	no	—
39	C₂³×C₃	C₂⁴	1	1	yes	characteristic
40	C₃×Q₈	—	4	8	no	—
41	C₂×C₃×C₄	—	2	4	no	—
42	C₂×C₃×C₄	—	2	4	no	—
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	—	2	4	no	—
45	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
46	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
47	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
48	C₂×Dic₃	—	2	8	no	—
49	C₂×Dic₃	—	2	16	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	C₂×Dic₃	1	8	yes	—
53	C₂²×C₄	—	3	8	no	—
54	C₂²×C₄	C₂²×D₃	1	2	yes	—
55	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₂²×C₄	C₂²×D₃	1	4	yes	—
57	C₄⋊C₄	—	12	4	no	—
58	C₄⋊C₄	C₂×Dic₃	1	16	yes	—
59	C₄²	—	12	4	no	—
60	C₄²	C₂×Dic₃	1	4	yes	—
61	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
62	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
63	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
64	C₃×C₄	—	2	8	no	—
65	C₃×C₄	—	2	2	no	—
66	C₃×C₄	—	2	4	no	—
67	Dic₃	—	4	8	no	—
68	C₂³	C₂³×D₃	1	1	yes	characteristic
69	Q₈	—	4	8	no	—
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	6	8	no	—
75	C₂×C₄	—	6	16	no	—
76	C₂×C₄	C₂²×Dic₃	1	4	yes	—
77	C₂×C₄	C₂²×Dic₃	1	2	yes	—
78	C₂×C₄	C₂²×Dic₃	1	8	yes	—
79	C₂×C₃	#518	1	1	yes	characteristic
80	C₂×C₃	#519	1	2	yes	—
81	C₂×C₃	#563	1	4	yes	—
82	C₂²	#1004	1	2	yes	—
83	C₂²	#1007	1	4	yes	—
84	C₂²	C₂³×Dic₃	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	8	no	—
87	C₄	—	2	4	no	—
88	C₄	—	12	8	no	—
89	C₃	#2381	1	1	yes	characteristic, complemented, Sylow
90	C₂	#5735	1	4	yes	—
91	C₂	#5798	1	1	yes	characteristic
92	C₂	#5812	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄⋊Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	#2381
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
7. C₃×C₄⋊Q₈	C₄	semidirect
31. C₄⋊Q₈	Dic₃	semidirect
89. C₃	#2381	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊Q₈	2
#5229	2
#4980	2
#4937	2
#2381	3