Subgroups — #11791860

Order 1152.

390 conjugacy classes of subgroups.

The subgroup lattice has 297 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	#10588802	C₂	1	1	yes	maximal, characteristic
3	M₁₆×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₉×#343	C₂	1	1	yes	maximal, characteristic
5	#10588234	C₂	1	1	yes	maximal, characteristic
6	#10588234	C₂	1	1	yes	maximal, characteristic
7	#10588207	C₂	1	1	yes	maximal, characteristic
8	#68974	—	3	1	no	maximal
9	C₂²×Dic₁₈	C₄	1	2	yes	—
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₉×M₁₆	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×Dic₉⋊C₄	C₄	1	2	yes	—
15	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂×#3667	C₂²	1	1	yes	characteristic
17	C₂×#3667	C₂²	1	1	yes	characteristic
18	#63450	—	2	1	no	—
19	#63450	—	2	1	no	—
20	#63450	—	2	1	no	—
21	#63450	—	2	1	no	—
22	#63375	—	2	1	no	—
23	Dic₉⋊C₈	—	2	1	no	—
24	Dic₉⋊C₈	—	2	1	no	—
25	C₈×Dic₉	—	2	1	no	—
26	C₂×#888	—	3	1	no	—
27	#5118	—	3	1	no	—
28	Dic₃×M₁₆	—	3	1	no	—
29	C₃×#343	D₃	1	1	yes	characteristic
30	#4554	—	3	1	no	—
31	#4554	—	3	1	no	—
32	#4530	—	3	1	no	—
33	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₉	C₂×C₄	1	2	yes	—
38	C₂×Dic₁₈	—	2	2	no	—
39	C₂×Dic₁₈	—	2	2	no	—
40	C₂×Dic₁₈	—	4	4	no	—
41	C₉×M₁₆	—	2	1	no	—
42	C₉×M₁₆	—	2	1	no	—
43	C₉×M₁₆	—	2	1	no	—
44	C₉×M₁₆	Q₈	1	2	yes	complemented
45	C₂×C₈×C₉	—	2	1	no	—
46	C₂×C₈×C₉	—	2	1	no	—
47	C₉×C₄⋊C₄	—	2	1	no	—
48	C₉×C₄⋊C₄	—	2	1	no	—
49	C₄⋊Dic₉	—	2	1	no	—
50	C₄⋊Dic₉	—	2	1	no	—
51	Dic₉⋊C₄	—	2	2	no	—
52	Dic₉⋊C₄	—	2	2	no	—
53	C₄×Dic₉	D₄	1	1	yes	characteristic
54	C₄×Dic₉	D₄	1	1	yes	characteristic
55	C₄×Dic₉	D₄	1	1	yes	characteristic
56	C₄×Dic₉	D₄	1	1	yes	characteristic
57	#3667	—	2	1	no	—
58	#3667	—	2	1	no	—
59	#3667	—	2	1	no	—
60	#3667	Q₈	1	2	yes	complemented
61	C₂×C₉⋊C₈	—	2	1	no	—
62	C₂×C₉⋊C₈	—	2	1	no	—
63	#1997	—	9	1	no	Sylow
64	C₂²×Dic₆	—	3	2	no	—
65	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
68	C₂×C₄⋊Dic₃	—	3	1	no	—
69	C₂×Dic₃⋊C₄	—	3	2	no	—
70	C₂×C₄×Dic₃	—	3	1	no	—
71	C₂×#208	—	3	1	no	—
72	C₂×#208	—	3	1	no	—
73	#888	—	6	1	no	—
74	#888	—	6	1	no	—
75	#888	—	6	1	no	—
76	#888	—	6	1	no	—
77	#815	—	6	1	no	—
78	Dic₃⋊C₈	—	6	1	no	—
79	Dic₃⋊C₈	—	6	1	no	—
80	C₈×Dic₃	—	6	1	no	—
81	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	4	2	no	—
84	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₉	#41	1	1	yes	characteristic
89	C₂×C₄×C₉	#41	1	1	yes	characteristic
90	C₂×Dic₉	—	2	2	no	—
91	C₂×Dic₉	—	2	2	no	—
92	C₂×Dic₉	—	2	2	no	—
93	C₂×Dic₉	—	4	2	no	—
94	C₂×Dic₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×Dic₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×Dic₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×Dic₉	C₂²⋊C₄	1	1	yes	characteristic
98	Dic₁₈	—	4	4	no	—
99	Dic₁₈	—	4	4	no	—
100	C₈×C₉	—	2	2	no	—
101	C₈×C₉	—	4	1	no	—
102	C₉⋊C₈	—	2	2	no	—
103	C₉⋊C₈	—	4	1	no	—
104	C₂×C₄⋊Q₈	—	9	1	no	—
105	#423	—	9	1	no	—
