Subgroups — #11775704

Order 1152.

351 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#529	C₂	1	1	yes	maximal, characteristic
2	C₄×#3667	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63362	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588220	C₂	1	2	yes	maximal, complemented
5	#10588217	C₂	1	2	yes	maximal, complemented
6	#71746	—	3	1	no	maximal
7	C₉×#126	—	2	1	no	—
8	C₉×#125	C₂²	1	1	yes	characteristic
9	C₉×#124	C₂²	1	1	yes	characteristic
10	C₉×#123	—	2	1	no	—
11	C₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
12	C₉×#121	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
14	C₄×D₄×C₉	—	2	1	no	—
15	C₄×D₄×C₉	C₄	1	1	yes	characteristic
16	C₉×#117	—	2	1	no	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₂×#3667	—	2	1	no	—
19	C₂²×C₉⋊C₈	—	2	1	no	—
20	#63362	—	2	1	no	—
21	#63362	C₂²	1	2	yes	—
22	#63362	C₂²	1	2	yes	—
23	C₄×C₉⋊C₈	—	2	1	no	—
24	C₃×#529	D₃	1	1	yes	characteristic
25	C₄×#208	—	3	1	no	—
26	C₂×#803	—	3	1	no	—
27	#4543	—	3	2	no	—
28	#4540	—	3	2	no	—
29	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₉	—	2	1	no	—
31	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	—	2	1	no	—
33	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
36	C₉×C₄⋊C₄	—	2	1	no	—
37	C₉×C₄⋊C₄	—	2	1	no	—
38	C₉×C₄⋊C₄	—	2	1	no	—
39	C₉×C₄⋊C₄	—	2	1	no	—
40	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₉×C₂²⋊C₄	—	2	1	no	—
43	C₉×C₂²⋊C₄	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₉	D₄	1	1	yes	characteristic
50	C₄²×C₉	D₄	1	1	yes	characteristic
51	C₄²×C₉	C₂³	1	1	yes	characteristic
52	#3667	—	2	4	no	—
53	C₂×C₉⋊C₈	—	2	4	no	—
54	C₂×C₉⋊C₈	—	2	2	no	—
55	C₂×C₉⋊C₈	—	2	2	no	—
56	#1524	—	9	1	no	Sylow
57	C₃×#126	—	2	1	no	—
58	C₃×#125	C₂×D₃	1	1	yes	characteristic
59	C₃×#124	C₂×D₃	1	1	yes	characteristic
60	C₃×#123	—	2	1	no	—
61	C₃×C₂²⋊Q₈	Dic₃	1	1	yes	characteristic
62	C₃×#121	Dic₃	1	1	yes	characteristic
63	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
64	C₃×C₄×D₄	—	2	1	no	—
65	C₃×C₄×D₄	Dic₃	1	1	yes	characteristic
66	C₃×#117	—	2	1	no	—
67	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
68	C₂×#208	—	6	1	no	—
69	C₂²×C₃⋊C₈	—	6	1	no	—
70	#803	—	3	2	no	—
71	#803	—	3	2	no	—
72	#803	—	6	1	no	—
73	C₄×C₃⋊C₈	—	6	1	no	—
74	C₂³×C₉	—	2	1	no	—
75	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₉	—	4	1	no	—
77	D₄×C₉	—	4	1	no	—
78	D₄×C₉	—	4	2	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	1	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	—	4	1	no	—
85	C₂×C₄×C₉	—	4	2	no	—
86	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
92	C₉⋊C₈	—	2	4	no	—
93	C₉⋊C₈	—	2	4	no	—
94	#529	D₉	1	1	yes	characteristic
95	C₄×M₁₆	—	9	1	no	—
96	C₂×#97	—	9	1	no	—
97	#330	—	9	2	no	—
98	#329	—	9	2	no	—
99	C₂×C₃×Q₈	C₂×Dic₃	1	1	yes	characteristic
100	C₂×C₃×D₄	—	2	1	no	—
101	C₂×C₃×D₄	C₂×Dic₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
104	C₂²×C₃×C₄	#67	1	1	yes	characteristic
105	C₂²×C₃×C₄	#67	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	2	1	no	—
107	C₃×C₄⋊C₄	—	2	1	no	—
108	C₃×C₄⋊C₄	—	2	1	no	—
109	C₃×C₄⋊C₄	—	2	1	no	—
110	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
111	C₃×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
112	C₃×C₂²⋊C₄	—	2	1	no	—
113	C₃×C₂²⋊C₄	—	2	1	no	—
114	C₃×C₂²⋊C₄	—	2	1	no	—
115	C₃×C₂²⋊C₄	—	2	1	no	—
116	C₃×C₂²⋊C₄	—	2	1	no	—
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
119	C₃×C₄²	#67	1	1	yes	characteristic
120	C₃×C₄²	#67	1	1	yes	characteristic
121	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
122	#208	—	6	4	no	—
123	C₂×C₃⋊C₈	—	6	4	no	—
124	C₂×C₃⋊C₈	—	6	2	no	—
125	C₂×C₃⋊C₈	—	6	2	no	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	—	2	1	no	—
128	C₂²×C₉	—	4	1	no	—
129	C₂²×C₉	—	4	1	no	—
130	C₂²×C₉	—	4	1	no	—
131	C₂²×C₉	#104	1	2	yes	—
132	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₉	—	2	1	no	—
