Subgroups — #11772490

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×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#63378	C₂	1	1	yes	maximal, characteristic
3	C₂×#63371	C₂	1	1	yes	maximal, characteristic
4	#67732	—	3	1	no	maximal
5	C₂×C₉×M₁₆	—	2	1	no	—
6	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₉	—	2	1	no	—
8	C₉×#108	—	2	1	no	—
9	C₉×#108	C₄	1	2	yes	complemented
10	#63378	—	2	1	no	—
11	#63378	C₄	1	2	yes	—
12	#63371	—	2	1	no	—
13	#63371	C₄	1	2	yes	complemented
14	C₂×C₉⋊C₁₆	—	2	1	no	—
15	C₂×C₃×#108	D₃	1	1	yes	characteristic
16	C₂×#818	—	3	1	no	—
17	C₂×#812	—	3	1	no	—
18	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
19	C₂²×Dic₉	—	4	1	no	—
20	C₉×M₁₆	—	2	2	no	—
21	C₉×M₁₆	—	4	1	no	—
22	C₂×C₈×C₉	—	4	1	no	—
23	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₉	Q₈	1	1	yes	characteristic
29	C₄⋊Dic₉	—	2	2	no	—
30	C₄⋊Dic₉	—	4	1	no	—
31	C₉⋊C₁₆	—	2	2	no	—
32	#1387	—	9	1	no	Sylow
33	C₂×C₃×M₁₆	—	2	1	no	—
34	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
35	C₂×C₄⋊Dic₃	—	6	1	no	—
36	C₃×#108	—	2	1	no	—
37	C₃×#108	Dic₃	1	2	yes	—
38	#818	—	3	2	no	—
39	#818	—	6	1	no	—
40	#812	—	3	2	no	—
41	#812	—	6	1	no	—
42	C₂×C₃⋊C₁₆	—	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	—	4	2	no	—
53	C₈×C₉	C₄²	1	1	yes	characteristic
54	C₈×C₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₈×C₉	C₄⋊C₄	1	1	yes	characteristic
56	C₈×C₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×M₃₂	—	9	1	no	—
58	C₂×#108	D₉	1	1	yes	characteristic
59	C₂×#107	—	9	1	no	—
60	C₂²×C₃×C₄	#67	1	1	yes	characteristic
61	C₂²×Dic₃	—	12	1	no	—
62	C₃×M₁₆	—	2	2	no	—
63	C₃×M₁₆	—	4	1	no	—
64	C₂×C₃×C₈	—	4	1	no	—
65	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
66	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
67	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
68	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
70	C₂×C₃×C₈	#67	1	1	yes	characteristic
71	C₄⋊Dic₃	—	6	2	no	—
72	C₄⋊Dic₃	—	12	1	no	—
73	C₃⋊C₁₆	—	6	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	—	2	1	no	—
85	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	18	1	no	—
87	M₃₂	—	9	2	no	—
88	M₃₂	—	18	1	no	—
89	C₂×C₁₆	—	18	1	no	—
90	#108	—	2	1	no	—
91	#108	Dic₉	1	2	yes	complemented
92	#107	—	9	2	no	—
93	#107	—	18	1	no	—
94	C₂³×C₃	#215	1	1	yes	characteristic
95	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
96	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
97	C₂×C₃×C₄	#213	1	1	yes	characteristic
98	C₂×C₃×C₄	#214	1	1	yes	characteristic
99	C₂×C₃×C₄	#216	1	1	yes	characteristic
100	C₂×C₃×C₄	#217	1	1	yes	characteristic
101	C₂×Dic₃	—	12	2	no	—
102	C₂×Dic₃	—	24	2	no	—
103	C₃×C₈	—	4	2	no	—
104	C₃×C₈	C₄×Dic₃	1	1	yes	characteristic
105	C₃×C₈	Dic₃⋊C₄	1	1	yes	characteristic
106	C₃×C₈	C₄⋊Dic₃	1	1	yes	characteristic
107	C₃×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
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	#611	1	1	yes	characteristic
115	M₁₆	—	2	2	no	—
116	M₁₆	—	4	1	no	—
117	C₂×C₈	—	4	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	D₃₆	1	1	yes	characteristic
122	C₂×C₈	C₂×Dic₉	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	—	18	2	no	—
127	C₂²×C₃	—	2	1	no	—
128	C₂²×C₃	—	2	1	no	—
129	C₂²×C₃	#808	1	1	yes	characteristic
130	C₂²×C₃	#809	1	1	yes	characteristic
131	C₂²×C₃	#835	1	1	yes	characteristic
132	C₃×C₄	#807	1	1	yes	characteristic
133	C₃×C₄	#810	1	1	yes	characteristic
134	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
135	C₃×C₄	#832	1	1	yes	characteristic
136	Dic₃	—	24	2	no	—
137	C₉	#1387	1	1	yes	characteristic, complemented, Sylow
138	C₂³	#3675	1	1	yes	characteristic
139	C₂×C₄	—	36	2	no	—
140	C₂×C₄	—	72	2	no	—
141	C₂×C₄	Dic₉⋊C₄	1	1	yes	characteristic
142	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
143	C₂×C₄	#3672	1	1	yes	characteristic
144	C₂×C₄	#3673	1	1	yes	characteristic
145	C₂×C₄	#3674	1	1	yes	characteristic
146	C₂×C₄	#3676	1	1	yes	characteristic
147	C₈	—	4	2	no	—
148	C₈	C₄×Dic₉	1	1	yes	characteristic
149	C₈	Dic₉⋊C₄	1	1	yes	characteristic
150	C₈	C₄⋊Dic₉	1	1	yes	characteristic
151	C₈	(C₂×D₉)⋊C₄	1	1	yes	characteristic
152	C₂×C₃	—	2	1	no	—
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	#4495	1	1	yes	characteristic
155	C₂×C₃	#4496	1	1	yes	characteristic
156	C₂×C₃	#4529	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#63367	1	1	yes	characteristic
160	C₂²	#63369	1	1	yes	characteristic
161	C₂²	#63395	1	1	yes	characteristic
162	C₄	—	72	2	no	—
163	C₄	#63366	1	1	yes	characteristic
164	C₄	#63368	1	1	yes	characteristic
165	C₄	C₉⋊#95	1	1	yes	characteristic
166	C₄	#63392	1	1	yes	characteristic
167	C₃	#67732	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	#10588172	1	1	yes	characteristic
171	C₂	#10588174	1	1	yes	characteristic
172	C₂	#10588206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63369
commutator	C₈×C₉	C₄²
Frattini	C₂²×C₃×C₈	C₂×D₃
Fitting	C₂×C₉×#108	C₂
socle	C₂²×C₃	#809
prime radical	C₂³×C₃	#215
cosocle	C₂²×C₈×C₉	C₂²
hypercenter	C₂×#108	D₉
hypocenter	C₉	#1387
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
9. C₉×#108	C₄	semidirect
13. #63371	C₄	semidirect
91. #108	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×#108	2
C₂×#63378	2
C₂×#63371	2
#67732	3