Subgroups — #11786356

Order 1152.

522 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#63389	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#10588200	C₂	1	2	yes	maximal, complemented
6	#69449	—	3	1	no	maximal
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic
9	C₂×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₉	C₂²	1	2	yes	—
11	C₂²×C₉⋊C₈	—	2	2	no	—
12	C₂²×C₉⋊C₈	—	2	4	no	—
13	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
16	C₉×#95	C₄	1	2	yes	—
17	#63389	—	2	2	no	—
18	C₄⋊(C₉⋊C₈)	—	2	4	no	—
19	C₂³×C₃⋊C₈	—	3	1	no	—
20	C₃×C₄×C₂²⋊C₄	D₃	1	1	yes	characteristic
21	C₂×#830	—	3	1	no	—
22	C₂×C₄⋊(C₃⋊C₈)	—	3	2	no	—
23	#4524	—	3	2	no	—
24	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	D₄	1	2	yes	—
31	C₂²×C₄×C₉	Q₈	1	2	yes	—
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₉×C₂²⋊C₄	—	2	2	no	—
34	C₉×C₂²⋊C₄	C₈	1	4	yes	complemented
35	C₄²×C₉	—	2	2	no	—
36	C₄²×C₉	—	2	2	no	—
37	C₂×C₉⋊C₈	—	2	8	no	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	2	no	—
40	C₂×C₉⋊C₈	—	2	2	no	—
41	C₂×C₉⋊C₈	—	2	1	no	—
42	C₂×C₉⋊C₈	—	2	1	no	—
43	C₂×C₉⋊C₈	—	2	2	no	—
44	C₂×C₉⋊C₈	D₄	1	4	yes	—
45	#1850	—	9	1	no	Sylow
46	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
47	C₂×C₃×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
48	C₂×C₃×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
50	C₂²×C₃⋊C₈	—	3	2	no	—
51	C₂²×C₃⋊C₈	—	3	1	no	—
52	C₂²×C₃⋊C₈	—	3	1	no	—
53	C₂²×C₃⋊C₈	—	6	4	no	—
54	C₂²×C₃⋊C₈	—	6	2	no	—
55	C₃×#95	Dic₃	1	2	yes	—
56	#830	—	6	2	no	—
57	C₄⋊(C₃⋊C₈)	—	6	4	no	—
58	C₂³×C₉	—	2	2	no	—
59	C₂³×C₉	—	2	2	no	—
60	C₂³×C₉	C₄⋊C₄	1	2	yes	—
61	C₂³×C₉	C₂×C₈	1	2	yes	—
62	C₂³×C₉	M₁₆	1	2	yes	—
63	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₉	C₂×C₈	1	4	yes	complemented
75	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₉	#41	1	2	yes	—
79	C₉⋊C₈	—	2	4	no	—
80	C₉⋊C₈	—	4	1	no	—
81	C₉⋊C₈	—	4	1	no	—
82	C₂³×C₈	—	9	1	no	—
83	C₂×C₄⋊C₈	—	9	2	no	—
84	C₂×C₂²⋊C₈	—	9	1	no	—
85	C₄×C₂²⋊C₄	D₉	1	1	yes	characteristic
86	#336	—	9	2	no	—
87	C₂⁴×C₃	C₂×Dic₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	Dic₆	1	2	yes	—
91	C₂²×C₃×C₄	D₁₂	1	2	yes	—
92	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
93	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
94	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₃×C₂²⋊C₄	—	2	2	no	—
97	C₃×C₂²⋊C₄	C₃⋊C₈	1	4	yes	—
98	C₃×C₄²	—	2	2	no	—
99	C₃×C₄²	—	2	2	no	—
100	C₂×C₃⋊C₈	—	3	4	no	—
101	C₂×C₃⋊C₈	—	6	8	no	—
102	C₂×C₃⋊C₈	—	6	2	no	—
103	C₂×C₃⋊C₈	—	6	2	no	—
104	C₂×C₃⋊C₈	—	6	1	no	—
105	C₂×C₃⋊C₈	—	6	1	no	—
106	C₂×C₃⋊C₈	—	6	2	no	—
107	C₂×C₃⋊C₈	—	6	2	no	—
108	C₂²×C₉	—	2	2	no	—
109	C₂²×C₉	—	2	2	no	—
110	C₂²×C₉	—	2	4	no	—
111	C₂²×C₉	—	2	4	no	—
112	C₂²×C₉	C₄⋊C₈	1	4	yes	complemented
113	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₉	C₄×D₄	1	2	yes	—
115	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
116	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂²×C₉	#131	1	2	yes	—
118	C₄×C₉	—	2	2	no	—
119	C₄×C₉	—	2	4	no	—
120	C₄×C₉	—	4	2	no	—
121	C₄×C₉	#120	1	1	yes	characteristic
122	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
123	C₄×C₉	#123	1	1	yes	characteristic
124	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
125	C₂²×C₈	—	9	2	no	—
126	C₂²×C₈	—	9	1	no	—
127	C₂²×C₈	—	9	1	no	—
128	C₂²×C₈	—	18	4	no	—
129	C₂²×C₈	—	18	2	no	—
130	C₂×C₂²⋊C₄	Dic₉	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	Dic₉	1	1	yes	characteristic
132	C₂×C₄²	C₂×D₉	1	2	yes	—
133	C₄⋊C₈	—	18	4	no	—
134	C₂²⋊C₈	—	18	2	no	—
135	#95	Dic₉	1	2	yes	—
136	C₂³×C₃	—	2	2	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	C₂×C₃⋊C₈	1	2	yes	—
139	C₂³×C₃	#208	1	2	yes	—
