Subgroups — #11775048

Order 1152.

426 conjugacy classes of subgroups.

The subgroup lattice has 270 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	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588605	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588214	C₂	1	4	yes	maximal, complemented
5	#71811	—	3	1	no	maximal
6	C₂×C₉×#41	—	2	1	no	—
7	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	C₉×#127	C₄	1	1	yes	characteristic, complemented
9	C₉×#124	—	2	1	no	—
10	C₉×#121	—	2	1	no	—
11	C₉×#121	C₂²	1	2	yes	—
12	C₄×D₄×C₉	—	2	1	no	—
13	C₄×D₄×C₉	C₄	1	2	yes	complemented
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₉	—	2	1	no	—
16	C₂×C₄×Dic₉	—	2	1	no	—
17	C₉⋊#95	C₂²	1	2	yes	—
18	#63389	C₂²	1	2	yes	—
19	C₄⋊(C₉⋊C₈)	—	2	1	no	—
20	C₄×C₉⋊C₈	—	2	1	no	—
21	C₃×#532	D₃	1	1	yes	characteristic
22	C₄×C₄⋊Dic₃	—	3	1	no	—
23	#4925	—	3	1	no	—
24	#4537	—	3	4	no	—
25	C₉×#41	—	4	2	no	—
26	C₉×#41	—	4	2	no	—
27	C₂×Q₈×C₉	—	2	1	no	—
28	C₂×D₄×C₉	—	2	2	no	—
29	C₂×D₄×C₉	—	2	1	no	—
30	C₂×D₄×C₉	—	2	1	no	—
31	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₉	—	2	2	no	—
37	C₉×C₄⋊C₄	—	2	1	no	—
38	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₉×C₂²⋊C₄	—	2	2	no	—
40	C₉×C₂²⋊C₄	—	2	2	no	—
41	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
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	C₄⋊Dic₉	—	2	4	no	—
46	C₄×Dic₉	—	4	2	no	—
47	C₂×C₉⋊C₈	—	2	2	no	—
48	#1502	—	9	1	no	Sylow
49	C₂×C₃×#41	—	2	1	no	—
50	C₃×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
51	C₃×#127	Dic₃	1	1	yes	characteristic
52	C₃×#124	—	2	1	no	—
53	C₃×#121	—	2	1	no	—
54	C₃×#121	C₂×D₃	1	2	yes	—
55	C₃×C₄×D₄	—	2	1	no	—
56	C₃×C₄×D₄	Dic₃	1	2	yes	—
57	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₂×C₄⋊Dic₃	—	6	1	no	—
59	C₂×C₄×Dic₃	—	6	1	no	—
60	C₃⋊#95	—	3	2	no	—
61	#830	—	3	2	no	—
62	C₄⋊(C₃⋊C₈)	—	6	1	no	—
63	C₄×C₃⋊C₈	—	6	1	no	—
64	C₂³×C₉	—	2	2	no	—
65	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₉	—	4	2	no	—
67	D₄×C₉	—	4	2	no	—
68	D₄×C₉	—	4	2	no	—
69	D₄×C₉	—	4	2	no	—
70	D₄×C₉	—	4	2	no	—
71	D₄×C₉	—	4	2	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	4	1	no	—
75	C₂×C₄×C₉	—	4	4	no	—
76	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₉	D₈	1	2	yes	—
81	C₂×C₄×C₉	SD₁₆	1	2	yes	—
82	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
84	C₂×Dic₉	—	2	4	no	—
85	C₂×Dic₉	—	4	2	no	—
86	C₂×Dic₉	—	4	2	no	—
87	C₉⋊C₈	—	4	2	no	—
88	#532	D₉	1	1	yes	characteristic
89	#431	—	9	1	no	—
90	C₄×C₄⋊C₄	—	9	1	no	—
91	#327	—	9	4	no	—
92	C₃×#41	—	4	2	no	—
93	C₃×#41	—	4	2	no	—
94	C₂×C₃×Q₈	—	2	1	no	—
95	C₂×C₃×D₄	—	2	2	no	—
96	C₂×C₃×D₄	—	2	1	no	—
97	C₂×C₃×D₄	—	2	1	no	—
98	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	#67	1	1	yes	characteristic
101	C₂²×C₃×C₄	#67	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×Dic₃	—	6	2	no	—
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	C₂×Dic₃	1	1	yes	characteristic
110	C₃×C₄²	#67	1	1	yes	characteristic
111	C₃×C₄²	#67	1	1	yes	characteristic
112	C₄⋊Dic₃	—	6	4	no	—
113	C₄×Dic₃	—	12	2	no	—
114	C₂×C₃⋊C₈	—	6	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
122	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
123	C₄×C₉	—	2	1	no	—
124	C₄×C₉	—	2	1	no	—
125	C₄×C₉	—	2	1	no	—
126	C₄×C₉	—	4	2	no	—
127	C₄×C₉	#102	1	2	yes	—
128	C₄×C₉	#102	1	2	yes	—
129	C₄×C₉	#104	1	2	yes	—
130	Dic₉	—	4	4	no	—
131	C₂×#41	—	2	1	no	—
132	C₄⋊Q₈	Dic₉	1	1	yes	characteristic, complemented
133	#127	Dic₉	1	1	yes	characteristic, complemented
134	#124	—	2	1	no	—
135	#121	—	2	1	no	—
136	#121	C₂×D₉	1	2	yes	—
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	Dic₉	1	2	yes	complemented
