Subgroups — #67537

Order 384.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5137	C₂	1	1	yes	maximal, characteristic
2	C₃×#349	C₂	1	1	yes	maximal, characteristic
3	C₃⋊#338	C₂	1	1	yes	maximal, characteristic
4	#1342	—	3	1	no	maximal, Sylow
5	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
6	#924	—	2	1	no	—
7	C₂×#208	—	2	1	no	—
8	C₂×Dic₁₂	C₄	1	1	yes	characteristic
9	C₃×M₃₂	—	2	1	no	—
10	#817	C₄	1	1	yes	characteristic
11	#816	—	2	1	no	—
12	#470	—	3	1	no	—
13	#349	D₃	1	1	yes	characteristic
14	#338	—	3	1	no	—
15	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
16	C₂×Dic₆	—	2	1	no	—
17	C₃×M₁₆	—	4	1	no	—
18	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₈	D₄	1	1	yes	characteristic
20	#217	—	4	1	no	—
21	C₄⋊Dic₃	—	2	1	no	—
22	Dic₃⋊C₄	—	4	1	no	—
23	#208	—	4	1	no	—
24	#208	—	4	1	no	—
25	C₂×C₃⋊C₈	—	2	2	no	—
26	Dic₁₂	—	4	1	no	—
27	C₃×C₁₆	—	4	1	no	—
28	C₂×Q₁₆	—	3	1	no	—
29	C₂×M₁₆	—	6	1	no	—
30	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
31	C₂²⋊Q₈	—	6	1	no	—
32	M₃₂	—	2	1	no	—
33	#106	—	3	1	no	—
34	#103	—	6	1	no	—
35	C₂³×C₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₃×C₄	—	2	1	no	—
37	C₂×C₃×C₄	—	2	1	no	—
38	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₃×C₄	D₈	1	1	yes	characteristic
40	C₂×Dic₃	—	4	1	no	—
41	C₂×Dic₃	—	4	1	no	—
42	Dic₆	—	4	1	no	—
43	Dic₆	—	4	1	no	—
44	C₃×C₈	—	2	1	no	—
45	C₃×C₈	—	2	1	no	—
46	C₃⋊C₈	—	4	2	no	—
47	C₂×Q₈	—	6	1	no	—
48	C₂²×C₄	D₁₂	1	1	yes	characteristic
49	Q₁₆	—	12	1	no	—
50	M₁₆	—	4	1	no	—
51	M₁₆	—	12	1	no	—
52	M₁₆	—	12	1	no	—
53	C₂×C₈	—	6	2	no	—
54	C₂×C₈	C₄×D₃	1	1	yes	characteristic
55	C₂×C₈	#67	1	1	yes	characteristic
56	C₄⋊C₄	—	6	1	no	—
57	C₄⋊C₄	—	12	1	no	—
58	C₂²⋊C₄	—	12	1	no	—
59	C₁₆	—	4	1	no	—
60	C₂²×C₃	—	2	1	no	—
61	C₂²×C₃	—	4	1	no	—
62	C₂²×C₃	#102	1	1	yes	characteristic
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	#99	1	1	yes	characteristic
65	C₃×C₄	#104	1	1	yes	characteristic
66	Dic₃	—	8	1	no	—
67	Dic₃	—	8	1	no	—
68	C₂³	#204	1	1	yes	characteristic
69	Q₈	—	12	1	no	—
70	Q₈	—	12	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	12	1	no	—
74	C₂×C₄	—	12	1	no	—
75	C₂×C₄	D₂₄	1	1	yes	characteristic
76	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	1	no	—
79	C₈	—	12	2	no	—
80	C₂×C₃	—	2	1	no	—
81	C₂×C₃	—	4	1	no	—
82	C₂×C₃	#327	1	1	yes	characteristic
83	C₂²	—	2	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	#821	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	24	1	no	—
88	C₄	—	24	1	no	—
89	C₄	#806	1	1	yes	characteristic
90	C₄	#825	1	1	yes	characteristic
91	C₃	#1342	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#4470	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4470
commutator	C₂×C₃×C₈	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₃
Fitting	C₃×#349	C₂
socle	C₂×C₃	#327
prime radical	C₂³×C₃	SD₁₆
cosocle	C₂×C₃×M₁₆	C₂²
hypercenter	#349	D₃
hypocenter	C₃	#1342
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
91. C₃	#1342	semidirect

Maximal subgroups

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

subgroup	index
#5137	2
C₃×#349	2
C₃⋊#338	2
#1342	3