Subgroups — C₂²×#63639 #11899359

Order 1152.

1200 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63639	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
4	C₂²×#812	—	3	4	no	maximal
5	C₂³×C₃²×C₄	C₄	1	1	yes	characteristic
6	C₂²×C₃²×C₈	C₄	1	6	yes	—
7	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×C₈	C₂²	1	6	yes	—
10	#63639	C₂²	1	16	yes	direct product factor
11	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	12	yes	complemented
12	C₂³×C₃×C₈	D₃	1	4	yes	—
13	C₂×#812	—	3	48	no	—
14	C₂²×C₃⋊C₁₆	—	3	8	no	—
15	C₂⁴×C₃²	C₈	1	1	yes	characteristic
16	C₂²×C₃²×C₄	C₈	1	6	yes	—
17	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
18	C₂²×C₃²×C₄	C₂×C₄	1	6	yes	—
19	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃²×C₈	C₂×C₄	1	3	yes	—
21	C₂×C₃²×C₈	C₂×C₄	1	12	yes	—
22	C₂×C₃²×C₈	C₂×C₄	1	6	yes	—
23	C₂×C₃²×C₈	C₂³	1	3	yes	—
24	C₂×C₃²×C₈	C₂³	1	4	yes	—
25	C₃⋊(C₃⋊C₁₆)	C₂³	1	8	yes	complemented
26	C₂²×M₃₂	—	9	1	no	Sylow
27	C₂³×C₃×C₄	Dic₃	1	4	yes	—
28	C₂²×C₃×C₈	Dic₃	1	4	yes	—
29	C₂²×C₃×C₈	Dic₃	1	24	yes	—
30	C₂²×C₃×C₈	C₂×D₃	1	24	yes	—
31	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
32	#812	—	3	64	no	—
33	C₂×C₃⋊C₁₆	—	3	48	no	—
34	C₂³×C₃²	—	2	4	no	—
35	C₂³×C₃²	C₂×C₈	1	6	yes	—
36	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
37	C₂×C₃²×C₄	C₂×C₈	1	12	yes	—
38	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
39	C₂×C₃²×C₄	C₂×C₈	1	6	yes	—
40	C₂×C₃²×C₄	C₂²×C₄	1	4	yes	—
41	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
42	C₃²×C₈	C₂²×C₄	1	3	yes	—
43	C₃²×C₈	C₂²×C₄	1	4	yes	—
44	C₃²×C₈	C₂⁴	1	1	yes	characteristic
45	C₂³×C₈	C₃⋊D₃	1	1	yes	characteristic
46	C₂×M₃₂	—	9	12	no	—
47	C₂²×C₁₆	—	9	2	no	—
48	C₂⁴×C₃	C₃⋊C₈	1	4	yes	—
49	C₂²×C₃×C₄	C₃⋊C₈	1	24	yes	—
50	C₂²×C₃×C₄	C₃⋊C₈	1	4	yes	—
51	C₂²×C₃×C₄	C₂×Dic₃	1	24	yes	—
52	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
53	C₂×C₃×C₈	C₂×Dic₃	1	48	yes	—
54	C₂×C₃×C₈	C₂×Dic₃	1	24	yes	—
55	C₂×C₃×C₈	C₂×Dic₃	1	12	yes	—
56	C₂×C₃×C₈	C₂²×D₃	1	12	yes	—
57	C₂×C₃×C₈	C₂²×D₃	1	16	yes	—
58	C₃⋊C₁₆	—	3	32	no	—
59	C₂²×C₃²	—	2	12	no	—
60	C₂²×C₃²	M₃₂	1	4	yes	complemented
61	C₂²×C₃²	C₂²×C₈	1	4	yes	—
62	C₂²×C₃²	C₂²×C₈	1	3	yes	—
63	C₃²×C₄	C₂²×C₈	1	3	yes	—
64	C₃²×C₄	C₂²×C₈	1	4	yes	—
65	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
66	C₂³×C₄	C₃⋊Dic₃	1	1	yes	characteristic
67	C₂²×C₈	C₃⋊Dic₃	1	1	yes	characteristic
68	C₂²×C₈	C₃⋊Dic₃	1	6	yes	—
69	C₂²×C₈	C₂×C₃⋊D₃	1	6	yes	—
70	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
71	M₃₂	—	9	16	no	—
72	C₂×C₁₆	—	9	12	no	—
73	C₂³×C₃	—	2	16	no	—
74	C₂³×C₃	C₂×C₃⋊C₈	1	24	yes	—
75	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
76	C₂×C₃×C₄	C₂×C₃⋊C₈	1	48	yes	—
77	C₂×C₃×C₄	C₂×C₃⋊C₈	1	24	yes	—
78	C₂×C₃×C₄	C₂×C₃⋊C₈	1	12	yes	—
79	C₂×C₃×C₄	C₂²×Dic₃	1	16	yes	—
80	C₂×C₃×C₄	C₂²×Dic₃	1	12	yes	—
81	C₃×C₈	C₂²×Dic₃	1	12	yes	—
82	C₃×C₈	C₂²×Dic₃	1	16	yes	—
83	C₃×C₈	C₂³×D₃	1	4	yes	—
84	C₂×C₃²	—	2	4	no	—
