Subgroups — C₂²×#63645 #11858398

Order 1152.

1800 conjugacy classes of subgroups.

The subgroup lattice has 144 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	C₂²×C₄⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
3	C₂×#63645	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#818	—	3	4	no	maximal
5	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
6	C₂³×C₃⋊Dic₃	—	2	2	no	—
7	C₂²×C₃²×C₈	C₄	1	8	yes	complemented
8	C₂²×C₃²×C₈	C₂²	1	6	yes	—
9	C₂×C₄⋊(C₃⋊Dic₃)	—	2	6	no	—
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	12	yes	—
11	#63645	C₂²	1	16	yes	direct product factor
12	C₂³×C₃×C₈	D₃	1	4	yes	—
13	C₂²×C₄⋊Dic₃	—	3	8	no	—
14	C₂×#818	—	3	48	no	—
15	C₂⁴×C₃²	D₄	1	1	yes	characteristic
16	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
17	C₂²×C₃²×C₄	D₄	1	3	yes	—
18	C₂²×C₃²×C₄	Q₈	1	4	yes	—
19	C₂²×C₃²×C₄	C₂³	1	3	yes	—
20	C₂²×C₃⋊Dic₃	—	2	12	no	—
21	C₂²×C₃⋊Dic₃	—	4	8	no	—
22	C₂×C₃²×C₈	C₂×C₄	1	24	yes	complemented
23	C₂×C₃²×C₈	C₂³	1	4	yes	—
24	C₄⋊(C₃⋊Dic₃)	—	2	12	no	—
25	C₄⋊(C₃⋊Dic₃)	C₂³	1	8	yes	—
26	C₂²×#107	—	9	1	no	Sylow
27	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
28	C₂³×Dic₃	—	6	8	no	—
29	C₂²×C₃×C₈	Dic₃	1	32	yes	complemented
30	C₂²×C₃×C₈	C₂×D₃	1	24	yes	—
31	C₂×C₄⋊Dic₃	—	3	48	no	—
32	C₂×C₄⋊Dic₃	—	6	24	no	—
33	#818	—	3	64	no	—
34	C₂³×C₃²	C₄⋊C₄	1	4	yes	—
35	C₂³×C₃²	D₈	1	4	yes	—
36	C₂³×C₃²	Q₁₆	1	4	yes	—
37	C₂³×C₃²	C₂×D₄	1	3	yes	—
38	C₂×C₃²×C₄	C₄⋊C₄	1	12	yes	—
39	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
40	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
41	C₂×C₃²×C₄	C₂×Q₈	1	6	yes	—
42	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
43	C₂×C₃⋊Dic₃	—	2	8	no	—
44	C₂×C₃⋊Dic₃	—	4	24	no	—
45	C₃²×C₈	C₂²×C₄	1	8	yes	complemented
46	C₂³×C₈	C₃⋊D₃	1	1	yes	characteristic
47	C₂²×C₄⋊C₄	—	9	2	no	—
48	C₂×#107	—	9	12	no	—
49	C₂⁴×C₃	D₁₂	1	4	yes	—
50	C₂²×C₃×C₄	Dic₆	1	16	yes	—
51	C₂²×C₃×C₄	D₁₂	1	12	yes	—
52	C₂²×C₃×C₄	C₂×Dic₃	1	16	yes	—
53	C₂²×C₃×C₄	C₂²×D₃	1	12	yes	—
54	C₂²×Dic₃	—	6	48	no	—
55	C₂²×Dic₃	—	12	32	no	—
56	C₂×C₃×C₈	C₂×Dic₃	1	96	yes	complemented
57	C₂×C₃×C₈	C₂²×D₃	1	16	yes	—
58	C₄⋊Dic₃	—	3	32	no	—
59	C₄⋊Dic₃	—	6	48	no	—
60	C₂²×C₃²	#107	1	16	yes	complemented
61	C₂²×C₃²	C₂×C₄⋊C₄	1	6	yes	—
62	C₂²×C₃²	C₂×D₈	1	6	yes	—
63	C₂²×C₃²	C₂×Q₁₆	1	6	yes	—
64	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
65	C₃²×C₄	C₂×C₄⋊C₄	1	6	yes	—
66	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
67	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
68	C₃⋊Dic₃	—	4	8	no	—
69	C₂³×C₄	—	18	2	no	—
70	C₂³×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
71	C₂²×C₈	C₃⋊Dic₃	1	8	yes	complemented
72	C₂²×C₈	C₂×C₃⋊D₃	1	6	yes	—
73	C₂×C₄⋊C₄	—	9	12	no	—
74	C₂×C₄⋊C₄	—	18	6	no	—
75	#107	—	9	16	no	—
76	C₂³×C₃	D₂₄	1	16	yes	—
77	C₂³×C₃	Dic₁₂	1	16	yes	—
78	C₂³×C₃	C₄⋊Dic₃	1	16	yes	—
79	C₂³×C₃	C₂×D₁₂	1	12	yes	—
80	C₂×C₃×C₄	C₄⋊Dic₃	1	48	yes	—
81	C₂×C₃×C₄	C₂×Dic₆	1	24	yes	—
82	C₂×C₃×C₄	C₂×D₁₂	1	12	yes	—
83	C₂×C₃×C₄	C₂²×Dic₃	1	24	yes	—
84	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
85	C₂×Dic₃	—	6	32	no	—
86	C₂×Dic₃	—	12	96	no	—
87	C₃×C₈	C₂²×Dic₃	1	32	yes	complemented
88	C₂×C₃²	C₂×#107	1	12	yes	complemented
