Subgroups — #11776126

Order 1152.

384 conjugacy classes of subgroups.

The subgroup lattice has 261 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	#10588605	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588224	C₂	1	2	yes	maximal, complemented
5	#10588221	C₂	1	2	yes	maximal, complemented
6	#71804	—	3	1	no	maximal
7	C₂×C₉×#41	—	2	1	no	—
8	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₉×#127	C₂²	1	1	yes	characteristic
10	C₉×#124	—	2	1	no	—
11	C₉×#121	—	2	1	no	—
12	C₉×#121	C₄	1	2	yes	—
13	C₄×D₄×C₉	—	2	1	no	—
14	C₄×D₄×C₉	C₄	1	2	yes	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₂²×C₉⋊C₈	—	2	1	no	—
17	#63389	—	2	1	no	—
18	C₄⋊(C₉⋊C₈)	—	2	1	no	—
19	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
21	C₄×C₉⋊C₈	—	2	1	no	—
22	C₃×#532	D₃	1	1	yes	characteristic
23	#4925	—	3	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
25	#4547	—	3	2	no	—
26	#4544	—	3	2	no	—
27	C₉×#41	—	4	2	no	—
28	C₉×#41	—	4	2	no	—
29	C₂×Q₈×C₉	—	2	1	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₉	—	2	2	no	—
35	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
38	C₉×C₄⋊C₄	—	2	1	no	—
39	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₉×C₂²⋊C₄	—	2	2	no	—
41	C₉×C₂²⋊C₄	—	2	2	no	—
42	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₉	D₄	1	1	yes	characteristic
44	C₄²×C₉	D₄	1	1	yes	characteristic
45	C₄²×C₉	C₂³	1	1	yes	characteristic
46	C₂×C₉⋊C₈	—	2	2	no	—
47	C₂×C₉⋊C₈	—	2	2	no	—
48	C₂×C₉⋊C₈	—	4	2	no	—
49	#1540	—	9	1	no	Sylow
50	C₂×C₃×#41	—	2	1	no	—
51	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
52	C₃×#127	C₂×D₃	1	1	yes	characteristic
53	C₃×#124	—	2	1	no	—
54	C₃×#121	—	2	1	no	—
55	C₃×#121	Dic₃	1	2	yes	—
56	C₃×C₄×D₄	—	2	1	no	—
57	C₃×C₄×D₄	Dic₃	1	2	yes	—
58	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₃⋊C₈	—	6	1	no	—
60	#830	—	6	1	no	—
61	C₄⋊(C₃⋊C₈)	—	3	2	no	—
62	C₄⋊(C₃⋊C₈)	—	3	2	no	—
63	C₄⋊(C₃⋊C₈)	—	6	1	no	—
64	C₄×C₃⋊C₈	—	6	1	no	—
65	C₂³×C₉	—	2	2	no	—
66	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	Q₈×C₉	—	4	1	no	—
68	Q₈×C₉	—	4	1	no	—
69	D₄×C₉	—	4	2	no	—
70	D₄×C₉	—	4	4	no	—
71	D₄×C₉	—	4	1	no	—
72	D₄×C₉	—	4	1	no	—
73	D₄×C₉	—	4	2	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	4	1	no	—
77	C₂×C₄×C₉	—	4	4	no	—
78	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₉	D₈	1	2	yes	—
82	C₂×C₄×C₉	SD₁₆	1	2	yes	—
83	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₉⋊C₈	—	4	2	no	—
87	C₉⋊C₈	—	4	2	no	—
88	#532	D₉	1	1	yes	characteristic
89	#431	—	9	1	no	—
90	C₂×C₄⋊C₈	—	9	1	no	—
91	#332	—	9	2	no	—
92	#331	—	9	2	no	—
93	C₃×#41	—	4	2	no	—
94	C₃×#41	—	4	2	no	—
95	C₂×C₃×Q₈	—	2	1	no	—
96	C₂×C₃×D₄	—	2	2	no	—
97	C₂×C₃×D₄	—	2	1	no	—
98	C₂×C₃×D₄	—	2	1	no	—
99	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
100	C₂²×C₃×C₄	—	2	2	no	—
101	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	#67	1	1	yes	characteristic
103	C₂²×C₃×C₄	#67	1	1	yes	characteristic
104	C₃×C₄⋊C₄	—	2	1	no	—
105	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
106	C₃×C₂²⋊C₄	—	2	2	no	—
107	C₃×C₂²⋊C₄	—	2	2	no	—
108	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
109	C₃×C₄²	#67	1	1	yes	characteristic
110	C₃×C₄²	#67	1	1	yes	characteristic
111	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃⋊C₈	—	6	2	no	—
113	C₂×C₃⋊C₈	—	6	2	no	—
114	C₂×C₃⋊C₈	—	12	2	no	—
115	C₂²×C₉	—	2	1	no	—
116	C₂²×C₉	—	2	1	no	—
117	C₂²×C₉	—	4	2	no	—
118	C₂²×C₉	—	4	2	no	—
119	C₂²×C₉	—	4	2	no	—
120	C₂²×C₉	#102	1	2	yes	—
121	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₉	—	2	1	no	—
123	C₄×C₉	—	2	1	no	—
124	C₄×C₉	—	2	1	no	—
125	C₄×C₉	—	4	2	no	—
126	C₄×C₉	#102	1	2	yes	—
127	C₄×C₉	C₂×D₈	1	1	yes	characteristic
128	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₄×C₉	#135	1	2	yes	—
130	C₂×#41	—	2	1	no	—
131	C₂²×C₈	—	18	1	no	—
132	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
