Subgroups — #11794670

Order 1152.

357 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#391	C₂	1	1	yes	maximal, characteristic
2	C₂×#63395	C₂	1	2	yes	maximal
3	#10588615	C₂	1	1	yes	maximal, characteristic
4	#10588206	C₂	1	2	yes	maximal
5	#10588200	C₂	1	1	yes	maximal, characteristic
6	#69670	—	3	1	no	maximal
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₉	—	2	1	no	—
11	C₂×C₄⋊Dic₉	—	2	1	no	—
12	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
14	C₉×#95	—	2	1	no	—
15	C₉×#95	—	2	1	no	—
16	#63395	—	2	4	no	—
17	C₃×#391	D₃	1	1	yes	characteristic
18	C₂×#835	—	3	2	no	—
19	#4935	—	3	1	no	—
20	#4529	—	3	2	no	—
21	#4524	—	3	1	no	—
22	C₂×Q₈×C₉	—	2	1	no	—
23	C₂×Q₈×C₉	—	2	2	no	—
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	D₄	1	2	yes	—
26	C₂²×C₄×C₉	—	2	1	no	—
27	C₂²×C₄×C₉	—	2	1	no	—
28	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₉	—	2	2	no	—
32	C₉×C₄⋊C₄	—	2	2	no	—
33	C₄²×C₉	—	2	2	no	—
34	C₄⋊Dic₉	—	2	2	no	—
35	C₄⋊Dic₉	—	2	2	no	—
36	C₄⋊Dic₉	—	4	2	no	—
37	C₄⋊Dic₉	—	4	2	no	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	2	no	—
40	C₂×C₉⋊C₈	—	2	2	no	—
41	#2072	—	9	1	no	Sylow
42	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₃	—	3	2	no	—
46	C₂×C₄⋊Dic₃	—	6	1	no	—
47	C₂×C₄⋊Dic₃	—	6	1	no	—
48	C₂²×C₃⋊C₈	—	3	2	no	—
49	C₃×#95	—	2	1	no	—
50	C₃×#95	—	2	1	no	—
51	#835	—	6	4	no	—
52	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
53	Q₈×C₉	—	2	4	no	—
54	Q₈×C₉	—	4	2	no	—
55	C₂×C₄×C₉	—	2	1	no	—
56	C₂×C₄×C₉	—	2	1	no	—
57	C₂×C₄×C₉	—	2	1	no	—
58	C₂×C₄×C₉	—	2	1	no	—
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	4	1	no	—
61	C₂×C₄×C₉	—	4	1	no	—
62	C₂×C₄×C₉	—	4	1	no	—
63	C₂×C₄×C₉	—	4	1	no	—
64	C₂×C₄×C₉	—	4	1	no	—
65	C₂×C₄×C₉	SD₁₆	1	2	yes	—
66	C₂×C₄×C₉	Q₁₆	1	2	yes	—
67	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₉	#41	1	1	yes	characteristic
69	C₂×C₄×C₉	#41	1	2	yes	—
70	C₂×C₄×C₉	#41	1	2	yes	—
71	C₂×Dic₉	—	4	2	no	—
72	C₂×Dic₉	—	4	2	no	—
73	C₂×Dic₉	—	4	2	no	—
74	C₉⋊C₈	—	4	2	no	—
75	C₂×#103	—	9	2	no	—
76	#391	D₉	1	1	yes	characteristic
77	#384	—	9	1	no	—
78	#340	—	9	2	no	—
79	#336	—	9	1	no	—
80	C₂×C₃×Q₈	—	2	1	no	—
81	C₂×C₃×Q₈	—	2	2	no	—
82	C₂×C₃×Q₈	—	2	2	no	—
83	C₂×C₃×Q₈	D₁₂	1	2	yes	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	#67	1	1	yes	characteristic
87	C₂²×C₃×C₄	#67	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂²×Dic₃	—	6	2	no	—
90	C₃×C₄⋊C₄	—	2	2	no	—
91	C₃×C₄²	—	2	2	no	—
92	C₄⋊Dic₃	—	6	2	no	—
93	C₄⋊Dic₃	—	6	2	no	—
94	C₄⋊Dic₃	—	12	2	no	—
95	C₄⋊Dic₃	—	12	2	no	—
96	C₂×C₃⋊C₈	—	6	2	no	—
97	C₂×C₃⋊C₈	—	6	2	no	—
98	C₂×C₃⋊C₈	—	6	2	no	—
99	C₂²×C₉	#121	1	1	yes	characteristic
100	C₂²×C₉	#123	1	1	yes	characteristic
101	C₂²×C₉	#124	1	1	yes	characteristic
102	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
104	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂²×C₉	#137	1	1	yes	characteristic
106	C₄×C₉	—	2	2	no	—
107	C₄×C₉	—	2	2	no	—
108	C₄×C₉	—	4	2	no	—
109	C₄×C₉	—	8	1	no	—
110	C₄×C₉	#123	1	2	yes	—
111	C₄×C₉	#126	1	2	yes	—
112	Dic₉	—	8	2	no	—
113	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
114	C₂²×C₈	—	9	2	no	—
115	C₂×C₄⋊C₄	—	9	2	no	—
116	C₂×C₄⋊C₄	—	18	1	no	—
117	C₂×C₄⋊C₄	—	18	1	no	—
118	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
119	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
120	#103	—	18	4	no	—
121	#95	—	2	1	no	—
122	#95	—	2	1	no	—
123	C₂³×C₃	C₂×#67	1	1	yes	characteristic
124	C₃×Q₈	—	2	4	no	—
125	C₃×Q₈	—	4	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	4	1	no	—
132	C₂×C₃×C₄	—	4	1	no	—
133	C₂×C₃×C₄	—	4	1	no	—
