Finite Group Databasebeta

← C2×D3×#625 #11724413

Subgroups — C2×D3×#625 #11724413

Order 864.

1122 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D32C212yesmaximal, complemented
2C2×#6295C211yesmaximal, characteristic, complemented
3C2×#6290C212yesmaximal, complemented
4C2×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C2×C3×#625C211yesmaximal, characteristic, complemented
6D3×#625C218yesmaximal, direct product factor
7C2×D3×#67—32nomaximal
8C22×#625—31nomaximal
9C22×C3×C3⋊D3C2212yescomplemented
10C22×C32×D3C2211yescharacteristic
11C22×C32×D3C2212yescomplemented
12C2×C3×D32—28no—
13C2×C3×D32C2218yescomplemented
14C2×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
15C2×C3×C3⋊Dic3C2211yescharacteristic, complemented
16#6295C2214yescomplemented
17#6290C2218yescomplemented
18D3×C3⋊Dic3C2214yescomplemented
19C3×#625C2214yescomplemented
20C23×C3×D3—32no—
21C23×C3×D3—32no—
22C22×D32D312yescomplemented
23C2×#638—32no—
24C22×C3⋊Dic3—31no—
25C2×C3×#67—32no—
26D3×#67—316no—
27C2×#626—32no—
28C2×#625—32no—
29C2×#625—32no—
30C2×#625—31no—
31C2×#625—38no—
32C2×#625—62no—
33C2×#625D311yescharacteristic, direct product factor
34C2×D3×Dic3—32no—
35C2×D3×Dic3—61no—
36C22×C33C2311yescharacteristic
37C2×C3×C3⋊D3—24no—
38C2×C3×C3⋊D3C2314yescomplemented
39C2×C32×D3—24no—
40C2×C32×D3D414yescomplemented
41C2×C32×D3C2312yes—
42C2×C32×D3C2314yescomplemented
43C3×D32—216no—
44C3⋊(C3⋊Dic3)C2312yescomplemented
45C3×C3⋊Dic3C2312yescomplemented
46C22×#67—92no—
47C2×D3×D4—91no—
48C23×C32—31no—
49C23×C32—32no—
50C22×C3⋊D3C2×D312yescomplemented
51C22×C3×D3—21no—
52C22×C3×D3—22no—
53C22×C3×D3—32no—
54C22×C3×D3—32no—
55C22×C3×D3—38no—
56C22×C3×D3—38no—
57C22×C3×D3—62no—
58C22×C3×D3—68no—
59C22×C3×D3—68no—
60C22×C3×D3C2×D312yes—
61C22×C3×D3C2×D312yescomplemented
62C22×C3×D3C2×D312yescomplemented
63C2×D32—28no—
64C2×D32C2×D318yescomplemented
65#638—38no—
66C2×C3⋊Dic3—32no—
67C2×C3⋊Dic3—31no—
68C2×C3⋊Dic3—34no—
69C2×C3⋊Dic3—61no—
70C2×C3⋊Dic3—62no—
71C2×C3⋊Dic3—121no—
72C2×C3⋊Dic3C2×D311yescharacteristic, complemented
73C3×#67—38no—
74C2×C3×Dic3—32no—
75C2×C3×Dic3—61no—
76#626—38no—
77#625—38no—
78#625—38no—
79#625—34no—
80#625—68no—
81#625C2×D314yesdirect product factor
82D3×Dic3—38no—
83D3×Dic3—64no—
84C2×C33C2×D412yescomplemented
85C2×C33C2411yescharacteristic
86C3×C3⋊D3—24no—
87C32×D3—24no—
88C32×D3C2×D414yescomplemented
89C24×C3—92no—
90C23×D3—32no—
91C23×D3—32no—
92C2×C3×D4—91no—
93C2×#67—32no—
94C2×#67—92no—
95C2×#67—916no—
96C2×#67—92no—
97C2×#67—92no—
98C2×#67—92no—
99C2×#67—182no—
100C22×Dic3—92no—
101C22×Dic3—181no—
102D3×D4—98no—
103C2×D12—91no—
104C2×C4×D3—91no—
105C22×C32—21no—
106C22×C32—22no—
