Subgroups — #11859468

Order 1152.

750 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×#63510	C₂	1	2	yes	maximal, complemented
3	#10588912	C₂	1	2	yes	maximal, complemented
4	#10588689	C₂	1	4	yes	maximal, complemented
5	D₄×C₉⋊C₈	C₂	1	4	yes	maximal, complemented
6	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#83275	—	3	1	no	maximal
9	C₂×C₉×#41	C₂²	1	2	yes	—
10	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₉×#127	C₄	1	1	yes	characteristic
12	C₉×#124	C₄	1	2	yes	—
13	C₉×#121	C₄	1	4	yes	—
14	C₄×D₄×C₉	C₂²	1	4	yes	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	#63510	—	2	8	no	—
17	C₂×#3667	C₂²	1	4	yes	—
18	C₂×#3667	C₂²	1	2	yes	—
19	C₂²×C₉⋊C₈	C₂²	1	4	yes	—
20	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
22	#63389	C₂²	1	8	yes	—
23	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	—
24	C₄×C₉⋊C₈	C₂²	1	2	yes	—
25	C₄×C₉⋊C₈	C₂²	1	2	yes	—
26	C₃×#532	D₃	1	1	yes	characteristic
27	C₂×#948	—	3	2	no	—
28	#5234	—	3	2	no	—
29	#5012	—	3	4	no	—
30	D₄×C₃⋊C₈	—	3	4	no	—
31	#4924	—	3	1	no	—
32	C₂×C₄×C₃⋊C₈	—	3	1	no	—
33	C₉×#41	—	2	4	no	—
34	C₉×#41	—	2	4	no	—
35	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
37	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
43	C₉×C₂²⋊C₄	C₂×C₄	1	8	yes	—
44	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₉	C₂³	1	2	yes	—
47	#3667	—	2	4	no	—
48	#3667	—	2	8	no	—
49	C₂×C₉⋊C₈	—	2	2	no	—
50	C₂×C₉⋊C₈	—	2	8	no	—
51	C₂×C₉⋊C₈	D₄	1	4	yes	complemented
52	C₂×C₉⋊C₈	C₂³	1	2	yes	—
53	C₂×C₉⋊C₈	C₂³	1	4	yes	—
54	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
56	#2931	—	9	1	no	Sylow
57	C₂×C₃×#41	C₂×D₃	1	2	yes	—
58	C₃×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
59	C₃×#127	Dic₃	1	1	yes	characteristic
60	C₃×#124	Dic₃	1	2	yes	—
61	C₃×#121	Dic₃	1	4	yes	—
62	C₃×C₄×D₄	C₂×D₃	1	4	yes	—
63	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
64	#948	—	6	8	no	—
65	C₂×#208	—	3	2	no	—
66	C₂×#208	—	3	4	no	—
67	C₂²×C₃⋊C₈	—	3	1	no	—
68	C₂²×C₃⋊C₈	—	3	1	no	—
69	C₂²×C₃⋊C₈	—	3	4	no	—
70	#830	—	3	8	no	—
71	C₄⋊(C₃⋊C₈)	—	3	4	no	—
72	C₄×C₃⋊C₈	—	3	2	no	—
73	C₄×C₃⋊C₈	—	3	2	no	—
74	C₂³×C₉	C₂²×C₄	1	4	yes	—
75	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₉	—	2	4	no	—
77	D₄×C₉	—	2	8	no	—
78	D₄×C₉	—	2	4	no	—
79	D₄×C₉	—	2	8	no	—
80	C₂×C₄×C₉	—	2	4	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	4	no	—
83	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₉	#41	1	2	yes	—
89	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
90	C₉⋊C₈	—	2	4	no	—
91	C₉⋊C₈	—	2	2	no	—
92	C₉⋊C₈	C₂×D₄	1	2	yes	—
93	C₉⋊C₈	C₂×D₄	1	2	yes	—
94	C₂×#131	—	9	2	no	—
95	#532	D₉	1	1	yes	characteristic
96	#436	—	9	4	no	—
97	C₈×D₄	—	9	4	no	—
98	#423	—	9	1	no	—
99	#408	—	9	2	no	—
100	C₂×C₄×C₈	—	9	1	no	—
101	C₃×#41	—	2	4	no	—
102	C₃×#41	—	2	4	no	—
103	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
104	C₂×C₃×D₄	C₂×Dic₃	1	4	yes	—
105	C₂×C₃×D₄	C₂×Dic₃	1	4	yes	—
106	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
107	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
108	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
109	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
111	C₃×C₂²⋊C₄	C₂×Dic₃	1	8	yes	—
112	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
113	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
114	C₃×C₄²	C₂²×D₃	1	2	yes	—
115	#208	—	6	8	no	—
116	#208	—	6	4	no	—
117	C₂×C₃⋊C₈	—	3	4	no	—
118	C₂×C₃⋊C₈	—	3	1	no	—
119	C₂×C₃⋊C₈	—	3	1	no	—
120	C₂×C₃⋊C₈	—	3	2	no	—
121	C₂×C₃⋊C₈	—	3	4	no	—
122	C₂×C₃⋊C₈	—	6	8	no	—
123	C₂×C₃⋊C₈	—	6	2	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	—	2	4	no	—
126	C₂²×C₉	—	2	8	no	—
127	C₂²×C₉	C₄×D₄	1	2	yes	—
128	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₉	—	2	4	no	—
130	C₄×C₉	—	2	2	no	—
131	C₄×C₉	C₄×D₄	1	2	yes	—
132	C₄×C₉	#131	1	4	yes	complemented
133	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₉	C₂×#41	1	1	yes	characteristic
135	C₂×#41	C₂×D₉	1	2	yes	—
136	#131	—	18	8	no	—
137	C₂×M₁₆	—	9	2	no	—