134	C₂×C₃×C₄	—	4	1	no	—
135	C₂×C₃×C₄	—	4	1	no	—
136	C₂×C₃×C₄	#215	1	2	yes	—
137	C₂×C₃×C₄	#216	1	2	yes	—
138	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
139	C₂×C₃×C₄	#235	1	1	yes	characteristic
140	C₂×C₃×C₄	#237	1	2	yes	—
141	C₂×C₃×C₄	#237	1	2	yes	—
142	C₂×Dic₃	—	12	2	no	—
143	C₂×Dic₃	—	12	2	no	—
144	C₂×Dic₃	—	12	2	no	—
145	C₃⋊C₈	—	12	2	no	—
146	C₂×C₉	#397	1	1	yes	characteristic
147	C₂×C₉	#460	1	1	yes	characteristic
148	C₂×C₉	#462	1	1	yes	characteristic
149	C₂×C₉	#483	1	1	yes	characteristic
150	C₂×C₉	#484	1	1	yes	characteristic
151	C₂×C₉	#487	1	1	yes	characteristic
152	C₂×C₉	#489	1	1	yes	characteristic
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	D₃₆	1	2	yes	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	18	2	no	—
160	C₂²×C₄	#611	1	1	yes	characteristic
161	C₂²×C₄	#611	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
163	C₂×C₈	—	18	2	no	—
164	C₂×C₈	—	18	2	no	—
165	C₂×C₈	—	18	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	18	2	no	—
168	C₄⋊C₄	—	18	2	no	—
169	C₄⋊C₄	—	36	2	no	—
170	C₄⋊C₄	—	36	2	no	—
171	C₄²	—	2	2	no	—
172	C₂²×C₃	#930	1	1	yes	characteristic
173	C₂²×C₃	#935	1	1	yes	characteristic
174	C₂²×C₃	#936	1	1	yes	characteristic
175	C₂²×C₃	C₂×#215	1	1	yes	characteristic
176	C₂²×C₃	C₂×#216	1	1	yes	characteristic
177	C₂²×C₃	#949	1	1	yes	characteristic
178	C₂²×C₃	#951	1	1	yes	characteristic
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	—	8	1	no	—
183	C₃×C₄	#879	1	2	yes	—
184	C₃×C₄	#886	1	2	yes	—
185	Dic₃	—	24	2	no	—
186	C₉	#2072	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₂×#611	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	4	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	36	2	no	—
201	C₂×C₄	—	36	2	no	—
202	C₂×C₄	—	36	2	no	—
203	C₂×C₄	#3674	1	2	yes	—
204	C₂×C₄	#3675	1	2	yes	—
205	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
206	C₂×C₄	#3697	1	1	yes	characteristic
207	C₂×C₄	#3699	1	2	yes	—
208	C₂×C₄	#3699	1	2	yes	—
209	C₈	—	36	2	no	—
210	C₂×C₃	#4953	1	1	yes	characteristic
211	C₂×C₃	#5027	1	1	yes	characteristic
212	C₂×C₃	#5031	1	1	yes	characteristic
213	C₂×C₃	#5043	1	1	yes	characteristic
214	C₂×C₃	#5044	1	1	yes	characteristic
215	C₂×C₃	#5055	1	1	yes	characteristic
216	C₂×C₃	#5079	1	1	yes	characteristic
217	C₂²	#63492	1	1	yes	characteristic
218	C₂²	#63497	1	1	yes	characteristic
219	C₂²	#63498	1	1	yes	characteristic
220	C₂²	C₂×#3674	1	1	yes	characteristic
221	C₂²	C₂×#3675	1	1	yes	characteristic
222	C₂²	#63511	1	1	yes	characteristic
223	C₂²	#63512	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	8	1	no	—
228	C₄	—	72	2	no	—
229	C₄	#63441	1	2	yes	—
230	C₄	#63448	1	2	yes	—
231	C₃	#69670	1	1	yes	characteristic
232	C₂	#10588633	1	1	yes	characteristic
233	C₂	#10588707	1	1	yes	characteristic
234	C₂	#10588710	1	1	yes	characteristic
235	C₂	#10588723	1	1	yes	characteristic
236	C₂	#10588724	1	1	yes	characteristic
237	C₂	#10588734	1	1	yes	characteristic
238	C₂	#10588759	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator, cosocle	C₂²×C₄×C₉	C₂³
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#391	C₂
socle, prime radical	C₂³×C₃	C₂×#67
hypercenter	#391	D₉
hypocenter	C₉	#2072
LCS	G ▷ C₂²×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

Complements

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

N	H ≅ G/N	type
186. C₉	#2072	semidirect

Maximal subgroups

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

subgroup	index
C₉×#391	2
C₂×#63395	2
#10588615	2
#10588206	2
#10588200	2
#69670	3