Subgroups — #11895459

Order 1152.

504 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10589669	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	#10588862	C₂	1	2	yes	maximal
5	#10588707	C₂	1	1	yes	maximal, characteristic
6	#10588703	C₂	1	1	yes	maximal, characteristic
7	#10588555	C₂	1	2	yes	maximal
8	#10588472	C₂	1	2	yes	maximal
9	#10588470	C₂	1	2	yes	maximal
10	#10588361	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₁₈	C₂	1	1	yes	maximal, characteristic
12	#79699	—	3	1	no	maximal
13	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
16	Q₈×Dic₉	C₂²	1	2	yes	—
17	#63506	C₂²	1	2	yes	—
18	C₂×#3674	C₂²	1	2	yes	—
19	#63451	—	2	2	no	—
20	#63450	—	2	2	no	—
21	#63450	C₂²	1	2	yes	—
22	#63449	C₂²	1	2	yes	—
23	#63432	—	2	1	no	—
24	#63431	—	2	1	no	—
25	#63431	C₂²	1	1	yes	characteristic
26	#63431	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₈	—	2	1	no	—
28	C₄×Dic₁₈	—	2	1	no	—
29	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
30	C₉×#107	C₂²	1	1	yes	characteristic
31	C₉×#103	C₂²	1	1	yes	characteristic
32	C₉×#103	C₂²	1	1	yes	characteristic
33	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
34	#63395	C₂²	1	2	yes	—
35	#63376	C₂²	1	2	yes	—
36	#63376	C₂²	1	2	yes	—
37	Dic₉⋊C₈	C₂²	1	2	yes	—
38	C₈×Dic₉	C₂²	1	2	yes	—
39	#63368	C₂²	1	2	yes	—
40	#63366	C₂²	1	2	yes	—
41	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
42	#5567	—	3	1	no	—
43	Q₈×Dic₆	—	3	1	no	—
44	C₃×C₄×Q₁₆	D₃	1	1	yes	characteristic
45	#5182	—	3	2	no	—
46	#5031	—	3	1	no	—
47	#5021	—	3	1	no	—
48	#4875	—	3	2	no	—
49	#4792	—	3	2	no	—
50	#4790	—	3	2	no	—
51	#4681	—	3	1	no	—
52	C₈×Dic₆	—	3	1	no	—
53	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₈	—	2	1	no	—
56	C₂×Dic₁₈	—	2	2	no	—
57	C₂×Dic₁₈	D₄	1	1	yes	characteristic
58	C₂×Dic₁₈	C₂³	1	2	yes	—
59	C₉×Q₁₆	—	2	2	no	—
60	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
62	C₉×C₄⋊C₄	—	2	1	no	—
63	C₉×C₄⋊C₄	—	2	1	no	—
64	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₄²×C₉	—	2	1	no	—
67	C₄²×C₉	—	2	1	no	—
68	C₄²×C₉	C₂³	1	1	yes	characteristic
69	#3674	—	2	4	no	—
70	C₄⋊Dic₉	—	2	1	no	—
71	C₄⋊Dic₉	—	2	1	no	—
72	C₄⋊Dic₉	—	2	1	no	—
73	C₄⋊Dic₉	—	2	2	no	—
74	C₄⋊Dic₉	D₄	1	1	yes	characteristic
75	C₄⋊Dic₉	C₂³	1	2	yes	—
76	Dic₉⋊C₄	—	2	2	no	—
77	Dic₉⋊C₄	—	2	2	no	—
78	Dic₉⋊C₄	—	2	2	no	—
79	Dic₉⋊C₄	—	2	2	no	—
80	Dic₉⋊C₄	D₄	1	2	yes	—
81	C₄×Dic₉	—	2	2	no	—
82	C₄×Dic₉	—	2	2	no	—
83	C₄×Dic₉	C₂³	1	2	yes	—
84	C₂×C₉⋊C₈	C₂³	1	2	yes	—
85	#3363	—	9	1	no	Sylow
86	C₂×C₃×Q₁₆	C₂×D₃	1	1	yes	characteristic
87	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
88	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
89	Q₈×Dic₃	—	3	2	no	—
90	#944	—	3	2	no	—
91	C₂×#216	—	3	2	no	—
92	#889	—	6	2	no	—
93	#888	—	3	2	no	—
94	#888	—	6	2	no	—
95	#887	—	3	2	no	—
96	#870	—	6	1	no	—
97	#869	—	3	1	no	—
98	#869	—	3	1	no	—
99	#869	—	6	1	no	—
100	C₄×Dic₆	—	3	1	no	—
101	C₄×Dic₆	—	6	1	no	—
102	C₄×Dic₆	—	6	1	no	—
103	C₃×#107	C₂×D₃	1	1	yes	characteristic
104	C₃×#103	C₂×D₃	1	1	yes	characteristic
105	C₃×#103	C₂×D₃	1	1	yes	characteristic
106	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
107	#835	—	3	2	no	—