85	C₂×C₃²	C₂×M₃₂	1	6	yes	complemented
86	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
87	C₂⁴	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
88	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	6	yes	—
89	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₃⋊Dic₃	1	6	yes	—
91	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
92	C₂×C₈	C₂×C₃⋊Dic₃	1	6	yes	—
93	C₂×C₈	C₂×C₃⋊Dic₃	1	12	yes	—
94	C₂×C₈	C₂×C₃⋊Dic₃	1	3	yes	—
95	C₂×C₈	C₂²×C₃⋊D₃	1	3	yes	—
96	C₂×C₈	C₂²×C₃⋊D₃	1	4	yes	—
97	C₁₆	—	9	8	no	—
98	C₂²×C₃	—	2	48	no	—
99	C₂²×C₃	#812	1	16	yes	complemented
100	C₂²×C₃	C₂²×C₃⋊C₈	1	16	yes	—
101	C₂²×C₃	C₂²×C₃⋊C₈	1	12	yes	—
102	C₃×C₄	C₂²×C₃⋊C₈	1	12	yes	—
103	C₃×C₄	C₂²×C₃⋊C₈	1	16	yes	—
104	C₃×C₄	C₂³×Dic₃	1	4	yes	—
105	C₃²	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
106	C₂³	—	2	4	no	—
107	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	6	yes	—
108	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
109	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	12	yes	—
110	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	6	yes	—
111	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
112	C₂×C₄	C₂²×C₃⋊Dic₃	1	4	yes	—
113	C₂×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
114	C₈	C₂²×C₃⋊Dic₃	1	4	yes	—
115	C₈	C₂²×C₃⋊Dic₃	1	3	yes	—
116	C₈	C₂³×C₃⋊D₃	1	1	yes	characteristic
117	C₂×C₃	—	2	16	no	—
118	C₂×C₃	C₂×#812	1	24	yes	complemented
119	C₂×C₃	C₂³×C₃⋊C₈	1	4	yes	—
120	C₂²	—	2	12	no	—
121	C₂²	#63639	1	4	yes	direct product factor
122	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	4	yes	—
123	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	3	yes	—
124	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	3	yes	—
125	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	4	yes	—
126	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
127	C₃	C₂²×#812	1	4	yes	complemented
128	C₂	—	2	4	no	—
129	C₂	C₂×#63639	1	6	yes	direct product factor
130	C₂	C₂³×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₈
Frattini	C₈	C₂³×C₃⋊D₃
Fitting	C₂³×C₃²×C₈	C₂
socle	C₂³×C₃²	C₂×C₈
prime radical	C₂⁴×C₃²	C₈
cosocle	C₃²×C₈	C₂⁴
hypercenter	C₂³×C₈	C₃⋊D₃
hypocenter	C₃²	C₂²×M₃₂
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#63639	C₂	direct (N × H)
3. C₂²×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
10. #63639	C₂²	direct (N × H)
11. C₂×C₃⋊(C₃⋊C₁₆)	C₂²	semidirect
25. C₃⋊(C₃⋊C₁₆)	C₂³	semidirect
60. C₂²×C₃²	M₃₂	semidirect
85. C₂×C₃²	C₂×M₃₂	semidirect
99. C₂²×C₃	#812	semidirect
105. C₃²	C₂²×M₃₂	semidirect
118. C₂×C₃	C₂×#812	semidirect
121. C₂²	#63639	direct (N × H)
127. C₃	C₂²×#812	semidirect
129. C₂	C₂×#63639	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₈	2
C₂×#63639	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₂²×#812	3

Subgroups — C22×#63639 #11899359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63639 #11899359