Subgroups — #11791675

Order 1152.

450 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63450	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
3	#10588655	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	1	yes	maximal, characteristic
5	#10588229	C₂	1	1	yes	maximal, characteristic
6	#10588206	C₂	1	1	yes	maximal, characteristic
7	#10588206	C₂	1	1	yes	maximal, characteristic
8	#68803	—	3	1	no	maximal
9	C₂²×Dic₁₈	—	2	1	no	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₉⋊C₄	—	2	1	no	—
16	C₂×Dic₉⋊C₄	—	2	1	no	—
17	C₂×Dic₉⋊C₄	—	2	1	no	—
18	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
19	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	#63450	—	2	2	no	—
21	#63450	C₄	1	4	yes	complemented
22	C₈×Dic₉	—	2	2	no	—
23	C₂×#888	—	3	1	no	—
24	C₂×C₈×Dic₃	—	3	1	no	—
25	#4975	—	3	1	no	—
26	C₃×#340	D₃	1	1	yes	characteristic
27	#4551	—	3	1	no	—
28	#4529	—	3	1	no	—
29	#4529	—	3	1	no	—
30	C₂²×C₄×C₉	—	2	1	no	—
31	C₂²×C₄×C₉	—	2	1	no	—
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂²×Dic₉	—	2	1	no	—
34	C₂²×Dic₉	—	2	1	no	—
35	C₂²×Dic₉	D₄	1	1	yes	characteristic
36	C₂²×Dic₉	D₄	1	1	yes	characteristic
37	C₂×Dic₁₈	—	2	4	no	—
38	C₂×Dic₁₈	—	4	2	no	—
39	C₂×Dic₁₈	—	4	2	no	—
40	C₂×C₈×C₉	—	2	1	no	—
41	C₂×C₈×C₉	—	2	1	no	—
42	C₂×C₈×C₉	Q₈	1	2	yes	complemented
43	C₉×C₄⋊C₄	—	2	1	no	—
44	C₉×C₄⋊C₄	—	2	1	no	—
45	C₉×C₄⋊C₄	—	2	1	no	—
46	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₉	—	2	1	no	—
48	C₄⋊Dic₉	—	2	1	no	—
49	C₄⋊Dic₉	—	2	1	no	—
50	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
51	Dic₉⋊C₄	—	2	4	no	—
52	Dic₉⋊C₄	—	4	2	no	—
53	Dic₉⋊C₄	—	4	2	no	—
54	C₄×Dic₉	C₂×C₄	1	2	yes	—
55	C₄×Dic₉	D₄	1	2	yes	—
56	C₂×C₉⋊C₈	—	2	1	no	—
57	C₂×C₉⋊C₈	—	2	1	no	—
58	C₂×C₉⋊C₈	Q₈	1	2	yes	complemented
59	#1990	—	9	1	no	Sylow
60	C₂²×Dic₆	—	6	1	no	—
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
63	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
64	C₂×C₄⋊Dic₃	—	3	1	no	—
65	C₂×C₄⋊Dic₃	—	3	1	no	—
66	C₂×Dic₃⋊C₄	—	6	1	no	—
67	C₂×Dic₃⋊C₄	—	6	1	no	—
68	C₂×Dic₃⋊C₄	—	6	1	no	—
69	C₂×C₄×Dic₃	—	3	1	no	—
70	C₂²×C₃⋊C₈	—	3	1	no	—
71	#888	—	3	4	no	—
72	#888	—	6	2	no	—
73	C₈×Dic₃	—	6	2	no	—
74	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	4	1	no	—
77	C₂×C₄×C₉	—	4	1	no	—
78	C₂×C₄×C₉	—	4	1	no	—
79	C₂×C₄×C₉	—	4	1	no	—
80	C₂×C₄×C₉	—	4	1	no	—
81	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₉	#41	1	1	yes	characteristic
86	C₂×C₄×C₉	#41	1	1	yes	characteristic
87	C₂×Dic₉	—	2	2	no	—
88	C₂×Dic₉	—	4	1	no	—
89	C₂×Dic₉	—	4	1	no	—
90	C₂×Dic₉	—	4	1	no	—
91	C₂×Dic₉	—	4	1	no	—
92	C₂×Dic₉	—	4	1	no	—
93	C₂×Dic₉	C₂²⋊C₄	1	2	yes	—
94	C₂×Dic₉	C₂²⋊C₄	1	2	yes	—
95	C₂×Dic₉	D₈	1	2	yes	—
96	C₂×Dic₉	SD₁₆	1	2	yes	—
97	C₂×Dic₉	SD₁₆	1	2	yes	—
98	C₂×Dic₉	Q₁₆	1	2	yes	—
99	Dic₁₈	—	4	8	no	—
100	C₈×C₉	—	2	2	no	—
101	C₉⋊C₈	—	2	2	no	—
102	C₂×C₄⋊Q₈	—	9	1	no	—
103	C₂×C₄×C₈	—	9	1	no	—
104	#384	—	9	1	no	—
105	#340	—	9	1	no	—
106	#340	—	9	1	no	—
107	#340	—	9	1	no	—
108	#340	D₉	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂²×Dic₃	—	3	1	no	—
113	C₂²×Dic₃	—	3	1	no	—
114	C₂²×Dic₃	—	6	1	no	—
115	C₂²×Dic₃	—	6	1	no	—
