Subgroups — C₂×#10588892 #11800982

Order 1152.

1848 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3676	C₂	1	2	yes	maximal
3	C₂²×C₄⋊Dic₉	C₂	1	2	yes	maximal, complemented
4	#10588892	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₉⋊#95	C₂	1	2	yes	maximal, complemented
6	C₂×#5212	—	3	1	no	maximal
7	C₂⁵×C₉	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₉	—	2	8	no	—
9	C₂³×C₄×C₉	C₄	1	8	yes	complemented
10	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₉	C₂²	1	4	yes	—
12	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂³×Dic₉	C₂²	1	4	yes	—
14	C₂×#3676	—	2	8	no	—
15	C₂×#3676	C₂²	1	8	yes	—
16	C₂×C₄⋊Dic₉	—	2	8	no	—
17	C₂×C₄⋊Dic₉	C₂²	1	8	yes	complemented
18	C₉⋊#95	C₂²	1	8	yes	complemented
19	C₂⁴×C₃×C₄	D₃	1	1	yes	characteristic
20	C₂²×#217	—	3	2	no	—
21	C₂²×C₄⋊Dic₃	—	3	2	no	—
22	#5212	—	3	8	no	—
23	C₂×C₃⋊#95	—	3	2	no	—
24	C₂⁴×C₉	—	2	4	no	—
25	C₂⁴×C₉	—	2	4	no	—
26	C₂⁴×C₉	C₂×C₄	1	4	yes	—
27	C₂⁴×C₉	D₄	1	4	yes	—
28	C₂⁴×C₉	Q₈	1	4	yes	—
29	C₂⁴×C₉	C₂³	1	2	yes	—
30	C₂⁴×C₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₉	—	2	4	no	—
32	C₂²×C₄×C₉	—	2	32	no	—
33	C₂²×C₄×C₉	—	2	16	no	—
34	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₉	C₂×C₄	1	16	yes	complemented
37	C₂²×C₄×C₉	D₄	1	8	yes	—
38	C₂²×C₄×C₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₉	C₂³	1	2	yes	—
40	C₂²×Dic₉	—	2	8	no	—
41	C₂²×Dic₉	—	2	8	no	—
42	C₂²×Dic₉	—	2	8	no	—
43	C₂²×Dic₉	C₂³	1	8	yes	—
44	#3676	—	2	16	no	—
45	C₄⋊Dic₉	—	2	16	no	—
46	C₂×#380	—	9	1	no	Sylow
47	C₂⁵×C₃	C₂×D₃	1	1	yes	characteristic
48	C₂³×C₃×C₄	—	2	8	no	—
49	C₂³×C₃×C₄	Dic₃	1	8	yes	—
50	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
51	C₂³×C₃×C₄	C₂×D₃	1	4	yes	—
52	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
53	C₂³×Dic₃	—	3	4	no	—
54	C₂×#217	—	3	8	no	—
55	C₂×#217	—	6	8	no	—
56	C₂×C₄⋊Dic₃	—	3	8	no	—
57	C₂×C₄⋊Dic₃	—	6	8	no	—
58	C₃⋊#95	—	3	8	no	—
59	C₂³×C₉	—	2	8	no	—
60	C₂³×C₉	—	2	8	no	—
61	C₂³×C₉	—	2	16	no	—
62	C₂³×C₉	—	2	16	no	—
63	C₂³×C₉	—	2	4	no	—
64	C₂³×C₉	—	2	4	no	—
65	C₂³×C₉	C₄⋊C₄	1	16	yes	complemented
66	C₂³×C₉	C₂²×C₄	1	4	yes	—
67	C₂³×C₉	C₂²×C₄	1	2	yes	—
68	C₂³×C₉	C₂×D₄	1	4	yes	—
69	C₂³×C₉	C₂×D₄	1	2	yes	—
70	C₂³×C₉	C₂×D₄	1	4	yes	—
71	C₂³×C₉	C₂×Q₈	1	4	yes	—
72	C₂³×C₉	C₂×Q₈	1	2	yes	—
73	C₂³×C₉	#41	1	4	yes	—
74	C₂³×C₉	C₂⁴	1	1	yes	characteristic
75	C₂×C₄×C₉	—	2	8	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	—	2	32	no	—
