Subgroups — #69849

Order 384.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#208	C₂	1	2	yes	maximal
3	#4455	C₂	1	1	yes	maximal, characteristic, complemented
4	#4454	C₂	1	2	yes	maximal, complemented
5	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#1455	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₁₆	C₄	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×#208	C₄	1	4	yes	—
10	C₃×#97	C₄	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#803	C₄	1	4	yes	—
14	C₄×C₃⋊C₈	C₂²	1	2	yes	—
15	C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	C₄×M₁₆	—	3	2	no	—
17	C₄×M₁₆	D₃	1	1	yes	characteristic
18	#322	—	3	2	no	—
19	#322	—	3	1	no	—
20	C₈²	—	3	1	no	—
21	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₃×M₁₆	—	2	4	no	—
24	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
26	C₃×C₄²	C₂×C₄	1	2	yes	—
27	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²	C₂³	1	1	yes	characteristic
29	#208	—	2	8	no	—
30	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
31	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
32	C₂×M₁₆	—	3	4	no	—
33	C₂×M₁₆	Dic₃	1	2	yes	—
34	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
35	#97	—	3	4	no	—
36	#97	Dic₃	1	2	yes	—
37	C₄×C₈	—	3	2	no	—
38	C₄×C₈	—	3	2	no	—
39	C₄×C₈	C₂×D₃	1	1	yes	characteristic
40	C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂³×C₃	C₄²	1	1	yes	characteristic
42	C₂×C₃×C₄	—	2	4	no	—
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
45	C₂×C₃×C₄	C₄²	1	2	yes	—
46	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
47	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₈	—	2	2	no	—
49	C₃×C₈	M₁₆	1	4	yes	complemented
50	C₃⋊C₈	—	2	4	no	—
51	C₃⋊C₈	M₁₆	1	8	yes	complemented
52	C₂²×C₄	C₄×D₃	1	2	yes	—
53	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
54	M₁₆	—	2	4	no	—
55	M₁₆	—	6	8	no	—
56	C₂×C₈	—	3	4	no	—
57	C₂×C₈	—	3	4	no	—
58	C₂×C₈	C₂×Dic₃	1	2	yes	—
59	C₂×C₈	C₂×Dic₃	1	2	yes	—
60	C₄²	C₄×D₃	1	2	yes	—
61	C₄²	C₂×Dic₃	1	1	yes	characteristic
62	C₄²	C₂²×D₃	1	1	yes	characteristic
63	C₂²×C₃	—	2	1	no	—
64	C₂²×C₃	—	2	2	no	—
65	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
66	C₃×C₄	—	2	1	no	—
67	C₃×C₄	—	2	2	no	—
68	C₃×C₄	C₂×M₁₆	1	4	yes	—
69	C₃×C₄	C₂×M₁₆	1	2	yes	—
70	C₂³	C₄×Dic₃	1	1	yes	characteristic
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
74	C₂×C₄	C₄×Dic₃	1	2	yes	—
75	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
76	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	3	8	no	—
79	C₈	—	6	4	no	—
80	C₈	#208	1	4	yes	complemented
81	C₂×C₃	—	4	1	no	—
82	C₂×C₃	C₄×M₁₆	1	1	yes	characteristic
83	C₂×C₃	C₄×M₁₆	1	2	yes	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	2	no	—
86	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	2	2	no	—
89	C₄	D₃×M₁₆	1	4	yes	—
90	C₄	C₂×#208	1	2	yes	—
91	C₃	#1455	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	4	1	no	—
93	C₂	C₄×#208	1	1	yes	characteristic
94	C₂	Dic₃×M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator, socle	C₂²×C₃	C₂×C₄²
Fitting	C₃×C₄×M₁₆	C₂
prime radical	C₂³×C₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	#1455
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #4455	C₂	semidirect
4. #4454	C₂	semidirect
5. C₈×C₃⋊C₈	C₂	semidirect
49. C₃×C₈	M₁₆	semidirect
51. C₃⋊C₈	M₁₆	semidirect
80. C₈	#208	semidirect
91. C₃	#1455	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×M₁₆	2
C₄×#208	2
#4455	2
#4454	2
C₈×C₃⋊C₈	2
#1455	3