133	#127	C₂×D₉	1	1	yes	characteristic
134	#124	—	2	1	no	—
135	#121	—	2	1	no	—
136	#121	Dic₉	1	2	yes	—
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	Dic₉	1	2	yes	—
139	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
140	C₄⋊C₈	—	9	2	no	—
141	C₄⋊C₈	—	9	2	no	—
142	C₄⋊C₈	—	18	1	no	—
143	C₂²⋊C₈	—	18	1	no	—
144	C₄×C₈	—	18	1	no	—
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	#217	1	1	yes	characteristic
147	C₃×Q₈	—	4	1	no	—
148	C₃×Q₈	—	4	1	no	—
149	C₃×D₄	—	4	2	no	—
150	C₃×D₄	—	4	4	no	—
151	C₃×D₄	—	4	1	no	—
152	C₃×D₄	—	4	1	no	—
153	C₃×D₄	—	4	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	4	no	—
158	C₂×C₃×C₄	#213	1	2	yes	—
159	C₂×C₃×C₄	#214	1	2	yes	—
160	C₂×C₃×C₄	#217	1	1	yes	characteristic
161	C₂×C₃×C₄	#217	1	1	yes	characteristic
162	C₂×C₃×C₄	#217	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
166	C₃⋊C₈	—	12	2	no	—
167	C₃⋊C₈	—	12	2	no	—
168	C₂×C₉	—	2	2	no	—
169	C₂×C₉	—	8	2	no	—
170	C₂×C₉	#413	1	1	yes	characteristic
171	C₂×C₉	C₂×#102	1	1	yes	characteristic
172	C₂×C₉	#416	1	1	yes	characteristic
173	#41	—	4	2	no	—
174	#41	—	4	2	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	C₂×Dic₉	1	2	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
182	C₂²×C₄	#611	1	1	yes	characteristic
183	C₂²×C₄	#611	1	1	yes	characteristic
184	C₂×C₈	—	18	2	no	—
185	C₂×C₈	—	18	2	no	—
186	C₂×C₈	—	36	2	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	C₂×Dic₉	1	1	yes	characteristic
192	C₄²	#611	1	1	yes	characteristic
193	C₄²	#611	1	1	yes	characteristic
194	C₄²	C₂²×D₉	1	1	yes	characteristic
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	—	4	2	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	#832	1	2	yes	—
201	C₂²×C₃	C₂×#217	1	1	yes	characteristic
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	4	2	no	—
206	C₃×C₄	#832	1	2	yes	—
207	C₃×C₄	C₂×#213	1	1	yes	characteristic
208	C₃×C₄	C₂×#214	1	1	yes	characteristic
209	C₃×C₄	#950	1	2	yes	—
210	C₉	#1540	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	2	2	no	—
212	C₂³	#3676	1	1	yes	characteristic
213	Q₈	—	4	1	no	—
214	Q₈	—	4	1	no	—
215	D₄	—	4	2	no	—
216	D₄	—	4	4	no	—
217	D₄	—	4	1	no	—
218	D₄	—	4	1	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	#3672	1	2	yes	—
225	C₂×C₄	#3673	1	2	yes	—
226	C₂×C₄	#3676	1	1	yes	characteristic
227	C₂×C₄	#3676	1	1	yes	characteristic
228	C₂×C₄	#3676	1	1	yes	characteristic
229	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
230	C₂×C₄	C₂×#611	1	1	yes	characteristic
231	C₂×C₄	C₂×#611	1	1	yes	characteristic
232	C₈	—	36	2	no	—
233	C₈	—	36	2	no	—
234	C₂×C₃	—	2	2	no	—
235	C₂×C₃	—	8	2	no	—
236	C₂×C₃	C₂×#832	1	1	yes	characteristic
237	C₂×C₃	#5233	1	1	yes	characteristic
238	C₂×C₃	#5237	1	1	yes	characteristic
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	#63392	1	2	yes	—
245	C₂²	C₂×#3676	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	#63392	1	2	yes	—
251	C₄	C₂×#3672	1	1	yes	characteristic
252	C₄	C₂×#3673	1	1	yes	characteristic
253	C₄	#63513	1	2	yes	—
254	C₃	#71804	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	—	8	2	no	—
257	C₂	C₂×#63392	1	1	yes	characteristic
258	C₂	#10588915	1	1	yes	characteristic
259	C₂	#10588917	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3676
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#532	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₃×#532	D₃
cosocle	C₄²×C₉	C₂³
hypercenter	#532	D₉
hypocenter	C₉	#1540
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
2. #10588605	C₂	semidirect
3. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
4. #10588224	C₂	semidirect
5. #10588221	C₂	semidirect
210. C₉	#1540	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
#10588605	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588224	2
#10588221	2
#71804	3