107C22×C32—34no—
108C22×C32—32no—
109C22×C32—34no—
110C22×C32—41no—
111C22×C32—64no—
112C22×C32C22×D311yescharacteristic
113C22×C32C22×D312yes—
114C2×C3⋊D3—24no—
115C2×C3⋊D3C22×D314yescomplemented
116C2×C3×D3—24no—
117C2×C3×D3—24no—
118C2×C3×D3—24no—
119C2×C3×D3—22no—
120C2×C3×D3—24no—
121C2×C3×D3—34no—
122C2×C3×D3—34no—
123C2×C3×D3—44no—
124C2×C3×D3—64no—
125C2×C3×D3—616no—
126C2×C3×D3—64no—
127C2×C3×D3—64no—
128C2×C3×D3—616no—
129C2×C3×D3—124no—
130C2×C3×D3#6718yescomplemented
131C2×C3×D3C22×D314yes—
132C2×C3×D3C22×D314yescomplemented
133C2×C3×D3C22×D314yescomplemented
134D32—216no—
135C3⋊Dic3—34no—
136C3⋊Dic3—32no—
137C3⋊Dic3—62no—
138C3⋊Dic3—64no—
139C3⋊Dic3—122no—
140C3⋊Dic3C22×D312yescomplemented
141C3×Dic3—34no—
142C3×Dic3—62no—
143C22×D4—271noSylow
144C33C22×D411yescharacteristic, complemented, Sylow
145C23×C3—32no—
146C23×C3—32no—
147C23×C3—32no—
148C23×C3—61no—
149C23×C3—62no—
150C23×C3—98no—
151C23×C3—92no—
152C23×C3—188no—
153C22×D3—32no—
154C22×D3—32no—
155C22×D3—38no—
156C22×D3—38no—
157C22×D3—62no—
158C22×D3—68no—
159C22×D3—68no—
160C22×D3D3211yescharacteristic
161C22×D3D3212yescomplemented
162C3×D4—94no—
163C2×C3×C4—91no—
164#67—38no—
165#67—98no—
166#67—98no—
167#67—98no—
168#67—98no—
169#67—188no—
170C2×Dic3—32no—
171C2×Dic3—61no—
172C2×Dic3—98no—
173C2×Dic3—91no—
174C2×Dic3—92no—
175C2×Dic3—184no—
176C2×Dic3—181no—
177C2×Dic3—182no—
178C2×Dic3—361no—
179D12—94no—
180C4×D3—94no—
181C2×C32—21no—
182C2×C32—22no—
183C2×C32—22no—
184C2×C32—24no—
185C2×C32—34no—
186C2×C32—42no—
187C2×C32—41no—
188C2×C32—64no—
189C2×C32D3×D412yescomplemented
190C2×C32C2×#6714yescomplemented
191C2×C32C23×D311yescharacteristic
192C2×C32C23×D312yes—
193C3⋊D3—24no—
194C3×D3—24no—
195C3×D3—24no—
196C3×D3—24no—
197C3×D3—44no—
198C3×D3—64no—
199C3×D3—64no—
200C3×D3—124no—
201C3×D3C2×#6718yescomplemented
202C24—92no—
203C2×D4—91no—
204C2×D4—271no—
205C2×D4—278no—
206C2×D4—272no—
207C22×C4—271no—
208C22×C3—21no—
209C22×C3—22no—
210C22×C3—34no—
211C22×C3—38no—
212C22×C3—34no—
213C22×C3—34no—
214C22×C3—41no—
215C22×C3—62no—
216C22×C3—64no—
217C22×C3—64no—
218C22×C3—64no—
219C22×C3—64no—
220C22×C3—94no—
221C22×C3—124no—
222C22×C3—1816no—
223C22×C3—184no—
224C22×C3C2×D3211yescharacteristic
225C22×C3C2×D3212yes—
226C2×D3—24no—
227C2×D3—34no—
228C2×D3—34no—
229C2×D3—64no—
230C2×D3—616no—
231C2×D3—64no—
232C2×D3—64no—
233C2×D3—616no—
234C2×D3—124no—
235C2×D3#62514yesdirect product factor
236C2×D3C2×D3212yes—
237C2×D3C2×D3214yescomplemented
238C3×C4—92no—
239Dic3—34no—
240Dic3—62no—
241Dic3—92no—
242Dic3—94no—
243Dic3—182no—
244Dic3—184no—
245Dic3—362no—
246C32—21no—
247C32—22no—
248C32—41no—
