Subgroups — C₂²×#63500 #11900267

Order 1152.

3792 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#611	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄×D₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#3676	C₂	1	2	yes	maximal, complemented
5	C₂×#63500	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#938	—	3	1	no	maximal
8	C₂⁵×C₉	C₂²	1	2	yes	—
9	C₂⁴×D₉	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	—	2	8	no	—
11	C₂²×D₄×C₉	C₂²	1	12	yes	complemented
12	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂²×#611	—	2	16	no	—
14	C₂²×#611	C₂²	1	24	yes	complemented
15	C₂³×Dic₉	C₂²	1	1	yes	characteristic
16	C₂³×Dic₉	C₂²	1	2	yes	—
17	C₂²×C₄×D₉	—	2	8	no	—
18	C₂²×C₄×D₉	C₂²	1	12	yes	complemented
19	C₂×#3676	C₂²	1	24	yes	complemented
20	#63500	C₂²	1	64	yes	direct product factor
21	C₂×C₄⋊Dic₉	C₂²	1	12	yes	complemented
22	C₂³×C₃×D₄	D₃	1	1	yes	characteristic
23	C₂³×#67	—	3	2	no	—
24	C₂³×C₄×D₃	—	3	1	no	—
25	C₂²×#217	—	3	2	no	—
26	C₂×#938	—	3	24	no	—
27	C₂²×C₄⋊Dic₃	—	3	1	no	—
28	C₂⁴×C₉	—	2	8	no	—
29	C₂⁴×C₉	—	2	8	no	—
30	C₂⁴×C₉	—	2	8	no	—
31	C₂⁴×C₉	C₂³	1	12	yes	—
32	C₂⁴×C₉	C₂³	1	1	yes	characteristic
33	C₂³×D₉	—	2	4	no	—
34	C₂³×D₉	—	2	4	no	—
35	C₂³×D₉	D₄	1	8	yes	complemented
36	C₂³×D₉	C₂³	1	6	yes	—
37	C₂×D₄×C₉	—	2	48	no	—
38	C₂×D₄×C₉	C₂³	1	16	yes	complemented
39	C₂²×C₄×C₉	D₄	1	8	yes	complemented
40	C₂²×C₄×C₉	C₂³	1	6	yes	—
41	C₂×#611	—	2	96	no	—
42	C₂×#611	C₂³	1	32	yes	complemented
43	C₂²×Dic₉	—	2	4	no	—
44	C₂²×Dic₉	—	2	8	no	—
45	C₂²×Dic₉	C₂³	1	6	yes	—
46	C₂²×Dic₉	C₂³	1	12	yes	—
47	C₂×C₄×D₉	—	2	48	no	—
48	C₂×C₄×D₉	C₂³	1	16	yes	complemented
49	#3676	C₂³	1	32	yes	complemented
50	C₄⋊Dic₉	C₂³	1	16	yes	complemented
51	C₂²×#121	—	9	1	no	Sylow
52	C₂⁵×C₃	C₂×D₃	1	2	yes	—
53	C₂⁴×D₃	—	3	1	no	—
54	C₂²×C₃×D₄	—	2	8	no	—
55	C₂²×C₃×D₄	C₂×D₃	1	12	yes	—
56	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂²×#67	—	3	24	no	—
58	C₂²×#67	—	6	16	no	—
59	C₂³×Dic₃	—	3	1	no	—
60	C₂³×Dic₃	—	3	2	no	—
61	C₂²×C₄×D₃	—	3	12	no	—
62	C₂²×C₄×D₃	—	6	8	no	—
63	C₂×#217	—	3	24	no	—
64	#938	—	3	64	no	—
65	C₂×C₄⋊Dic₃	—	3	12	no	—
66	C₂³×C₉	—	2	24	no	—
67	C₂³×C₉	—	2	24	no	—
68	C₂³×C₉	—	2	24	no	—
69	C₂³×C₉	—	4	32	no	—
70	C₂³×C₉	C₂×D₄	1	4	yes	—
71	C₂³×C₉	C₂×D₄	1	4	yes	—
