Subgroups — #11786391

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3667	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	#69450	—	3	1	no	maximal
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₂²⋊C₄	C₄	1	2	yes	—
9	C₂×C₄²×C₉	C₂²	1	2	yes	—
10	C₂×#3667	—	2	2	no	—
11	C₂×#3667	—	2	4	no	—
12	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
15	C₉×#95	C₄	1	1	yes	characteristic
16	C₉×#95	C₄	1	1	yes	characteristic
17	#63389	—	2	2	no	—
18	C₄⋊(C₉⋊C₈)	—	2	4	no	—
19	C₂²×#208	—	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	4	no	—
34	C₄²×C₉	—	2	2	no	—
35	C₄²×C₉	—	2	2	no	—
36	#3667	—	2	8	no	—
37	C₂×C₉⋊C₈	—	2	2	no	—
38	C₂×C₉⋊C₈	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	2	no	—
40	C₂×C₉⋊C₈	—	2	1	no	—
41	C₂×C₉⋊C₈	—	2	1	no	—
42	C₂×C₉⋊C₈	—	2	2	no	—
43	C₂×C₉⋊C₈	D₄	1	4	yes	—
44	#1851	—	9	1	no	Sylow
45	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×C₂²⋊C₄	Dic₃	1	2	yes	—
47	C₂×C₃×C₄²	C₂×D₃	1	2	yes	—
48	C₂×#208	—	6	4	no	—
49	C₂×#208	—	6	2	no	—
50	C₂²×C₃⋊C₈	—	3	2	no	—
51	C₂²×C₃⋊C₈	—	3	1	no	—
52	C₂²×C₃⋊C₈	—	3	1	no	—
53	C₃×#95	Dic₃	1	1	yes	characteristic
54	C₃×#95	Dic₃	1	1	yes	characteristic
55	#830	—	6	2	no	—
56	C₄⋊(C₃⋊C₈)	—	6	4	no	—
57	C₂³×C₉	—	2	2	no	—
58	C₂³×C₉	—	2	2	no	—
59	C₂³×C₉	—	2	1	no	—
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	C₄⋊C₄	1	2	yes	—
62	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	—	2	2	no	—
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	4	no	—
68	C₂×C₄×C₉	—	2	2	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	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₉	M₁₆	1	4	yes	complemented
74	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₉	#41	1	2	yes	—
78	C₉⋊C₈	—	2	4	no	—
79	C₉⋊C₈	—	4	1	no	—
80	C₉⋊C₈	—	4	1	no	—
81	C₂²×M₁₆	—	9	1	no	—
82	C₂×C₄⋊C₈	—	9	2	no	—
83	C₂×C₂²⋊C₈	—	9	1	no	—
84	C₄×C₂²⋊C₄	D₉	1	1	yes	characteristic
85	#336	—	9	2	no	—
86	C₂⁴×C₃	C₂×Dic₃	1	1	yes	characteristic
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	Dic₆	1	2	yes	—
90	C₂²×C₃×C₄	D₁₂	1	2	yes	—
91	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
92	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
93	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₃×C₂²⋊C₄	—	2	4	no	—
96	C₃×C₄²	—	2	2	no	—
97	C₃×C₄²	—	2	2	no	—
98	#208	—	6	8	no	—
99	C₂×C₃⋊C₈	—	3	4	no	—
100	C₂×C₃⋊C₈	—	6	2	no	—
101	C₂×C₃⋊C₈	—	6	2	no	—
102	C₂×C₃⋊C₈	—	6	1	no	—
103	C₂×C₃⋊C₈	—	6	1	no	—
104	C₂×C₃⋊C₈	—	6	2	no	—
105	C₂×C₃⋊C₈	—	6	2	no	—
106	C₂²×C₉	—	2	2	no	—
107	C₂²×C₉	—	2	2	no	—
108	C₂²×C₉	—	2	2	no	—
109	C₂²×C₉	—	4	2	no	—
110	C₂²×C₉	—	4	2	no	—
111	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₉	C₄×D₄	1	2	yes	—
113	C₂²×C₉	C₂×M₁₆	1	2	yes	—
114	C₂²×C₉	#131	1	1	yes	characteristic
115	C₂²×C₉	#131	1	1	yes	characteristic
116	C₄×C₉	—	2	2	no	—
117	C₄×C₉	—	2	4	no	—
118	C₄×C₉	—	4	2	no	—
119	C₄×C₉	#120	1	1	yes	characteristic
120	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
121	C₄×C₉	#123	1	1	yes	characteristic
122	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
123	C₂×M₁₆	—	18	4	no	—
124	C₂×M₁₆	—	18	2	no	—
125	C₂²×C₈	—	9	2	no	—
126	C₂²×C₈	—	9	1	no	—
127	C₂²×C₈	—	9	1	no	—
128	C₂×C₂²⋊C₄	Dic₉	1	2	yes	—
129	C₂×C₄²	C₂×D₉	1	2	yes	—
130	C₄⋊C₈	—	18	4	no	—
131	C₂²⋊C₈	—	18	2	no	—
132	#95	Dic₉	1	1	yes	characteristic
133	#95	Dic₉	1	1	yes	characteristic
134	C₂³×C₃	—	2	2	no	—
135	C₂³×C₃	—	2	2	no	—
