Subgroups — C₂×#10588522 #11859262

Order 1152.

828 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	C₂²×#3663	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₉)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588522	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4838	—	3	1	no	maximal
8	C₂²×D₃₆	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₉	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₉)⋊C₄	C₄	1	2	yes	—
13	C₂×C₄⋊Dic₉	C₄	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×#3663	—	2	8	no	—
17	C₂×#3663	C₂²	1	8	yes	complemented
18	C₄×D₃₆	C₄	1	4	yes	—
19	C₄×D₃₆	C₂²	1	4	yes	—
20	C₉×C₄⋊C₈	C₂²	1	4	yes	complemented
21	(C₂×D₉)⋊C₈	C₂²	1	8	yes	complemented
22	C₄×C₉⋊C₈	C₂²	1	4	yes	complemented
23	C₂²×ΓL(1,𝔽₂₅)	—	3	2	no	—
24	C₂×C₄×D₁₂	—	3	1	no	—
25	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
26	C₂×(C₂×D₃)⋊C₈	—	3	2	no	—
27	C₂×C₄×C₃⋊C₈	—	3	1	no	—
28	#4838	—	3	8	no	—
29	C₂³×D₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂²×Dic₉	C₂×C₄	1	2	yes	—
34	C₂×D₃₆	—	2	4	no	—
35	C₂×D₃₆	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₉	—	2	4	no	—
37	C₂×C₄×D₉	—	2	4	no	—
38	C₂×C₄×D₉	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₉	C₂³	1	4	yes	—
40	C₂×C₈×C₉	—	2	4	no	—
41	C₂×C₈×C₉	C₂³	1	4	yes	—
42	C₄²×C₉	C₂×C₄	1	2	yes	—
43	C₄²×C₉	C₂³	1	2	yes	—
44	(C₂×D₉)⋊C₄	C₂×C₄	1	8	yes	—
45	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
46	C₂×C₉⋊C₈	—	2	2	no	—
47	C₂×C₉⋊C₈	D₄	1	4	yes	complemented
48	C₂×C₉⋊C₈	C₂³	1	2	yes	—
49	C₂×C₉⋊C₈	C₂³	1	2	yes	—
50	#3663	—	2	16	no	—
51	C₂×#436	—	9	1	no	Sylow
52	C₂²×D₁₂	—	3	1	no	—
53	C₂²×C₄×D₃	—	3	2	no	—
54	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×(C₂×D₃)⋊C₄	—	3	2	no	—
57	C₂×C₄⋊Dic₃	—	3	1	no	—
58	C₂²×C₃⋊C₈	—	3	1	no	—
59	C₂²×C₃⋊C₈	—	3	1	no	—
60	C₂×ΓL(1,𝔽₂₅)	—	3	8	no	—
61	C₂×ΓL(1,𝔽₂₅)	—	6	8	no	—
62	C₄×D₁₂	—	3	4	no	—
63	C₄×D₁₂	—	3	4	no	—
64	C₃×C₄⋊C₈	C₂×D₃	1	4	yes	—
65	(C₂×D₃)⋊C₈	—	3	8	no	—
66	C₄×C₃⋊C₈	—	3	4	no	—
67	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₉	—	2	4	no	—
69	C₂²×D₉	—	2	4	no	—
70	C₂²×D₉	—	2	4	no	—
71	C₂²×D₉	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	M₁₆	1	4	yes	complemented
74	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₉	#41	1	2	yes	—
79	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₉	—	2	4	no	—
81	C₂×Dic₉	C₂²×C₄	1	4	yes	—
82	D₃₆	—	2	8	no	—
83	C₄×D₉	—	2	8	no	—
84	C₄×D₉	—	2	4	no	—
85	C₄×D₉	—	2	4	no	—
86	C₈×C₉	—	2	4	no	—
87	C₉⋊C₈	—	2	2	no	—
88	C₉⋊C₈	C₂×D₄	1	4	yes	complemented
89	C₂²×M₁₆	—	9	2	no	—
90	C₂×C₄×D₄	—	9	1	no	—
91	#436	—	9	8	no	—
92	C₂×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
93	C₂×C₂²⋊C₈	—	9	2	no	—
94	C₂×C₄×C₈	—	9	1	no	—
95	C₂³×D₃	—	3	2	no	—
96	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂²×Dic₃	—	3	2	no	—
100	C₂×D₁₂	—	3	4	no	—