249C32C2×D3×D411yescharacteristic, complemented
250C32C22×#6712yescomplemented
251C23—31no—
252C23—32no—
253C23—98no—
254C23—92no—
255C23—188no—
256D4—94no—
257D4—274no—
258D4—278no—
259C2×C4—91no—
260C2×C4—274no—
261C2×C4—271no—
262C2×C3—21no—
263C2×C3—22no—
264C2×C3—24no—
265C2×C3—22no—
266C2×C3—38no—
267C2×C3—41no—
268C2×C3—42no—
269C2×C3—64no—
270C2×C3—64no—
271C2×C3—64no—
272C2×C3—124no—
273C2×C3—184no—
274C2×C3C2×#62512yescomplemented
275C2×C3D3×#6714yescomplemented
276C2×C3C22×D3211yescharacteristic
277C2×C3C22×D3212yes—
278D3—24no—
279D3—64no—
280D3—64no—
281D3—124no—
282D3C2×#62514yesdirect product factor
283C22—32no—
284C22—34no—
285C22—34no—
286C22—64no—
287C22—94no—
288C22—1816no—
289C22—184no—
290C22D3311yescharacteristic
291C4—92no—
292C4—272no—
293C3—22no—
294C3—21no—
295C3—41no—
296C3C22×#62511yescharacteristic, complemented
297C3C2×D3×#6712yescomplemented
298C2—34no—
299C2—64no—
300C2—184no—
301C2D3×#62512yesdirect product factor
302C2C2×D3311yescharacteristic
subgroupG/H
center, hypercenterC22D33
commutator, cosocleC2×C33C24
FrattiniC2C2×D33
Fitting, socleC22×C33C23
hypocenterC33C22×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×D32C2semidirect
2. C2×#6295C2semidirect
3. C2×#6290C2semidirect
4. C2×D3×C3⋊Dic3C2semidirect
5. C2×C3×#625C2semidirect
6. D3×#625C2direct (N × H)
9. C22×C3×C3⋊D3C22semidirect
11. C22×C32×D3C22semidirect
13. C2×C3×D32C22semidirect
14. C2×C3⋊(C3⋊Dic3)C22semidirect
15. C2×C3×C3⋊Dic3C22semidirect
16. #6295C22semidirect
17. #6290C22semidirect
18. D3×C3⋊Dic3C22semidirect
19. C3×#625C22semidirect
22. C22×D32D3semidirect
33. C2×#625D3direct (N × H)
38. C2×C3×C3⋊D3C23semidirect
40. C2×C32×D3D4semidirect
42. C2×C32×D3C23semidirect
44. C3⋊(C3⋊Dic3)C23semidirect
45. C3×C3⋊Dic3C23semidirect
50. C22×C3⋊D3C2×D3semidirect
61. C22×C3×D3C2×D3semidirect
62. C22×C3×D3C2×D3semidirect
64. C2×D32C2×D3semidirect
72. C2×C3⋊Dic3C2×D3semidirect
81. #625C2×D3direct (N × H)
84. C2×C33C2×D4semidirect
88. C32×D3C2×D4semidirect
115. C2×C3⋊D3C22×D3semidirect
130. C2×C3×D3#67semidirect
132. C2×C3×D3C22×D3semidirect
133. C2×C3×D3C22×D3semidirect
140. C3⋊Dic3C22×D3semidirect
144. C33C22×D4semidirect
161. C22×D3D32semidirect
189. C2×C32D3×D4semidirect
190. C2×C32C2×#67semidirect
201. C3×D3C2×#67semidirect
235. C2×D3#625direct (N × H)
237. C2×D3C2×D32semidirect
249. C32C2×D3×D4semidirect
250. C32C22×#67semidirect
274. C2×C3C2×#625semidirect
275. C2×C3D3×#67semidirect
282. D3C2×#625direct (N × H)
296. C3C22×#625semidirect
297. C3C2×D3×#67semidirect
301. C2D3×#625direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D322
C2×#62952
C2×#62902
C2×D3×C3⋊Dic32
C2×C3×#6252
D3×#6252
C2×D3×#673
C22×#6253