Subgroups — #73308

Order 384.

504 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#869	C₂	1	2	yes	maximal
2	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
3	#4936	C₂	1	4	yes	maximal
4	#4929	C₂	1	8	yes	maximal
5	#2862	—	3	1	no	maximal, Sylow
6	C₂²×Dic₆	C₂²	1	4	yes	—
7	C₂×C₃×C₄²	C₂²	1	4	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
12	#869	—	2	8	no	—
13	C₃⋊#95	C₂²	1	16	yes	—
14	C₂×C₄⋊Q₈	—	3	2	no	—
15	#391	—	3	8	no	—
16	#382	—	3	4	no	—
17	C₄³	D₃	1	1	yes	characteristic
18	C₂²×C₃×C₄	C₂³	1	4	yes	—
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂²×C₃×C₄	C₂³	1	2	yes	—
21	C₂²×Dic₃	C₂³	1	8	yes	—
22	C₂×Dic₆	—	2	8	no	—
23	C₂×Dic₆	—	2	16	no	—
24	C₃×C₄²	—	2	8	no	—
25	C₃×C₄²	D₄	1	4	yes	—
26	C₃×C₄²	Q₈	1	8	yes	complemented
27	C₄⋊Dic₃	—	2	8	no	—
28	C₄⋊Dic₃	—	2	8	no	—
29	C₂²×Q₈	—	3	4	no	—
30	C₄⋊Q₈	—	6	8	no	—
31	C₂×C₄⋊C₄	—	3	4	no	—
32	C₂×C₄⋊C₄	—	3	4	no	—
33	C₂×C₄²	C₂×D₃	1	4	yes	—
34	C₂×C₄²	C₂×D₃	1	2	yes	—
35	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
36	#95	—	3	16	no	—
37	C₂³×C₃	C₂⁴	1	1	yes	characteristic
38	C₂×C₃×C₄	—	2	8	no	—
39	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
40	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
41	C₂×C₃×C₄	C₂×Q₈	1	8	yes	—
42	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
43	C₂×C₃×C₄	#41	1	8	yes	—
44	C₂×Dic₃	—	2	8	no	—
45	C₂×Dic₃	—	2	16	no	—
46	Dic₆	—	4	16	no	—
47	C₂×Q₈	—	6	8	no	—
48	C₂×Q₈	—	6	16	no	—
49	C₂²×C₄	—	3	8	no	—
50	C₂²×C₄	C₂²×D₃	1	4	yes	—
51	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
52	C₂²×C₄	C₂²×D₃	1	2	yes	—
53	C₄⋊C₄	—	6	8	no	—
54	C₄⋊C₄	—	6	8	no	—
55	C₄²	—	2	8	no	—
56	C₄²	Dic₆	1	8	yes	complemented
57	C₄²	D₁₂	1	4	yes	—
58	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
59	C₂²×C₃	C₂²×Q₈	1	2	yes	—
60	C₂²×C₃	C₂×#41	1	4	yes	—
61	C₃×C₄	—	2	8	no	—
62	C₃×C₄	#124	1	8	yes	—
63	C₃×C₄	C₄⋊Q₈	1	4	yes	—
64	Dic₃	—	4	8	no	—
65	C₂³	C₂³×D₃	1	1	yes	characteristic
66	Q₈	—	12	16	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	6	8	no	—
69	C₂×C₄	—	6	16	no	—
70	C₂×C₄	C₂×Dic₆	1	8	yes	—
71	C₂×C₄	C₂×Dic₆	1	4	yes	—
72	C₂×C₄	C₂×D₁₂	1	2	yes	—
73	C₂×C₄	C₂×D₁₂	1	4	yes	—
74	C₂×C₄	#235	1	8	yes	—
75	C₂×C₃	C₂×#124	1	2	yes	—
76	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₃	#533	1	4	yes	—
78	C₂²	C₂²×Dic₆	1	2	yes	—
79	C₂²	C₂²×D₁₂	1	1	yes	characteristic
80	C₂²	C₂×#235	1	4	yes	—
81	C₄	—	2	8	no	—
82	C₄	—	12	8	no	—
83	C₄	#869	1	4	yes	—
84	C₄	#875	1	8	yes	—
85	C₃	#2862	1	1	yes	characteristic, complemented, Sylow
86	C₂	C₂×#869	1	1	yes	characteristic
87	C₂	#5470	1	4	yes	—
88	C₂	C₂×#875	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×C₄³	C₂
hypercenter	C₄³	D₃
hypocenter	C₃	#2862
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
26. C₃×C₄²	Q₈	semidirect
56. C₄²	Dic₆	semidirect
85. C₃	#2862	semidirect

Maximal subgroups

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

subgroup	index
C₂×#869	2
C₃×C₄³	2
#4936	2
#4929	2
#2862	3