101	C₂×D₁₂	—	6	4	no	—
102	C₂×C₄×D₃	—	3	4	no	—
103	C₂×C₄×D₃	—	3	4	no	—
104	C₂×C₄×D₃	—	6	4	no	—
105	C₂×C₄×D₃	—	6	4	no	—
106	C₂×C₃×C₈	—	2	4	no	—
107	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
108	C₃×C₄²	C₄×D₃	1	2	yes	—
109	C₃×C₄²	C₂²×D₃	1	2	yes	—
110	(C₂×D₃)⋊C₄	—	3	8	no	—
111	C₄⋊Dic₃	—	3	4	no	—
112	C₂×C₃⋊C₈	—	3	4	no	—
113	C₂×C₃⋊C₈	—	3	2	no	—
114	C₂×C₃⋊C₈	—	3	2	no	—
115	C₂×C₃⋊C₈	—	6	2	no	—
116	ΓL(1,𝔽₂₅)	—	6	16	no	—
117	C₂²×C₉	C₄×D₄	1	2	yes	—
118	C₂²×C₉	C₂×M₁₆	1	2	yes	—
119	C₂²×C₉	#131	1	2	yes	—
120	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₉	—	2	4	no	—
122	C₂×D₉	—	2	4	no	—
123	C₂×D₉	—	2	4	no	—
124	C₂×D₉	—	4	8	no	—
125	C₄×C₉	—	2	2	no	—
126	C₄×C₉	C₄×D₄	1	2	yes	—
127	C₄×C₉	C₂×M₁₆	1	4	yes	complemented
128	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₉	C₂×#41	1	1	yes	characteristic
130	Dic₉	—	2	4	no	—
131	C₂²×D₄	—	9	1	no	—
132	C₂³×C₄	—	9	2	no	—
133	C₂×M₁₆	—	9	8	no	—
134	C₂×M₁₆	—	18	8	no	—
135	C₂²×C₈	—	9	1	no	—
136	C₂²×C₈	—	9	1	no	—
137	C₂²×C₈	C₂×D₉	1	2	yes	—
138	C₄×D₄	—	9	4	no	—
139	C₄×D₄	—	9	4	no	—
140	C₂×C₄⋊C₄	—	9	1	no	—
141	C₂×C₂²⋊C₄	—	9	2	no	—
142	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
143	C₄⋊C₈	C₂×D₉	1	4	yes	complemented
144	C₂²⋊C₈	—	9	8	no	—
145	C₄×C₈	—	9	4	no	—
146	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂²×D₃	—	3	4	no	—
148	C₂²×D₃	—	6	4	no	—
149	C₂²×D₃	—	6	4	no	—
150	C₂²×D₃	—	6	4	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
153	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
156	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
157	C₂×C₃×C₄	#239	1	2	yes	—
158	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
159	C₂×Dic₃	—	3	4	no	—
160	C₂×Dic₃	—	6	4	no	—
161	D₁₂	—	6	8	no	—
162	C₄×D₃	—	6	8	no	—
163	C₄×D₃	—	6	4	no	—
164	C₄×D₃	—	6	4	no	—
165	C₃×C₈	—	2	4	no	—
166	C₃⋊C₈	—	3	4	no	—
167	C₃⋊C₈	—	6	2	no	—
168	C₂×C₉	#436	1	4	yes	complemented
169	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
171	C₂×C₉	C₂×#131	1	1	yes	characteristic
172	D₉	—	4	4	no	—
173	C₂⁴	—	9	2	no	—
174	C₂×D₄	—	9	4	no	—
175	C₂×D₄	—	18	4	no	—
176	C₂²×C₄	—	9	2	no	—
177	C₂²×C₄	—	9	4	no	—
178	C₂²×C₄	—	9	4	no	—
179	C₂²×C₄	—	18	4	no	—
180	C₂²×C₄	—	18	4	no	—
181	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
182	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
184	M₁₆	—	18	16	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	9	4	no	—
187	C₂×C₈	—	9	2	no	—
188	C₂×C₈	—	9	2	no	—
189	C₂×C₈	—	18	2	no	—
190	C₂×C₈	C₂²×D₉	1	4	yes	—
191	C₄⋊C₄	—	9	4	no	—
192	C₂²⋊C₄	—	9	8	no	—
193	C₄²	C₄×D₉	1	2	yes	—
194	C₄²	C₂²×D₉	1	2	yes	—
195	C₂²×C₃	#893	1	2	yes	—
196	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
197	C₂²×C₃	#907	1	2	yes	—
198	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
199	C₂×D₃	—	6	4	no	—
200	C₂×D₃	—	6	4	no	—
