Subgroups — #11859521

Order 1152.

726 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×#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	#10588688	C₂	1	8	yes	maximal, complemented
5	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#63362	C₂	1	1	yes	maximal, characteristic, complemented
7	#83276	—	3	1	no	maximal
8	C₂×C₉×#41	C₂²	1	2	yes	—
9	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₉×#127	C₄	1	1	yes	characteristic
11	C₉×#124	C₄	1	2	yes	—
12	C₉×#121	C₄	1	4	yes	—
13	C₄×D₄×C₉	C₂²	1	4	yes	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	#63510	—	2	8	no	—
16	C₂×#3667	C₂²	1	4	yes	—
17	C₂×#3667	C₂²	1	2	yes	—
18	C₂²×C₉⋊C₈	C₂²	1	4	yes	—
19	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	#63389	C₂²	1	8	yes	—
22	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	—
23	#63362	C₂²	1	4	yes	—
24	C₃×#532	D₃	1	1	yes	characteristic
25	C₂×#948	—	3	2	no	—
26	#5234	—	3	2	no	—
27	#5011	—	3	8	no	—
28	#4924	—	3	1	no	—
29	C₂×#803	—	3	1	no	—
30	C₉×#41	—	2	4	no	—
31	C₉×#41	—	2	4	no	—
32	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
34	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₉	C₂³	1	4	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
40	C₉×C₂²⋊C₄	C₂×C₄	1	8	yes	—
41	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₉	C₂³	1	2	yes	—
44	#3667	—	2	4	no	—
45	#3667	—	2	8	no	—
46	C₂×C₉⋊C₈	—	2	2	no	—
47	C₂×C₉⋊C₈	—	2	8	no	—
48	C₂×C₉⋊C₈	D₄	1	4	yes	complemented
49	C₂×C₉⋊C₈	C₂³	1	2	yes	—
50	C₂×C₉⋊C₈	C₂³	1	4	yes	—
51	C₂×C₉⋊C₈	C₂³	1	2	yes	—
52	#2932	—	9	1	no	Sylow
53	C₂×C₃×#41	C₂×D₃	1	2	yes	—
54	C₃×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
55	C₃×#127	Dic₃	1	1	yes	characteristic
56	C₃×#124	Dic₃	1	2	yes	—
57	C₃×#121	Dic₃	1	4	yes	—
58	C₃×C₄×D₄	C₂×D₃	1	4	yes	—
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	#948	—	6	8	no	—
61	C₂×#208	—	3	2	no	—
62	C₂×#208	—	3	4	no	—
63	C₂²×C₃⋊C₈	—	3	1	no	—
64	C₂²×C₃⋊C₈	—	3	1	no	—
65	C₂²×C₃⋊C₈	—	3	4	no	—
66	#830	—	3	8	no	—
67	C₄⋊(C₃⋊C₈)	—	3	4	no	—
68	#803	—	3	4	no	—
69	C₂³×C₉	C₂²×C₄	1	4	yes	—
70	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₉	—	2	4	no	—
72	D₄×C₉	—	2	8	no	—
73	D₄×C₉	—	2	4	no	—
74	D₄×C₉	—	4	4	no	—
75	C₂×C₄×C₉	—	2	4	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	4	2	no	—
78	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₉	#41	1	2	yes	—
84	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
85	C₉⋊C₈	—	2	4	no	—
86	C₉⋊C₈	—	2	2	no	—
87	C₉⋊C₈	C₂×D₄	1	4	yes	—
88	C₂×#131	—	9	2	no	—
89	#532	D₉	1	1	yes	characteristic
90	#435	—	9	8	no	—
91	#423	—	9	1	no	—
92	#408	—	9	2	no	—
93	C₂×#97	—	9	1	no	—
94	C₃×#41	—	2	4	no	—
95	C₃×#41	—	2	4	no	—
96	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
97	C₂×C₃×D₄	C₂×Dic₃	1	4	yes	—
98	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
99	C₂×C₃×D₄	C₂×Dic₃	1	4	yes	—
100	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
101	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
104	C₃×C₂²⋊C₄	C₂×Dic₃	1	8	yes	—
105	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
106	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
107	C₃×C₄²	C₂²×D₃	1	2	yes	—
108	#208	—	6	8	no	—
109	#208	—	6	4	no	—
110	C₂×C₃⋊C₈	—	3	4	no	—
111	C₂×C₃⋊C₈	—	3	2	no	—
112	C₂×C₃⋊C₈	—	3	2	no	—
113	C₂×C₃⋊C₈	—	3	4	no	—
114	C₂×C₃⋊C₈	—	6	8	no	—
115	C₂×C₃⋊C₈	—	6	2	no	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	—	2	4	no	—
118	C₂²×C₉	—	2	8	no	—
119	C₂²×C₉	C₄×D₄	1	2	yes	—
120	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₉	—	2	4	no	—
122	C₄×C₉	—	2	2	no	—
123	C₄×C₉	—	2	2	no	—
124	C₄×C₉	C₄×D₄	1	2	yes	—
