Subgroups — #73408

Order 384.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#376	C₂	1	1	yes	maximal, characteristic
2	#4929	C₂	1	3	yes	maximal
3	#4928	C₂	1	3	yes	maximal
4	C₃⋊#401	C₂	1	2	yes	maximal
5	#4658	C₂	1	6	yes	maximal
6	#2645	—	3	1	no	maximal, Sylow
7	C₂²×Dic₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²	C₂²	1	3	yes	—
9	C₂×C₄⋊Dic₃	C₂²	1	3	yes	—
10	C₂×Dic₃⋊C₄	C₂²	1	6	yes	—
11	C₃×#95	C₂²	1	3	yes	—
12	C₃×#95	C₂²	1	1	yes	characteristic
13	C₃⋊#95	C₂²	1	6	yes	—
14	C₃⋊#95	C₂²	1	6	yes	—
15	C₃⋊#95	C₂²	1	6	yes	—
16	#401	—	3	2	no	—
17	#400	—	3	6	no	—
18	#391	—	3	3	no	—
19	#389	—	3	3	no	—
20	#376	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄	C₂³	1	3	yes	—
22	C₂²×C₃×C₄	C₂³	1	3	yes	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃	C₂³	1	2	yes	—
25	C₂²×Dic₃	C₂³	1	6	yes	—
26	C₂×Dic₆	—	2	6	no	—
27	C₃×C₄²	—	2	6	no	—
28	C₄⋊Dic₃	—	2	6	no	—
29	Dic₃⋊C₄	—	4	6	no	—
30	C₂²×Q₈	—	3	1	no	—
31	C₂×C₄⋊C₄	—	3	3	no	—
32	C₂×C₄⋊C₄	—	3	6	no	—
33	C₂×C₄²	C₂×D₃	1	3	yes	—
34	#95	—	3	6	no	—
35	#95	—	3	6	no	—
36	#95	—	3	6	no	—
37	#95	C₂×D₃	1	3	yes	—
38	#95	C₂×D₃	1	1	yes	characteristic
39	C₂³×C₃	C₂⁴	1	1	yes	characteristic
40	C₂×C₃×C₄	—	2	3	no	—
41	C₂×C₃×C₄	—	2	3	no	—
42	C₂×C₃×C₄	—	2	3	no	—
43	C₂×C₃×C₄	—	2	6	no	—
44	C₂×C₃×C₄	—	2	3	no	—
45	C₂×C₃×C₄	#41	1	6	yes	—
46	C₂×Dic₃	—	2	6	no	—
47	C₂×Dic₃	—	2	6	no	—
48	C₂×Dic₃	—	2	6	no	—
49	C₂×Dic₃	—	2	6	no	—
50	Dic₆	—	4	4	no	—
51	C₂×Q₈	—	6	6	no	—
52	C₂²×C₄	—	3	2	no	—
53	C₂²×C₄	—	3	6	no	—
54	C₂²×C₄	C₂²×D₃	1	3	yes	—
55	C₂²×C₄	C₂²×D₃	1	3	yes	—
56	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
57	C₄⋊C₄	—	6	6	no	—
58	C₄⋊C₄	—	12	6	no	—
59	C₄²	—	2	6	no	—
60	C₂²×C₃	C₂×#41	1	3	yes	—
61	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
62	C₂²×C₃	2₋¹⁺⁴	1	3	yes	—
63	C₃×C₄	—	4	3	no	—
64	C₃×C₄	—	4	3	no	—
65	C₃×C₄	#126	1	4	yes	—
66	Dic₃	—	4	2	no	—
67	Dic₃	—	4	6	no	—
68	C₂³	C₂³×D₃	1	1	yes	characteristic
69	Q₈	—	12	4	no	—
70	C₂×C₄	—	2	3	no	—
71	C₂×C₄	—	2	3	no	—
72	C₂×C₄	—	2	3	no	—
73	C₂×C₄	—	2	6	no	—
74	C₂×C₄	—	2	3	no	—
75	C₂×C₄	—	6	6	no	—
76	C₂×C₄	—	6	6	no	—
77	C₂×C₄	—	6	6	no	—
78	C₂×C₄	—	6	6	no	—
79	C₂×C₄	#235	1	6	yes	—
80	C₂×C₃	C₂×#126	1	1	yes	characteristic
81	C₂×C₃	#541	1	3	yes	—
82	C₂×C₃	#542	1	3	yes	—
83	C₂²	C₂×#235	1	3	yes	—
84	C₂²	#1009	1	1	yes	characteristic
85	C₂²	#1010	1	3	yes	—
86	C₄	—	4	3	no	—
87	C₄	—	4	3	no	—
88	C₄	—	12	2	no	—
89	C₄	—	12	6	no	—
90	C₄	#876	1	4	yes	—
91	C₃	#2645	1	1	yes	characteristic, complemented, Sylow
92	C₂	C₂×#876	1	1	yes	characteristic
93	C₂	#5518	1	3	yes	—
94	C₂	#5534	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#376	C₂
hypercenter	#376	D₃
hypocenter	C₃	#2645
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #376

Complements

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

N	H ≅ G/N	type
91. C₃	#2645	semidirect

Maximal subgroups

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

subgroup	index
C₃×#376	2
#4929	2
#4928	2
C₃⋊#401	2
#4658	2
#2645	3