201	C₂×D₃	—	6	4	no	—
202	C₂×D₃	—	12	8	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	#893	1	2	yes	—
205	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
206	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
207	C₃×C₄	C₂×#239	1	1	yes	characteristic
208	Dic₃	—	6	4	no	—
209	C₉	C₂×#436	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	9	4	no	—
211	C₂³	—	18	4	no	—
212	C₂³	—	18	4	no	—
213	C₂³	—	18	4	no	—
214	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
215	D₄	—	18	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	9	4	no	—
218	C₂×C₄	—	18	4	no	—
219	C₂×C₄	—	18	4	no	—
220	C₂×C₄	—	18	8	no	—
221	C₂×C₄	—	18	4	no	—
222	C₂×C₄	#3663	1	4	yes	complemented
223	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
224	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
226	C₂×C₄	D₄×D₉	1	2	yes	—
227	C₂×C₄	#3701	1	2	yes	—
228	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	—	9	4	no	—
231	C₈	—	18	2	no	—
232	C₂×C₃	#4838	1	4	yes	—
233	C₂×C₃	C₂×#893	1	1	yes	characteristic
234	C₂×C₃	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
235	C₂×C₃	C₂×#907	1	1	yes	characteristic
236	D₃	—	12	4	no	—
237	C₂²	—	18	4	no	—
238	C₂²	—	18	4	no	—
239	C₂²	—	18	4	no	—
240	C₂²	—	36	8	no	—
241	C₂²	#63455	1	2	yes	—
242	C₂²	C₂×#3663	1	2	yes	—
243	C₂²	#63469	1	2	yes	—
244	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	18	4	no	—
247	C₄	#63455	1	2	yes	—
248	C₄	C₂×#3663	1	4	yes	complemented
249	C₄	C₂×D₄×D₉	1	1	yes	characteristic
250	C₄	C₂×#3701	1	1	yes	characteristic
251	C₃	C₂×#4838	1	1	yes	characteristic
252	C₂	—	36	4	no	—
253	C₂	#10588522	1	4	yes	direct product factor
254	C₂	C₂×#63455	1	1	yes	characteristic
255	C₂	C₂²×#3663	1	1	yes	characteristic
256	C₂	C₂×#63469	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×C₄⋊C₈	C₂
socle	C₂³×C₃	C₂×C₄×D₃
prime radical	C₂²×D₃₆	C₄
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₉
hypocenter	C₉	C₂×#436
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#3663	C₂	semidirect
3. C₂×C₉×C₄⋊C₈	C₂	semidirect
4. C₂×(C₂×D₉)⋊C₈	C₂	semidirect
5. C₂×C₄×C₉⋊C₈	C₂	semidirect
6. #10588522	C₂	direct (N × H)
17. C₂×#3663	C₂²	semidirect
20. C₉×C₄⋊C₈	C₂²	semidirect
21. (C₂×D₉)⋊C₈	C₂²	semidirect
22. C₄×C₉⋊C₈	C₂²	semidirect
47. C₂×C₉⋊C₈	D₄	semidirect
73. C₂×C₄×C₉	M₁₆	semidirect
88. C₉⋊C₈	C₂×D₄	semidirect
92. C₂×C₄⋊C₈	D₉	semidirect
127. C₄×C₉	C₂×M₁₆	semidirect
143. C₄⋊C₈	C₂×D₉	semidirect
168. C₂×C₉	#436	semidirect
209. C₉	C₂×#436	semidirect
222. C₂×C₄	#3663	semidirect
248. C₄	C₂×#3663	semidirect
253. C₂	#10588522	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×#3663	2
C₂×C₄×D₃₆	2
C₂×C₉×C₄⋊C₈	2
C₂×(C₂×D₉)⋊C₈	2
C₂×C₄×C₉⋊C₈	2
#10588522	2
C₂×#4838	3

Subgroups — C2×#10588522 #11859262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588522 #11859262