138	C₂×M₁₆	—	9	4	no	—
139	C₂²×C₈	—	9	1	no	—
140	C₂²×C₈	—	9	4	no	—
141	C₂²×C₈	—	9	1	no	—
142	C₄⋊Q₈	Dic₉	1	1	yes	characteristic
143	#127	Dic₉	1	1	yes	characteristic
144	#124	Dic₉	1	2	yes	—
145	#121	Dic₉	1	4	yes	—
146	C₄×D₄	C₂×D₉	1	4	yes	—
147	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
148	C₄⋊C₈	—	9	4	no	—
149	C₂²⋊C₈	—	9	8	no	—
150	C₄×C₈	—	9	2	no	—
151	C₄×C₈	—	9	2	no	—
152	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
153	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
154	C₃×Q₈	—	2	4	no	—
155	C₃×D₄	—	2	8	no	—
156	C₃×D₄	—	2	4	no	—
157	C₃×D₄	—	2	8	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
162	C₂×C₃×C₄	#237	1	2	yes	—
163	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
164	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
165	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
167	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
168	C₃⋊C₈	—	3	2	no	—
169	C₃⋊C₈	—	3	2	no	—
170	C₃⋊C₈	—	6	4	no	—
171	C₃⋊C₈	—	6	2	no	—
172	C₂×C₉	—	2	2	no	—
173	C₂×C₉	—	4	4	no	—
174	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×C₉	C₂×#131	1	2	yes	—
176	#41	—	2	4	no	—
177	#41	—	2	4	no	—
178	C₂×Q₈	C₂×Dic₉	1	2	yes	—
179	C₂×D₄	C₂×Dic₉	1	4	yes	—
180	C₂×D₄	C₂×Dic₉	1	4	yes	—
181	C₂×D₄	C₂×Dic₉	1	2	yes	—
182	C₂²×C₄	C₂×Dic₉	1	2	yes	—
183	C₂²×C₄	C₂²×D₉	1	4	yes	—
184	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
185	M₁₆	—	18	8	no	—
186	M₁₆	—	18	4	no	—
187	C₂×C₈	—	9	4	no	—
188	C₂×C₈	—	9	1	no	—
189	C₂×C₈	—	9	1	no	—
190	C₂×C₈	—	9	2	no	—
191	C₂×C₈	—	9	4	no	—
192	C₂×C₈	—	18	8	no	—
193	C₂×C₈	—	18	2	no	—
194	C₄⋊C₄	C₂×Dic₉	1	4	yes	—
195	C₂²⋊C₄	C₂×Dic₉	1	8	yes	—
196	C₄²	C₂×Dic₉	1	1	yes	characteristic
197	C₄²	C₂×Dic₉	1	1	yes	characteristic
198	C₄²	C₂²×D₉	1	2	yes	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	2	8	no	—
202	C₂²×C₃	D₄×Dic₃	1	2	yes	—
203	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	D₄×Dic₃	1	2	yes	—
207	C₃×C₄	#948	1	4	yes	—
208	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
209	C₃×C₄	C₂×#237	1	1	yes	characteristic
210	C₉	#2931	1	1	yes	characteristic, complemented, Sylow
211	C₂³	C₂²×Dic₉	1	4	yes	—
212	C₂³	C₂²×Dic₉	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	D₄	—	2	8	no	—
215	D₄	—	2	4	no	—
216	D₄	—	2	8	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	D₄×D₉	1	2	yes	—
221	C₂×C₄	#3699	1	2	yes	—
222	C₂×C₄	C₂²×Dic₉	1	2	yes	—
223	C₂×C₄	C₂²×Dic₉	1	4	yes	—
224	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
225	C₂×C₄	C₂²×Dic₉	1	2	yes	—
226	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
227	C₈	—	9	2	no	—
228	C₈	—	9	2	no	—
229	C₈	—	18	4	no	—
230	C₈	—	18	2	no	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	—	4	4	no	—
233	C₂×C₃	C₂×D₄×Dic₃	1	1	yes	characteristic
234	C₂×C₃	C₂×#948	1	2	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	8	no	—
238	C₂²	D₄×Dic₉	1	2	yes	—
239	C₂²	C₂³×Dic₉	1	1	yes	characteristic
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	D₄×Dic₉	1	2	yes	—
243	C₄	#63510	1	4	yes	complemented
244	C₄	C₂×D₄×D₉	1	1	yes	characteristic
245	C₄	C₂×#3699	1	1	yes	characteristic
246	C₃	#83275	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	4	4	no	—
249	C₂	C₂×D₄×Dic₉	1	1	yes	characteristic
250	C₂	C₂×#63510	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#532	C₂
socle	C₂²×C₃	C₂³×Dic₃
prime radical	C₃×#532	D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	#532	D₉
hypocenter	C₉	#2931
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
2. C₂×#63510	C₂	semidirect
3. #10588912	C₂	semidirect
4. #10588689	C₂	semidirect
5. D₄×C₉⋊C₈	C₂	semidirect
6. #10588604	C₂	semidirect
7. C₂×C₄×C₉⋊C₈	C₂	semidirect
51. C₂×C₉⋊C₈	D₄	semidirect
132. C₄×C₉	#131	semidirect
210. C₉	#2931	semidirect
243. C₄	#63510	semidirect

Maximal subgroups

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

subgroup	index
C₉×#532	2
C₂×#63510	2
#10588912	2
#10588689	2
D₄×C₉⋊C₈	2
#10588604	2
C₂×C₄×C₉⋊C₈	2
#83275	3