78	C₂×C₄×C₉	C₂²⋊C₄	1	16	yes	complemented
79	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
80	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₉	C₂×D₄	1	8	yes	—
82	C₂×Dic₉	—	2	8	no	—
83	C₂×Dic₉	—	2	8	no	—
84	C₂×Dic₉	—	2	8	no	—
85	C₂×Dic₉	—	4	16	no	—
86	C₂⁴×C₄	D₉	1	1	yes	characteristic
87	C₂²×C₄⋊C₄	—	9	2	no	—
88	C₂²×C₂²⋊C₄	—	9	2	no	—
89	#380	—	9	8	no	—
90	C₂×#95	—	9	2	no	—
91	C₂⁴×C₃	—	2	4	no	—
92	C₂⁴×C₃	—	2	4	no	—
93	C₂⁴×C₃	Dic₆	1	4	yes	—
94	C₂⁴×C₃	D₁₂	1	4	yes	—
95	C₂⁴×C₃	C₂×Dic₃	1	4	yes	—
96	C₂⁴×C₃	C₂²×D₃	1	2	yes	—
97	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	2	32	no	—
100	C₂²×C₃×C₄	—	2	16	no	—
101	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
102	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
103	C₂²×C₃×C₄	C₂×Dic₃	1	16	yes	—
104	C₂²×C₃×C₄	#67	1	8	yes	—
105	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
106	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
107	C₂²×Dic₃	—	3	8	no	—
108	C₂²×Dic₃	—	6	8	no	—
109	C₂²×Dic₃	—	6	8	no	—
110	C₂²×Dic₃	—	6	8	no	—
111	#217	—	6	16	no	—
112	C₄⋊Dic₃	—	6	16	no	—
113	C₂²×C₉	—	2	4	no	—
114	C₂²×C₉	—	2	4	no	—
115	C₂²×C₉	—	2	8	no	—
116	C₂²×C₉	—	2	8	no	—
117	C₂²×C₉	—	2	32	no	—
118	C₂²×C₉	C₂×C₂²⋊C₄	1	4	yes	—
119	C₂²×C₉	C₂×C₄⋊C₄	1	4	yes	—
120	C₂²×C₉	C₂×C₄⋊C₄	1	8	yes	complemented
121	C₂²×C₉	#117	1	4	yes	—
122	C₂²×C₉	#121	1	8	yes	—
123	C₂²×C₉	C₂²⋊Q₈	1	8	yes	—
124	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₉	C₂²×D₄	1	2	yes	—
127	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
128	C₂²×C₉	C₂×#41	1	2	yes	—
129	C₄×C₉	—	2	4	no	—
130	C₄×C₉	C₂×C₂²⋊C₄	1	8	yes	complemented
131	Dic₉	—	4	8	no	—
132	C₂⁵	C₂×D₉	1	1	yes	characteristic
133	C₂³×C₄	—	2	8	no	—
134	C₂³×C₄	—	9	4	no	—
135	C₂³×C₄	Dic₉	1	8	yes	complemented
136	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
137	C₂³×C₄	C₂×D₉	1	4	yes	—
138	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	9	8	no	—
140	C₂×C₄⋊C₄	—	18	8	no	—
141	C₂×C₂²⋊C₄	—	9	8	no	—
142	C₂×C₂²⋊C₄	—	18	8	no	—
143	#95	—	9	8	no	—
144	C₂³×C₃	—	2	16	no	—
145	C₂³×C₃	—	2	8	no	—
146	C₂³×C₃	—	2	16	no	—
147	C₂³×C₃	—	2	8	no	—
148	C₂³×C₃	—	2	4	no	—
149	C₂³×C₃	—	2	4	no	—
150	C₂³×C₃	C₄⋊Dic₃	1	16	yes	—
151	C₂³×C₃	C₂×Dic₆	1	4	yes	—
152	C₂³×C₃	C₂×Dic₆	1	2	yes	—
153	C₂³×C₃	C₂×D₁₂	1	4	yes	—
154	C₂³×C₃	C₂×D₁₂	1	2	yes	—
155	C₂³×C₃	#235	1	4	yes	—
