Subgroups — #69886

Order 384.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#405	C₂	1	1	yes	maximal, characteristic
2	#4553	C₂	1	2	yes	maximal
3	#4466	C₂	1	2	yes	maximal, complemented
4	#4463	C₂	1	2	yes	maximal, complemented
5	#2166	—	3	1	no	maximal, Sylow
6	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
8	C₃×#117	C₄	1	1	yes	characteristic
9	C₃×#97	C₄	1	1	yes	characteristic
10	C₃×#97	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₄	1	1	yes	characteristic
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#812	C₂²	1	4	yes	—
14	#405	D₃	1	1	yes	characteristic
15	#364	—	3	2	no	—
16	#349	—	3	2	no	—
17	#347	—	3	2	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₃×M₁₆	—	2	2	no	—
20	C₂×C₃×C₈	—	2	2	no	—
21	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₃×C₄⋊C₄	—	2	1	no	—
26	C₃×C₂²⋊C₄	—	2	1	no	—
27	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃⋊C₁₆	—	2	4	no	—
30	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
31	C₂²×C₈	Dic₃	1	1	yes	characteristic
32	#117	Dic₃	1	1	yes	characteristic
33	M₃₂	—	3	4	no	—
34	#97	Dic₃	1	1	yes	characteristic
35	#97	C₂×D₃	1	1	yes	characteristic
36	C₄×C₈	Dic₃	1	1	yes	characteristic
37	C₄×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂³×C₃	M₁₆	1	1	yes	characteristic
39	C₂×C₃×C₄	—	2	1	no	—
40	C₂×C₃×C₄	—	2	1	no	—
41	C₂×C₃×C₄	—	2	1	no	—
42	C₂×C₃×C₄	—	2	1	no	—
43	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₃×C₈	—	2	1	no	—
46	C₃×C₈	—	2	1	no	—
47	C₃×C₈	M₁₆	1	2	yes	—
48	C₃×C₈	#41	1	2	yes	—
49	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
50	M₁₆	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
53	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
54	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
55	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
56	C₄⋊C₄	—	2	1	no	—
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂×Dic₃	1	1	yes	characteristic
59	C₄²	C₂×Dic₃	1	1	yes	characteristic
60	C₁₆	—	6	4	no	—
61	C₂²×C₃	—	2	1	no	—
62	C₂²×C₃	—	4	1	no	—
63	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
64	C₃×C₄	—	2	1	no	—
65	C₃×C₄	—	4	1	no	—
66	C₃×C₄	—	4	1	no	—
67	C₃×C₄	#117	1	1	yes	characteristic
68	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
69	C₂³	#208	1	1	yes	characteristic
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	#208	1	1	yes	characteristic
75	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₈	#208	1	2	yes	—
79	C₈	#235	1	2	yes	—
80	C₂×C₃	—	2	1	no	—
81	C₂×C₃	—	4	1	no	—
82	C₂×C₃	#432	1	1	yes	characteristic
83	C₂²	—	2	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	C₂×#208	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	4	1	no	—
89	C₄	C₂×#208	1	1	yes	characteristic
90	C₄	#926	1	1	yes	characteristic
91	C₃	#2166	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#4926	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#926
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#405	C₂
socle	C₂×C₃	#432
prime radical	C₂³×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#405	D₃
hypocenter	C₃	#2166
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #97 ◁ #405

Complements

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

N	H ≅ G/N	type
3. #4466	C₂	semidirect
4. #4463	C₂	semidirect
91. C₃	#2166	semidirect

Maximal subgroups

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

subgroup	index
C₃×#405	2
#4553	2
#4466	2
#4463	2
#2166	3