Subgroups — #11772487

Order 1152.

204 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#63378	C₂	1	1	yes	maximal, characteristic
3	C₂×#63372	C₂	1	1	yes	maximal, characteristic
4	#67817	—	3	1	no	maximal
5	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊Dic₉	—	2	1	no	—
7	C₂×#3667	—	2	1	no	—
8	C₉×M₃₂	—	2	1	no	—
9	C₉×M₃₂	C₄	1	2	yes	complemented
10	C₂×C₉×C₁₆	—	2	1	no	—
11	#63378	—	2	1	no	—
12	#63378	C₄	1	2	yes	—
13	#63372	—	2	1	no	—
14	#63372	C₄	1	2	yes	complemented
15	C₂×C₃×M₃₂	D₃	1	1	yes	characteristic
16	C₂×#819	—	3	1	no	—
17	C₂×#818	—	3	1	no	—
18	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
19	C₂²×Dic₉	—	4	1	no	—
20	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₉	D₄	1	1	yes	characteristic
25	C₂×C₈×C₉	Q₈	1	1	yes	characteristic
26	C₄⋊Dic₉	—	2	2	no	—
27	C₄⋊Dic₉	—	4	1	no	—
28	#3667	—	2	2	no	—
29	#3667	—	4	1	no	—
30	C₂×C₉⋊C₈	—	4	1	no	—
31	C₉×C₁₆	—	2	2	no	—
32	#1387	—	9	1	no	Sylow
33	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
34	C₂×C₄⋊Dic₃	—	6	1	no	—
35	C₂×#208	—	6	1	no	—
36	C₃×M₃₂	—	2	1	no	—
37	C₃×M₃₂	Dic₃	1	2	yes	—
38	C₂×C₃×C₁₆	—	2	1	no	—
39	#819	—	3	2	no	—
40	#819	—	6	1	no	—
41	#818	—	3	2	no	—
42	#818	—	6	1	no	—
43	C₂³×C₉	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₉	D₈	1	1	yes	characteristic
48	C₂×C₄×C₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₄×C₉	Q₁₆	1	1	yes	characteristic
50	C₂×Dic₉	—	4	2	no	—
51	C₂×Dic₉	—	8	2	no	—
52	C₈×C₉	C₄²	1	1	yes	characteristic
53	C₈×C₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₈×C₉	C₄⋊C₄	1	1	yes	characteristic
55	C₈×C₉	C₄⋊C₄	1	1	yes	characteristic
56	C₉⋊C₈	—	4	2	no	—
57	C₂×M₃₂	D₉	1	1	yes	characteristic
58	C₂×#108	—	9	1	no	—
59	C₂×#107	—	9	1	no	—
60	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
61	C₂²×Dic₃	—	12	1	no	—
62	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
63	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
64	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
66	C₂×C₃×C₈	#67	1	1	yes	characteristic
67	C₂×C₃×C₈	#67	1	1	yes	characteristic
68	C₄⋊Dic₃	—	6	2	no	—
69	C₄⋊Dic₃	—	12	1	no	—
70	#208	—	6	2	no	—
71	#208	—	12	1	no	—
72	C₂×C₃⋊C₈	—	12	1	no	—
73	C₃×C₁₆	—	2	2	no	—
74	C₂²×C₉	—	2	1	no	—
75	C₂²×C₉	—	2	1	no	—
76	C₂²×C₉	#102	1	1	yes	characteristic
77	C₂²×C₉	#103	1	1	yes	characteristic
78	C₂²×C₉	#106	1	1	yes	characteristic
79	C₄×C₉	#95	1	1	yes	characteristic
80	C₄×C₉	#102	1	1	yes	characteristic
81	C₄×C₉	#103	1	1	yes	characteristic
82	C₄×C₉	#107	1	1	yes	characteristic
83	Dic₉	—	8	2	no	—
84	C₂×M₁₆	—	18	1	no	—
85	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	18	1	no	—
87	M₃₂	—	2	1	no	—
88	M₃₂	Dic₉	1	2	yes	complemented
89	C₂×C₁₆	—	2	1	no	—
90	#108	—	9	2	no	—
91	#108	—	18	1	no	—
92	#107	—	9	2	no	—
93	#107	—	18	1	no	—
94	C₂³×C₃	#204	1	1	yes	characteristic
95	C₂×C₃×C₄	#204	1	1	yes	characteristic
96	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
97	C₂×C₃×C₄	Dic₁₂	1	1	yes	characteristic
98	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
99	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