106	C₄×M₁₆	—	9	1	no	—
107	#343	—	9	1	no	—
108	#343	—	9	1	no	—
109	#343	—	9	1	no	—
110	#343	D₉	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
112	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂²×Dic₃	—	3	1	no	—
114	C₂²×Dic₃	—	3	1	no	—
115	C₂²×Dic₃	—	3	2	no	—
116	C₂×Dic₆	—	6	2	no	—
117	C₂×Dic₆	—	6	2	no	—
118	C₂×Dic₆	—	12	4	no	—
119	C₃×M₁₆	—	2	1	no	—
120	C₃×M₁₆	—	2	1	no	—
121	C₃×M₁₆	—	2	1	no	—
122	C₃×M₁₆	Dic₆	1	2	yes	—
123	C₂×C₃×C₈	—	2	1	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₃×C₄⋊C₄	—	2	1	no	—
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₄⋊Dic₃	—	6	1	no	—
128	C₄⋊Dic₃	—	6	1	no	—
129	Dic₃⋊C₄	—	6	2	no	—
130	Dic₃⋊C₄	—	6	2	no	—
131	C₄×Dic₃	—	3	1	no	—
132	C₄×Dic₃	—	3	1	no	—
133	C₄×Dic₃	—	3	1	no	—
134	C₄×Dic₃	—	3	1	no	—
135	#208	—	3	2	no	—
136	#208	—	6	1	no	—
137	#208	—	6	1	no	—
138	#208	—	6	1	no	—
139	C₂×C₃⋊C₈	—	6	1	no	—
140	C₂×C₃⋊C₈	—	6	1	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	—	2	1	no	—
143	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₉	C₄×Q₈	1	1	yes	characteristic
146	C₄×C₉	—	4	2	no	—
147	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
148	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
149	C₄×C₉	#124	1	1	yes	characteristic
150	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
151	Dic₉	—	2	1	no	—
152	Dic₉	—	2	1	no	—
153	Dic₉	—	4	2	no	—
154	Dic₉	#100	1	2	yes	—
155	Dic₉	#101	1	2	yes	—
156	C₂²×Q₈	—	9	2	no	—
157	C₂×M₁₆	—	9	1	no	—
158	C₂×M₁₆	—	9	1	no	—
159	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
160	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
161	C₄⋊Q₈	—	18	1	no	—
162	C₄⋊Q₈	—	18	1	no	—
163	C₄⋊Q₈	—	18	1	no	—
164	C₄⋊Q₈	—	18	1	no	—
165	C₂×C₄⋊C₄	—	9	2	no	—
166	C₂×C₄⋊C₄	—	9	1	no	—
167	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
168	C₂×C₄²	—	9	1	no	—
169	C₄⋊C₈	—	18	1	no	—
170	C₄⋊C₈	—	18	1	no	—
171	#97	—	18	1	no	—
172	C₄×C₈	—	18	1	no	—
173	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	4	2	no	—
176	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
177	C₂×C₃×C₄	#235	1	1	yes	characteristic
178	C₂×C₃×C₄	D₃×D₄	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	1	no	—
183	C₂×Dic₃	—	3	1	no	—
184	C₂×Dic₃	—	3	1	no	—
185	C₂×Dic₃	—	3	1	no	—
186	C₂×Dic₃	—	6	2	no	—
187	C₂×Dic₃	—	6	2	no	—
188	C₂×Dic₃	—	6	2	no	—
189	C₂×Dic₃	—	12	2	no	—
190	Dic₆	—	12	4	no	—
191	Dic₆	—	12	4	no	—
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	—	4	1	no	—
194	C₃⋊C₈	—	6	2	no	—
195	C₃⋊C₈	—	12	1	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	#391	1	1	yes	characteristic
199	C₂×C₉	C₂×#100	1	1	yes	characteristic
200	C₂×C₉	C₂×#101	1	1	yes	characteristic
201	C₂×Q₈	—	18	2	no	—
202	C₂×Q₈	—	18	2	no	—
203	C₂×Q₈	—	36	4	no	—
204	C₂²×C₄	—	9	1	no	—
205	C₂²×C₄	—	9	1	no	—
206	C₂²×C₄	—	9	2	no	—
207	C₂²×C₄	C₄×D₉	1	2	yes	—
208	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
209	M₁₆	—	2	1	no	—