116	C₂×Dic₆	—	6	4	no	—
117	C₂×Dic₆	—	12	2	no	—
118	C₂×Dic₆	—	12	2	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	Dic₆	1	2	yes	—
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	—	2	1	no	—
124	C₃×C₄⋊C₄	—	2	1	no	—
125	C₃×C₄⋊C₄	C₄×D₃	1	2	yes	—
126	C₄⋊Dic₃	—	3	2	no	—
127	C₄⋊Dic₃	—	6	1	no	—
128	C₄⋊Dic₃	—	6	1	no	—
129	C₄⋊Dic₃	—	6	1	no	—
130	Dic₃⋊C₄	—	6	4	no	—
131	Dic₃⋊C₄	—	12	2	no	—
132	Dic₃⋊C₄	—	12	2	no	—
133	C₄×Dic₃	—	3	2	no	—
134	C₄×Dic₃	—	3	2	no	—
135	C₂×C₃⋊C₈	—	3	2	no	—
136	C₂×C₃⋊C₈	—	6	1	no	—
137	C₂×C₃⋊C₈	—	6	1	no	—
138	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₉	#124	1	1	yes	characteristic
140	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
142	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
143	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
144	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
145	C₄×C₉	—	4	2	no	—
146	C₄×C₉	—	8	1	no	—
147	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
148	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
149	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
150	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
151	Dic₉	—	4	2	no	—
152	Dic₉	—	8	1	no	—
153	Dic₉	#102	1	4	yes	—
154	Dic₉	#103	1	4	yes	—
155	C₂²×Q₈	—	18	1	no	—
156	C₂²×C₈	—	9	1	no	—
157	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
158	C₄⋊Q₈	—	9	4	no	—
159	C₄⋊Q₈	—	18	2	no	—
160	C₂×C₄⋊C₄	—	9	1	no	—
161	C₂×C₄⋊C₄	—	9	1	no	—
162	C₂×C₄⋊C₄	—	18	1	no	—
163	C₂×C₄⋊C₄	—	18	1	no	—
164	C₂×C₄⋊C₄	—	18	1	no	—
165	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
167	C₂×C₄²	—	9	1	no	—
168	C₄×C₈	—	18	2	no	—
169	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	4	1	no	—
172	C₂×C₃×C₄	—	4	1	no	—
173	C₂×C₃×C₄	—	4	1	no	—
174	C₂×C₃×C₄	—	4	1	no	—
175	C₂×C₃×C₄	—	4	1	no	—
176	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	#235	1	1	yes	characteristic
179	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	#237	1	1	yes	characteristic
181	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
182	C₂×Dic₃	—	3	2	no	—
183	C₂×Dic₃	—	3	2	no	—
184	C₂×Dic₃	—	3	2	no	—
185	C₂×Dic₃	—	3	2	no	—
186	C₂×Dic₃	—	3	2	no	—
187	C₂×Dic₃	—	3	2	no	—
188	C₂×Dic₃	—	6	2	no	—
189	C₂×Dic₃	—	12	1	no	—
190	C₂×Dic₃	—	12	1	no	—
191	C₂×Dic₃	—	12	1	no	—
192	C₂×Dic₃	—	12	1	no	—
193	C₂×Dic₃	—	12	1	no	—
194	Dic₆	—	12	8	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃⋊C₈	—	6	2	no	—
197	C₂×C₉	#391	1	1	yes	characteristic
198	C₂×C₉	C₂×#102	1	1	yes	characteristic
199	C₂×C₉	C₂×#103	1	1	yes	characteristic
200	C₂×C₉	#486	1	1	yes	characteristic
201	C₂×C₉	#487	1	1	yes	characteristic
202	C₂×C₉	#498	1	1	yes	characteristic
203	C₂×C₉	#500	1	1	yes	characteristic
204	C₂×Q₈	—	18	4	no	—
205	C₂×Q₈	—	36	2	no	—
206	C₂×Q₈	—	36	2	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	9	1	no	—
210	C₂²×C₄	—	9	1	no	—
211	C₂²×C₄	—	18	1	no	—
212	C₂²×C₄	—	18	1	no	—
213	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	9	2	no	—
217	C₂×C₈	—	18	1	no	—
218	C₂×C₈	—	18	1	no	—
219	C₂×C₈	Dic₁₈	1	2	yes	complemented
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	9	2	no	—
