Subgroups — #11791397

Order 1152.

504 conjugacy classes of subgroups.

The subgroup lattice has 279 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×#63396	C₂	1	4	yes	maximal, complemented
3	C₄×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#3667	C₂	1	1	yes	maximal, characteristic, complemented
5	#71818	—	3	1	no	maximal
6	C₂×C₉×#41	C₂²	1	2	yes	—
7	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	C₉×#127	C₄	1	1	yes	characteristic, complemented
9	C₉×#124	C₄	1	2	yes	complemented
10	C₉×#121	—	2	1	no	—
11	C₉×#121	—	2	1	no	—
12	C₄×D₄×C₉	—	2	1	no	—
13	C₄×D₄×C₉	—	2	1	no	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₉	—	2	1	no	—
16	C₂×C₄×Dic₉	C₂²	1	2	yes	—
17	C₂×#3667	C₂²	1	2	yes	—
18	#63396	—	2	8	no	—
19	C₉⋊#95	—	2	1	no	—
20	#63362	—	2	1	no	—
21	C₄×C₉⋊C₈	—	2	1	no	—
22	C₃×#532	D₃	1	1	yes	characteristic
23	C₂×#837	—	3	4	no	—
24	C₄×C₄⋊Dic₃	—	3	1	no	—
25	C₄×#208	—	3	1	no	—
26	C₉×#41	—	2	4	no	—
27	C₉×#41	—	2	4	no	—
28	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	—	2	1	no	—
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₉	—	2	2	no	—
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₉	—	2	2	no	—
38	C₉×C₄⋊C₄	—	2	1	no	—
39	C₉×C₄⋊C₄	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₉	D₄	1	1	yes	characteristic
45	C₄²×C₉	D₄	1	1	yes	characteristic
46	C₄⋊Dic₉	—	2	4	no	—
47	C₄×Dic₉	—	2	4	no	—
48	#3667	—	2	2	no	—
49	#3667	D₄	1	4	yes	complemented
50	C₂×C₉⋊C₈	—	2	2	no	—
51	#1980	—	9	1	no	Sylow
52	C₂×C₃×#41	C₂×D₃	1	2	yes	—
53	C₃×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
54	C₃×#127	Dic₃	1	1	yes	characteristic
55	C₃×#124	Dic₃	1	2	yes	—
56	C₃×#121	—	2	1	no	—
57	C₃×#121	—	2	1	no	—
58	C₃×C₄×D₄	—	2	1	no	—
59	C₃×C₄×D₄	—	2	1	no	—
60	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
61	C₂×C₄⋊Dic₃	—	6	1	no	—
62	C₂×C₄×Dic₃	—	3	2	no	—
63	C₂×#208	—	3	2	no	—
64	#837	—	6	8	no	—
65	C₃⋊#95	—	6	1	no	—
66	#803	—	6	1	no	—
67	C₄×C₃⋊C₈	—	6	1	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₉	—	2	4	no	—
71	D₄×C₉	—	2	4	no	—
72	D₄×C₉	—	4	2	no	—
73	D₄×C₉	—	4	2	no	—
74	D₄×C₉	—	4	2	no	—
75	D₄×C₉	—	4	2	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	4	1	no	—
79	C₂×C₄×C₉	—	4	4	no	—
80	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₉	#41	1	2	yes	—
88	C₂×Dic₉	—	2	4	no	—
89	C₂×Dic₉	—	4	2	no	—
90	C₂×Dic₉	—	4	2	no	—
91	C₉⋊C₈	—	2	4	no	—
92	#532	D₉	1	1	yes	characteristic
93	C₂×#104	—	9	4	no	—
94	C₄×M₁₆	—	9	1	no	—
95	C₄×C₄⋊C₄	—	9	1	no	—
96	C₃×#41	—	2	4	no	—
97	C₃×#41	—	2	4	no	—
98	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
99	C₂×C₃×D₄	—	2	2	no	—
100	C₂×C₃×D₄	—	2	1	no	—
101	C₂×C₃×D₄	—	2	1	no	—
102	C₂×C₃×D₄	C₂×Dic₃	1	2	yes	—
103	C₂²×C₃×C₄	—	2	2	no	—
104	C₂²×C₃×C₄	#67	1	1	yes	characteristic
105	C₂²×C₃×C₄	#67	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂²×Dic₃	—	6	2	no	—
108	C₃×C₄⋊C₄	—	2	1	no	—
109	C₃×C₄⋊C₄	—	2	1	no	—
110	C₃×C₂²⋊C₄	—	2	2	no	—
111	C₃×C₂²⋊C₄	—	2	2	no	—
112	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
113	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
114	C₃×C₄²	#67	1	1	yes	characteristic
115	C₃×C₄²	#67	1	1	yes	characteristic
116	C₄⋊Dic₃	—	6	4	no	—
117	C₄×Dic₃	—	6	4	no	—
118	#208	—	3	4	no	—
119	#208	—	6	2	no	—
120	C₂×C₃⋊C₈	—	6	2	no	—
121	C₂²×C₉	—	2	1	no	—
122	C₂²×C₉	—	2	1	no	—
123	C₂²×C₉	—	4	2	no	—
124	C₂²×C₉	—	4	2	no	—
125	C₂²×C₉	—	4	2	no	—
126	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₉	#124	1	1	yes	characteristic
128	C₂²×C₉	#127	1	1	yes	characteristic
129	C₄×C₉	—	2	1	no	—
130	C₄×C₉	—	2	1	no	—
131	C₄×C₉	—	2	1	no	—
132	C₄×C₉	—	4	2	no	—
133	C₄×C₉	#104	1	2	yes	complemented
134	C₄×C₉	C₄×D₄	1	2	yes	—
135	C₄×C₉	#121	1	2	yes	—
136	Dic₉	—	4	4	no	—
137	C₂×#41	C₂×D₉	1	2	yes	—
138	C₂×M₁₆	—	9	2	no	—
139	C₄⋊Q₈	Dic₉	1	1	yes	characteristic, complemented
140	#127	Dic₉	1	1	yes	characteristic, complemented
141	#124	Dic₉	1	2	yes	complemented
142	#121	—	2	1	no	—
143	#121	—	2	1	no	—
144	C₄×D₄	—	2	1	no	—