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	9	2	no	—
213	M₁₆	—	18	1	no	—
214	M₁₆	—	18	1	no	—
215	M₁₆	—	18	1	no	—
216	M₁₆	Dic₁₈	1	2	yes	complemented
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	18	1	no	—
220	C₂×C₈	—	18	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	18	2	no	—
224	C₄⋊C₄	—	18	2	no	—
225	C₄⋊C₄	—	18	1	no	—
226	C₄⋊C₄	—	18	1	no	—
227	C₄²	—	9	1	no	—
228	C₄²	—	9	1	no	—
229	C₄²	—	9	1	no	—
230	C₄²	—	9	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
234	C₂²×C₃	D₃×C₂²⋊C₄	1	1	yes	characteristic
235	C₂²×C₃	#887	1	1	yes	characteristic
236	C₃×C₄	—	4	2	no	—
237	C₃×C₄	#878	1	1	yes	characteristic
238	C₃×C₄	#885	1	1	yes	characteristic
239	C₃×C₄	#888	1	1	yes	characteristic
240	C₃×C₄	#896	1	1	yes	characteristic
241	Dic₃	—	3	2	no	—
242	Dic₃	—	3	2	no	—
243	Dic₃	—	6	1	no	—
244	Dic₃	—	6	1	no	—
245	Dic₃	—	12	2	no	—
246	C₉	#1997	1	1	yes	characteristic, complemented, Sylow
247	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
248	Q₈	—	36	4	no	—
249	Q₈	—	36	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	9	1	no	—
253	C₂×C₄	—	9	1	no	—
254	C₂×C₄	—	9	1	no	—
255	C₂×C₄	—	9	1	no	—
256	C₂×C₄	—	18	2	no	—
257	C₂×C₄	—	18	2	no	—
258	C₂×C₄	—	18	2	no	—
259	C₂×C₄	—	36	2	no	—
260	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
261	C₂×C₄	#3697	1	1	yes	characteristic
262	C₂×C₄	D₄×D₉	1	1	yes	characteristic
263	C₂×C₄	D₄×D₉	1	1	yes	characteristic
264	C₂×C₄	#3699	1	1	yes	characteristic
265	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	4	1	no	—
268	C₈	—	18	2	no	—
269	C₈	—	36	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	#4646	1	1	yes	characteristic
273	C₂×C₃	D₃×#100	1	1	yes	characteristic
274	C₂×C₃	D₃×#101	1	1	yes	characteristic
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₁₈	1	1	yes	characteristic
278	C₂²	C₂²⋊C₄×D₉	1	1	yes	characteristic
279	C₂²	#63449	1	1	yes	characteristic
280	C₄	—	4	2	no	—
281	C₄	—	9	2	no	—
282	C₄	—	9	2	no	—
283	C₄	—	18	1	no	—
284	C₄	—	18	1	no	—
285	C₄	—	36	2	no	—
286	C₄	#63440	1	1	yes	characteristic
287	C₄	#63447	1	1	yes	characteristic
288	C₄	#63450	1	1	yes	characteristic
289	C₄	#63458	1	1	yes	characteristic
290	C₃	#68974	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	#10588326	1	1	yes	characteristic
294	C₂	D₉×#100	1	1	yes	characteristic
295	C₂	D₉×#101	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₉
commutator	C₂³×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#343	C₂
socle	C₂²×C₃	D₃×C₂²⋊C₄
prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#343	D₉
hypocenter	C₉	#1997
LCS	G ▷ C₂³×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

Complements

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

N	H ≅ G/N	type
44. C₉×M₁₆	Q₈	semidirect
60. #3667	Q₈	semidirect
216. M₁₆	Dic₁₈	semidirect
246. C₉	#1997	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
#10588802	2
M₁₆×Dic₉	2
C₉×#343	2
#10588234	2
#10588207	2
#68974	3