Subgroups — #11774607

Order 1152.

369 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	#10588606	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588605	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588212	C₂	1	2	yes	maximal, complemented
5	D₄⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
6	#71640	—	3	1	no	maximal
7	C₂×C₉×#41	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₄×D₄×C₉	—	2	1	no	—
11	C₄×D₄×C₉	—	2	1	no	—
12	C₄×D₄×C₉	C₄	1	1	yes	characteristic
13	C₄×D₄×C₉	C₂²	1	1	yes	characteristic
14	C₉×#117	—	2	1	no	—
15	C₉×#117	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₉	—	2	1	no	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	#63389	—	2	1	no	—
19	#63389	—	2	1	no	—
20	C₄⋊(C₉⋊C₈)	—	2	1	no	—
21	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
22	#63362	—	2	1	no	—
23	C₄×C₉⋊C₈	C₂²	1	2	yes	—
24	C₃×C₄×#41	D₃	1	1	yes	characteristic
25	#4926	—	3	1	no	—
26	#4925	—	3	1	no	—
27	#4535	—	3	2	no	—
28	D₄⋊(C₃⋊C₈)	—	3	2	no	—
29	C₉×#41	—	2	2	no	—
30	C₉×#41	—	2	2	no	—
31	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	—	2	1	no	—
37	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
40	C₉×C₄⋊C₄	—	2	1	no	—
41	C₉×C₄⋊C₄	—	2	1	no	—
42	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₉×C₂²⋊C₄	—	2	1	no	—
45	C₉×C₂²⋊C₄	—	2	1	no	—
46	C₉×C₂²⋊C₄	—	2	1	no	—
47	C₄²×C₉	—	2	1	no	—
48	C₄²×C₉	—	2	1	no	—
49	C₄²×C₉	—	2	1	no	—
50	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₉	D₄	1	1	yes	characteristic
52	C₄²×C₉	D₄	1	1	yes	characteristic
53	C₄²×C₉	C₂³	1	1	yes	characteristic
54	C₂×C₉⋊C₈	—	2	2	no	—
55	C₂×C₉⋊C₈	—	2	2	no	—
56	#1488	—	9	1	no	Sylow
57	C₂×C₃×#41	Dic₃	1	1	yes	characteristic
58	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
59	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
60	C₃×C₄×D₄	—	2	1	no	—
61	C₃×C₄×D₄	—	2	1	no	—
62	C₃×C₄×D₄	Dic₃	1	1	yes	characteristic
63	C₃×C₄×D₄	C₂×D₃	1	1	yes	characteristic
64	C₃×#117	—	2	1	no	—
65	C₃×#117	Dic₃	1	1	yes	characteristic
66	C₂×C₃×C₄²	—	2	1	no	—
67	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
68	#830	—	6	1	no	—
69	#830	—	6	1	no	—
70	C₄⋊(C₃⋊C₈)	—	3	2	no	—
71	C₄⋊(C₃⋊C₈)	—	6	1	no	—
72	#803	—	6	1	no	—
73	C₄×C₃⋊C₈	—	3	2	no	—
74	C₂³×C₉	—	2	1	no	—
75	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₉	—	2	1	no	—
77	Q₈×C₉	M₁₆	1	2	yes	—
78	D₄×C₉	—	2	4	no	—
79	D₄×C₉	—	2	1	no	—
80	D₄×C₉	M₁₆	1	2	yes	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	4	1	no	—
89	C₂×C₄×C₉	—	4	2	no	—
90	C₂×C₄×C₉	—	4	2	no	—
91	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₉⋊C₈	—	4	2	no	—
98	C₉⋊C₈	—	4	2	no	—
99	C₄×#41	D₉	1	1	yes	characteristic
100	#432	—	9	1	no	—
101	#431	—	9	1	no	—
102	Q₈⋊C₈	—	9	2	no	—
103	D₄⋊C₈	—	9	2	no	—
104	C₃×#41	—	2	2	no	—
105	C₃×#41	—	2	2	no	—
106	C₂×C₃×Q₈	C₂×Dic₃	1	1	yes	characteristic
107	C₂×C₃×D₄	—	2	1	no	—
108	C₂×C₃×D₄	C₂×Dic₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	2	1	no	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	#67	1	1	yes	characteristic
114	C₂²×C₃×C₄	#67	1	1	yes	characteristic
115	C₃×C₄⋊C₄	—	2	1	no	—
116	C₃×C₄⋊C₄	—	2	1	no	—
117	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
118	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
119	C₃×C₂²⋊C₄	—	2	1	no	—
120	C₃×C₂²⋊C₄	—	2	1	no	—
121	C₃×C₂²⋊C₄	—	2	1	no	—
122	C₃×C₄²	—	2	1	no	—
123	C₃×C₄²	—	2	1	no	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
126	C₃×C₄²	#67	1	1	yes	characteristic
127	C₃×C₄²	#67	1	1	yes	characteristic
128	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
129	C₂×C₃⋊C₈	—	6	2	no	—
130	C₂×C₃⋊C₈	—	6	2	no	—
131	C₂²×C₉	—	2	1	no	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	1	no	—
134	C₂²×C₉	—	2	1	no	—
135	C₂²×C₉	—	4	1	no	—
136	C₂²×C₉	—	4	1	no	—
137	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₉	—	2	1	no	—
139	C₄×C₉	—	2	2	no	—
140	C₄×C₉	—	2	1	no	—
141	C₄×C₉	—	2	1	no	—
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	—	4	1	no	—