100	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
101	C₂×Dic₃	—	12	2	no	—
102	C₂×Dic₃	—	24	2	no	—
103	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
104	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
105	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
106	C₃×C₈	#217	1	1	yes	characteristic
107	C₃⋊C₈	—	12	2	no	—
108	C₂×C₉	—	2	1	no	—
109	C₂×C₉	—	2	1	no	—
110	C₂×C₉	#340	1	1	yes	characteristic
111	C₂×C₉	#361	1	1	yes	characteristic
112	C₂×C₉	#362	1	1	yes	characteristic
113	C₂²×C₄	—	36	1	no	—
114	C₂²×C₄	D₃₆	1	1	yes	characteristic
115	M₁₆	—	18	2	no	—
116	M₁₆	—	36	1	no	—
117	C₂×C₈	—	36	1	no	—
118	C₂×C₈	Dic₁₈	1	1	yes	characteristic
119	C₂×C₈	C₄×D₉	1	1	yes	characteristic
120	C₂×C₈	C₄×D₉	1	1	yes	characteristic
121	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
122	C₂×C₈	#611	1	1	yes	characteristic
123	C₂×C₈	#611	1	1	yes	characteristic
124	C₄⋊C₄	—	18	2	no	—
125	C₄⋊C₄	—	36	1	no	—
126	C₁₆	—	2	2	no	—
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	—	2	1	no	—
129	C₂²×C₃	#816	1	1	yes	characteristic
130	C₂²×C₃	#817	1	1	yes	characteristic
131	C₂²×C₃	#821	1	1	yes	characteristic
132	C₃×C₄	#816	1	1	yes	characteristic
133	C₃×C₄	#818	1	1	yes	characteristic
134	C₃×C₄	#821	1	1	yes	characteristic
135	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
136	Dic₃	—	24	2	no	—
137	C₉	#1387	1	1	yes	characteristic, complemented, Sylow
138	C₂³	#3664	1	1	yes	characteristic
139	C₂×C₄	—	36	2	no	—
140	C₂×C₄	—	72	2	no	—
141	C₂×C₄	Dic₃₆	1	1	yes	characteristic
142	C₂×C₄	#3664	1	1	yes	characteristic
143	C₂×C₄	D₇₂	1	1	yes	characteristic
144	C₂×C₄	C₄⋊Dic₉	1	1	yes	characteristic
145	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
146	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
147	C₈	—	36	2	no	—
148	C₈	C₄×Dic₉	1	1	yes	characteristic
149	C₈	Dic₉⋊C₄	1	1	yes	characteristic
150	C₈	Dic₉⋊C₄	1	1	yes	characteristic
151	C₈	#3676	1	1	yes	characteristic
152	C₂×C₃	—	2	1	no	—
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	#4516	1	1	yes	characteristic
155	C₂×C₃	#4517	1	1	yes	characteristic
156	C₂×C₃	#4551	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#63376	1	1	yes	characteristic
160	C₂²	#63377	1	1	yes	characteristic
161	C₂²	#63380	1	1	yes	characteristic
162	C₄	—	72	2	no	—
163	C₄	#63376	1	1	yes	characteristic
164	C₄	#63378	1	1	yes	characteristic
165	C₄	#63380	1	1	yes	characteristic
166	C₄	C₉⋊#95	1	1	yes	characteristic
167	C₃	#67817	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	#10588193	1	1	yes	characteristic
171	C₂	#10588195	1	1	yes	characteristic
172	C₂	#10588229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63380
commutator	C₈×C₉	C₄²
Frattini	C₂²×C₃×C₈	C₂×D₃
Fitting	C₂×C₉×M₃₂	C₂
socle	C₂²×C₃	#821
prime radical	C₂³×C₃	#204
cosocle	C₂²×C₈×C₉	C₂²
hypercenter	C₂×M₃₂	D₉
hypocenter	C₉	#1387
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
9. C₉×M₃₂	C₄	semidirect
14. #63372	C₄	semidirect
88. M₃₂	Dic₉	semidirect
137. C₉	#1387	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×M₃₂	2
C₂×#63378	2
C₂×#63372	2
#67817	3