156	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
157	C₂³×C₃	C₂²×Dic₃	1	2	yes	—
158	C₂³×C₃	C₂×#67	1	4	yes	—
159	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
160	C₂×C₃×C₄	—	2	8	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	2	32	no	—
163	C₂×C₃×C₄	#217	1	16	yes	—
164	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
165	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
166	C₂×C₃×C₄	C₂×#67	1	8	yes	—
167	C₂×Dic₃	—	6	8	no	—
168	C₂×Dic₃	—	6	8	no	—
169	C₂×Dic₃	—	6	8	no	—
170	C₂×Dic₃	—	12	16	no	—
171	C₂×C₉	—	2	8	no	—
172	C₂×C₉	#380	1	8	yes	complemented
173	C₂×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₉	C₂×#117	1	1	yes	characteristic
176	C₂×C₉	C₂×#121	1	2	yes	—
177	C₂×C₉	C₂×C₂²⋊Q₈	1	2	yes	—
178	C₂⁴	—	2	4	no	—
179	C₂⁴	—	2	4	no	—
180	C₂⁴	Dic₁₈	1	4	yes	—
181	C₂⁴	D₃₆	1	4	yes	—
182	C₂⁴	C₂×Dic₉	1	4	yes	—
183	C₂⁴	C₂²×D₉	1	2	yes	—
184	C₂⁴	C₂²×D₉	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	—	9	8	no	—
189	C₂²×C₄	—	18	8	no	—
190	C₂²×C₄	—	18	8	no	—
191	C₂²×C₄	—	18	8	no	—
192	C₂²×C₄	C₂×Dic₉	1	4	yes	—
193	C₂²×C₄	C₂×Dic₉	1	4	yes	—
194	C₂²×C₄	C₂×Dic₉	1	16	yes	complemented
195	C₂²×C₄	#611	1	8	yes	—
196	C₂²×C₄	C₂²×D₉	1	2	yes	—
197	C₂²×C₄	C₂²×D₉	1	2	yes	—
198	C₄⋊C₄	—	18	16	no	—
199	C₂²⋊C₄	—	18	16	no	—
200	C₂²×C₃	—	2	32	no	—
201	C₂²×C₃	—	2	8	no	—
202	C₂²×C₃	—	2	4	no	—
203	C₂²×C₃	—	2	8	no	—
204	C₂²×C₃	—	2	4	no	—
205	C₂²×C₃	#924	1	8	yes	—
206	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
207	C₂²×C₃	C₂×C₄⋊Dic₃	1	8	yes	—
208	C₂²×C₃	#926	1	4	yes	—
209	C₂²×C₃	#930	1	8	yes	—
210	C₂²×C₃	C₂×#217	1	4	yes	—
211	C₂²×C₃	C₂²×Dic₆	1	1	yes	characteristic
212	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
213	C₂²×C₃	C₂×#235	1	2	yes	—
214	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
215	C₂²×C₃	C₂²×#67	1	2	yes	—
216	C₃×C₄	—	2	4	no	—
217	C₃×C₄	C₂×#217	1	8	yes	—
218	Dic₃	—	12	8	no	—
219	C₉	C₂×#380	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	2	8	no	—
221	C₂³	—	2	16	no	—
222	C₂³	—	2	8	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	16	no	—
226	C₂³	C₄⋊Dic₉	1	16	yes	complemented
227	C₂³	C₂×Dic₁₈	1	4	yes	—
228	C₂³	C₂×Dic₁₈	1	2	yes	—
229	C₂³	C₂×D₃₆	1	4	yes	—
230	C₂³	C₂×D₃₆	1	2	yes	—
231	C₂³	#3697	1	4	yes	—
232	C₂³	C₂²×Dic₉	1	4	yes	—
233	C₂³	C₂²×Dic₉	1	2	yes	—
234	C₂³	C₂×#611	1	4	yes	—
235	C₂³	C₂³×D₉	1	1	yes	characteristic
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	32	no	—
