Subgroups — #11786350

Order 1152.

522 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₄×#3676	C₂	1	1	yes	maximal, characteristic
3	C₂×#63389	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₉⋊C₈	C₂	1	2	yes	maximal, complemented
5	#10588228	C₂	1	2	yes	maximal, complemented
6	#71974	—	3	1	no	maximal
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	—	2	2	no	—
9	C₂²×C₈×C₉	—	2	4	no	—
10	C₂²×C₈×C₉	C₂²	1	2	yes	—
11	C₂×#3676	C₄	1	1	yes	characteristic
12	C₂×#3676	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₉	C₂²	1	2	yes	—
14	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₉⋊#95	C₄	1	2	yes	—
17	#63389	—	2	2	no	—
18	Dic₉⋊C₈	—	2	4	no	—
19	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
20	C₄×#217	—	3	1	no	—
21	C₂×#830	—	3	1	no	—
22	C₂×Dic₃⋊C₈	—	3	2	no	—
23	#4550	—	3	2	no	—
24	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	D₄	1	2	yes	—
29	C₂²×C₄×C₉	Q₈	1	2	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₉	C₂×C₄	1	2	yes	—
32	C₂²×Dic₉	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₉	—	2	8	no	—
34	C₂×C₈×C₉	—	2	2	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	D₄	1	4	yes	—
37	#3676	—	2	2	no	—
38	#3676	C₈	1	4	yes	complemented
39	C₄×Dic₉	—	2	2	no	—
40	C₄×Dic₉	—	2	2	no	—
41	C₂×C₉⋊C₈	—	2	2	no	—
42	C₂×C₉⋊C₈	—	2	1	no	—
43	C₂×C₉⋊C₈	—	2	1	no	—
44	C₂×C₉⋊C₈	—	2	2	no	—
45	#1850	—	9	1	no	Sylow
46	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
47	C₂²×C₃×C₈	—	2	2	no	—
48	C₂²×C₃×C₈	—	2	4	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
50	C₂×#217	—	3	1	no	—
51	C₂×#217	—	3	1	no	—
52	C₂×C₄×Dic₃	—	3	2	no	—
53	C₂²×C₃⋊C₈	—	3	1	no	—
54	C₂²×C₃⋊C₈	—	3	1	no	—
55	C₃⋊#95	—	3	2	no	—
56	#830	—	6	2	no	—
57	Dic₃⋊C₈	—	6	4	no	—
58	C₂³×C₉	—	2	2	no	—
59	C₂³×C₉	—	2	2	no	—
60	C₂³×C₉	C₄⋊C₄	1	2	yes	—
61	C₂³×C₉	C₂×C₈	1	2	yes	—
62	C₂³×C₉	M₁₆	1	2	yes	—
63	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	4	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₉	#41	1	2	yes	—
73	C₂×Dic₉	—	2	2	no	—
74	C₂×Dic₉	—	2	2	no	—
75	C₂×Dic₉	—	2	2	no	—
76	C₂×Dic₉	—	2	2	no	—
77	C₂×Dic₉	—	2	2	no	—
78	C₂×Dic₉	C₂×C₈	1	4	yes	complemented
79	C₈×C₉	—	2	4	no	—
80	C₉⋊C₈	—	4	1	no	—
81	C₉⋊C₈	—	4	1	no	—
82	C₂³×C₈	D₉	1	1	yes	characteristic
83	C₂×C₄⋊C₈	—	9	2	no	—
84	C₂×C₂²⋊C₈	—	9	1	no	—
85	C₄×C₂²⋊C₄	—	9	1	no	—
86	#336	—	9	2	no	—
87	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	Dic₆	1	2	yes	—
91	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
92	C₂²×C₃×C₄	#67	1	2	yes	—
93	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
94	C₂²×Dic₃	—	3	2	no	—
95	C₂²×Dic₃	—	3	2	no	—
96	C₂×C₃×C₈	—	2	8	no	—
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	#67	1	4	yes	—
100	#217	—	3	4	no	—
101	#217	—	6	2	no	—
102	C₄×Dic₃	—	6	2	no	—
103	C₄×Dic₃	—	6	2	no	—
104	C₂×C₃⋊C₈	—	6	2	no	—
105	C₂×C₃⋊C₈	—	6	2	no	—
106	C₂×C₃⋊C₈	—	6	1	no	—
107	C₂×C₃⋊C₈	—	6	1	no	—
108	C₂²×C₉	—	2	2	no	—
109	C₂²×C₉	—	2	2	no	—
110	C₂²×C₉	—	2	4	no	—
111	C₂²×C₉	—	2	4	no	—
112	C₂²×C₉	C₄⋊C₈	1	4	yes	complemented
113	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₉	C₄×D₄	1	2	yes	—
115	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
116	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂²×C₉	#131	1	2	yes	—
118	C₄×C₉	—	2	2	no	—
119	C₄×C₉	#120	1	1	yes	characteristic
120	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
121	C₄×C₉	#123	1	1	yes	characteristic
122	Dic₉	—	2	4	no	—
123	Dic₉	—	4	2	no	—
124	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	9	1	no	—
128	C₂²×C₈	—	9	1	no	—
129	C₂²×C₈	C₂×D₉	1	2	yes	—
130	C₂×C₂²⋊C₄	—	9	1	no	—
131	C₂×C₂²⋊C₄	—	9	1	no	—
132	C₂×C₄²	—	9	2	no	—
133	C₄⋊C₈	—	18	4	no	—
134	C₂²⋊C₈	—	18	2	no	—
135	#95	—	9	2	no	—
136	C₂³×C₃	—	2	2	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	D₃×C₈	1	2	yes	—
