Subgroups — #13147424

Order 1728.

250 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11721869	C₂	1	1	yes	maximal, characteristic
2	#11720779	C₂	1	3	yes	maximal, complemented
3	C₂×#88410	C₂	1	3	yes	maximal, complemented
4	#41×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	#88538	C₄	1	3	yes	—
6	C₂×#6180	C₂²	1	3	yes	—
7	#88410	C₂²	1	1	yes	characteristic
8	#88410	C₂²	1	3	yes	—
9	C₃×#948	—	6	2	no	—
10	Q₈×H(3,𝔽₃)	C₈	1	1	yes	characteristic
11	D₄×H(3,𝔽₃)	C₈	1	3	yes	—
12	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
13	#6180	C₂×C₄	1	3	yes	—
14	#6180	C₂³	1	1	yes	characteristic
15	C₃×#504	—	9	1	no	maximal
16	C₃²×#41	—	2	2	no	—
17	C₃×#208	—	6	6	no	—
18	C₂×C₃×C₃⋊C₈	—	6	6	no	—
19	C₂²×H(3,𝔽₃)	C₂×C₈	1	3	yes	—
20	C₄×H(3,𝔽₃)	C₂×C₈	1	3	yes	—
21	C₄×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
22	C₃×#131	—	9	1	no	—
23	#948	—	6	2	no	—
24	C₃×M₃₂	—	9	3	no	—
25	C₂×C₃×C₁₆	—	9	3	no	—
26	C₃²×Q₈	—	2	2	no	—
27	C₃²×D₄	—	2	6	no	—
28	C₂×C₃²×C₄	—	2	6	no	—
29	C₃×C₃⋊C₈	—	6	6	no	—
30	C₃×C₃⋊C₈	—	6	2	no	—
31	#504	—	9	1	no	Sylow
32	C₂×H(3,𝔽₃)	—	2	3	no	—
33	C₂×H(3,𝔽₃)	C₂²×C₈	1	1	yes	characteristic
34	C₃×#41	—	6	2	no	—
35	C₃×#41	#157	1	1	yes	characteristic
36	C₃×M₁₆	—	9	3	no	—
37	C₂×C₃×C₈	—	9	3	no	—
38	#208	—	6	6	no	—
39	C₂×C₃⋊C₈	—	6	6	no	—
40	C₃×C₁₆	—	9	1	no	—
41	C₃×C₁₆	—	9	3	no	—
42	C₂²×C₃²	—	2	6	no	—
43	C₃²×C₄	—	2	2	no	—
44	C₃²×C₄	—	2	6	no	—
45	#131	—	9	1	no	—
46	M₃₂	—	9	3	no	—
47	C₂×C₁₆	—	9	3	no	—
48	H(3,𝔽₃)	#504	1	1	yes	characteristic, complemented, Sylow
49	C₃×Q₈	—	6	2	no	—
50	C₃×Q₈	#622	1	1	yes	characteristic
51	C₃×D₄	—	6	6	no	—
52	C₃×D₄	#622	1	3	yes	—
53	C₂×C₃×C₄	—	6	6	no	—
54	C₂×C₃×C₄	C₂×#157	1	3	yes	—
55	C₃×C₈	—	9	1	no	—
56	C₃×C₈	—	9	3	no	—
57	C₃⋊C₈	—	6	2	no	—
58	C₃⋊C₈	—	6	6	no	—
59	C₂×C₃²	—	2	2	no	—
60	C₂×C₃²	—	4	6	no	—
61	#41	#1088	1	1	yes	characteristic
62	M₁₆	—	9	3	no	—
63	C₂×C₈	—	9	3	no	—
64	C₁₆	—	9	1	no	—
65	C₁₆	—	9	3	no	—
66	C₂²×C₃	—	6	6	no	—
67	C₂²×C₃	C₂×#622	1	3	yes	—
68	C₃×C₄	—	6	6	no	—
69	C₃×C₄	—	6	2	no	—
70	C₃×C₄	C₂×#622	1	3	yes	—
71	C₃×C₄	C₂²×#157	1	1	yes	characteristic
72	C₃²	—	2	2	no	—
73	Q₈	#6188	1	1	yes	characteristic
74	D₄	#6188	1	3	yes	—
75	C₂×C₄	C₂×#1088	1	3	yes	—
76	C₈	—	9	1	no	—
77	C₈	—	9	3	no	—
78	C₂×C₃	—	2	3	no	—
79	C₂×C₃	—	6	2	no	—
80	C₂×C₃	—	12	6	no	—
81	C₂×C₃	C₂²×#622	1	1	yes	characteristic
82	C₂²	C₂×#6188	1	3	yes	—
83	C₄	C₂×#6188	1	3	yes	—
84	C₄	C₂²×#1088	1	1	yes	characteristic
85	C₃	—	6	2	no	—
86	C₃	#10593838	1	1	yes	characteristic
87	C₂	—	2	3	no	—
88	C₂	C₂²×#6188	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₄	C₂²×#157
commutator	C₂×H(3,𝔽₃)	C₂²×C₈
Fitting, prime radical	#41×H(3,𝔽₃)	C₄
socle	C₂×C₃	C₂²×#622
cosocle	#6180	C₂³
hypercenter	C₃×#41	#157
hypocenter	H(3,𝔽₃)	#504
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₃×#41

Complements

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

N	H ≅ G/N	type
2. #11720779	C₂	semidirect
3. C₂×#88410	C₂	semidirect
48. H(3,𝔽₃)	#504	semidirect

Maximal subgroups

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

subgroup	index
#11721869	2
#11720779	2
C₂×#88410	2
C₃×#504	9