Subgroups — #67689

Order 384.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#408	C₂	1	1	yes	maximal, characteristic
2	#4549	C₂	1	2	yes	maximal, complemented
3	#1315	—	3	1	no	maximal, Sylow
4	C₂×C₃×#41	C₄	1	1	yes	characteristic
5	C₂×C₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₃×C₂²⋊C₈	—	2	2	no	—
8	C₂×C₃⋊C₁₆	—	2	2	no	—
9	#408	D₃	1	1	yes	characteristic
10	C₂²⋊C₁₆	—	3	2	no	—
11	C₃×#41	—	4	1	no	—
12	C₃×#41	—	4	1	no	—
13	C₂×C₃×Q₈	C₈	1	1	yes	characteristic
14	C₂×C₃×D₄	—	2	1	no	—
15	C₂×C₃×D₄	C₈	1	1	yes	characteristic
16	C₂²×C₃×C₄	—	2	1	no	—
17	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
18	C₃×M₁₆	—	4	1	no	—
19	C₂×C₃×C₈	—	2	1	no	—
20	C₂×C₃×C₈	—	4	1	no	—
21	C₂×C₃×C₈	D₄	1	2	yes	—
22	C₃⋊C₁₆	—	4	2	no	—
23	C₂×#41	Dic₃	1	1	yes	characteristic
24	C₂×M₁₆	Dic₃	1	1	yes	characteristic
25	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
26	C₂×C₁₆	—	6	2	no	—
27	C₂²⋊C₈	—	2	2	no	—
28	C₂³×C₃	—	2	1	no	—
29	C₂³×C₃	M₁₆	1	1	yes	characteristic
30	C₃×Q₈	—	4	1	no	—
31	C₃×D₄	—	4	1	no	—
32	C₃×D₄	—	4	2	no	—
33	C₂×C₃×C₄	—	2	1	no	—
34	C₂×C₃×C₄	—	2	1	no	—
35	C₂×C₃×C₄	—	2	1	no	—
36	C₂×C₃×C₄	—	4	1	no	—
37	C₂×C₃×C₄	—	4	1	no	—
38	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
40	C₃×C₈	—	2	2	no	—
41	C₃×C₈	—	4	1	no	—
42	#41	—	4	1	no	—
43	#41	—	4	1	no	—
44	C₂×Q₈	C₃⋊C₈	1	1	yes	characteristic
45	C₂×D₄	—	2	1	no	—
46	C₂×D₄	C₃⋊C₈	1	1	yes	characteristic
47	C₂²×C₄	—	2	1	no	—
48	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
49	M₁₆	—	4	1	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	#67	1	2	yes	—
53	C₁₆	—	12	2	no	—
54	C₂²×C₃	—	2	1	no	—
55	C₂²×C₃	—	2	2	no	—
56	C₂²×C₃	—	4	1	no	—
57	C₂²×C₃	—	4	2	no	—
58	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₃×C₄	—	2	1	no	—
60	C₃×C₄	—	4	1	no	—
61	C₃×C₄	#99	1	1	yes	characteristic
62	C₃×C₄	#100	1	1	yes	characteristic
63	C₂³	—	2	1	no	—
64	C₂³	#208	1	1	yes	characteristic
65	Q₈	—	4	1	no	—
66	D₄	—	4	1	no	—
67	D₄	—	4	2	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
74	C₂×C₄	#217	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	4	1	no	—
77	C₂×C₃	—	4	1	no	—
78	C₂×C₃	—	8	1	no	—
79	C₂×C₃	#323	1	1	yes	characteristic
80	C₂×C₃	#350	1	2	yes	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	4	2	no	—
85	C₂²	#830	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₄	#833	1	1	yes	characteristic
89	C₄	#834	1	1	yes	characteristic
90	C₃	#1315	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	4	1	no	—
92	C₂	—	8	1	no	—
93	C₂	#4525	1	1	yes	characteristic
94	C₂	#4561	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂×C₃×C₄	C₂×C₈
Frattini	C₂²×C₈	C₂×D₃
Fitting	C₃×#408	C₂
socle	C₂²×C₃	C₂²⋊C₈
prime radical	C₂×C₃×#41	C₄
cosocle	C₂²×C₃×C₈	C₂²
hypercenter	#408	D₃
hypocenter	C₃	#1315
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #408

Complements

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

N	H ≅ G/N	type
2. #4549	C₂	semidirect
90. C₃	#1315	semidirect

Maximal subgroups

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

subgroup	index
C₃×#408	2
#4549	2
#1315	3