134	C₄×C₉	—	2	1	no	—
135	C₄×C₉	—	2	1	no	—
136	C₄×C₉	—	2	1	no	—
137	C₄×C₉	—	2	1	no	—
138	C₄×C₉	—	4	1	no	—
139	C₄×C₉	—	4	1	no	—
140	C₄×C₉	—	4	1	no	—
141	C₄×C₉	#104	1	2	yes	—
142	C₂×M₁₆	—	18	1	no	—
143	C₂²×C₈	—	18	1	no	—
144	#126	—	2	1	no	—
145	#125	C₂×D₉	1	1	yes	characteristic
146	#124	C₂×D₉	1	1	yes	characteristic
147	#123	—	2	1	no	—
148	C₂²⋊Q₈	Dic₉	1	1	yes	characteristic
149	#121	Dic₉	1	1	yes	characteristic
150	C₄×Q₈	Dic₉	1	1	yes	characteristic
151	C₄×D₄	—	2	1	no	—
152	C₄×D₄	Dic₉	1	1	yes	characteristic
153	#117	—	2	1	no	—
154	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
155	#97	—	9	2	no	—
156	#97	—	9	2	no	—
157	#97	—	18	1	no	—
158	C₄×C₈	—	18	1	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	#217	1	1	yes	characteristic
161	C₃×Q₈	—	4	1	no	—
162	C₃×D₄	—	4	1	no	—
163	C₃×D₄	—	4	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	4	1	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	#217	1	1	yes	characteristic
172	C₂×C₃×C₄	#217	1	1	yes	characteristic
173	C₂×C₃×C₄	#217	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
177	C₃⋊C₈	—	6	4	no	—
178	C₃⋊C₈	—	6	4	no	—
179	C₂×C₉	—	2	2	no	—
180	C₂×C₉	—	8	1	no	—
181	C₂×C₉	#413	1	1	yes	characteristic
182	C₂×C₉	C₂×#104	1	1	yes	characteristic
183	C₂×C₉	C₂×#104	1	1	yes	characteristic
184	C₂×Q₈	C₂×Dic₉	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂×Dic₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
189	C₂²×C₄	#611	1	1	yes	characteristic
190	C₂²×C₄	#611	1	1	yes	characteristic
191	M₁₆	—	18	4	no	—
192	C₂×C₈	—	18	4	no	—
193	C₂×C₈	—	18	2	no	—
194	C₂×C₈	—	18	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	C₂×Dic₉	1	1	yes	characteristic
200	C₄⋊C₄	C₂×Dic₉	1	1	yes	characteristic
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	C₂×Dic₉	1	1	yes	characteristic
208	C₄²	#611	1	1	yes	characteristic
209	C₄²	#611	1	1	yes	characteristic
210	C₄²	C₂²×D₉	1	1	yes	characteristic
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	#837	1	2	yes	—
217	C₂²×C₃	C₂×#217	1	1	yes	characteristic
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	4	1	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	—	4	1	no	—
226	C₃×C₄	#837	1	2	yes	—
227	C₉	#1524	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	2	1	no	—
229	C₂³	#3676	1	1	yes	characteristic
230	Q₈	—	4	1	no	—
231	D₄	—	4	1	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	#3676	1	1	yes	characteristic
241	C₂×C₄	#3676	1	1	yes	characteristic
242	C₂×C₄	#3676	1	1	yes	characteristic
243	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
244	C₂×C₄	C₂×#611	1	1	yes	characteristic
245	C₂×C₄	C₂×#611	1	1	yes	characteristic
246	C₈	—	18	4	no	—
247	C₈	—	18	4	no	—
248	C₂×C₃	—	2	2	no	—
249	C₂×C₃	—	8	1	no	—
250	C₂×C₃	C₂×#837	1	1	yes	characteristic
251	C₂×C₃	C₂×#837	1	1	yes	characteristic
252	C₂×C₃	#5237	1	1	yes	characteristic
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	#63396	1	2	yes	—
259	C₂²	C₂×#3676	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	#63396	1	2	yes	—
269	C₃	#71746	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	8	1	no	—
272	C₂	C₂×#63396	1	1	yes	characteristic
273	C₂	C₂×#63396	1	1	yes	characteristic
274	C₂	#10588915	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3676
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#529	C₂
socle	C₂²×C₃	C₂×#217
prime radical	C₃×#121	Dic₃
cosocle	C₄²×C₉	C₂³
hypercenter	#529	D₉
hypocenter	C₉	#1524
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
2. C₄×#3667	C₂	semidirect
3. C₂×#63362	C₂	semidirect
4. #10588220	C₂	semidirect
5. #10588217	C₂	semidirect
227. C₉	#1524	semidirect

Maximal subgroups

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

subgroup	index
C₉×#529	2
C₄×#3667	2
C₂×#63362	2
#10588220	2
#10588217	2
#71746	3