144	C₄×C₉	—	4	1	no	—
145	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
146	C₄×C₉	C₂×M₁₆	1	1	yes	characteristic
147	C₄×C₉	#135	1	2	yes	—
148	C₂×#41	Dic₉	1	1	yes	characteristic
149	C₄×Q₈	Dic₉	1	1	yes	characteristic
150	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
151	C₄×D₄	—	2	1	no	—
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	Dic₉	1	1	yes	characteristic
154	C₄×D₄	C₂×D₉	1	1	yes	characteristic
155	#117	—	2	1	no	—
156	#117	Dic₉	1	1	yes	characteristic
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
159	C₄⋊C₈	—	9	2	no	—
160	C₄⋊C₈	—	18	1	no	—
161	C₂²⋊C₈	—	18	1	no	—
162	C₂²⋊C₈	—	18	1	no	—
163	#97	—	18	1	no	—
164	C₄×C₈	—	9	2	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	#217	1	1	yes	characteristic
167	C₃×Q₈	—	2	1	no	—
168	C₃×Q₈	#208	1	2	yes	—
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	2	1	no	—
171	C₃×D₄	#208	1	2	yes	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	#217	1	1	yes	characteristic
183	C₂×C₃×C₄	#217	1	1	yes	characteristic
184	C₂×C₃×C₄	#217	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
188	C₃⋊C₈	—	12	2	no	—
189	C₃⋊C₈	—	12	2	no	—
190	C₂×C₉	—	4	1	no	—
191	C₂×C₉	—	4	2	no	—
192	C₂×C₉	#407	1	1	yes	characteristic
193	C₂×C₉	#416	1	1	yes	characteristic
194	C₂×C₉	#421	1	1	yes	characteristic
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	C₂×Dic₉	1	1	yes	characteristic
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₂×Dic₉	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
204	C₂²×C₄	#611	1	1	yes	characteristic
205	C₂²×C₄	#611	1	1	yes	characteristic
206	C₂×C₈	—	18	2	no	—
207	C₂×C₈	—	18	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₂×Dic₉	1	1	yes	characteristic
211	C₄⋊C₄	C₂×Dic₉	1	1	yes	characteristic
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₂×Dic₉	1	1	yes	characteristic
219	C₄²	#611	1	1	yes	characteristic
220	C₄²	#611	1	1	yes	characteristic
221	C₄²	C₂²×D₉	1	1	yes	characteristic
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	4	1	no	—
227	C₂²×C₃	—	4	1	no	—
228	C₂²×C₃	C₂×#217	1	1	yes	characteristic
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	4	1	no	—
235	C₃×C₄	—	4	1	no	—
236	C₃×C₄	C₂×#208	1	1	yes	characteristic
237	C₃×C₄	C₂×#208	1	1	yes	characteristic
238	C₃×C₄	#950	1	2	yes	—
239	C₉	#1488	1	1	yes	characteristic, complemented, Sylow
240	C₂³	—	2	1	no	—
241	C₂³	#3676	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	#3667	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	#3667	1	2	yes	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	#3676	1	1	yes	characteristic
258	C₂×C₄	#3676	1	1	yes	characteristic
259	C₂×C₄	#3676	1	1	yes	characteristic
260	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
261	C₂×C₄	C₂×#611	1	1	yes	characteristic
262	C₂×C₄	C₂×#611	1	1	yes	characteristic
263	C₈	—	36	2	no	—
264	C₈	—	36	2	no	—
265	C₂×C₃	—	4	1	no	—
266	C₂×C₃	—	4	2	no	—
267	C₂×C₃	#5207	1	1	yes	characteristic
268	C₂×C₃	#5233	1	1	yes	characteristic
269	C₂×C₃	#5236	1	1	yes	characteristic
270	C₂²	—	2	1	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	C₂×#3676	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	C₂×#3667	1	1	yes	characteristic
285	C₄	C₂×#3667	1	1	yes	characteristic
286	C₄	#63513	1	2	yes	—
287	C₃	#71640	1	1	yes	characteristic
288	C₂	—	4	1	no	—
289	C₂	—	4	2	no	—
290	C₂	#10588887	1	1	yes	characteristic
291	C₂	#10588914	1	1	yes	characteristic
292	C₂	#10588917	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₄×C₉×#41	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₂×C₃×#41	Dic₃
cosocle	C₄²×C₉	C₂³
hypercenter	C₄×#41	D₉
hypocenter	C₉	#1488
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #10588606	C₂	semidirect
3. #10588605	C₂	semidirect
4. #10588212	C₂	semidirect
5. D₄⋊(C₉⋊C₈)	C₂	semidirect
239. C₉	#1488	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×#41	2
#10588606	2
#10588605	2
#10588212	2
D₄⋊(C₉⋊C₈)	2
#71640	3