Finite Group Databasebeta

← #16855626

Subgroups — #16855626

Order 3968.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#14521560C211yesmaximal, characteristic
2C2×#11758503C211yesmaximal, characteristic, complemented
3#14521391C212yesmaximal, complemented
4#14521385C211yesmaximal, characteristic, complemented
5#14521380C211yesmaximal, characteristic, complemented
6#14521374C212yesmaximal, complemented
7#14521368C211yesmaximal, characteristic, complemented
8#14521363C211yesmaximal, characteristic, complemented
9#14521087C212yesmaximal, complemented
10#14521046C212yesmaximal, complemented
11C2×C31×C22⋊C8C211yesmaximal, characteristic, complemented
12C23×C4×C31C2211yescharacteristic
13C2×#92495C2211yescharacteristic
14C22×C8×C31C2212yes—
15#11758530—21no—
16C2×C22⋊Dic31—21no—
17#11758507C2211yescharacteristic
18#11758507C2211yescharacteristic
19#11758506—21no—
20C4×C22⋊D31—21no—
21C4×C22⋊D31—21no—
22#11758503—21no—
23#11758503—21no—
24#11758503C2212yes—
25#11758503C2212yes—
26#11758503C2211yescharacteristic
27C2×C4⋊Dic31C2211yescharacteristic
28C2×C4⋊Dic31C2211yescharacteristic
29#11758501C2211yescharacteristic
30#11758501C2211yescharacteristic
31C2×C4×Dic31—21no—
32C31×C22⋊C8C2211yescharacteristic
33C31×C22⋊C8C2211yescharacteristic
34C31×C22⋊C8C2212yes—
35#11758409C2212yescomplemented
36#11758409C2212yescomplemented
37#11758406C2212yescomplemented
38#11758406C2212yescomplemented
39#11758405C2212yescomplemented
40#11758405C2212yescomplemented
41#11758404C2212yescomplemented
42#11758404C2212yescomplemented
43C24×C31D411yescharacteristic
44C22×C4×C31—24no—
45C22×C4×C31D411yescharacteristic
46C22×C4×C31D411yescharacteristic
47C22×C4×C31D411yescharacteristic
48C22×C4×C31C2311yescharacteristic
49C22×C4×C31C2311yescharacteristic
50C22×C4×C31C2311yescharacteristic
51C2×C22⋊D31—21no—
52C2×C22⋊D31—21no—
53C2×C22⋊D31—21no—
54C22×Dic31—21no—
55C22×Dic31—21no—
56#92495—24no—
57C2×D124C2311yescharacteristic
58C2×C4×D31—21no—
59C2×Dic62C2311yescharacteristic
60C2×C8×C31—24no—
61C2×C8×C31C2312yes—
62C2×C8×C31C2312yes—
63C22⋊Dic31—21no—
64C22⋊Dic31—21no—
65C22⋊Dic31—22no—
66C22⋊Dic31—22no—
67C22⋊Dic31—21no—
68(C2×D31)⋊C4—21no—
69(C2×D31)⋊C4—21no—
70C4⋊Dic31—21no—
71C4⋊Dic31—21no—
72C4⋊Dic31C2312yes—
73C4⋊Dic31C2312yes—
74C4⋊Dic31C2311yescharacteristic
75C4⋊Dic31C2311yescharacteristic
76Dic31⋊C4—21no—
77Dic31⋊C4—21no—
78C4×Dic31—22no—
79C4×Dic31—22no—
80C4×Dic31—21no—
81C23×C31—22no—
82C23×C31—22no—
83C23×C31—41no—
84C23×C31C2×D411yescharacteristic
85C23×C31C2×D411yescharacteristic
86C23×C31C2×D411yescharacteristic
87C22×D31—21no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2×D411yescharacteristic
