Subgroups — #11785950

Order 1152.

390 conjugacy classes of subgroups.

The subgroup lattice has 255 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₉⋊C₈	C₂	1	2	yes	maximal
2	C₄²×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#336	C₂	1	1	yes	maximal, characteristic
4	#10588228	C₂	1	2	yes	maximal
5	#10588200	C₂	1	1	yes	maximal, characteristic
6	#68605	—	3	1	no	maximal
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
16	Dic₉⋊C₈	—	2	4	no	—
17	C₂×Dic₃⋊C₈	—	3	2	no	—
18	C₄²×Dic₃	—	3	1	no	—
19	C₃×#336	D₃	1	1	yes	characteristic
20	#4550	—	3	2	no	—
21	#4524	—	3	1	no	—
22	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₉	—	2	2	no	—
30	C₂×C₈×C₉	—	2	2	no	—
31	C₂×C₈×C₉	—	2	2	no	—
32	C₄²×C₉	—	2	2	no	—
33	C₄×Dic₉	—	2	2	no	—
34	C₄×Dic₉	—	2	2	no	—
35	C₄×Dic₉	—	2	2	no	—
36	C₄×Dic₉	—	2	2	no	—
37	C₄×Dic₉	C₈	1	4	yes	complemented
38	C₄×Dic₉	D₄	1	2	yes	—
39	C₄×Dic₉	Q₈	1	2	yes	—
40	C₂×C₉⋊C₈	—	2	2	no	—
41	C₂×C₉⋊C₈	—	2	2	no	—
42	C₂×C₉⋊C₈	—	2	2	no	—
43	#1831	—	9	1	no	Sylow
44	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
45	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄×Dic₃	—	3	1	no	—
47	C₂×C₄×Dic₃	—	3	1	no	—
48	C₂×C₄×Dic₃	—	3	1	no	—
49	C₂×C₄×Dic₃	—	3	1	no	—
50	C₂×C₄×Dic₃	—	3	1	no	—
51	C₂×C₄×Dic₃	—	3	1	no	—
52	C₂²×C₃⋊C₈	—	3	2	no	—
53	Dic₃⋊C₈	—	6	4	no	—
54	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉	—	2	1	no	—
56	C₂×C₄×C₉	—	2	1	no	—
57	C₂×C₄×C₉	—	2	1	no	—
58	C₂×C₄×C₉	—	2	1	no	—
59	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
60	C₂×C₄×C₉	M₁₆	1	2	yes	—
61	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₉	#41	1	2	yes	—
64	C₂×C₄×C₉	#41	1	2	yes	—
65	C₂×Dic₉	—	2	1	no	—
66	C₂×Dic₉	—	2	1	no	—
67	C₂×Dic₉	—	2	1	no	—
68	C₂×Dic₉	—	2	1	no	—
69	C₂×Dic₉	—	2	1	no	—
70	C₂×Dic₉	—	2	1	no	—
71	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
72	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
73	C₂×Dic₉	C₂×C₈	1	2	yes	—
74	C₂×Dic₉	C₂×C₈	1	2	yes	—
75	C₂×Dic₉	M₁₆	1	2	yes	—
76	C₂×Dic₉	M₁₆	1	2	yes	—
77	C₈×C₉	—	4	2	no	—
78	C₉⋊C₈	—	4	2	no	—
79	C₂×C₄⋊C₈	—	9	2	no	—
80	C₄³	—	9	1	no	—
81	#336	—	9	2	no	—
82	#336	—	9	1	no	—
83	#336	D₉	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂²×Dic₃	—	3	1	no	—
88	C₂²×Dic₃	—	3	1	no	—
89	C₂²×Dic₃	—	3	1	no	—
90	C₂²×Dic₃	—	3	1	no	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₃×C₄²	—	2	2	no	—
95	C₄×Dic₃	—	3	4	no	—
96	C₄×Dic₃	—	3	2	no	—
97	C₄×Dic₃	—	3	2	no	—
98	C₄×Dic₃	—	6	2	no	—
99	C₄×Dic₃	—	6	2	no	—
100	C₄×Dic₃	—	6	2	no	—
101	C₄×Dic₃	—	6	2	no	—
102	C₂×C₃⋊C₈	—	6	2	no	—
103	C₂×C₃⋊C₈	—	6	2	no	—
104	C₂×C₃⋊C₈	—	6	2	no	—
105	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₉	#117	1	1	yes	characteristic
107	C₂²×C₉	#117	1	1	yes	characteristic
108	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
109	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
112	C₄×C₉	—	2	2	no	—
113	C₄×C₉	—	2	2	no	—
114	C₄×C₉	#124	1	2	yes	—
115	C₄×C₉	#125	1	2	yes	—
116	Dic₉	—	2	2	no	—
117	Dic₉	—	2	2	no	—
118	Dic₉	—	2	2	no	—
119	Dic₉	C₄⋊C₈	1	4	yes	—
120	C₂²×C₈	—	9	2	no	—
121	C₂²×C₈	C₂×D₉	1	2	yes	—
122	C₂×C₄²	—	9	1	no	—
123	C₂×C₄²	—	9	1	no	—
124	C₂×C₄²	—	9	1	no	—
125	C₂×C₄²	—	9	1	no	—
126	C₂×C₄²	—	9	1	no	—
127	C₂×C₄²	—	9	1	no	—
128	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
129	C₄⋊C₈	—	18	4	no	—
130	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
136	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
137	C₂×C₃×C₄	#235	1	2	yes	—
138	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	#237	1	2	yes	—
