Subgroups — #11783035

Order 1152.

984 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#63642	C₂	1	2	yes	maximal
3	C₂×C₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
4	#10592631	C₂	1	1	yes	maximal, characteristic
5	C₂×#63630	C₂	1	1	yes	maximal, characteristic
6	#69749	—	3	4	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
11	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
12	C₃²×C₄×C₈	C₄	1	4	yes	complemented
13	#63658	C₄	1	4	yes	—
14	#63642	C₄	1	8	yes	complemented
15	C₈×C₃⋊Dic₃	C₄	1	8	yes	complemented
16	#63630	C₄	1	4	yes	complemented
17	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
18	C₂×#815	—	3	8	no	—
19	C₂×C₈×Dic₃	—	3	8	no	—
20	#4932	—	3	4	no	—
21	C₂×#803	—	3	4	no	—
22	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
25	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
26	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
27	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
29	C₃²×C₄²	C₂×C₄	1	2	yes	—
30	C₃²×C₄²	C₂×C₄	1	2	yes	—
31	C₄×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
32	C₄×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
33	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
34	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
35	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
36	#1726	—	9	1	no	Sylow
37	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
38	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
39	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
40	C₂×C₄×Dic₃	—	3	8	no	—
41	C₂²×C₃⋊C₈	—	3	8	no	—
42	C₃×C₄×C₈	Dic₃	1	16	yes	complemented
43	C₃⋊#95	—	3	16	no	—
44	#815	—	3	32	no	—
45	C₈×Dic₃	—	3	32	no	—
46	#803	—	3	16	no	—
47	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	—	2	4	no	—
49	C₂×C₃²×C₄	C₄²	1	4	yes	—
50	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
52	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
53	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃⋊Dic₃	—	2	8	no	—
55	C₂×C₃⋊Dic₃	C₄²	1	8	yes	—
56	C₃²×C₈	C₄²	1	4	yes	—
57	C₃²×C₈	C₄²	1	4	yes	—
58	C₃⋊(C₃⋊C₈)	C₄²	1	8	yes	—
59	C₂×#97	—	9	2	no	—
60	C₂×#97	—	9	1	no	—
61	C₂×C₄×C₈	—	9	2	no	—
62	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
63	#376	—	9	1	no	—
64	C₂²×C₃×C₄	C₄×D₃	1	8	yes	—
65	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
66	C₂²×Dic₃	—	3	16	no	—
67	C₂×C₃×C₈	C₄×D₃	1	16	yes	—
68	C₂×C₃×C₈	C₄×D₃	1	16	yes	—
69	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
70	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
71	C₃×C₄²	C₄×D₃	1	8	yes	—
72	C₃×C₄²	C₂×Dic₃	1	8	yes	—
73	C₄×Dic₃	—	3	16	no	—
74	C₄×Dic₃	—	3	16	no	—
75	C₂×C₃⋊C₈	—	3	16	no	—
76	C₂×C₃⋊C₈	—	3	16	no	—
77	C₂×C₃⋊C₈	—	3	16	no	—
78	C₂²×C₃²	C₂×C₄²	1	2	yes	—
79	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
80	C₂²×C₃²	#131	1	4	yes	—
81	C₃²×C₄	—	2	4	no	—
82	C₃²×C₄	C₂×C₄²	1	2	yes	—
83	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
84	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
85	C₃⋊Dic₃	—	2	8	no	—
86	C₂²×C₈	—	9	2	no	—
87	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
88	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
89	C₂×C₄²	—	9	2	no	—
90	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
91	#97	—	9	8	no	—
92	#97	—	9	4	no	—
93	C₄×C₈	—	9	8	no	—
94	C₄×C₈	C₃⋊Dic₃	1	4	yes	complemented
95	#95	—	9	4	no	—
96	C₂³×C₃	C₂×C₄×D₃	1	4	yes	—
97	C₂×C₃×C₄	—	2	16	no	—
98	C₂×C₃×C₄	C₄×Dic₃	1	16	yes	—
99	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
100	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
101	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
102	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
103	C₂×Dic₃	—	3	32	no	—
104	C₂×Dic₃	—	6	32	no	—
105	C₃×C₈	C₄×Dic₃	1	16	yes	—
106	C₃×C₈	C₄×Dic₃	1	16	yes	—
107	C₃⋊C₈	—	3	32	no	—
108	C₂×C₃²	C₄³	1	1	yes	characteristic
109	C₂×C₃²	#405	1	4	yes	—
110	C₂×C₃²	#405	1	2	yes	—
111	C₂²×C₄	—	9	4	no	—
112	C₂²×C₄	C₄×C₃⋊D₃	1	2	yes	—
113	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
114	C₂×C₈	—	9	4	no	—
115	C₂×C₈	—	9	4	no	—
116	C₂×C₈	—	9	4	no	—
117	C₂×C₈	C₄×C₃⋊D₃	1	4	yes	—
118	C₂×C₈	C₄×C₃⋊D₃	1	4	yes	—
119	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
120	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
121	C₄²	—	9	4	no	—
122	C₄²	—	9	4	no	—
123	C₄²	C₄×C₃⋊D₃	1	2	yes	—
124	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
125	C₂²×C₃	C₄²×D₃	1	8	yes	—
126	C₂²×C₃	#901	1	16	yes	—
127	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
128	C₃×C₄	—	2	16	no	—
129	C₃×C₄	C₄²×D₃	1	8	yes	—
130	C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
131	C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
132	Dic₃	—	6	32	no	—
133	C₃²	#1726	1	1	yes	characteristic, complemented, Sylow
134	C₂³	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	9	8	no	—
137	C₂×C₄	—	18	8	no	—
138	C₂×C₄	C₄×C₃⋊Dic₃	1	4	yes	—
139	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
141	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
142	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
143	C₈	—	9	8	no	—
144	C₈	C₄×C₃⋊Dic₃	1	4	yes	—
145	C₈	C₄×C₃⋊Dic₃	1	4	yes	—
146	C₂×C₃	#4689	1	16	yes	—
147	C₂×C₃	C₄²×Dic₃	1	4	yes	—
148	C₂×C₃	#5101	1	8	yes	—
149	C₂²	C₄²×C₃⋊D₃	1	2	yes	—
150	C₂²	#64110	1	4	yes	—
151	C₂²	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	18	8	no	—
154	C₄	C₄²×C₃⋊D₃	1	2	yes	—
155	C₄	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
156	C₄	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
157	C₃	#69749	1	4	yes	complemented
158	C₂	#10592388	1	4	yes	—
159	C₂	C₄²×C₃⋊Dic₃	1	1	yes	characteristic
160	C₂	#10592800	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₄³
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#1726
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₃²×C₄×C₈	C₄	semidirect
14. #63642	C₄	semidirect
15. C₈×C₃⋊Dic₃	C₄	semidirect
16. #63630	C₄	semidirect
42. C₃×C₄×C₈	Dic₃	semidirect
94. C₄×C₈	C₃⋊Dic₃	semidirect
133. C₃²	#1726	semidirect
157. C₃	#69749	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×#63642	2
C₂×C₈×C₃⋊Dic₃	2
#10592631	2
C₂×#63630	2
#69749	3