Subgroups — #69805

Order 384.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#803	C₂	1	1	yes	maximal, characteristic
4	#4550	C₂	1	4	yes	maximal
5	#1745	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃	C₄	1	4	yes	—
9	C₂×C₄×Dic₃	C₂²	1	2	yes	—
10	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
11	C₃×#97	C₄	1	4	yes	complemented
12	#803	C₄	1	4	yes	complemented
13	C₂×#97	—	3	1	no	—
14	C₂×#97	D₃	1	1	yes	characteristic
15	C₄³	—	3	1	no	—
16	#336	—	3	4	no	—
17	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃	C₂×C₄	1	4	yes	—
20	C₂×C₃×C₈	—	2	4	no	—
21	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
22	C₃×C₄²	C₂×C₄	1	2	yes	—
23	C₃×C₄²	C₂×C₄	1	2	yes	—
24	C₄×Dic₃	—	2	4	no	—
25	C₄×Dic₃	—	2	8	no	—
26	C₂×C₃⋊C₈	—	2	4	no	—
27	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₈	—	3	2	no	—
29	C₂²×C₈	C₂×D₃	1	2	yes	—
30	C₂×C₄²	—	3	4	no	—
31	C₂×C₄²	—	3	2	no	—
32	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
33	#97	—	3	4	no	—
34	#97	Dic₃	1	4	yes	complemented
35	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄	—	2	4	no	—
37	C₂×C₃×C₄	C₄²	1	4	yes	—
38	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄	#41	1	4	yes	—
41	C₂×Dic₃	—	2	4	no	—
42	C₂×Dic₃	—	2	4	no	—
43	C₂×Dic₃	M₁₆	1	8	yes	—
44	C₃×C₈	—	2	4	no	—
45	C₃⋊C₈	—	2	4	no	—
46	C₂²×C₄	—	3	4	no	—
47	C₂²×C₄	C₄×D₃	1	2	yes	—
48	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	—	3	4	no	—
51	C₂×C₈	—	6	4	no	—
52	C₂×C₈	C₂×Dic₃	1	4	yes	—
53	C₄²	—	6	8	no	—
54	C₄²	—	6	4	no	—
55	C₄²	C₄×D₃	1	2	yes	—
56	C₄²	C₂×Dic₃	1	2	yes	—
57	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
58	C₂²×C₃	#117	1	2	yes	—
59	C₂²×C₃	C₂×M₁₆	1	4	yes	—
60	C₃×C₄	—	2	4	no	—
61	C₃×C₄	#117	1	2	yes	—
62	C₃×C₄	#117	1	2	yes	—
63	Dic₃	—	2	8	no	—
64	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	3	8	no	—
67	C₂×C₄	—	6	4	no	—
68	C₂×C₄	—	6	4	no	—
69	C₂×C₄	C₄×Dic₃	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
71	C₂×C₄	#235	1	4	yes	—
72	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
73	C₈	—	2	4	no	—
74	C₈	—	6	4	no	—
75	C₂×C₃	#376	1	1	yes	characteristic
76	C₂×C₃	C₄×M₁₆	1	2	yes	—
77	C₂×C₃	#432	1	4	yes	—
78	C₂²	#872	1	2	yes	—
79	C₂²	D₃×M₁₆	1	4	yes	—
80	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	6	8	no	—
83	C₄	#872	1	2	yes	—
84	C₄	#926	1	2	yes	—
85	C₃	#1745	1	1	yes	characteristic, complemented, Sylow
86	C₂	#4705	1	4	yes	—
87	C₂	#4932	1	1	yes	characteristic
88	C₂	Dic₃×M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₂×C₃×#97	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#97	D₃
hypocenter	C₃	#1745
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
11. C₃×#97	C₄	semidirect
12. #803	C₄	semidirect
34. #97	Dic₃	semidirect
85. C₃	#1745	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#97	2
C₄²×Dic₃	2
C₂×#803	2
#4550	2
#1745	3