139	C₂³×C₃	ΓL(1,𝔽₂₅)	1	2	yes	—
140	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
141	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
147	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
148	C₂×C₃×C₄	#235	1	2	yes	—
149	C₂×C₃×C₄	C₂×#67	1	2	yes	—
150	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
151	C₂×Dic₃	—	3	4	no	—
152	C₂×Dic₃	—	6	2	no	—
153	C₂×Dic₃	—	6	2	no	—
154	C₂×Dic₃	—	6	2	no	—
155	C₂×Dic₃	—	6	2	no	—
156	C₂×Dic₃	—	6	2	no	—
157	C₃×C₈	—	2	4	no	—
158	C₃⋊C₈	—	12	1	no	—
159	C₃⋊C₈	—	12	1	no	—
160	C₂×C₉	—	2	4	no	—
161	C₂×C₉	#385	1	1	yes	characteristic
162	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₉	#424	1	1	yes	characteristic
164	C₂×C₉	C₈×D₄	1	2	yes	—
165	C₂×C₉	#435	1	2	yes	—
166	C₂⁴	C₄×D₉	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	9	2	no	—
170	C₂²×C₄	—	9	2	no	—
171	C₂²×C₄	Dic₁₈	1	2	yes	—
172	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
173	C₂²×C₄	#611	1	2	yes	—
174	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	18	2	no	—
179	C₂×C₈	—	18	1	no	—
180	C₂×C₈	—	18	1	no	—
181	C₂×C₈	—	18	2	no	—
182	C₂×C₈	#611	1	4	yes	—
183	C₂²⋊C₄	—	9	4	no	—
184	C₂²⋊C₄	—	18	2	no	—
185	C₄²	—	18	2	no	—
186	C₄²	—	18	2	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	Dic₃⋊C₈	1	4	yes	—
192	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
193	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
194	C₂²×C₃	#901	1	2	yes	—
195	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
196	C₂²×C₃	C₄×#67	1	2	yes	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	#924	1	2	yes	—
199	C₃×C₄	#929	1	1	yes	characteristic
200	C₃×C₄	#953	1	1	yes	characteristic
201	Dic₃	—	6	4	no	—
202	Dic₃	—	12	2	no	—
203	C₉	#1850	1	1	yes	characteristic, complemented, Sylow
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₈×D₉	1	2	yes	—
207	C₂³	#3663	1	2	yes	—
208	C₂³	Dic₉⋊C₄	1	2	yes	—
209	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	9	4	no	—
215	C₂×C₄	—	18	2	no	—
216	C₂×C₄	—	18	2	no	—
217	C₂×C₄	—	18	2	no	—
218	C₂×C₄	—	18	2	no	—
219	C₂×C₄	—	18	2	no	—
220	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
221	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
222	C₂×C₄	#3697	1	2	yes	—
223	C₂×C₄	C₂×#611	1	1	yes	characteristic
224	C₂×C₄	C₂×#611	1	2	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	36	1	no	—
227	C₈	—	36	1	no	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	C₂×Dic₃⋊C₈	1	1	yes	characteristic
230	C₂×C₃	#5102	1	1	yes	characteristic
231	C₂×C₃	C₈×#67	1	2	yes	—
232	C₂×C₃	#5111	1	2	yes	—
233	C₂×C₃	#5210	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	Dic₉⋊C₈	1	4	yes	complemented
239	C₂²	C₂×C₈×D₉	1	1	yes	characteristic
240	C₂²	C₂×#3663	1	1	yes	characteristic
241	C₂²	#63464	1	2	yes	—
242	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
243	C₂²	C₄×#611	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	18	4	no	—
246	C₄	—	36	2	no	—
247	C₄	#63486	1	2	yes	—
248	C₄	#63491	1	1	yes	characteristic
249	C₄	#63515	1	1	yes	characteristic
250	C₃	#71974	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	C₂×Dic₉⋊C₈	1	1	yes	characteristic
253	C₂	#10588783	1	1	yes	characteristic
254	C₂	C₈×#611	1	2	yes	—
255	C₂	#10588792	1	2	yes	—
256	C₂	#10588890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂³×C₈×C₉	C₂
socle	C₂³×C₃	C₂×C₄×D₃
prime radical	C₂⁴×C₃	C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂³×C₈	D₉
hypocenter	C₉	#1850
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₉⋊C₈	C₂	semidirect
5. #10588228	C₂	semidirect
38. #3676	C₈	semidirect
78. C₂×Dic₉	C₂×C₈	semidirect
112. C₂²×C₉	C₄⋊C₈	semidirect
203. C₉	#1850	semidirect
238. C₂²	Dic₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₄×#3676	2
C₂×#63389	2
C₂×Dic₉⋊C₈	2
#10588228	2
#71974	3