89	C₂×C₃²	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
91	C₂×C₃²	C₂²×Q₁₆	1	1	yes	characteristic
92	C₂⁴	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
93	C₂²×C₄	—	18	12	no	—
94	C₂²×C₄	—	36	8	no	—
95	C₂²×C₄	#634	1	4	yes	—
96	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	3	yes	—
97	C₂²×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
98	C₂²×C₄	C₂²×C₃⋊D₃	1	3	yes	—
99	C₂×C₈	C₂×C₃⋊Dic₃	1	24	yes	complemented
100	C₂×C₈	C₂²×C₃⋊D₃	1	4	yes	—
101	C₄⋊C₄	—	9	8	no	—
102	C₄⋊C₄	—	18	12	no	—
103	C₂²×C₃	#818	1	64	yes	complemented
104	C₂²×C₃	C₂×D₂₄	1	24	yes	—
105	C₂²×C₃	C₂×Dic₁₂	1	24	yes	—
106	C₂²×C₃	C₂×C₄⋊Dic₃	1	24	yes	—
107	C₂²×C₃	C₂²×D₁₂	1	4	yes	—
108	C₃×C₄	C₂×C₄⋊Dic₃	1	24	yes	—
109	C₃×C₄	C₂²×Dic₆	1	4	yes	—
110	C₃×C₄	C₂³×Dic₃	1	4	yes	—
111	Dic₃	—	12	32	no	—
112	C₃²	C₂²×#107	1	1	yes	characteristic, complemented, Sylow
113	C₂³	#3745	1	4	yes	—
114	C₂³	#3746	1	4	yes	—
115	C₂³	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
116	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
117	C₂×C₄	—	18	8	no	—
118	C₂×C₄	—	36	24	no	—
119	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	12	yes	—
120	C₂×C₄	C₂×#634	1	6	yes	—
121	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
122	C₂×C₄	C₂²×C₃⋊Dic₃	1	6	yes	—
123	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
124	C₈	C₂²×C₃⋊Dic₃	1	8	yes	complemented
125	C₂×C₃	C₂×#818	1	48	yes	complemented
126	C₂×C₃	C₂²×D₂₄	1	4	yes	—
127	C₂×C₃	C₂²×Dic₁₂	1	4	yes	—
128	C₂×C₃	C₂²×C₄⋊Dic₃	1	4	yes	—
129	C₂²	#63645	1	16	yes	direct product factor
130	C₂²	C₂×#3745	1	6	yes	—
131	C₂²	C₂×#3746	1	6	yes	—
132	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	6	yes	—
133	C₂²	C₂²×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
134	C₄	—	36	8	no	—
135	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	6	yes	—
136	C₄	C₂²×#634	1	1	yes	characteristic
137	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
138	C₃	C₂²×#818	1	4	yes	complemented
139	C₂	C₂×#63645	1	12	yes	direct product factor
140	C₂	C₂²×#3745	1	1	yes	characteristic
141	C₂	C₂²×#3746	1	1	yes	characteristic
142	C₂	C₂²×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₃⋊D₃
Fitting	C₂³×C₃²×C₈	C₂
socle, prime radical	C₂⁴×C₃²	D₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	C₂³×C₈	C₃⋊D₃
hypocenter	C₃²	C₂²×#107
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#63645	C₂	direct (N × H)
7. C₂²×C₃²×C₈	C₄	semidirect
11. #63645	C₂²	direct (N × H)
22. C₂×C₃²×C₈	C₂×C₄	semidirect
29. C₂²×C₃×C₈	Dic₃	semidirect
45. C₃²×C₈	C₂²×C₄	semidirect
56. C₂×C₃×C₈	C₂×Dic₃	semidirect
60. C₂²×C₃²	#107	semidirect
71. C₂²×C₈	C₃⋊Dic₃	semidirect
87. C₃×C₈	C₂²×Dic₃	semidirect
88. C₂×C₃²	C₂×#107	semidirect
99. C₂×C₈	C₂×C₃⋊Dic₃	semidirect
103. C₂²×C₃	#818	semidirect
112. C₃²	C₂²×#107	semidirect
124. C₈	C₂²×C₃⋊Dic₃	semidirect
125. C₂×C₃	C₂×#818	semidirect
129. C₂²	#63645	direct (N × H)
138. C₃	C₂²×#818	semidirect
139. C₂	C₂×#63645	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
C₂²×C₄⋊(C₃⋊Dic₃)	2
C₂×#63645	2
C₂²×#818	3

Subgroups — C22×#63645 #11858398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63645 #11858398