140	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
141	C₂×Dic₃	—	3	2	no	—
142	C₂×Dic₃	—	3	2	no	—
143	C₂×Dic₃	—	3	2	no	—
144	C₂×Dic₃	—	3	2	no	—
145	C₂×Dic₃	—	3	2	no	—
146	C₂×Dic₃	—	3	2	no	—
147	C₂×Dic₃	—	6	1	no	—
148	C₂×Dic₃	—	6	1	no	—
149	C₂×Dic₃	—	6	1	no	—
150	C₂×Dic₃	—	6	1	no	—
151	C₂×Dic₃	—	6	1	no	—
152	C₂×Dic₃	—	6	1	no	—
153	C₃×C₈	—	4	2	no	—
154	C₃⋊C₈	—	12	2	no	—
155	C₂×C₉	#382	1	1	yes	characteristic
156	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
157	C₂×C₉	#423	1	1	yes	characteristic
158	C₂×C₉	#431	1	1	yes	characteristic
159	C₂×C₉	#431	1	1	yes	characteristic
160	C₂×C₉	#432	1	1	yes	characteristic
161	C₂×C₉	#432	1	1	yes	characteristic
162	C₂²×C₄	—	9	1	no	—
163	C₂²×C₄	—	9	1	no	—
164	C₂²×C₄	—	9	1	no	—
165	C₂²×C₄	—	9	1	no	—
166	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
167	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	18	2	no	—
173	C₂×C₈	—	18	2	no	—
174	C₂×C₈	—	18	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	9	4	no	—
177	C₄²	—	9	2	no	—
178	C₄²	—	9	2	no	—
179	C₄²	—	18	2	no	—
180	C₄²	—	18	2	no	—
181	C₄²	—	18	2	no	—
182	C₄²	—	18	2	no	—
183	C₂²×C₃	#872	1	1	yes	characteristic
184	C₂²×C₃	#877	1	1	yes	characteristic
185	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
187	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
188	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
189	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	#885	1	2	yes	—
193	C₃×C₄	#889	1	2	yes	—
194	Dic₃	—	3	4	no	—
195	Dic₃	—	6	2	no	—
196	Dic₃	—	6	2	no	—
197	Dic₃	—	6	2	no	—
198	C₉	#1831	1	1	yes	characteristic, complemented, Sylow
199	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	9	2	no	—
205	C₂×C₄	—	9	2	no	—
206	C₂×C₄	—	9	2	no	—
207	C₂×C₄	—	9	2	no	—
208	C₂×C₄	—	9	2	no	—
209	C₂×C₄	—	9	2	no	—
210	C₂×C₄	—	18	1	no	—
211	C₂×C₄	—	18	1	no	—
212	C₂×C₄	—	18	1	no	—
213	C₂×C₄	—	18	1	no	—
214	C₂×C₄	—	18	1	no	—
215	C₂×C₄	—	18	1	no	—
216	C₂×C₄	C₈×D₉	1	2	yes	—
217	C₂×C₄	#3663	1	2	yes	—
218	C₂×C₄	#3697	1	2	yes	—
219	C₂×C₄	D₄×D₉	1	1	yes	characteristic
220	C₂×C₄	#3699	1	2	yes	—
221	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
222	C₈	—	4	2	no	—
223	C₈	—	36	2	no	—
224	C₂×C₃	#4650	1	1	yes	characteristic
225	C₂×C₃	#4685	1	1	yes	characteristic
226	C₂×C₃	#4705	1	1	yes	characteristic
227	C₂×C₃	#4718	1	1	yes	characteristic
228	C₂×C₃	#4719	1	1	yes	characteristic
229	C₂×C₃	D₃×C₄⋊C₈	1	1	yes	characteristic
230	C₂×C₃	#4837	1	1	yes	characteristic
231	C₂²	#63434	1	1	yes	characteristic
232	C₂²	#63439	1	1	yes	characteristic
233	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
234	C₂²	C₂×C₈×D₉	1	1	yes	characteristic
235	C₂²	C₂×#3663	1	1	yes	characteristic
236	C₂²	M₁₆×D₉	1	1	yes	characteristic
237	C₂²	M₁₆×D₉	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	9	4	no	—
241	C₄	—	18	2	no	—
242	C₄	—	18	2	no	—
243	C₄	—	18	2	no	—
244	C₄	#63447	1	2	yes	—
245	C₄	#63451	1	2	yes	—
246	C₃	#68605	1	1	yes	characteristic
247	C₂	#10588330	1	1	yes	characteristic
248	C₂	#10588367	1	1	yes	characteristic
249	C₂	#10588386	1	1	yes	characteristic
250	C₂	#10588398	1	1	yes	characteristic
251	C₂	#10588399	1	1	yes	characteristic
252	C₂	D₉×C₄⋊C₈	1	1	yes	characteristic
253	C₂	#10588521	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#336	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#336	D₉
hypocenter	C₉	#1831
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
37. C₄×Dic₉	C₈	semidirect
198. C₉	#1831	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₉⋊C₈	2
C₄²×Dic₉	2
C₉×#336	2
#10588228	2
#10588200	2
#68605	3