Subgroups — #67525

Order 384.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5102	C₂	1	1	yes	maximal, characteristic
2	C₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#4549	C₂	1	1	yes	maximal, characteristic
4	#1323	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
6	#926	C₄	1	1	yes	characteristic
7	C₂×#208	C₄	1	1	yes	characteristic
8	C₂×C₃×C₁₆	—	2	1	no	—
9	Dic₃⋊C₈	—	2	1	no	—
10	Dic₃⋊C₈	—	2	1	no	—
11	C₂×C₃⋊C₁₆	—	2	1	no	—
12	#424	—	3	1	no	—
13	C₂²⋊C₁₆	—	3	1	no	—
14	C₂²⋊C₁₆	D₃	1	1	yes	characteristic
15	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₈	—	4	1	no	—
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃×C₈	D₄	1	1	yes	characteristic
19	#217	—	2	1	no	—
20	C₄⋊Dic₃	C₈	1	1	yes	characteristic
21	C₄⋊Dic₃	C₈	1	1	yes	characteristic
22	C₄×Dic₃	—	2	1	no	—
23	#208	—	4	1	no	—
24	C₂×C₃⋊C₈	—	2	1	no	—
25	C₃×C₁₆	—	4	1	no	—
26	C₃⋊C₁₆	—	4	1	no	—
27	C₂×M₁₆	—	3	1	no	—
28	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
29	#117	—	3	1	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	6	1	no	—
32	C₄⋊C₈	—	6	1	no	—
33	C₄⋊C₈	—	6	1	no	—
34	C₂³×C₃	M₁₆	1	1	yes	characteristic
35	C₂×C₃×C₄	—	2	1	no	—
36	C₂×C₃×C₄	—	2	1	no	—
37	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₃	—	2	1	no	—
40	C₂×Dic₃	—	2	1	no	—
41	C₃×C₈	—	2	1	no	—
42	C₃×C₈	—	2	1	no	—
43	C₃⋊C₈	—	4	1	no	—
44	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
45	M₁₆	—	12	1	no	—
46	C₂×C₈	—	4	1	no	—
47	C₂×C₈	—	6	1	no	—
48	C₂×C₈	D₁₂	1	1	yes	characteristic
49	C₂×C₈	#67	1	1	yes	characteristic
50	C₄⋊C₄	—	3	1	no	—
51	C₄⋊C₄	—	3	1	no	—
52	C₂²⋊C₄	—	6	1	no	—
53	C₄²	—	6	1	no	—
54	C₁₆	—	4	1	no	—
55	C₁₆	—	12	1	no	—
56	C₂²×C₃	—	2	1	no	—
57	C₂²×C₃	—	2	1	no	—
58	C₂²×C₃	—	2	1	no	—
59	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	#99	1	1	yes	characteristic
62	C₃×C₄	#101	1	1	yes	characteristic
63	Dic₃	—	4	1	no	—
64	Dic₃	—	4	1	no	—
65	C₂³	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	6	1	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	D₃×C₈	1	1	yes	characteristic
71	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	—	12	1	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂×C₃	#324	1	1	yes	characteristic
77	C₂×C₃	#350	1	1	yes	characteristic
78	C₂×C₃	#350	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	(C₂×D₃)⋊C₈	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	12	1	no	—
85	C₄	—	12	1	no	—
86	C₄	#806	1	1	yes	characteristic
87	C₄	#824	1	1	yes	characteristic
88	C₃	#1323	1	1	yes	characteristic, complemented, Sylow
89	C₂	—	4	1	no	—
90	C₂	#4469	1	1	yes	characteristic
91	C₂	#4508	1	1	yes	characteristic
92	C₂	#4515	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₈
Frattini	C₂²×C₈	C₂×D₃
Fitting	C₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₃	C₂²⋊C₈
prime radical	C₂³×C₃	M₁₆
cosocle	C₂²×C₃×C₈	C₂²
hypercenter	C₂²⋊C₁₆	D₃
hypocenter	C₃	#1323
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
88. C₃	#1323	semidirect

Maximal subgroups

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

subgroup	index
#5102	2
C₃×C₂²⋊C₁₆	2
#4549	2
#1323	3