239	C₂×C₄	—	18	8	no	—
240	C₂×C₄	—	18	8	no	—
241	C₂×C₄	—	18	8	no	—
242	C₂×C₄	—	36	16	no	—
243	C₂×C₄	#3676	1	16	yes	complemented
244	C₂×C₄	C₂²×Dic₉	1	4	yes	—
245	C₂×C₄	C₂²×Dic₉	1	4	yes	—
246	C₂×C₄	C₂×#611	1	8	yes	—
247	C₂×C₃	—	2	8	no	—
248	C₂×C₃	#5212	1	8	yes	—
249	C₂×C₃	C₂×#924	1	2	yes	—
250	C₂×C₃	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
251	C₂×C₃	C₂×#926	1	1	yes	characteristic
252	C₂×C₃	C₂×#930	1	2	yes	—
253	C₂×C₃	C₂²×#217	1	1	yes	characteristic
254	C₂²	—	2	8	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	32	no	—
258	C₂²	—	2	4	no	—
259	C₂²	#63486	1	8	yes	—
260	C₂²	C₂×C₄⋊Dic₉	1	4	yes	—
261	C₂²	C₂×C₄⋊Dic₉	1	8	yes	complemented
262	C₂²	#63488	1	4	yes	—
263	C₂²	#63492	1	8	yes	—
264	C₂²	C₂×#3676	1	4	yes	—
265	C₂²	C₂²×Dic₁₈	1	1	yes	characteristic
266	C₂²	C₂²×D₃₆	1	1	yes	characteristic
267	C₂²	C₂×#3697	1	2	yes	—
268	C₂²	C₂³×Dic₉	1	1	yes	characteristic
269	C₂²	C₂²×#611	1	2	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	36	8	no	—
272	C₄	C₂×#3676	1	8	yes	complemented
273	C₃	C₂×#5212	1	1	yes	characteristic
274	C₂	—	2	8	no	—
275	C₂	#10588892	1	8	yes	direct product factor
276	C₂	C₂×#63486	1	2	yes	—
277	C₂	C₂²×C₄⋊Dic₉	1	1	yes	characteristic
278	C₂	C₂×#63488	1	1	yes	characteristic
279	C₂	C₂×#63492	1	2	yes	—
280	C₂	C₂²×#3676	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂³×C₃	C₂³×D₃
Fitting	C₂⁴×C₄×C₉	C₂
socle	C₂⁴×C₃	C₂²×D₃
prime radical	C₂⁵×C₃	C₂×D₃
cosocle	C₂³×C₉	C₂⁴
hypercenter	C₂⁴×C₄	D₉
hypocenter	C₉	C₂×#380
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊Dic₉	C₂	semidirect
4. #10588892	C₂	direct (N × H)
5. C₂×C₉⋊#95	C₂	semidirect
9. C₂³×C₄×C₉	C₄	semidirect
17. C₂×C₄⋊Dic₉	C₂²	semidirect
18. C₉⋊#95	C₂²	semidirect
36. C₂²×C₄×C₉	C₂×C₄	semidirect
65. C₂³×C₉	C₄⋊C₄	semidirect
78. C₂×C₄×C₉	C₂²⋊C₄	semidirect
120. C₂²×C₉	C₂×C₄⋊C₄	semidirect
130. C₄×C₉	C₂×C₂²⋊C₄	semidirect
135. C₂³×C₄	Dic₉	semidirect
172. C₂×C₉	#380	semidirect
194. C₂²×C₄	C₂×Dic₉	semidirect
219. C₉	C₂×#380	semidirect
226. C₂³	C₄⋊Dic₉	semidirect
243. C₂×C₄	#3676	semidirect
261. C₂²	C₂×C₄⋊Dic₉	semidirect
272. C₄	C₂×#3676	semidirect
275. C₂	#10588892	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₉	2
C₂²×#3676	2
C₂²×C₄⋊Dic₉	2
#10588892	2
C₂×C₉⋊#95	2
C₂×#5212	3

Subgroups — C2×#10588892 #11800982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588892 #11800982