72	C₂³×C₉	#41	1	4	yes	—
73	C₂³×C₉	C₂⁴	1	8	yes	—
74	C₂³×C₉	C₂⁴	1	3	yes	—
75	C₂²×D₉	—	2	32	no	—
76	C₂²×D₉	—	2	12	no	—
77	C₂²×D₉	—	2	12	no	—
78	C₂²×D₉	C₂×D₄	1	24	yes	complemented
79	C₂²×D₉	C₂⁴	1	4	yes	—
80	D₄×C₉	—	2	32	no	—
81	C₂×C₄×C₉	C₂×D₄	1	24	yes	complemented
82	C₂×C₄×C₉	C₂⁴	1	4	yes	—
83	#611	—	2	64	no	—
84	C₂×Dic₉	—	2	12	no	—
85	C₂×Dic₉	—	2	24	no	—
86	C₂×Dic₉	C₂⁴	1	8	yes	—
87	C₂×Dic₉	C₂⁴	1	4	yes	—
88	C₄×D₉	—	2	32	no	—
89	C₂³×D₄	—	9	2	no	—
90	C₂³×D₄	D₉	1	1	yes	characteristic, complemented
91	C₂⁴×C₄	—	9	1	no	—
92	C₂×#121	—	9	24	no	—
93	C₂²×C₄⋊C₄	—	9	1	no	—
94	C₂²×C₂²⋊C₄	—	9	2	no	—
95	C₂⁴×C₃	—	2	8	no	—
96	C₂⁴×C₃	—	2	8	no	—
97	C₂⁴×C₃	—	2	8	no	—
98	C₂⁴×C₃	C₂²×D₃	1	12	yes	—
99	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
100	C₂³×D₃	—	3	8	no	—
101	C₂³×D₃	—	3	6	no	—
102	C₂³×D₃	—	6	4	no	—
103	C₂³×D₃	—	6	4	no	—
104	C₂×C₃×D₄	—	2	48	no	—
105	C₂×C₃×D₄	C₂²×D₃	1	16	yes	—
106	C₂²×C₃×C₄	#67	1	8	yes	—
107	C₂²×C₃×C₄	C₂²×D₃	1	6	yes	—
108	C₂×#67	—	3	32	no	—
109	C₂×#67	—	6	96	no	—
110	C₂²×Dic₃	—	3	12	no	—
111	C₂²×Dic₃	—	3	6	no	—
112	C₂²×Dic₃	—	6	8	no	—
113	C₂²×Dic₃	—	6	4	no	—
114	C₂×C₄×D₃	—	3	16	no	—
115	C₂×C₄×D₃	—	6	48	no	—
116	#217	—	3	32	no	—
117	C₄⋊Dic₃	—	3	16	no	—
118	C₂²×C₉	—	2	8	no	—
119	C₂²×C₉	—	2	8	no	—
120	C₂²×C₉	—	2	8	no	—
121	C₂²×C₉	—	4	48	no	—
122	C₂²×C₉	#121	1	16	yes	complemented
123	C₂²×C₉	C₂²×D₄	1	6	yes	—
124	C₂²×C₉	C₂²×D₄	1	6	yes	—
125	C₂²×C₉	C₂×#41	1	6	yes	—
126	C₂²×C₉	C₂⁵	1	1	yes	characteristic
127	C₂×D₉	—	2	48	no	—
128	C₂×D₉	—	2	4	no	—
129	C₂×D₉	—	2	4	no	—
130	C₂×D₉	C₂²×D₄	1	8	yes	complemented
131	C₄×C₉	C₂²×D₄	1	8	yes	complemented
132	Dic₉	—	2	4	no	—
133	Dic₉	—	2	8	no	—
134	C₂⁵	—	9	1	no	—
135	C₂⁵	C₂×D₉	1	2	yes	—
136	C₂²×D₄	—	2	8	no	—
137	C₂²×D₄	—	9	24	no	—
138	C₂²×D₄	—	18	16	no	—
139	C₂²×D₄	C₂×D₉	1	12	yes	complemented
140	C₂³×C₄	—	9	1	no	—
141	C₂³×C₄	—	9	2	no	—
142	C₂³×C₄	—	9	12	no	—
143	C₂³×C₄	—	18	8	no	—
144	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
145	#121	—	9	64	no	—
146	C₂×C₄⋊C₄	—	9	12	no	—
147	C₂×C₂²⋊C₄	—	9	24	no	—
148	C₂³×C₃	—	2	24	no	—
149	C₂³×C₃	—	2	24	no	—
150	C₂³×C₃	—	2	24	no	—
151	C₂³×C₃	—	4	32	no	—
152	C₂³×C₃	D₃×D₄	1	4	yes	—
153	C₂³×C₃	#237	1	4	yes	—
154	C₂³×C₃	C₂×#67	1	4	yes	—
155	C₂³×C₃	C₂³×D₃	1	8	yes	—