125	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₉	C₂×#41	1	1	yes	characteristic
127	C₂×#41	C₂×D₉	1	2	yes	—
128	#131	—	18	8	no	—
129	C₂×M₁₆	—	9	2	no	—
130	C₂×M₁₆	—	9	4	no	—
131	C₂²×C₈	—	9	1	no	—
132	C₂²×C₈	—	9	4	no	—
133	C₂²×C₈	—	9	1	no	—
134	C₄⋊Q₈	Dic₉	1	1	yes	characteristic
135	#127	Dic₉	1	1	yes	characteristic
136	#124	Dic₉	1	2	yes	—
137	#121	Dic₉	1	4	yes	—
138	C₄×D₄	C₂×D₉	1	4	yes	—
139	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
140	C₄⋊C₈	—	9	4	no	—
141	C₂²⋊C₈	—	9	8	no	—
142	#97	—	9	4	no	—
143	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
144	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
145	C₃×Q₈	—	2	4	no	—
146	C₃×D₄	—	2	8	no	—
147	C₃×D₄	—	2	4	no	—
148	C₃×D₄	—	4	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
153	C₂×C₃×C₄	#237	1	2	yes	—
154	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
155	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
156	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
157	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
159	C₃⋊C₈	—	3	4	no	—
160	C₃⋊C₈	—	6	4	no	—
161	C₃⋊C₈	—	6	2	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	—	4	4	no	—
164	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×C₉	#568	1	2	yes	—
166	#41	—	2	4	no	—
167	#41	—	2	4	no	—
168	C₂×Q₈	C₂×Dic₉	1	2	yes	—
169	C₂×D₄	C₂×Dic₉	1	4	yes	—
170	C₂×D₄	C₂×Dic₉	1	2	yes	—
171	C₂×D₄	C₂×Dic₉	1	4	yes	—
172	C₂²×C₄	C₂×Dic₉	1	2	yes	—
173	C₂²×C₄	C₂²×D₉	1	4	yes	—
174	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
175	M₁₆	—	18	8	no	—
176	M₁₆	—	18	4	no	—
177	C₂×C₈	—	9	4	no	—
178	C₂×C₈	—	9	2	no	—
179	C₂×C₈	—	9	2	no	—
180	C₂×C₈	—	9	4	no	—
181	C₂×C₈	—	18	8	no	—
182	C₂×C₈	—	18	2	no	—
183	C₄⋊C₄	C₂×Dic₉	1	4	yes	—
184	C₂²⋊C₄	C₂×Dic₉	1	8	yes	—
185	C₄²	C₂×Dic₉	1	1	yes	characteristic
186	C₄²	C₂×Dic₉	1	1	yes	characteristic
187	C₄²	C₂²×D₉	1	2	yes	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	8	no	—
191	C₂²×C₃	D₄×Dic₃	1	2	yes	—
192	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	D₄×Dic₃	1	2	yes	—
197	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
198	C₃×C₄	C₂×#237	1	1	yes	characteristic
199	C₉	#2932	1	1	yes	characteristic, complemented, Sylow
200	C₂³	C₂²×Dic₉	1	4	yes	—
201	C₂³	C₂²×Dic₉	1	1	yes	characteristic
202	Q₈	—	2	4	no	—
203	D₄	—	2	8	no	—
204	D₄	—	2	4	no	—
205	D₄	—	4	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	D₄×D₉	1	2	yes	—
210	C₂×C₄	#3699	1	2	yes	—
211	C₂×C₄	C₂²×Dic₉	1	2	yes	—
212	C₂×C₄	C₂²×Dic₉	1	2	yes	—
213	C₂×C₄	C₂²×Dic₉	1	4	yes	—
214	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
215	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
216	C₈	—	9	4	no	—
217	C₈	—	18	4	no	—
218	C₈	—	18	2	no	—
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	—	4	4	no	—
221	C₂×C₃	C₂×D₄×Dic₃	1	1	yes	characteristic
222	C₂×C₃	#5819	1	2	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	8	no	—
226	C₂²	D₄×Dic₉	1	2	yes	—
227	C₂²	C₂³×Dic₉	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	D₄×Dic₉	1	2	yes	—
232	C₄	C₂×D₄×D₉	1	1	yes	characteristic
233	C₄	C₂×#3699	1	1	yes	characteristic
234	C₃	#83276	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	4	no	—
237	C₂	C₂×D₄×Dic₉	1	1	yes	characteristic
238	C₂	#10589921	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	#2932
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. #10588688	C₂	semidirect
5. #10588604	C₂	semidirect
6. C₂×#63362	C₂	semidirect
48. C₂×C₉⋊C₈	D₄	semidirect
199. C₉	#2932	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
#10588688	2
#10588604	2
C₂×#63362	2
#83276	3