139	C₂×C₄⋊C₄	—	18	1	no	—
140	C₂×C₄²	—	18	1	no	—
141	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
142	C₄⋊C₈	—	18	1	no	—
143	C₂²⋊C₈	—	9	2	no	—
144	C₄×C₈	—	18	1	no	—
145	#95	—	9	2	no	—
146	C₂³×C₃	—	2	2	no	—
147	C₂³×C₃	C₂×#67	1	1	yes	characteristic
148	C₃×Q₈	—	4	2	no	—
149	C₃×D₄	—	4	2	no	—
150	C₃×D₄	—	4	2	no	—
151	C₃×D₄	—	4	2	no	—
152	C₃×D₄	—	4	2	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	#217	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
166	C₂×Dic₃	—	6	4	no	—
167	C₂×Dic₃	—	12	2	no	—
168	C₂×Dic₃	—	12	2	no	—
169	C₃⋊C₈	—	12	2	no	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	—	2	1	no	—
172	C₂×C₉	—	8	2	no	—
173	C₂×C₉	#410	1	1	yes	characteristic
174	C₂×C₉	C₂×#102	1	1	yes	characteristic
175	C₂×C₉	C₂×#104	1	1	yes	characteristic
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	C₂×Dic₉	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	18	2	no	—
185	C₂²×C₄	#611	1	1	yes	characteristic
186	C₂²×C₄	#611	1	1	yes	characteristic
187	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
188	C₂×C₈	—	18	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	18	4	no	—
191	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₄²	—	36	2	no	—
195	C₄²	C₂×Dic₉	1	1	yes	characteristic
196	C₄²	C₂×Dic₉	1	1	yes	characteristic
197	C₄²	#611	1	1	yes	characteristic
198	C₄²	#611	1	1	yes	characteristic
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	4	2	no	—
202	C₂²×C₃	—	4	2	no	—
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	C₂×#213	1	1	yes	characteristic
205	C₂²×C₃	C₂×#214	1	1	yes	characteristic
206	C₂²×C₃	C₂×#217	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	4	2	no	—
211	C₃×C₄	#832	1	2	yes	—
212	C₃×C₄	#832	1	2	yes	—
213	C₃×C₄	#837	1	2	yes	—
214	Dic₃	—	12	4	no	—
215	C₉	#1502	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	2	2	no	—
217	C₂³	C₂×#611	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	18	4	no	—
229	C₂×C₄	—	36	2	no	—
230	C₂×C₄	—	36	2	no	—
231	C₂×C₄	#3672	1	2	yes	—
232	C₂×C₄	#3673	1	2	yes	—
233	C₂×C₄	#3676	1	1	yes	characteristic
234	C₂×C₄	#3676	1	1	yes	characteristic
235	C₂×C₄	#3676	1	1	yes	characteristic
236	C₂×C₄	#3676	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
238	C₂×C₄	C₂×#611	1	1	yes	characteristic
239	C₈	—	36	2	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	8	2	no	—
243	C₂×C₃	C₂×#832	1	1	yes	characteristic
244	C₂×C₃	C₂×#837	1	1	yes	characteristic
245	C₂×C₃	#5240	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	C₂×#3672	1	1	yes	characteristic
252	C₂²	C₂×#3673	1	1	yes	characteristic
253	C₂²	C₂×#3676	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	36	4	no	—
259	C₄	#63392	1	2	yes	—
260	C₄	#63392	1	2	yes	—
261	C₄	#63396	1	2	yes	—
262	C₃	#71811	1	1	yes	characteristic
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	8	2	no	—
266	C₂	C₂×#63392	1	1	yes	characteristic
267	C₂	C₂×#63396	1	1	yes	characteristic
268	C₂	#10588920	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3676
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#532	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₃×#532	D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#532	D₉
hypocenter	C₉	#1502
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. C₄×C₄⋊Dic₉	C₂	semidirect
3. #10588605	C₂	semidirect
4. #10588214	C₂	semidirect
7. C₉×C₄⋊Q₈	C₄	semidirect
8. C₉×#127	C₄	semidirect
13. C₄×D₄×C₉	C₄	semidirect
132. C₄⋊Q₈	Dic₉	semidirect
133. #127	Dic₉	semidirect
138. C₄×D₄	Dic₉	semidirect
215. C₉	#1502	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
C₄×C₄⋊Dic₉	2
#10588605	2
#10588214	2
#71811	3