Subgroups — #11783411

Order 1152.

582 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3667	C₂	1	1	yes	maximal, characteristic
4	#10588234	C₂	1	4	yes	maximal, complemented
5	#71987	—	3	1	no	maximal
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₉×M₁₆	—	2	2	no	—
8	C₂×C₉×M₁₆	—	2	1	no	—
9	C₂×C₉×M₁₆	C₄	1	4	yes	complemented
10	C₂×C₉×M₁₆	C₂²	1	2	yes	—
11	C₂²×C₈×C₉	—	2	1	no	—
12	C₂×#3676	C₄	1	2	yes	—
13	#63488	—	2	1	no	—
14	#63488	—	2	1	no	—
15	#63488	—	2	1	no	—
16	#63488	—	2	1	no	—
17	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
18	C₂×C₄×Dic₉	C₄	1	2	yes	—
19	C₂×#3667	—	2	2	no	—
20	C₂×#3667	—	2	1	no	—
21	C₂×#3667	C₄	1	4	yes	complemented
22	C₂×#3667	C₂²	1	2	yes	—
23	C₂²×C₉⋊C₈	—	2	1	no	—
24	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
25	C₂×#926	—	3	1	no	—
26	C₂²×#208	—	3	1	no	—
27	#4554	—	3	4	no	—
28	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	D₄	1	2	yes	—
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₉	C₂×C₄	1	4	yes	—
40	C₉×M₁₆	—	2	4	no	—
41	C₉×M₁₆	—	2	2	no	—
42	C₉×M₁₆	C₂×C₄	1	4	yes	—
43	C₂×C₈×C₉	—	2	4	no	—
44	C₂×C₈×C₉	—	2	2	no	—
45	#3676	—	2	2	no	—
46	#3676	—	2	2	no	—
47	C₄⋊Dic₉	—	2	2	no	—
48	C₄⋊Dic₉	—	2	2	no	—
49	C₄×Dic₉	—	2	2	no	—
50	C₄×Dic₉	—	2	2	no	—
51	#3667	—	2	4	no	—
52	#3667	—	2	2	no	—
53	#3667	C₂×C₄	1	4	yes	—
54	C₂×C₉⋊C₈	—	2	4	no	—
55	C₂×C₉⋊C₈	—	2	2	no	—
56	#1742	—	9	1	no	Sylow
57	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×M₁₆	—	2	2	no	—
59	C₂×C₃×M₁₆	—	2	1	no	—
60	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
61	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
62	C₂²×C₃×C₈	—	2	1	no	—
63	C₂×#217	—	3	2	no	—
64	#926	—	6	1	no	—
65	#926	—	6	1	no	—
66	#926	—	6	1	no	—
67	#926	—	6	1	no	—
68	C₂×C₄⋊Dic₃	—	3	2	no	—
69	C₂×C₄×Dic₃	—	3	2	no	—
70	C₂×#208	—	3	4	no	—
71	C₂×#208	—	3	2	no	—
72	C₂×#208	—	6	1	no	—
73	C₂×#208	—	6	2	no	—
74	C₂²×C₃⋊C₈	—	6	1	no	—
75	C₂³×C₉	—	4	2	no	—
76	C₂³×C₉	C₄²	1	2	yes	—
77	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂³×C₉	C₄⋊C₄	1	2	yes	—
80	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₉	C₄²	1	2	yes	—
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	2	yes	—
85	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
99	C₂×Dic₉	—	2	4	no	—
100	C₂×Dic₉	—	4	4	no	—
101	C₈×C₉	—	2	4	no	—
102	C₉⋊C₈	—	2	4	no	—
103	C₂²×M₁₆	—	9	1	no	—
104	C₂²×M₁₆	D₉	1	1	yes	characteristic
105	C₂×#117	—	9	1	no	—
106	#343	—	9	4	no	—
107	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	Dic₆	1	2	yes	—
109	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
111	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
112	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
114	C₂²×C₃×C₄	#67	1	1	yes	characteristic
115	C₂²×C₃×C₄	#67	1	1	yes	characteristic
116	C₂²×C₃×C₄	#67	1	2	yes	—
117	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
118	C₂²×Dic₃	—	3	4	no	—
119	C₃×M₁₆	—	2	4	no	—
120	C₃×M₁₆	—	2	2	no	—
121	C₃×M₁₆	C₂×Dic₃	1	4	yes	—
122	C₂×C₃×C₈	—	2	4	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	#217	—	6	2	no	—
125	#217	—	6	2	no	—
126	C₄⋊Dic₃	—	6	2	no	—
127	C₄⋊Dic₃	—	6	2	no	—
128	C₄×Dic₃	—	6	2	no	—
129	C₄×Dic₃	—	6	2	no	—
130	#208	—	3	4	no	—
131	#208	—	6	4	no	—
132	#208	—	6	2	no	—
133	C₂×C₃⋊C₈	—	6	4	no	—
134	C₂×C₃⋊C₈	—	6	2	no	—
135	C₂²×C₉	—	2	2	no	—
136	C₂²×C₉	—	2	2	no	—
137	C₂²×C₉	—	2	2	no	—
138	C₂²×C₉	—	4	2	no	—
139	C₂²×C₉	—	4	2	no	—
140	C₂²×C₉	#95	1	2	yes	—
141	C₂²×C₉	#95	1	2	yes	—
142	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
143	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₉	#95	1	2	yes	—
146	C₄×C₉	#95	1	2	yes	—
147	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
151	Dic₉	—	4	4	no	—
152	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	9	4	no	—
156	C₂×M₁₆	—	9	2	no	—
157	C₂×M₁₆	—	18	1	no	—
158	C₂×M₁₆	—	18	2	no	—
159	C₂×M₁₆	Dic₉	1	4	yes	complemented
160	C₂×M₁₆	C₂×D₉	1	2	yes	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	18	1	no	—
163	#117	—	18	1	no	—
164	#117	—	18	1	no	—
165	#117	—	18	1	no	—
166	#117	—	18	1	no	—
167	C₂×C₄⋊C₄	—	9	2	no	—
168	C₂×C₂²⋊C₄	—	9	2	no	—
169	C₂×C₄²	—	9	2	no	—
170	C₂³×C₃	—	4	2	no	—
171	C₂³×C₃	C₄×Dic₃	1	2	yes	—
172	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
173	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
175	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
177	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
178	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
179	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
180	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