136	C₂³×C₃	—	2	1	no	—
137	C₂³×C₃	—	2	1	no	—
138	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
139	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	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	#208	1	4	yes	—
150	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
151	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
153	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
154	C₂×C₃×C₄	#237	1	2	yes	—
155	C₃⋊C₈	—	6	4	no	—
156	C₃⋊C₈	—	12	1	no	—
157	C₃⋊C₈	—	12	1	no	—
158	C₂×C₉	—	4	2	no	—
159	C₂×C₉	#385	1	1	yes	characteristic
160	C₂×C₉	#423	1	1	yes	characteristic
161	C₂×C₉	#424	1	1	yes	characteristic
162	C₂×C₉	#435	1	2	yes	—
163	C₂×C₉	#436	1	2	yes	—
164	C₂⁴	C₂×Dic₉	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	Dic₁₈	1	2	yes	—
168	C₂²×C₄	D₃₆	1	2	yes	—
169	C₂²×C₄	C₂×Dic₉	1	2	yes	—
170	C₂²×C₄	C₂×Dic₉	1	2	yes	—
171	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
173	M₁₆	—	18	8	no	—
174	C₂×C₈	—	9	4	no	—
175	C₂×C₈	—	18	2	no	—
176	C₂×C₈	—	18	2	no	—
177	C₂×C₈	—	18	1	no	—
178	C₂×C₈	—	18	1	no	—
179	C₂×C₈	—	18	2	no	—
180	C₂×C₈	—	18	2	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	4	2	no	—
188	C₂²×C₃	—	4	2	no	—
189	C₂²×C₃	C₂×#208	1	2	yes	—
190	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
191	C₂²×C₃	D₄×Dic₃	1	2	yes	—
192	C₂²×C₃	#948	1	1	yes	characteristic
193	C₂²×C₃	#948	1	1	yes	characteristic
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	4	no	—
196	C₃×C₄	—	4	2	no	—
197	C₃×C₄	#878	1	2	yes	—
198	C₃×C₄	#882	1	1	yes	characteristic
199	C₃×C₄	#886	1	1	yes	characteristic
200	C₉	#1851	1	1	yes	characteristic, complemented, Sylow
201	C₂³	—	2	2	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	2	no	—
205	C₂³	C₄⋊Dic₉	1	2	yes	—
206	C₂³	C₂²×Dic₉	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	#3667	1	4	yes	complemented
217	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
218	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
219	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
220	C₂×C₄	D₄×D₉	1	2	yes	—
221	C₂×C₄	#3699	1	2	yes	—
222	C₈	—	18	4	no	—
223	C₈	—	36	1	no	—
224	C₈	—	36	1	no	—
225	C₂×C₃	—	4	2	no	—
226	C₂×C₃	#4924	1	1	yes	characteristic
227	C₂×C₃	#4950	1	1	yes	characteristic
228	C₂×C₃	#4999	1	1	yes	characteristic
229	C₂×C₃	#5011	1	2	yes	—
230	C₂×C₃	#5012	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	C₂×#3667	1	2	yes	—
237	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
238	C₂²	D₄×Dic₉	1	2	yes	—
239	C₂²	#63510	1	1	yes	characteristic
240	C₂²	#63510	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	—	4	2	no	—
244	C₄	#63440	1	2	yes	—
245	C₄	#63444	1	1	yes	characteristic
246	C₄	#63448	1	1	yes	characteristic
247	C₃	#69450	1	1	yes	characteristic
248	C₂	—	4	2	no	—
249	C₂	#10588604	1	1	yes	characteristic
250	C₂	#10588630	1	1	yes	characteristic
251	C₂	#10588677	1	1	yes	characteristic
252	C₂	#10588688	1	2	yes	—
253	C₂	#10588689	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂³×C₉	C₂²×C₄
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	#1851
LCS	G ▷ C₂³×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
73. C₂×C₄×C₉	M₁₆	semidirect
200. C₉	#1851	semidirect
216. C₂×C₄	#3667	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#3667	2
C₄×C₉×C₂²⋊C₄	2
C₂×#63389	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588200	2
#69450	3