Subgroups — #67515

Order 384.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4688	C₂	1	1	yes	maximal, characteristic
2	C₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#1330	—	3	1	no	maximal, Sylow
5	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
6	C₄×D₁₂	C₄	1	1	yes	characteristic
7	C₂×C₃×C₁₆	—	2	1	no	—
8	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	(C₂×D₃)⋊C₈	—	2	1	no	—
10	C₂×C₃⋊C₁₆	—	2	1	no	—
11	C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
12	#436	—	3	1	no	—
13	C₄⋊C₁₆	—	3	1	no	—
14	#346	D₃	1	1	yes	characteristic, complemented
15	C₂×D₁₂	C₈	1	1	yes	characteristic
16	C₂×C₄×D₃	—	2	1	no	—
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃×C₈	D₄	1	1	yes	characteristic
19	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
20	(C₂×D₃)⋊C₄	—	2	1	no	—
21	C₄⋊Dic₃	C₈	1	1	yes	characteristic
22	C₂×C₃⋊C₈	—	2	1	no	—
23	ΓL(1,𝔽₂₅)	—	4	1	no	—
24	ΓL(1,𝔽₂₅)	—	4	1	no	—
25	C₃×C₁₆	—	2	2	no	—
26	C₃⋊C₁₆	—	4	1	no	—
27	C₂×M₁₆	—	6	1	no	—
28	C₄×D₄	—	3	1	no	—
29	C₂×C₁₆	—	2	1	no	—
30	C₂×C₁₆	—	6	1	no	—
31	C₄⋊C₈	—	3	1	no	—
32	C₂²⋊C₈	—	6	1	no	—
33	C₄×C₈	C₂×D₃	1	1	yes	characteristic
34	C₂²×D₃	—	2	1	no	—
35	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
37	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
38	C₂×Dic₃	—	2	1	no	—
39	D₁₂	—	4	1	no	—
40	C₄×D₃	—	4	1	no	—
41	C₄×D₃	—	4	1	no	—
42	C₃×C₈	—	2	1	no	—
43	C₃×C₈	D₈	1	1	yes	characteristic
44	C₃×C₈	SD₁₆	1	1	yes	characteristic
45	C₃⋊C₈	—	4	1	no	—
46	C₂×D₄	—	3	1	no	—
47	C₂²×C₄	—	6	1	no	—
48	M₁₆	—	12	1	no	—
49	M₁₆	—	12	1	no	—
50	C₂×C₈	—	6	1	no	—
51	C₂×C₈	D₁₂	1	1	yes	characteristic
52	C₂×C₈	#67	1	1	yes	characteristic
53	C₄⋊C₄	—	3	1	no	—
54	C₂²⋊C₄	—	6	1	no	—
55	C₄²	C₄×D₃	1	1	yes	characteristic
56	C₁₆	—	2	2	no	—
57	C₁₆	—	12	1	no	—
58	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×D₃	—	4	1	no	—
60	C₂×D₃	—	4	1	no	—
61	C₂×D₃	—	4	1	no	—
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	#102	1	1	yes	characteristic
65	C₃×C₄	#104	1	1	yes	characteristic
66	Dic₃	—	4	1	no	—
67	C₂³	—	6	1	no	—
68	D₄	—	12	1	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	—	12	1	no	—
71	C₂×C₄	—	12	1	no	—
72	C₂×C₄	D₃×C₈	1	1	yes	characteristic
73	C₂×C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
74	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	12	1	no	—
77	C₈	#204	1	1	yes	characteristic
78	C₈	D₂₄	1	1	yes	characteristic
79	C₂×C₃	D₄⋊C₈	1	1	yes	characteristic
80	C₂×C₃	#349	1	1	yes	characteristic
81	C₂×C₃	#350	1	1	yes	characteristic
82	D₃	—	8	1	no	—
83	C₂²	—	12	1	no	—
84	C₂²	—	12	1	no	—
85	C₂²	—	12	1	no	—
86	C₂²	(C₂×D₃)⋊C₈	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	2	1	no	—
89	C₄	—	12	1	no	—
90	C₄	#805	1	1	yes	characteristic
91	C₄	#821	1	1	yes	characteristic
92	C₃	#1330	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	24	1	no	—
94	C₂	#4459	1	1	yes	characteristic
95	C₂	#4514	1	1	yes	characteristic
96	C₂	#4515	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#346	C₂
socle	C₂²×C₃	C₂²⋊C₈
prime radical	C₂×D₁₂	C₈
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#346	D₃
hypocenter	C₃	#1330
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₃×#346	C₂	semidirect
3. C₄⋊(C₃⋊C₁₆)	C₂	semidirect
14. #346	D₃	semidirect
92. C₃	#1330	semidirect

Maximal subgroups

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

subgroup	index
#4688	2
C₃×#346	2
C₄⋊(C₃⋊C₁₆)	2
#1330	3