Subgroups — C₂×#10588605 #11858700

Order 1152.

792 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	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63389	C₂	1	2	yes	maximal
3	#10588605	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₉⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×#4925	—	3	1	no	maximal
7	C₂³×C₄×C₉	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₉	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₉	C₄	1	4	yes	—
12	C₂×C₄²×C₉	C₄	1	1	yes	characteristic
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉	C₂²	1	4	yes	—
16	C₂²×C₉⋊C₈	C₂²	1	4	yes	—
17	#63389	C₂²	1	8	yes	—
18	C₄⋊(C₉⋊C₈)	C₂²	1	8	yes	complemented
19	C₄×C₉⋊C₈	C₂²	1	8	yes	complemented
20	C₂²×C₃×C₄²	D₃	1	1	yes	characteristic
21	C₂×#830	—	3	2	no	—
22	#4925	—	3	8	no	—
23	C₂×C₄⋊(C₃⋊C₈)	—	3	2	no	—
24	C₂×C₄×C₃⋊C₈	—	3	2	no	—
25	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	—	2	4	no	—
27	C₂²×C₄×C₉	—	2	4	no	—
28	C₂²×C₄×C₉	C₈	1	8	yes	complemented
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₄²×C₉	C₂×C₄	1	4	yes	—
40	C₄²×C₉	C₂×C₄	1	4	yes	—
41	C₄²×C₉	C₂×C₄	1	2	yes	—
42	C₄²×C₉	C₂×C₄	1	2	yes	—
43	C₄²×C₉	C₂³	1	2	yes	—
44	C₄²×C₉	C₂³	1	2	yes	—
45	C₂×C₉⋊C₈	—	2	8	no	—
46	C₂×C₉⋊C₈	C₂³	1	8	yes	—
47	C₂×#431	—	9	1	no	Sylow
48	C₂³×C₃×C₄	Dic₃	1	1	yes	characteristic
49	C₂³×C₃×C₄	Dic₃	1	1	yes	characteristic
50	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
52	C₂×C₃×C₄²	Dic₃	1	4	yes	—
53	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₃⋊C₈	—	3	4	no	—
58	#830	—	3	8	no	—
59	C₄⋊(C₃⋊C₈)	—	3	8	no	—
60	C₄×C₃⋊C₈	—	3	8	no	—
61	C₂³×C₉	—	2	2	no	—
62	C₂³×C₉	—	2	2	no	—
63	C₂³×C₉	C₂×C₈	1	4	yes	—
64	C₂³×C₉	C₂²×C₄	1	2	yes	—
65	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	8	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	C₂×C₈	1	4	yes	—
72	C₂×C₄×C₉	C₂×C₈	1	4	yes	—
73	C₂×C₄×C₉	C₂×C₈	1	16	yes	complemented
74	C₂×C₄×C₉	M₁₆	1	4	yes	—
75	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₉	#41	1	4	yes	—
82	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
83	C₉⋊C₈	—	2	8	no	—
84	C₂²×C₄²	D₉	1	1	yes	characteristic
85	#431	—	9	8	no	—
86	C₂×C₄⋊C₈	—	9	2	no	—
87	C₂×C₂²⋊C₈	—	9	2	no	—
88	C₂×C₄×C₈	—	9	2	no	—
89	C₂⁴×C₃	C₂×Dic₃	1	1	yes	characteristic
90	C₂²×C₃×C₄	—	2	4	no	—
91	C₂²×C₃×C₄	—	2	4	no	—
92	C₂²×C₃×C₄	C₃⋊C₈	1	8	yes	—
93	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
95	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
97	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
100	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₃×C₄²	C₂×Dic₃	1	4	yes	—
104	C₃×C₄²	C₂×Dic₃	1	4	yes	—
105	C₃×C₄²	C₂×Dic₃	1	2	yes	—
106	C₃×C₄²	C₂×Dic₃	1	2	yes	—
107	C₃×C₄²	C₂²×D₃	1	2	yes	—
108	C₃×C₄²	C₂²×D₃	1	2	yes	—
109	C₂×C₃⋊C₈	—	3	8	no	—
110	C₂×C₃⋊C₈	—	6	8	no	—
111	C₂²×C₉	—	2	2	no	—
112	C₂²×C₉	—	2	2	no	—
113	C₂²×C₉	—	2	8	no	—
114	C₂²×C₉	#117	1	2	yes	—
115	C₂²×C₉	C₂²×C₈	1	4	yes	—
116	C₂²×C₉	C₂²×C₈	1	2	yes	—
117	C₂²×C₉	C₂×M₁₆	1	2	yes	—
118	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₉	—	2	2	no	—
120	C₄×C₉	—	2	2	no	—
121	C₄×C₉	#117	1	2	yes	—
122	C₄×C₉	C₂²×C₈	1	4	yes	—
123	C₄×C₉	C₂²×C₈	1	4	yes	—
124	C₄×C₉	C₂×M₁₆	1	4	yes	—
125	C₄×C₉	C₂×#41	1	2	yes	—
126	C₂³×C₄	Dic₉	1	1	yes	characteristic
127	C₂³×C₄	Dic₉	1	1	yes	characteristic
128	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
129	C₂²×C₈	—	9	4	no	—
130	C₂×C₄²	Dic₉	1	1	yes	characteristic
131	C₂×C₄²	Dic₉	1	1	yes	characteristic
132	C₂×C₄²	Dic₉	1	4	yes	—
133	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
134	C₂×C₄²	C₂×D₉	1	4	yes	—
135	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
136	C₄⋊C₈	—	9	8	no	—
137	C₂²⋊C₈	—	9	8	no	—
138	C₄×C₈	—	9	8	no	—
139	C₂³×C₃	—	2	2	no	—
140	C₂³×C₃	—	2	2	no	—
141	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
142	C₂³×C₃	C₂²×Dic₃	1	2	yes	—
143	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	8	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
150	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
151	C₂×C₃×C₄	C₂×C₃⋊C₈	1	16	yes	—
152	C₂×C₃×C₄	#208	1	4	yes	—
