Subgroups — #3804

Order 144.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#638	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#67	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#627	C₂	1	1	yes	maximal, characteristic, complemented
5	#626	C₂	1	1	yes	maximal, characteristic, complemented
6	#624	C₂	1	1	yes	maximal, characteristic, complemented
7	D₃×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#237	—	3	1	no	maximal
9	#235	—	3	1	no	maximal
10	C₂²×C₃²	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×D₃	C₂²	1	1	yes	characteristic
13	C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₃×Dic₃	C₂²	1	1	yes	characteristic
15	C₃×Dic₃	C₂²	1	1	yes	characteristic
16	C₃×Dic₃	C₂²	1	1	yes	characteristic
17	C₃×D₄	—	3	1	no	—
18	C₂×C₃×C₄	—	3	1	no	—
19	#67	—	3	1	no	—
20	#67	—	3	1	no	—
21	#67	—	3	1	no	—
22	#67	—	6	1	no	—
23	#67	D₃	1	1	yes	characteristic, complemented
24	C₂×Dic₃	—	3	1	no	—
25	C₂×Dic₃	D₃	1	1	yes	characteristic, complemented
26	D₁₂	—	3	1	no	—
27	C₄×D₃	—	3	1	no	—
28	C₄×D₃	—	3	1	no	—
29	C₄×D₃	—	3	1	no	—
30	Dic₆	—	3	1	no	—
31	Dic₆	—	3	1	no	—
32	C₂×C₃²	—	2	1	no	—
33	C₂×C₃²	C₂³	1	1	yes	characteristic
34	C₃⋊D₃	—	2	1	no	—
35	C₃×D₃	—	2	1	no	—
36	#41	—	9	1	no	Sylow
37	C₂²×C₃	—	2	1	no	—
38	C₂²×C₃	—	3	1	no	—
39	C₂²×C₃	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃	C₂×D₃	1	1	yes	characteristic
41	C₂×D₃	—	3	1	no	—
42	C₂×D₃	—	3	1	no	—
43	C₂×D₃	—	6	1	no	—
44	C₂×D₃	C₂×D₃	1	1	yes	characteristic
45	C₃×C₄	—	3	1	no	—
46	C₃×C₄	—	3	1	no	—
47	C₃×C₄	—	3	1	no	—
48	Dic₃	—	3	1	no	—
49	Dic₃	—	3	1	no	—
50	Dic₃	—	6	1	no	—
51	Dic₃	C₂×D₃	1	1	yes	characteristic
52	Dic₃	C₂×D₃	1	1	yes	characteristic
53	Dic₃	C₂×D₃	1	1	yes	characteristic
54	C₃²	#41	1	1	yes	characteristic, complemented, Sylow
55	Q₈	—	9	1	no	—
56	D₄	—	3	1	no	—
57	D₄	—	9	1	no	—
58	D₄	—	9	1	no	—
59	C₂×C₄	—	3	1	no	—
60	C₂×C₄	—	9	1	no	—
61	C₂×C₄	—	9	1	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	—	2	1	no	—
64	C₂×C₃	—	2	1	no	—
65	C₂×C₃	—	4	1	no	—
66	C₂×C₃	—	6	1	no	—
67	C₂×C₃	C₂²×D₃	1	1	yes	characteristic
68	C₂×C₃	C₂²×D₃	1	1	yes	characteristic
69	D₃	—	2	1	no	—
70	D₃	—	6	1	no	—
71	D₃	—	6	1	no	—
72	D₃	—	12	1	no	—
73	C₂²	—	3	1	no	—
74	C₂²	—	9	1	no	—
75	C₂²	D₃²	1	1	yes	characteristic
76	C₄	—	3	1	no	—
77	C₄	—	3	1	no	—
78	C₄	—	3	1	no	—
79	C₄	—	9	1	no	—
80	C₃	—	2	1	no	—
81	C₃	#235	1	1	yes	characteristic, complemented
82	C₃	#237	1	1	yes	characteristic, complemented
83	C₂	—	2	1	no	—
84	C₂	—	6	1	no	—
85	C₂	—	18	1	no	—
86	C₂	C₂×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃²
commutator, socle, cosocle	C₂×C₃²	C₂³
Fitting	C₂²×C₃²	C₂²
hypercenter	C₂²	D₃²
hypocenter	C₃²	#41
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. #638	C₂	semidirect
2. C₃×#67	C₂	semidirect
3. C₂×C₃×Dic₃	C₂	semidirect
4. #627	C₂	semidirect
5. #626	C₂	semidirect
6. #624	C₂	semidirect
7. D₃×Dic₃	C₂	semidirect
23. #67	D₃	semidirect
25. C₂×Dic₃	D₃	semidirect
54. C₃²	#41	semidirect
81. C₃	#235	semidirect
82. C₃	#237	semidirect

Maximal subgroups

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

subgroup	index
#638	2
C₃×#67	2
C₂×C₃×Dic₃	2
#627	2
#626	2
#624	2
D₃×Dic₃	2
#237	3
#235	3