Subgroups — #71823

Order 384.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic
2	#4926	C₂	1	2	yes	maximal
3	#4543	C₂	1	4	yes	maximal, complemented
4	#1537	—	3	1	no	maximal, Sylow
5	C₃×C₄⋊Q₈	C₄	1	2	yes	—
6	C₃×#125	C₂²	1	2	yes	—
7	C₃×C₂²⋊Q₈	—	2	2	no	—
8	C₃×C₄×Q₈	—	2	2	no	—
9	C₃×#117	C₄	1	2	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	#830	—	2	2	no	—
12	C₄⋊(C₃⋊C₈)	—	2	2	no	—
13	#803	C₂²	1	4	yes	—
14	#533	D₃	1	1	yes	characteristic
15	#432	—	3	2	no	—
16	#330	—	3	4	no	—
17	C₂×C₃×Q₈	—	2	2	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	D₄	1	2	yes	—
20	C₃×C₄⋊C₄	—	2	2	no	—
21	C₃×C₄⋊C₄	—	2	4	no	—
22	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
23	C₃×C₂²⋊C₄	—	2	2	no	—
24	C₃×C₄²	—	2	2	no	—
25	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²	D₄	1	2	yes	—
27	C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₂×C₃⋊C₈	—	2	4	no	—
29	C₄⋊Q₈	Dic₃	1	2	yes	—
30	#125	C₂×D₃	1	2	yes	—
31	C₂²⋊Q₈	—	2	2	no	—
32	C₄×Q₈	—	2	2	no	—
33	#117	Dic₃	1	2	yes	—
34	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
35	C₄⋊C₈	—	6	2	no	—
36	C₂²⋊C₈	—	6	2	no	—
37	#97	—	3	4	no	—
38	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₃×Q₈	—	4	4	no	—
40	C₂×C₃×C₄	—	2	2	no	—
41	C₂×C₃×C₄	—	2	4	no	—
42	C₂×C₃×C₄	—	4	2	no	—
43	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
44	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
47	C₃⋊C₈	—	4	4	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
50	C₂²×C₄	#67	1	2	yes	—
51	C₂×C₈	—	6	4	no	—
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	2	4	no	—
54	C₄⋊C₄	C₂×Dic₃	1	4	yes	—
55	C₂²⋊C₄	—	2	2	no	—
56	C₄²	—	2	2	no	—
57	C₄²	C₂×Dic₃	1	1	yes	characteristic
58	C₄²	#67	1	2	yes	—
59	C₄²	C₂²×D₃	1	1	yes	characteristic
60	C₂²×C₃	—	2	1	no	—
61	C₂²×C₃	—	2	2	no	—
62	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	—	2	2	no	—
65	C₃×C₄	—	2	2	no	—
66	C₃×C₄	—	4	4	no	—
67	C₃×C₄	#101	1	2	yes	—
68	C₂³	#217	1	1	yes	characteristic
69	Q₈	—	4	4	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	#217	1	2	yes	—
74	C₂×C₄	#217	1	1	yes	characteristic
75	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
76	C₂×C₄	C₂×#67	1	2	yes	—
77	C₈	—	12	4	no	—
78	C₂×C₃	—	4	1	no	—
79	C₂×C₃	C₂×#101	1	1	yes	characteristic
80	C₂×C₃	#421	1	2	yes	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	2	no	—
83	C₂²	C₂×#217	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	2	no	—
86	C₄	—	2	2	no	—
87	C₄	—	4	4	no	—
88	C₄	#836	1	2	yes	—
89	C₃	#1537	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	4	1	no	—
91	C₂	C₂×#836	1	1	yes	characteristic
92	C₂	#5236	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#217
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#533	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	#533	D₃
hypocenter	C₃	#1537
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
3. #4543	C₂	semidirect
89. C₃	#1537	semidirect

Maximal subgroups

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

subgroup	index
C₃×#533	2
#4926	2
#4543	2
#1537	3