Subgroups — #71401

Order 384.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#5101	C₂	1	1	yes	maximal, characteristic
3	C₂×#812	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	#2151	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
7	#926	C₄	1	1	yes	characteristic
8	C₂×#208	C₄	1	1	yes	characteristic
9	C₂×C₃×C₁₆	—	2	2	no	—
10	C₂×C₃×C₁₆	C₂²	1	2	yes	—
11	#815	C₄	1	2	yes	—
12	C₈×Dic₃	C₂²	1	2	yes	—
13	#812	—	2	2	no	—
14	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂×M₃₂	—	3	1	no	—
16	C₂²×C₁₆	D₃	1	1	yes	characteristic
17	#405	—	3	1	no	—
18	C₄⋊C₁₆	—	3	4	no	—
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₈	D₄	1	2	yes	—
22	C₂×C₃×C₈	Q₈	1	2	yes	—
23	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
24	#217	C₈	1	2	yes	—
25	C₄⋊Dic₃	C₈	1	2	yes	—
26	C₄×Dic₃	C₂×C₄	1	2	yes	—
27	#208	—	2	2	no	—
28	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₃×C₁₆	—	2	2	no	—
30	C₃⋊C₁₆	—	2	2	no	—
31	C₂×M₁₆	—	3	1	no	—
32	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
33	#117	—	3	1	no	—
34	M₃₂	—	6	2	no	—
35	C₂×C₁₆	—	2	2	no	—
36	C₂×C₁₆	—	3	2	no	—
37	C₂×C₁₆	C₂×D₃	1	2	yes	—
38	#97	—	3	2	no	—
39	C₄×C₈	—	3	2	no	—
40	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
43	C₂×C₃×C₄	M₁₆	1	2	yes	—
44	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₃	C₂×C₈	1	4	yes	—
46	C₃×C₈	C₄⋊C₄	1	2	yes	—
47	C₃×C₈	C₂×D₄	1	1	yes	characteristic
48	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₃⋊C₈	—	2	2	no	—
50	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
51	M₁₆	—	6	2	no	—
52	C₂×C₈	—	3	2	no	—
53	C₂×C₈	Dic₆	1	2	yes	—
54	C₂×C₈	C₄×D₃	1	1	yes	characteristic
55	C₂×C₈	#67	1	2	yes	—
56	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
57	C₄⋊C₄	—	3	2	no	—
58	C₂²⋊C₄	—	3	2	no	—
59	C₄²	—	3	2	no	—
60	C₁₆	—	2	2	no	—
61	C₁₆	—	6	2	no	—
62	C₂²×C₃	—	2	2	no	—
63	C₂²×C₃	C₄⋊C₈	1	2	yes	—
64	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
65	C₃×C₄	C₄⋊C₈	1	2	yes	—
66	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
68	Dic₃	—	2	4	no	—
69	C₂³	D₃×C₈	1	1	yes	characteristic
70	C₂×C₄	—	3	4	no	—
71	C₂×C₄	D₃×C₈	1	1	yes	characteristic
72	C₂×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
73	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
74	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
75	C₈	—	6	2	no	—
76	C₈	Dic₃⋊C₄	1	2	yes	—
77	C₈	C₂×Dic₆	1	1	yes	characteristic
78	C₈	C₂×#67	1	1	yes	characteristic
79	C₂×C₃	—	2	2	no	—
80	C₂×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₃	#504	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	Dic₃⋊C₈	1	2	yes	—
84	C₂²	C₂×D₃×C₈	1	1	yes	characteristic
85	C₄	—	6	4	no	—
86	C₄	Dic₃⋊C₈	1	2	yes	—
87	C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
88	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
89	C₃	#2151	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	2	no	—
91	C₂	#4901	1	2	yes	—
92	C₂	C₂×Dic₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₈
Fitting	C₂²×C₃×C₁₆	C₂
prime radical	C₂³×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	#2151
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
4. Dic₃⋊C₁₆	C₂	semidirect
89. C₃	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₁₆	2
#5101	2
C₂×#812	2
Dic₃⋊C₁₆	2
#2151	3