140	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
141	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
152	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
153	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
155	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
156	C₂×C₃×C₄	#237	1	2	yes	—
157	C₃⋊C₈	—	6	4	no	—
158	C₃⋊C₈	—	12	1	no	—
159	C₃⋊C₈	—	12	1	no	—
160	C₂×C₉	—	2	4	no	—
161	C₂×C₉	#385	1	1	yes	characteristic
162	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₉	#424	1	1	yes	characteristic
164	C₂×C₉	C₈×D₄	1	2	yes	—
165	C₂×C₉	#435	1	2	yes	—
166	C₂⁴	C₂×Dic₉	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	Dic₁₈	1	2	yes	—
170	C₂²×C₄	D₃₆	1	2	yes	—
171	C₂²×C₄	C₂×Dic₉	1	2	yes	—
172	C₂²×C₄	C₂×Dic₉	1	2	yes	—
173	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
175	C₂×C₈	—	9	4	no	—
176	C₂×C₈	—	18	8	no	—
177	C₂×C₈	—	18	2	no	—
178	C₂×C₈	—	18	2	no	—
179	C₂×C₈	—	18	1	no	—
180	C₂×C₈	—	18	1	no	—
181	C₂×C₈	—	18	2	no	—
182	C₂×C₈	—	18	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	C₉⋊C₈	1	4	yes	complemented
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	4	yes	—
192	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
193	C₂²×C₃	C₂×#208	1	1	yes	characteristic
194	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
195	C₂²×C₃	D₄×Dic₃	1	2	yes	—
196	C₂²×C₃	#948	1	2	yes	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	4	2	no	—
200	C₃×C₄	#878	1	2	yes	—
201	C₃×C₄	#882	1	1	yes	characteristic
202	C₃×C₄	#886	1	1	yes	characteristic
203	C₉	#1850	1	1	yes	characteristic, complemented, Sylow
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₂×C₉⋊C₈	1	2	yes	—
207	C₂³	#3667	1	2	yes	—
208	C₂³	C₄⋊Dic₉	1	2	yes	—
209	C₂³	C₂²×Dic₉	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₂×C₉⋊C₈	1	4	yes	complemented
220	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
221	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
222	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
223	C₂×C₄	D₄×D₉	1	2	yes	—
224	C₂×C₄	#3699	1	2	yes	—
225	C₈	—	18	4	no	—
226	C₈	—	36	1	no	—
227	C₈	—	36	1	no	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
230	C₂×C₃	#4950	1	1	yes	characteristic
231	C₂×C₃	#4999	1	1	yes	characteristic
232	C₂×C₃	D₄×C₃⋊C₈	1	2	yes	—
233	C₂×C₃	#5011	1	2	yes	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	C₄⋊(C₉⋊C₈)	1	4	yes	complemented
239	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
240	C₂²	C₂×#3667	1	1	yes	characteristic
241	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
242	C₂²	D₄×Dic₉	1	2	yes	—
243	C₂²	#63510	1	2	yes	—
244	C₄	—	2	4	no	—
245	C₄	—	2	2	no	—
246	C₄	—	4	2	no	—
247	C₄	#63440	1	2	yes	—
248	C₄	#63444	1	1	yes	characteristic
249	C₄	#63448	1	1	yes	characteristic
250	C₃	#69449	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	C₂×C₄⋊(C₉⋊C₈)	1	1	yes	characteristic
253	C₂	#10588630	1	1	yes	characteristic
254	C₂	#10588677	1	1	yes	characteristic
255	C₂	D₄×C₉⋊C₈	1	2	yes	—
256	C₂	#10588688	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₄×C₉×C₂²⋊C₄	C₂
socle	C₂³×C₃	C₂²×Dic₃
prime radical	C₂⁴×C₃	C₂×Dic₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₄×C₂²⋊C₄	D₉
hypocenter	C₉	#1850
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
5. #10588200	C₂	semidirect
34. C₉×C₂²⋊C₄	C₈	semidirect
74. C₂×C₄×C₉	C₂×C₈	semidirect
112. C₂²×C₉	C₄⋊C₈	semidirect
184. C₂²⋊C₄	C₉⋊C₈	semidirect
203. C₉	#1850	semidirect
219. C₂×C₄	C₂×C₉⋊C₈	semidirect
238. C₂²	C₄⋊(C₉⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₉⋊C₈	2
C₄×C₉×C₂²⋊C₄	2
C₂×#63389	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588200	2
#69449	3