Subgroups — #71499

Order 384.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	#5183	C₂	1	2	yes	maximal
3	C₂×#829	C₂	1	2	yes	maximal
4	C₂×Dic₂₄	C₂	1	1	yes	maximal, characteristic
5	Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic
6	#2247	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
8	#944	—	2	2	no	—
9	C₂×#216	—	2	2	no	—
10	C₂×Dic₁₂	C₂²	1	2	yes	—
11	C₃×Q₃₂	—	2	2	no	—
12	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
13	#829	—	2	4	no	—
14	C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	Dic₂₄	—	2	2	no	—
17	C₂×Q₃₂	—	3	2	no	—
18	C₂×Q₃₂	—	3	1	no	—
19	C₂×Q₃₂	D₃	1	1	yes	characteristic
20	#494	—	3	2	no	—
21	C₄×C₁₆	—	3	1	no	—
22	C₂×C₃×Q₈	—	2	2	no	—
23	C₂×Dic₆	—	2	2	no	—
24	C₃×Q₁₆	—	2	2	no	—
25	C₃×Q₁₆	—	2	2	no	—
26	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
27	#216	—	4	4	no	—
28	Dic₃⋊C₄	—	4	4	no	—
29	C₄×Dic₃	D₄	1	1	yes	characteristic
30	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
31	Dic₁₂	—	2	2	no	—
32	Dic₁₂	—	2	2	no	—
33	C₃×C₁₆	D₄	1	2	yes	—
34	C₃⋊C₁₆	D₄	1	2	yes	—
35	C₂×Q₁₆	—	3	2	no	—
36	C₂×Q₁₆	—	6	2	no	—
37	C₂×Q₁₆	C₂×D₃	1	2	yes	—
38	C₄⋊Q₈	—	6	2	no	—
39	Q₃₂	—	2	2	no	—
40	Q₃₂	—	6	4	no	—
41	Q₃₂	—	6	2	no	—
42	C₂×C₁₆	—	3	1	no	—
43	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
44	C₄×C₈	—	3	1	no	—
45	C₃×Q₈	—	4	2	no	—
46	C₃×Q₈	—	4	2	no	—
47	C₂×C₃×C₄	—	4	2	no	—
48	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₃	—	4	2	no	—
50	C₂×Dic₃	D₈	1	2	yes	—
51	Dic₆	—	4	2	no	—
52	Dic₆	—	4	2	no	—
53	C₃×C₈	C₂×D₄	1	1	yes	characteristic
54	C₃×C₈	C₂×D₄	1	1	yes	characteristic
55	C₃⋊C₈	D₈	1	2	yes	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	6	2	no	—
58	Q₁₆	—	2	2	no	—
59	Q₁₆	—	2	2	no	—
60	Q₁₆	—	6	2	no	—
61	Q₁₆	—	6	2	no	—
62	Q₁₆	—	12	4	no	—
63	C₂×C₈	—	3	1	no	—
64	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
65	C₄⋊C₄	—	12	4	no	—
66	C₄²	—	3	1	no	—
67	C₁₆	—	3	2	no	—
68	C₁₆	#67	1	2	yes	—
69	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
70	C₃×C₄	—	8	2	no	—
71	C₃×C₄	#127	1	1	yes	characteristic
72	C₃×C₄	C₂×D₈	1	1	yes	characteristic
73	Dic₃	—	8	2	no	—
74	Dic₃	Q₃₂	1	4	yes	complemented
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	Q₈	—	12	2	no	—
78	Q₈	—	12	2	no	—
79	C₂×C₄	—	3	2	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	—	12	2	no	—
82	C₂×C₄	D₃×D₄	1	1	yes	characteristic
83	C₈	—	3	2	no	—
84	C₈	D₃×D₄	1	1	yes	characteristic
85	C₈	C₂×#67	1	1	yes	characteristic
86	C₂×C₃	#493	1	1	yes	characteristic
87	C₂×C₃	C₂×Q₃₂	1	2	yes	—
88	C₂²	D₃×D₈	1	1	yes	characteristic
89	C₄	—	3	4	no	—
90	C₄	—	8	2	no	—
91	C₄	—	24	2	no	—
92	C₄	D₃×D₈	1	1	yes	characteristic
93	C₄	#940	1	1	yes	characteristic
94	C₃	#2247	1	1	yes	characteristic, complemented, Sylow
95	C₂	D₃×Q₃₂	1	2	yes	—
96	C₂	#5151	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₈
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂×C₃×Q₃₂	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₃
hypocenter	C₃	#2247
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
74. Dic₃	Q₃₂	semidirect
94. C₃	#2247	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₃₂	2
#5183	2
C₂×#829	2
C₂×Dic₂₄	2
Dic₃×C₁₆	2
#2247	3