Subgroups — #11786251

Order 1152.

744 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×#63645	C₂	1	2	yes	maximal
3	C₂×#63644	C₂	1	2	yes	maximal
4	#10592635	C₂	1	2	yes	maximal
5	#69766	—	3	4	no	maximal
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
10	C₃²×C₄×C₈	C₄	1	4	yes	complemented
11	#63658	—	2	4	no	—
12	#63645	—	2	4	no	—
13	#63644	—	2	4	no	—
14	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
15	C₂×#818	—	3	8	no	—
16	C₂×#817	—	3	8	no	—
17	#4936	—	3	8	no	—
18	C₂²×C₃²×C₄	D₄	1	2	yes	—
19	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
20	C₂²×C₃⋊Dic₃	—	2	4	no	—
21	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃²×C₈	D₄	1	2	yes	—
24	C₂×C₃²×C₈	D₄	1	2	yes	—
25	C₂×C₃²×C₈	Q₈	1	2	yes	—
26	C₂×C₃²×C₈	Q₈	1	2	yes	—
27	C₃²×C₄²	C₂×C₄	1	2	yes	—
28	C₃²×C₄²	Q₈	1	2	yes	—
29	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
30	C₄⋊(C₃⋊Dic₃)	—	2	4	no	—
31	#1846	—	9	1	no	Sylow
32	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
33	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
34	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
35	C₂×C₄⋊Dic₃	—	3	16	no	—
36	C₃×C₄×C₈	Dic₃	1	16	yes	complemented
37	C₃⋊#95	—	6	16	no	—
38	#818	—	6	16	no	—
39	#817	—	6	16	no	—
40	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
41	C₂×C₃²×C₄	—	2	4	no	—
42	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
43	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₃⋊Dic₃	—	4	4	no	—
50	C₂×C₃⋊Dic₃	—	4	8	no	—
51	C₃²×C₈	C₄⋊C₄	1	4	yes	—
52	C₃²×C₈	C₄⋊C₄	1	4	yes	—
53	C₂×#107	—	9	2	no	—
54	C₂×#106	—	9	2	no	—
55	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
56	#382	—	9	2	no	—
57	C₂²×C₃×C₄	D₁₂	1	8	yes	—
58	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
59	C₂²×Dic₃	—	6	16	no	—
60	C₂×C₃×C₈	Dic₆	1	8	yes	—
61	C₂×C₃×C₈	Dic₆	1	8	yes	—
62	C₂×C₃×C₈	D₁₂	1	8	yes	—
63	C₂×C₃×C₈	D₁₂	1	8	yes	—
64	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
65	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
66	C₃×C₄²	Dic₆	1	8	yes	—
67	C₃×C₄²	C₂×Dic₃	1	8	yes	—
68	C₄⋊Dic₃	—	6	16	no	—
69	C₄⋊Dic₃	—	6	16	no	—
70	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₃²	#127	1	1	yes	characteristic
72	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
73	C₂²×C₃²	#135	1	4	yes	—
74	C₃²×C₄	—	2	4	no	—
75	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
78	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
79	C₃⋊Dic₃	—	8	4	no	—
80	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	9	4	no	—
83	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
84	#107	—	18	4	no	—
85	#106	—	18	4	no	—
86	C₄×C₈	C₃⋊Dic₃	1	4	yes	complemented
87	#95	—	18	4	no	—
88	C₂³×C₃	C₂×D₁₂	1	4	yes	—
89	C₂×C₃×C₄	—	2	16	no	—
90	C₂×C₃×C₄	C₄⋊Dic₃	1	16	yes	—
91	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
92	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
93	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
94	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
95	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
96	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
97	C₂×Dic₃	—	12	16	no	—
98	C₂×Dic₃	—	12	32	no	—
99	C₃×C₈	C₄⋊Dic₃	1	16	yes	—
100	C₃×C₈	C₄⋊Dic₃	1	16	yes	—
101	C₂×C₃²	#384	1	1	yes	characteristic
102	C₂×C₃²	#427	1	2	yes	—
103	C₂×C₃²	#495	1	2	yes	—
104	C₂×C₃²	#499	1	2	yes	—
105	C₂²×C₄	—	18	4	no	—
106	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
107	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
108	C₂×C₈	#634	1	2	yes	—
109	C₂×C₈	#634	1	2	yes	—
110	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
111	C₂×C₈	C₄⋊(C₃⋊D₃)	1	2	yes	—
112	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
113	C₂×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
114	C₄⋊C₄	—	18	4	no	—
115	C₄⋊C₄	—	18	4	no	—
116	C₄²	#634	1	2	yes	—
117	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
118	C₂²×C₃	#869	1	4	yes	—
119	C₂²×C₃	#874	1	4	yes	—
120	C₂²×C₃	#904	1	16	yes	—
121	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
122	C₃×C₄	—	2	16	no	—
123	C₃×C₄	#869	1	4	yes	—
124	C₃×C₄	#869	1	4	yes	—
125	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
126	C₃×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
127	Dic₃	—	24	16	no	—
128	C₃²	#1846	1	1	yes	characteristic, complemented, Sylow
129	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	36	4	no	—
132	C₂×C₄	—	36	8	no	—
133	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
134	C₂×C₄	C₂×#634	1	1	yes	characteristic
135	C₂×C₄	C₂×#634	1	1	yes	characteristic
136	C₂×C₄	C₂×#634	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
140	C₈	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
141	C₈	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
142	C₂×C₃	#4683	1	8	yes	—
143	C₂×C₃	#4696	1	8	yes	—
144	C₂×C₃	#4935	1	4	yes	—
145	C₂×C₃	#5106	1	8	yes	—
146	C₂²	#64078	1	1	yes	characteristic
147	C₂²	#64083	1	1	yes	characteristic
148	C₂²	#64113	1	4	yes	—
149	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
150	C₄	—	2	4	no	—
151	C₄	—	72	4	no	—
152	C₄	#64078	1	1	yes	characteristic
153	C₄	#64078	1	1	yes	characteristic
154	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
155	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
156	C₃	#69766	1	4	yes	complemented
157	C₂	#10592382	1	2	yes	—
158	C₂	#10592395	1	2	yes	—
159	C₂	#10592634	1	1	yes	characteristic
160	C₂	#10592805	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#1846
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. C₃²×C₄×C₈	C₄	semidirect
36. C₃×C₄×C₈	Dic₃	semidirect
86. C₄×C₈	C₃⋊Dic₃	semidirect
128. C₃²	#1846	semidirect
156. C₃	#69766	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×#63645	2
C₂×#63644	2
#10592635	2
#69766	3