108	#816	—	3	2	no	—
109	#816	—	3	2	no	—
110	Dic₃⋊C₈	—	3	2	no	—
111	C₈×Dic₃	—	3	2	no	—
112	#810	—	3	2	no	—
113	#807	—	3	2	no	—
114	C₄⋊(C₃⋊C₈)	—	3	1	no	—
115	Q₈×C₉	—	2	1	no	—
116	Q₈×C₉	—	2	1	no	—
117	Q₈×C₉	—	2	1	no	—
118	Q₈×C₉	#41	1	2	yes	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₉	—	2	2	no	—
127	C₂×Dic₉	—	2	2	no	—
128	C₂×Dic₉	C₂×D₄	1	2	yes	—
129	C₂×Dic₉	C₂×D₄	1	2	yes	—
130	Dic₁₈	—	2	2	no	—
131	Dic₁₈	—	2	2	no	—
132	Dic₁₈	—	8	2	no	—
133	Dic₁₈	—	8	1	no	—
134	Dic₁₈	Q₁₆	1	4	yes	complemented
135	C₈×C₉	—	2	1	no	—
136	C₈×C₉	—	2	1	no	—
137	C₉⋊C₈	—	2	2	no	—
138	Q₈²	—	9	1	no	—
139	#557	—	9	1	no	—
140	#487	—	9	2	no	—
141	#487	—	9	1	no	—
142	#477	—	9	1	no	—
143	#477	—	9	2	no	—
144	#462	—	9	2	no	—
145	#462	—	9	1	no	—
146	C₈×Q₈	—	9	1	no	—
147	C₄×Q₁₆	—	9	2	no	—
148	C₄×Q₁₆	D₉	1	1	yes	characteristic
149	C₂×C₃×Q₈	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₃×Q₈	C₂²×D₃	1	1	yes	characteristic
151	C₂×Dic₆	—	3	1	no	—
152	C₂×Dic₆	—	3	2	no	—
153	C₂×Dic₆	—	6	2	no	—
154	C₂×Dic₆	—	6	1	no	—
155	C₃×Q₁₆	—	2	2	no	—
156	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
157	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
158	C₃×C₄⋊C₄	—	2	1	no	—
159	C₃×C₄⋊C₄	—	2	1	no	—
160	C₃×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
161	C₃×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
162	C₃×C₄²	—	2	1	no	—
163	C₃×C₄²	—	2	1	no	—
164	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
165	#216	—	6	4	no	—
166	C₄⋊Dic₃	—	3	1	no	—
167	C₄⋊Dic₃	—	3	2	no	—
168	C₄⋊Dic₃	—	6	2	no	—
169	C₄⋊Dic₃	—	6	1	no	—
170	C₄⋊Dic₃	—	6	1	no	—
171	C₄⋊Dic₃	—	6	1	no	—
172	Dic₃⋊C₄	—	3	2	no	—
173	Dic₃⋊C₄	—	6	2	no	—
174	Dic₃⋊C₄	—	6	2	no	—
175	Dic₃⋊C₄	—	6	2	no	—
176	Dic₃⋊C₄	—	6	2	no	—
177	C₄×Dic₃	—	3	2	no	—
178	C₄×Dic₃	—	6	2	no	—
179	C₄×Dic₃	—	6	2	no	—
180	C₂×C₃⋊C₈	—	3	2	no	—
181	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
182	C₄×C₉	—	2	2	no	—
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	4	1	no	—
185	C₄×C₉	—	4	1	no	—
186	C₄×C₉	—	4	1	no	—
187	C₄×C₉	C₂×Q₁₆	1	2	yes	—
188	C₄×C₉	C₂×#41	1	1	yes	characteristic
189	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
190	Dic₉	—	2	2	no	—
191	Dic₉	—	4	2	no	—
192	Dic₉	—	4	2	no	—
193	Dic₉	C₂×Q₁₆	1	4	yes	—
194	C₂×Q₁₆	—	9	2	no	—
195	C₂×Q₁₆	C₂×D₉	1	1	yes	characteristic
196	C₄⋊Q₈	—	9	2	no	—
197	C₄⋊Q₈	—	9	2	no	—
198	C₄⋊Q₈	—	9	1	no	—
199	C₄⋊Q₈	—	9	1	no	—
200	C₄⋊Q₈	—	18	1	no	—
201	C₄⋊Q₈	—	18	2	no	—
202	#125	—	18	1	no	—
203	#125	—	18	2	no	—
204	C₄×Q₈	—	9	1	no	—
205	C₄×Q₈	—	9	2	no	—
206	C₄×Q₈	—	9	2	no	—
207	C₄×Q₈	—	18	1	no	—
208	C₄×Q₈	—	18	1	no	—
209	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
211	#107	—	9	2	no	—
212	#107	C₂×D₉	1	1	yes	characteristic
213	C₄⋊C₈	—	9	2	no	—
214	C₄⋊C₈	—	9	1	no	—
215	#103	—	9	2	no	—
216	#103	—	9	2	no	—
217	#103	—	9	2	no	—
218	#103	—	9	2	no	—
219	#103	C₂×D₉	1	1	yes	characteristic
220	#103	C₂×D₉	1	1	yes	characteristic
221	C₄×C₈	—	9	2	no	—
222	C₄×C₈	C₂×D₉	1	1	yes	characteristic
223	C₃×Q₈	—	2	1	no	—
224	C₃×Q₈	—	2	1	no	—
225	C₃×Q₈	—	2	1	no	—
226	C₃×Q₈	#235	1	2	yes	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
234	C₂×Dic₃	—	3	2	no	—