145	C₄×D₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	18	1	no	—
147	C₂×C₄²	—	9	2	no	—
148	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
149	#104	—	18	8	no	—
150	#97	—	18	1	no	—
151	C₄×C₈	—	18	1	no	—
152	#95	—	18	1	no	—
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	C₂×#67	1	1	yes	characteristic
155	C₃×Q₈	—	2	4	no	—
156	C₃×D₄	—	2	4	no	—
157	C₃×D₄	—	4	2	no	—
158	C₃×D₄	—	4	2	no	—
159	C₃×D₄	—	4	2	no	—
160	C₃×D₄	—	4	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	—	4	4	no	—
165	C₂×C₃×C₄	#217	1	1	yes	characteristic
166	C₂×C₃×C₄	#217	1	1	yes	characteristic
167	C₂×C₃×C₄	#217	1	1	yes	characteristic
168	C₂×C₃×C₄	#217	1	1	yes	characteristic
169	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
170	C₂×C₃×C₄	#237	1	2	yes	—
171	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
173	C₂×Dic₃	—	6	4	no	—
174	C₂×Dic₃	—	12	2	no	—
175	C₂×Dic₃	—	12	2	no	—
176	C₃⋊C₈	—	6	4	no	—
177	C₂×C₉	—	2	1	no	—
178	C₂×C₉	—	2	1	no	—
179	C₂×C₉	—	8	2	no	—
180	C₂×C₉	#390	1	1	yes	characteristic
181	C₂×C₉	C₂×#104	1	1	yes	characteristic
182	C₂×C₉	#421	1	1	yes	characteristic
183	#41	—	2	4	no	—
184	#41	—	2	4	no	—
185	C₂×Q₈	C₂×Dic₉	1	2	yes	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	C₂×Dic₉	1	2	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	18	2	no	—
192	C₂²×C₄	#611	1	1	yes	characteristic
193	C₂²×C₄	#611	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
195	M₁₆	—	9	4	no	—
196	M₁₆	—	18	2	no	—
197	C₂×C₈	—	18	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	18	4	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₄²	—	18	4	no	—
204	C₄²	C₂×Dic₉	1	1	yes	characteristic
205	C₄²	C₂×Dic₉	1	1	yes	characteristic
206	C₄²	#611	1	1	yes	characteristic
207	C₄²	#611	1	1	yes	characteristic
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	4	2	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	#936	1	1	yes	characteristic
214	C₂²×C₃	#940	1	1	yes	characteristic
215	C₂²×C₃	C₂×#217	1	1	yes	characteristic
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	#837	1	2	yes	—
221	C₃×C₄	D₄×Dic₃	1	2	yes	—
222	C₃×C₄	#938	1	2	yes	—
223	Dic₃	—	12	4	no	—
224	C₉	#1980	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	2	no	—
226	C₂³	C₂×#611	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	18	4	no	—
238	C₂×C₄	—	36	2	no	—
239	C₂×C₄	—	36	2	no	—
240	C₂×C₄	#3676	1	1	yes	characteristic
241	C₂×C₄	#3676	1	1	yes	characteristic
242	C₂×C₄	#3676	1	1	yes	characteristic
243	C₂×C₄	#3676	1	1	yes	characteristic
244	C₂×C₄	D₄×D₉	1	2	yes	—
245	C₂×C₄	#3699	1	2	yes	—
246	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
247	C₂×C₄	C₂×#611	1	1	yes	characteristic
248	C₈	—	18	4	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	8	2	no	—
252	C₂×C₃	#5221	1	1	yes	characteristic
253	C₂×C₃	C₂×#837	1	1	yes	characteristic
254	C₂×C₃	#5236	1	1	yes	characteristic
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#63498	1	1	yes	characteristic
261	C₂²	#63502	1	1	yes	characteristic
262	C₂²	C₂×#3676	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	36	4	no	—
268	C₄	#63396	1	2	yes	complemented
269	C₄	D₄×Dic₉	1	2	yes	—
270	C₄	#63500	1	2	yes	—
271	C₃	#71818	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	2	no	—
275	C₂	#10588901	1	1	yes	characteristic
276	C₂	C₂×#63396	1	1	yes	characteristic
277	C₂	#10588914	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
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	#1980
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×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. C₂×#63396	C₂	semidirect
3. C₄×C₄⋊Dic₉	C₂	semidirect
4. C₄×#3667	C₂	semidirect
7. C₉×C₄⋊Q₈	C₄	semidirect
8. C₉×#127	C₄	semidirect
9. C₉×#124	C₄	semidirect
49. #3667	D₄	semidirect
133. C₄×C₉	#104	semidirect
139. C₄⋊Q₈	Dic₉	semidirect
140. #127	Dic₉	semidirect
141. #124	Dic₉	semidirect
224. C₉	#1980	semidirect
268. C₄	#63396	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×#63396	2
C₄×C₄⋊Dic₉	2
C₄×#3667	2
#71818	3