Subgroups — #11786250

Order 1152.

372 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×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63378	C₂	1	2	yes	maximal
3	C₂×#63377	C₂	1	2	yes	maximal
4	#10588616	C₂	1	2	yes	maximal
5	#69766	—	3	1	no	maximal
6	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₉	C₂²	1	4	yes	—
10	C₄×C₈×C₉	C₄	1	4	yes	complemented
11	C₉⋊#95	—	2	4	no	—
12	#63378	—	2	4	no	—
13	#63377	—	2	4	no	—
14	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
15	C₂×#818	—	3	2	no	—
16	C₂×#817	—	3	2	no	—
17	#4936	—	3	2	no	—
18	C₂²×C₄×C₉	D₄	1	2	yes	—
19	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₉	—	2	4	no	—
21	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₉	D₄	1	2	yes	—
24	C₂×C₈×C₉	D₄	1	2	yes	—
25	C₂×C₈×C₉	Q₈	1	2	yes	—
26	C₂×C₈×C₉	Q₈	1	2	yes	—
27	C₄²×C₉	C₂×C₄	1	2	yes	—
28	C₄²×C₉	Q₈	1	2	yes	—
29	C₄⋊Dic₉	—	2	4	no	—
30	C₄⋊Dic₉	—	2	4	no	—
31	#1846	—	9	1	no	Sylow
32	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
35	C₂×C₄⋊Dic₃	—	3	4	no	—
36	C₃×C₄×C₈	Dic₃	1	4	yes	—
37	C₃⋊#95	—	6	4	no	—
38	#818	—	6	4	no	—
39	#817	—	6	4	no	—
40	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₉	—	2	4	no	—
42	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×Dic₉	—	4	4	no	—
50	C₂×Dic₉	—	4	8	no	—
51	C₈×C₉	C₄⋊C₄	1	4	yes	—
52	C₈×C₉	C₄⋊C₄	1	4	yes	—
53	C₂×#107	—	9	2	no	—
54	C₂×#106	—	9	2	no	—
55	C₂×C₄×C₈	D₉	1	1	yes	characteristic
56	#382	—	9	2	no	—
57	C₂²×C₃×C₄	D₁₂	1	2	yes	—
58	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
59	C₂²×Dic₃	—	6	4	no	—
60	C₂×C₃×C₈	Dic₆	1	2	yes	—
61	C₂×C₃×C₈	Dic₆	1	2	yes	—
62	C₂×C₃×C₈	D₁₂	1	2	yes	—
63	C₂×C₃×C₈	D₁₂	1	2	yes	—
64	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
65	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
66	C₃×C₄²	Dic₆	1	2	yes	—
67	C₃×C₄²	C₂×Dic₃	1	2	yes	—
68	C₄⋊Dic₃	—	6	4	no	—
69	C₄⋊Dic₃	—	6	4	no	—
70	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₉	#127	1	1	yes	characteristic
72	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
73	C₂²×C₉	#135	1	4	yes	—
74	C₄×C₉	—	2	4	no	—
75	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
79	Dic₉	—	8	4	no	—
80	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	9	4	no	—
83	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
84	#107	—	18	4	no	—
85	#106	—	18	4	no	—
86	C₄×C₈	Dic₉	1	4	yes	complemented
87	#95	—	18	4	no	—
88	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
89	C₂×C₃×C₄	—	2	4	no	—
90	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
91	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
97	C₂×Dic₃	—	12	4	no	—
98	C₂×Dic₃	—	12	8	no	—
99	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
100	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
101	C₂×C₉	#384	1	1	yes	characteristic
102	C₂×C₉	#427	1	2	yes	—
103	C₂×C₉	#495	1	2	yes	—
104	C₂×C₉	#499	1	2	yes	—
105	C₂²×C₄	—	18	4	no	—
106	C₂²×C₄	D₃₆	1	2	yes	—
107	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
108	C₂×C₈	Dic₁₈	1	2	yes	—
109	C₂×C₈	Dic₁₈	1	2	yes	—
110	C₂×C₈	D₃₆	1	2	yes	—
111	C₂×C₈	D₃₆	1	2	yes	—
112	C₂×C₈	C₂×Dic₉	1	2	yes	—
113	C₂×C₈	C₂×Dic₉	1	2	yes	—
114	C₄⋊C₄	—	18	4	no	—
115	C₄⋊C₄	—	18	4	no	—
116	C₄²	Dic₁₈	1	2	yes	—
117	C₄²	C₂×Dic₉	1	2	yes	—
118	C₂²×C₃	#869	1	1	yes	characteristic
119	C₂²×C₃	#874	1	1	yes	characteristic
120	C₂²×C₃	#904	1	4	yes	—
121	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	#869	1	1	yes	characteristic
124	C₃×C₄	#869	1	1	yes	characteristic
125	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
126	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
127	Dic₃	—	24	4	no	—
128	C₉	#1846	1	1	yes	characteristic, complemented, Sylow
129	C₂³	C₂×D₃₆	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⋊Dic₉	1	4	yes	—
134	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
137	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
138	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
140	C₈	C₄⋊Dic₉	1	4	yes	—
141	C₈	C₄⋊Dic₉	1	4	yes	—
142	C₂×C₃	#4683	1	2	yes	—
143	C₂×C₃	#4696	1	2	yes	—
144	C₂×C₃	#4935	1	1	yes	characteristic
145	C₂×C₃	#5106	1	2	yes	—
146	C₂²	#63431	1	1	yes	characteristic
147	C₂²	#63436	1	1	yes	characteristic
148	C₂²	#63467	1	4	yes	—
149	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
150	C₄	—	2	4	no	—
151	C₄	—	72	4	no	—
152	C₄	#63431	1	1	yes	characteristic
153	C₄	#63431	1	1	yes	characteristic
154	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
155	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
156	C₃	#69766	1	1	yes	characteristic
157	C₂	#10588363	1	2	yes	—
158	C₂	#10588378	1	2	yes	—
159	C₂	#10588615	1	1	yes	characteristic
160	C₂	#10588787	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₄×C₈×C₉	C₂
socle, prime radical	C₂³×C₃	C₂×D₁₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄×C₈	D₉
hypocenter	C₉	#1846
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
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. C₄×C₈×C₉	C₄	semidirect
86. C₄×C₈	Dic₉	semidirect
128. C₉	#1846	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×C₉	2
C₂×#63378	2
C₂×#63377	2
#10588616	2
#69766	3