Subgroups — C₂²×#88410 #13147375

Order 1728.

299 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6180	C₂	1	1	yes	maximal, characteristic
2	C₂×#88410	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
4	C₂×#6180	C₄	1	3	yes	—
5	C₂×#6180	C₂²	1	3	yes	—
6	#88410	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×C₃⋊C₈	—	6	2	no	—
8	C₂³×H(3,𝔽₃)	C₈	1	1	yes	characteristic
9	C₂×C₄×H(3,𝔽₃)	C₈	1	3	yes	—
10	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
11	#6180	C₂×C₄	1	3	yes	—
12	#6180	C₂³	1	1	yes	characteristic
13	C₂²×C₃×C₁₆	—	9	1	no	maximal
14	C₂²×C₃²×C₄	—	2	2	no	—
15	C₂×C₃×C₃⋊C₈	—	6	6	no	—
16	C₂×C₃×C₃⋊C₈	—	6	6	no	—
17	C₂²×H(3,𝔽₃)	C₁₆	1	4	yes	complemented
18	C₂²×H(3,𝔽₃)	C₂×C₈	1	3	yes	—
19	C₄×H(3,𝔽₃)	C₂×C₈	1	3	yes	—
20	C₄×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₈	—	9	1	no	—
22	C₂²×C₃⋊C₈	—	6	2	no	—
23	C₂×C₃×C₁₆	—	9	6	no	—
24	C₂³×C₃²	—	2	2	no	—
25	C₂×C₃²×C₄	—	2	6	no	—
26	C₂×C₃²×C₄	—	2	6	no	—
27	C₃×C₃⋊C₈	—	6	6	no	—
28	C₃×C₃⋊C₈	—	6	2	no	—
29	C₂²×C₁₆	—	9	1	no	Sylow
30	C₂×H(3,𝔽₃)	C₂×C₁₆	1	6	yes	complemented
31	C₂×H(3,𝔽₃)	C₂²×C₈	1	1	yes	characteristic
32	C₂²×C₃×C₄	—	6	2	no	—
33	C₂²×C₃×C₄	#157	1	1	yes	characteristic
34	C₂×C₃×C₈	—	9	3	no	—
35	C₂×C₃×C₈	—	9	3	no	—
36	C₂×C₃⋊C₈	—	6	6	no	—
37	C₂×C₃⋊C₈	—	6	6	no	—
38	C₃×C₁₆	—	9	4	no	—
39	C₂²×C₃²	—	2	8	no	—
40	C₂²×C₃²	—	2	6	no	—
41	C₃²×C₄	—	2	2	no	—
42	C₃²×C₄	—	2	6	no	—
43	C₂²×C₈	—	9	1	no	—
44	C₂×C₁₆	—	9	6	no	—
45	H(3,𝔽₃)	C₂²×C₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₂³×C₃	—	6	2	no	—
47	C₂³×C₃	#622	1	1	yes	characteristic
48	C₂×C₃×C₄	—	6	6	no	—
49	C₂×C₃×C₄	—	6	6	no	—
50	C₂×C₃×C₄	#622	1	3	yes	—
51	C₂×C₃×C₄	C₂×#157	1	3	yes	—
52	C₃×C₈	—	9	1	no	—
53	C₃×C₈	—	9	3	no	—
54	C₃⋊C₈	—	6	2	no	—
55	C₃⋊C₈	—	6	6	no	—
56	C₂×C₃²	—	2	12	no	—
57	C₂×C₃²	—	2	2	no	—
58	C₂²×C₄	#1088	1	1	yes	characteristic
59	C₂×C₈	—	9	3	no	—
60	C₂×C₈	—	9	3	no	—
61	C₁₆	—	9	4	no	—
62	C₂²×C₃	—	6	6	no	—
63	C₂²×C₃	—	6	8	no	—
64	C₂²×C₃	#3708	1	4	yes	—
65	C₂²×C₃	C₂×#622	1	3	yes	—
66	C₃×C₄	—	6	6	no	—
67	C₃×C₄	—	6	2	no	—
68	C₃×C₄	C₂×#622	1	3	yes	—
69	C₃×C₄	C₂²×#157	1	1	yes	characteristic
70	C₃²	—	2	2	no	—
71	C₂³	#6188	1	1	yes	characteristic
72	C₂×C₄	#6188	1	3	yes	—
73	C₂×C₄	C₂×#1088	1	3	yes	—
74	C₈	—	9	1	no	—
75	C₈	—	9	3	no	—
76	C₂×C₃	—	6	12	no	—
77	C₂×C₃	—	6	2	no	—
78	C₂×C₃	C₂×#3708	1	6	yes	—
79	C₂×C₃	C₂²×#622	1	1	yes	characteristic
80	C₂²	#88410	1	4	yes	direct product factor
81	C₂²	C₂×#6188	1	3	yes	—
82	C₄	C₂×#6188	1	3	yes	—
83	C₄	C₂²×#1088	1	1	yes	characteristic
84	C₃	—	6	2	no	—
85	C₃	C₂²×#3708	1	1	yes	characteristic
86	C₂	C₂×#88410	1	6	yes	direct product factor
87	C₂	C₂²×#6188	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₄	#157
commutator, hypocenter	H(3,𝔽₃)	C₂²×C₁₆
Frattini	C₃×C₄	C₂²×#157
Fitting	C₂²×C₄×H(3,𝔽₃)	C₄
socle	C₂³×C₃	#622
prime radical	C₂³×H(3,𝔽₃)	C₈
cosocle	#6180	C₂³
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88410	C₂	direct (N × H)
6. #88410	C₂²	direct (N × H)
17. C₂²×H(3,𝔽₃)	C₁₆	semidirect
30. C₂×H(3,𝔽₃)	C₂×C₁₆	semidirect
45. H(3,𝔽₃)	C₂²×C₁₆	semidirect
80. C₂²	#88410	direct (N × H)
86. C₂	C₂×#88410	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#6180	2
C₂×#88410	2
C₂²×C₃×C₁₆	9

Subgroups — C22×#88410 #13147375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#88410 #13147375