181	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
182	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
184	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
185	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₃×C₄	#217	1	2	yes	—
187	C₂×C₃×C₄	#217	1	2	yes	—
188	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
194	C₂×Dic₃	—	6	4	no	—
195	C₂×Dic₃	—	12	4	no	—
196	C₃×C₈	—	2	4	no	—
197	C₃⋊C₈	—	6	4	no	—
198	C₂×C₉	—	2	2	no	—
199	C₂×C₉	—	2	2	no	—
200	C₂×C₉	—	2	1	no	—
201	C₂×C₉	—	2	1	no	—
202	C₂×C₉	C₂×#95	1	1	yes	characteristic
203	C₂×C₉	#413	1	2	yes	—
204	C₂⁴	C₄×D₉	1	1	yes	characteristic
205	C₂²×C₄	—	9	4	no	—
206	C₂²×C₄	Dic₁₈	1	2	yes	—
207	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
208	C₂²×C₄	C₄×D₉	1	2	yes	—
209	C₂²×C₄	D₃₆	1	1	yes	characteristic
210	C₂²×C₄	D₃₆	1	1	yes	characteristic
211	C₂²×C₄	C₂×Dic₉	1	2	yes	—
212	C₂²×C₄	#611	1	1	yes	characteristic
213	C₂²×C₄	#611	1	1	yes	characteristic
214	C₂²×C₄	#611	1	2	yes	—
215	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	9	4	no	—
219	M₁₆	—	18	4	no	—
220	M₁₆	—	18	2	no	—
221	M₁₆	C₂×Dic₉	1	4	yes	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	18	4	no	—
225	C₂×C₈	—	18	2	no	—
226	C₄⋊C₄	—	18	2	no	—
227	C₄⋊C₄	—	18	2	no	—
228	C₂²⋊C₄	—	18	2	no	—
229	C₂²⋊C₄	—	18	2	no	—
230	C₄²	—	18	2	no	—
231	C₄²	—	18	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	4	2	no	—
236	C₂²×C₃	—	4	2	no	—
237	C₂²×C₃	C₃⋊#95	1	2	yes	—
238	C₂²×C₃	C₃⋊#95	1	2	yes	—
239	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
240	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
241	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
242	C₃×C₄	C₃⋊#95	1	2	yes	—
243	C₃×C₄	C₃⋊#95	1	2	yes	—
244	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
245	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
246	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
247	C₃×C₄	C₂×#217	1	1	yes	characteristic
248	Dic₃	—	12	4	no	—
249	C₉	#1742	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	4	2	no	—
251	C₂³	C₄×Dic₉	1	2	yes	—
252	C₂³	Dic₉⋊C₄	1	2	yes	—
253	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
254	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
255	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
256	C₂×C₄	—	18	4	no	—
257	C₂×C₄	—	36	4	no	—
258	C₂×C₄	C₄×Dic₉	1	2	yes	—
259	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
260	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
261	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
262	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
263	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
264	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
265	C₂×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
266	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
267	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
268	C₂×C₄	#3676	1	2	yes	—
269	C₂×C₄	#3676	1	2	yes	—
270	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
272	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
273	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
274	C₂×C₄	C₂×#611	1	1	yes	characteristic
275	C₂×C₄	C₂×#611	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	18	4	no	—
278	C₂×C₃	—	2	2	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	#5132	1	2	yes	—
283	C₂×C₃	C₂×C₃⋊#95	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	C₉⋊#95	1	2	yes	—
290	C₂²	C₉⋊#95	1	2	yes	—
291	C₂²	C₂×C₄×Dic₉	1	1	yes	characteristic
292	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
293	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
294	C₄	—	36	4	no	—
295	C₄	C₉⋊#95	1	2	yes	—
296	C₄	C₉⋊#95	1	2	yes	—
297	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
298	C₄	C₂×C₄⋊Dic₉	1	1	yes	characteristic
299	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
300	C₄	C₂×#3676	1	1	yes	characteristic
301	C₃	#71987	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	#10588814	1	2	yes	—
307	C₂	C₂×C₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₉)⋊C₄
commutator	C₂²×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂²×C₉×M₁₆	C₂
socle	C₂²×C₃	C₂×(C₂×D₃)⋊C₄
prime radical	C₂⁴×C₃	C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂²×M₁₆	D₉
hypocenter	C₉	#1742
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10588234	C₂	semidirect
9. C₂×C₉×M₁₆	C₄	semidirect
21. C₂×#3667	C₄	semidirect
159. C₂×M₁₆	Dic₉	semidirect
249. C₉	#1742	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×M₁₆	2
C₂×#63488	2
C₂²×#3667	2
#10588234	2
#71987	3