156	C₂³×C₃	C₂³×D₃	1	3	yes	—
157	C₂²×D₃	—	3	24	no	—
158	C₂²×D₃	—	3	4	no	—
159	C₂²×D₃	—	6	32	no	—
160	C₂²×D₃	—	6	12	no	—
161	C₂²×D₃	—	6	12	no	—
162	C₃×D₄	—	2	32	no	—
163	C₂×C₃×C₄	C₂×#67	1	24	yes	—
164	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
165	#67	—	6	64	no	—
166	C₂×Dic₃	—	3	8	no	—
167	C₂×Dic₃	—	3	4	no	—
168	C₂×Dic₃	—	6	24	no	—
169	C₂×Dic₃	—	6	12	no	—
170	C₄×D₃	—	6	32	no	—
171	C₂×C₉	—	4	8	no	—
172	C₂×C₉	C₂×#121	1	12	yes	complemented
173	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
174	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
175	C₂×C₉	C₂²×#41	1	1	yes	characteristic
176	D₉	—	2	8	no	—
177	C₂⁴	—	2	8	no	—
178	C₂⁴	—	2	8	no	—
179	C₂⁴	—	2	8	no	—
180	C₂⁴	—	9	8	no	—
181	C₂⁴	—	9	6	no	—
182	C₂⁴	—	18	4	no	—
183	C₂⁴	—	18	4	no	—
184	C₂⁴	C₂²×D₉	1	12	yes	—
185	C₂⁴	C₂²×D₉	1	1	yes	characteristic
186	C₂×D₄	—	2	48	no	—
187	C₂×D₄	—	9	32	no	—
188	C₂×D₄	—	18	96	no	—
189	C₂×D₄	C₂²×D₉	1	16	yes	complemented
190	C₂²×C₄	—	9	6	no	—
191	C₂²×C₄	—	9	12	no	—
192	C₂²×C₄	—	9	16	no	—
193	C₂²×C₄	—	18	8	no	—
194	C₂²×C₄	—	18	4	no	—
195	C₂²×C₄	—	18	48	no	—
196	C₂²×C₄	#611	1	8	yes	complemented
197	C₂²×C₄	C₂²×D₉	1	6	yes	—
198	C₄⋊C₄	—	9	16	no	—
199	C₂²⋊C₄	—	9	32	no	—
200	C₂²×C₃	—	2	8	no	—
201	C₂²×C₃	—	2	8	no	—
202	C₂²×C₃	—	2	8	no	—
203	C₂²×C₃	—	4	48	no	—
204	C₂²×C₃	#938	1	16	yes	—
205	C₂²×C₃	C₂×D₃×D₄	1	6	yes	—
206	C₂²×C₃	C₂×#237	1	6	yes	—
207	C₂²×C₃	C₂²×#67	1	6	yes	—
208	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
209	C₂×D₃	—	3	8	no	—
210	C₂×D₃	—	6	4	no	—
211	C₂×D₃	—	6	4	no	—
212	C₂×D₃	—	6	48	no	—
213	C₃×C₄	C₂²×#67	1	8	yes	—
214	Dic₃	—	6	8	no	—
215	Dic₃	—	6	4	no	—
216	C₉	C₂²×#121	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	2	24	no	—
218	C₂³	—	2	24	no	—
219	C₂³	—	2	24	no	—
220	C₂³	—	4	32	no	—
221	C₂³	—	9	4	no	—
222	C₂³	—	9	24	no	—
223	C₂³	—	18	32	no	—
224	C₂³	—	18	12	no	—
225	C₂³	—	18	12	no	—
226	C₂³	D₄×D₉	1	4	yes	—
227	C₂³	#3699	1	4	yes	—
228	C₂³	C₂×#611	1	4	yes	—
229	C₂³	C₂³×D₉	1	8	yes	—
230	C₂³	C₂³×D₉	1	3	yes	—
231	D₄	—	2	32	no	—
232	D₄	—	18	64	no	—
233	C₂×C₄	—	9	4	no	—
234	C₂×C₄	—	9	8	no	—
235	C₂×C₄	—	18	24	no	—
236	C₂×C₄	—	18	12	no	—
237	C₂×C₄	—	18	32	no	—
238	C₂×C₄	C₂×#611	1	24	yes	complemented
239	C₂×C₄	C₂³×D₉	1	4	yes	—
240	C₂×C₃	—	4	8	no	—