153	C₂×C₃×C₄	#235	1	4	yes	—
154	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
155	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
156	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
157	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
158	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
159	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
161	C₃⋊C₈	—	6	8	no	—
162	C₂×C₉	—	2	4	no	—
163	C₂×C₉	#431	1	4	yes	complemented
164	C₂×C₉	C₂×#117	1	1	yes	characteristic
165	C₂×C₉	C₂³×C₈	1	1	yes	characteristic
166	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
167	C₂⁴	C₂×Dic₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	C₉⋊C₈	1	8	yes	complemented
171	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₉	1	2	yes	—
173	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
174	C₂²×C₄	C₂×Dic₉	1	4	yes	—
175	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
176	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
177	C₂²×C₄	C₂×Dic₉	1	4	yes	—
178	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₉	1	2	yes	—
180	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
181	C₂×C₈	—	9	8	no	—
182	C₂×C₈	—	18	8	no	—
183	C₄²	C₂×Dic₉	1	4	yes	—
184	C₄²	C₂×Dic₉	1	4	yes	—
185	C₄²	C₂×Dic₉	1	2	yes	—
186	C₄²	C₂×Dic₉	1	2	yes	—
187	C₄²	C₂²×D₉	1	2	yes	—
188	C₄²	C₂²×D₉	1	2	yes	—
189	C₂²×C₃	—	2	8	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
193	C₂²×C₃	C₂²×C₃⋊C₈	1	2	yes	—
194	C₂²×C₃	C₂×#208	1	2	yes	—
195	C₂²×C₃	#926	1	2	yes	—
196	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	C₂²×C₃⋊C₈	1	4	yes	—
200	C₃×C₄	C₂²×C₃⋊C₈	1	4	yes	—
201	C₃×C₄	C₂×#208	1	4	yes	—
202	C₃×C₄	#926	1	2	yes	—
203	C₃×C₄	C₂×#235	1	2	yes	—
204	C₉	C₂×#431	1	1	yes	characteristic, complemented, Sylow
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	C₂×C₉⋊C₈	1	4	yes	—
208	C₂³	C₂²×Dic₉	1	2	yes	—
209	C₂³	C₂²×Dic₉	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	C₂×C₉⋊C₈	1	4	yes	—
216	C₂×C₄	C₂×C₉⋊C₈	1	4	yes	—
217	C₂×C₄	C₂×C₉⋊C₈	1	16	yes	complemented
218	C₂×C₄	#3667	1	4	yes	—
219	C₂×C₄	#3697	1	4	yes	—
220	C₂×C₄	C₂²×Dic₉	1	2	yes	—
221	C₂×C₄	C₂²×Dic₉	1	2	yes	—
222	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
223	C₂×C₄	C₂²×Dic₉	1	2	yes	—
224	C₂×C₄	C₂²×Dic₉	1	2	yes	—
225	C₂×C₄	C₂²×Dic₉	1	2	yes	—
226	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
227	C₈	—	18	8	no	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	#4925	1	4	yes	—
230	C₂×C₃	C₂³×C₃⋊C₈	1	1	yes	characteristic
231	C₂×C₃	C₂²×#208	1	1	yes	characteristic
232	C₂×C₃	C₂×#926	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	2	no	—
236	C₂²	C₂²×C₉⋊C₈	1	4	yes	—
237	C₂²	C₂²×C₉⋊C₈	1	2	yes	—
238	C₂²	C₂×#3667	1	2	yes	—
239	C₂²	#63488	1	2	yes	—
240	C₂²	C₂³×Dic₉	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	C₂²×C₉⋊C₈	1	4	yes	—
244	C₄	C₂²×C₉⋊C₈	1	4	yes	—
245	C₄	C₂×#3667	1	4	yes	—
246	C₄	#63488	1	2	yes	—
247	C₄	C₂×#3697	1	2	yes	—
248	C₃	C₂×#4925	1	1	yes	characteristic
249	C₂	—	2	4	no	—
250	C₂	#10588605	1	4	yes	direct product factor
251	C₂	C₂³×C₉⋊C₈	1	1	yes	characteristic
252	C₂	C₂²×#3667	1	1	yes	characteristic
253	C₂	C₂×#63488	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₉
commutator	C₂×C₉	C₂³×C₈
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂²×C₄²×C₉	C₂
socle	C₂³×C₃	C₂²×Dic₃
prime radical	C₂⁴×C₃	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂²×C₄²	D₉
hypocenter	C₉	C₂×#431
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. #10588605	C₂	direct (N × H)
4. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
5. C₂×C₄×C₉⋊C₈	C₂	semidirect
18. C₄⋊(C₉⋊C₈)	C₂²	semidirect
19. C₄×C₉⋊C₈	C₂²	semidirect
28. C₂²×C₄×C₉	C₈	semidirect
73. C₂×C₄×C₉	C₂×C₈	semidirect
163. C₂×C₉	#431	semidirect
170. C₂²×C₄	C₉⋊C₈	semidirect
204. C₉	C₂×#431	semidirect
217. C₂×C₄	C₂×C₉⋊C₈	semidirect
250. C₂	#10588605	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₉	2
C₂×#63389	2
#10588605	2
C₂×C₄⋊(C₉⋊C₈)	2
C₂×C₄×C₉⋊C₈	2
C₂×#4925	3

Subgroups — C2×#10588605 #11858700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588605 #11858700