235	C₂×Dic₃	—	3	2	no	—
236	C₂×Dic₃	—	6	2	no	—
237	C₂×Dic₃	—	6	2	no	—
238	Dic₆	—	3	4	no	—
239	Dic₆	—	6	2	no	—
240	Dic₆	—	6	2	no	—
241	Dic₆	—	24	2	no	—
242	Dic₆	—	24	1	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	—	2	1	no	—
245	C₃⋊C₈	—	6	2	no	—
246	C₂×C₉	#548	1	1	yes	characteristic
247	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
248	C₂×C₉	#577	1	1	yes	characteristic
249	C₂×Q₈	—	9	1	no	—
250	C₂×Q₈	—	9	2	no	—
251	C₂×Q₈	—	18	2	no	—
252	C₂×Q₈	—	18	1	no	—
253	C₂×Q₈	C₂²×D₉	1	1	yes	characteristic
254	C₂×Q₈	C₂²×D₉	1	1	yes	characteristic
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	18	4	no	—
257	C₂×C₈	—	9	2	no	—
258	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
259	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	9	2	no	—
263	C₄⋊C₄	—	9	1	no	—
264	C₄⋊C₄	—	9	2	no	—
265	C₄⋊C₄	—	18	2	no	—
266	C₄⋊C₄	—	18	2	no	—
267	C₄⋊C₄	—	18	2	no	—
268	C₄⋊C₄	—	18	2	no	—
269	C₄⋊C₄	—	18	2	no	—
270	C₄⋊C₄	—	18	1	no	—
271	C₄⋊C₄	—	18	1	no	—
272	C₄⋊C₄	—	18	1	no	—
273	C₄⋊C₄	C₂²×D₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₉	1	1	yes	characteristic
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	9	2	no	—
278	C₄²	—	18	2	no	—
279	C₄²	—	18	2	no	—
280	C₄²	C₂²×D₉	1	1	yes	characteristic
281	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	4	1	no	—
285	C₃×C₄	—	4	1	no	—
286	C₃×C₄	—	4	1	no	—
287	C₃×C₄	D₃×Q₁₆	1	2	yes	—
288	C₃×C₄	C₂×#235	1	1	yes	characteristic
289	C₃×C₄	#1010	1	1	yes	characteristic
290	Dic₃	—	3	4	no	—
291	Dic₃	—	6	2	no	—
292	Dic₃	—	12	2	no	—
293	Dic₃	—	12	2	no	—
294	C₉	#3363	1	1	yes	characteristic, complemented, Sylow
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	9	4	no	—
299	Q₈	—	18	2	no	—
300	Q₈	—	18	2	no	—
301	Q₈	—	72	1	no	—
302	Q₈	—	72	2	no	—
303	Q₈	#3697	1	2	yes	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	9	2	no	—
309	C₂×C₄	—	9	2	no	—
310	C₂×C₄	—	18	2	no	—
311	C₂×C₄	—	18	2	no	—
312	C₂×C₄	D₄×D₉	1	1	yes	characteristic
313	C₂×C₄	D₄×D₉	1	1	yes	characteristic
314	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	18	2	no	—
318	C₂×C₃	#5552	1	1	yes	characteristic
319	C₂×C₃	C₂×D₃×Q₁₆	1	1	yes	characteristic
320	C₂×C₃	#5770	1	1	yes	characteristic
321	C₂²	C₂×D₄×D₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	9	4	no	—
328	C₄	—	18	2	no	—
329	C₄	—	36	2	no	—
330	C₄	—	36	2	no	—
331	C₄	Q₁₆×D₉	1	2	yes	—
332	C₄	C₂×#3697	1	1	yes	characteristic
333	C₄	#63716	1	1	yes	characteristic
334	C₃	#79699	1	1	yes	characteristic
335	C₂	#10589654	1	1	yes	characteristic
336	C₂	C₂×Q₁₆×D₉	1	1	yes	characteristic
337	C₂	#10589872	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₉
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₄×C₉×Q₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₃×D₄
hypercenter	C₄×Q₁₆	D₉
hypocenter	C₉	#3363
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
134. Dic₁₈	Q₁₆	semidirect
294. C₉	#3363	semidirect

Maximal subgroups

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

subgroup	index
#10589669	2
Q₈×Dic₁₈	2
C₄×C₉×Q₁₆	2
#10588862	2
#10588707	2
#10588703	2
#10588555	2
#10588472	2
#10588470	2
#10588361	2
C₈×Dic₁₈	2
#79699	3