241	C₂×C₃	C₂×#938	1	12	yes	—
242	C₂×C₃	C₂²×D₃×D₄	1	1	yes	characteristic
243	C₂×C₃	C₂²×#237	1	1	yes	characteristic
244	C₂×C₃	C₂³×#67	1	1	yes	characteristic
245	D₃	—	6	8	no	—
246	C₂²	—	2	8	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	4	48	no	—
250	C₂²	—	9	8	no	—
251	C₂²	—	18	4	no	—
252	C₂²	—	18	4	no	—
253	C₂²	—	18	48	no	—
254	C₂²	#63500	1	16	yes	direct product factor
255	C₂²	C₂×D₄×D₉	1	6	yes	—
256	C₂²	C₂×#3699	1	6	yes	—
257	C₂²	C₂²×#611	1	6	yes	—
258	C₂²	C₂⁴×D₉	1	1	yes	characteristic
259	C₄	—	18	8	no	—
260	C₄	—	18	4	no	—
261	C₄	C₂²×#611	1	8	yes	complemented
262	C₃	C₂²×#938	1	1	yes	characteristic
263	C₂	—	4	8	no	—
264	C₂	—	18	8	no	—
265	C₂	C₂×#63500	1	12	yes	direct product factor
266	C₂	C₂²×D₄×D₉	1	1	yes	characteristic
267	C₂	C₂²×#3699	1	1	yes	characteristic
268	C₂	C₂³×#611	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂⁵
Frattini	C₂²×C₃	C₂⁴×D₃
Fitting	C₂³×D₄×C₉	C₂
socle	C₂⁴×C₃	C₂²×D₃
hypercenter	C₂³×D₄	D₉
hypocenter	C₉	C₂²×#121
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×C₉	C₂	semidirect
2. C₂³×#611	C₂	semidirect
3. C₂³×C₄×D₉	C₂	semidirect
4. C₂²×#3676	C₂	semidirect
5. C₂×#63500	C₂	direct (N × H)
6. C₂²×C₄⋊Dic₉	C₂	semidirect
11. C₂²×D₄×C₉	C₂²	semidirect
14. C₂²×#611	C₂²	semidirect
18. C₂²×C₄×D₉	C₂²	semidirect
19. C₂×#3676	C₂²	semidirect
20. #63500	C₂²	direct (N × H)
21. C₂×C₄⋊Dic₉	C₂²	semidirect
35. C₂³×D₉	D₄	semidirect
38. C₂×D₄×C₉	C₂³	semidirect
39. C₂²×C₄×C₉	D₄	semidirect
42. C₂×#611	C₂³	semidirect
48. C₂×C₄×D₉	C₂³	semidirect
49. #3676	C₂³	semidirect
50. C₄⋊Dic₉	C₂³	semidirect
78. C₂²×D₉	C₂×D₄	semidirect
81. C₂×C₄×C₉	C₂×D₄	semidirect
90. C₂³×D₄	D₉	semidirect
122. C₂²×C₉	#121	semidirect
130. C₂×D₉	C₂²×D₄	semidirect
131. C₄×C₉	C₂²×D₄	semidirect
139. C₂²×D₄	C₂×D₉	semidirect
172. C₂×C₉	C₂×#121	semidirect
189. C₂×D₄	C₂²×D₉	semidirect
196. C₂²×C₄	#611	semidirect
216. C₉	C₂²×#121	semidirect
238. C₂×C₄	C₂×#611	semidirect
254. C₂²	#63500	direct (N × H)
261. C₄	C₂²×#611	semidirect
265. C₂	C₂×#63500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₉	2
C₂³×#611	2
C₂³×C₄×D₉	2
C₂²×#3676	2
C₂×#63500	2
C₂²×C₄⋊Dic₉	2
C₂²×#938	3

Subgroups — C22×#63500 #11900267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63500 #11900267