224	C₄⋊C₄	—	18	4	no	—
225	C₄⋊C₄	—	18	1	no	—
226	C₄⋊C₄	—	18	1	no	—
227	C₄⋊C₄	—	18	1	no	—
228	C₄⋊C₄	—	36	2	no	—
229	C₄⋊C₄	—	36	2	no	—
230	C₄⋊C₄	C₄×D₉	1	2	yes	—
231	C₄²	—	9	2	no	—
232	C₄²	—	9	2	no	—
233	C₂²×C₃	D₃×C₂²⋊C₄	1	1	yes	characteristic
234	C₂²×C₃	#885	1	1	yes	characteristic
235	C₂²×C₃	#888	1	1	yes	characteristic
236	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
237	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
238	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
239	C₂²×C₃	D₃×Q₁₆	1	1	yes	characteristic
240	C₃×C₄	—	4	2	no	—
241	C₃×C₄	—	8	1	no	—
242	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
243	C₃×C₄	#878	1	1	yes	characteristic
244	C₃×C₄	#887	1	1	yes	characteristic
245	C₃×C₄	#896	1	1	yes	characteristic
246	Dic₃	—	3	4	no	—
247	Dic₃	—	3	4	no	—
248	Dic₃	—	12	2	no	—
249	Dic₃	—	24	1	no	—
250	C₉	#1990	1	1	yes	characteristic, complemented, Sylow
251	C₂³	D₄×D₉	1	1	yes	characteristic
252	Q₈	—	36	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	9	2	no	—
260	C₂×C₄	—	9	2	no	—
261	C₂×C₄	—	9	2	no	—
262	C₂×C₄	—	9	2	no	—
263	C₂×C₄	—	9	2	no	—
264	C₂×C₄	—	9	2	no	—
265	C₂×C₄	—	18	2	no	—
266	C₂×C₄	—	36	1	no	—
267	C₂×C₄	—	36	1	no	—
268	C₂×C₄	—	36	1	no	—
269	C₂×C₄	—	36	1	no	—
270	C₂×C₄	—	36	1	no	—
271	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
273	C₂×C₄	#3697	1	1	yes	characteristic
274	C₂×C₄	D₄×D₉	1	1	yes	characteristic
275	C₂×C₄	#3699	1	1	yes	characteristic
276	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	18	2	no	—
279	C₂×C₃	#4646	1	1	yes	characteristic
280	C₂×C₃	#4760	1	1	yes	characteristic
281	C₂×C₃	D₃×#102	1	1	yes	characteristic
282	C₂×C₃	#4792	1	1	yes	characteristic
283	C₂×C₃	D₃×#103	1	1	yes	characteristic
284	C₂×C₃	#4855	1	1	yes	characteristic
285	C₂×C₃	#4874	1	1	yes	characteristic
286	C₂²	C₂²⋊C₄×D₉	1	1	yes	characteristic
287	C₂²	#63447	1	1	yes	characteristic
288	C₂²	#63450	1	1	yes	characteristic
289	C₂²	D₈×D₉	1	1	yes	characteristic
290	C₂²	SD₁₆×D₉	1	1	yes	characteristic
291	C₂²	SD₁₆×D₉	1	1	yes	characteristic
292	C₂²	Q₁₆×D₉	1	1	yes	characteristic
293	C₄	—	4	2	no	—
294	C₄	—	8	1	no	—
295	C₄	—	9	4	no	—
296	C₄	—	9	4	no	—
297	C₄	—	36	2	no	—
298	C₄	—	72	1	no	—
299	C₄	C₄×Dic₁₈	1	1	yes	characteristic
300	C₄	#63440	1	1	yes	characteristic
301	C₄	#63449	1	1	yes	characteristic
302	C₄	#63458	1	1	yes	characteristic
303	C₃	#68803	1	1	yes	characteristic
304	C₂	#10588326	1	1	yes	characteristic
305	C₂	#10588440	1	1	yes	characteristic
306	C₂	D₉×#102	1	1	yes	characteristic
307	C₂	#10588472	1	1	yes	characteristic
308	C₂	D₉×#103	1	1	yes	characteristic
309	C₂	#10588535	1	1	yes	characteristic
310	C₂	#10588554	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#340	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#340	D₉
hypocenter	C₉	#1990
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
21. #63450	C₄	semidirect
42. C₂×C₈×C₉	Q₈	semidirect
58. C₂×C₉⋊C₈	Q₈	semidirect
219. C₂×C₈	Dic₁₈	semidirect
250. C₉	#1990	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63450	2
C₂×C₈×Dic₉	2
#10588655	2
C₉×#340	2
#10588229	2
#10588206	2
#68803	3