96C2×C4×C31#4114yes—
97C2×C4×C31C2411yescharacteristic
98C22⋊D31—42no—
99C22⋊D31—42no—
100C22⋊D31—42no—
101C2×Dic31—22no—
102C2×Dic31—22no—
103C2×Dic31—21no—
104C2×Dic31—21no—
105C2×Dic31—21no—
106C2×Dic31—42no—
107C2×Dic31—42no—
108D124—22no—
109C4×D31—42no—
110Dic62—22no—
111C8×C31—22no—
112C8×C31—22no—
113#3091—311nomaximal, Sylow
114C22×C31—21no—
115C22×C31—21no—
116C22×C31—21no—
117C22×C31—21no—
118C22×C31—21no—
119C22×C31—21no—
120C22×C31—21no—
121C22×C31—44no—
122C22×C31#12312yes—
123C22×C31#13512yes—
124C22×C31C22×D411yescharacteristic
125C2×D31—41no—
126C2×D31—41no—
127C2×D31—41no—
128C4×C31—21no—
129C4×C31—21no—
130C4×C31#12312yes—
131C4×C31C2×#4112yes—
132Dic31—41no—
133Dic31—42no—
134Dic31—41no—
135Dic31—42no—
136Dic31—41no—
137#525—311no—
138C2×#117—311no—
139#485—312no—
140#484—311no—
141#483—311no—
142#482—312no—
143#481—311no—
144#480—311no—
145#427—312no—
146#416—312no—
147C2×C22⋊C8D3111yescharacteristic, complemented
148C2×C31—22no—
149C2×C31—22no—
150C2×C31—41no—
151C2×C31C2×#12311yescharacteristic
152C2×C31C2×#13511yescharacteristic
153C2×C31#57511yescharacteristic
154D31—81no—
155C2×#41—311no—
156C23×C4C2×D3111yescharacteristic
157C22×C8C2×D3112yes—
158#123—621no—
159C22⋊Q8—311no—
160C22⋊Q8—311no—
161#121—311no—
162#121—311no—
163#120—621no—
164C4×D4—621no—
165C4×D4—621no—
166#117—312no—
167#117—312no—
168#117—311no—
169#117—621no—
170#117—621no—
171C2×C4⋊C4—311no—
172C2×C4⋊C4—311no—
173C2×C22⋊C4—621no—
174C2×C42—621no—
175#107—312no—
176#107—312no—
177#106—312no—
178#106—312no—
179#103—312no—
180#103—312no—
181#102—312no—
182#102—312no—
183C22⋊C8C2×D3111yescharacteristic
184C22⋊C8C2×D3111yescharacteristic
185C22⋊C8C2×D3112yes—
186C31#309111yescharacteristic, complemented, Sylow
187C24D12411yescharacteristic
188#41—624no—
189C2×Q8—311no—
190C2×D4—311no—
191C2×D4—621no—
192C2×D4—621no—
193C2×D4—621no—
194C22×C4—24no—
195C22×C4—621no—
196C22×C4—621no—
197C22×C4—621no—
198C22×C4D12411yescharacteristic
199C22×C4D12411yescharacteristic
200C22×C4D12411yescharacteristic
201C22×C4C22×D3111yescharacteristic
202C22×C4C22×D3111yescharacteristic
203C22×C4C22×D3111yescharacteristic
204C2×C8—24no—
205C2×C8C22×D3112yes—
206C2×C8C22×D3112yes—
207C4⋊C4—312no—
208C4⋊C4—312no—
209C4⋊C4—311no—
210C4⋊C4—311no—
211C4⋊C4—621no—
212C4⋊C4—621no—
213C4⋊C4—621no—
214C4⋊C4—621no—
215C22⋊C4—622no—
216C22⋊C4—622no—
217C22⋊C4—621no—
218C22⋊C4—621no—
219C22⋊C4—621no—
220C22⋊C4—621no—
221C22⋊C4—621no—
222C42—622no—
223C42—622no—
224C42—621no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—621no—
229C23C2×D12411yescharacteristic
230C23C2×D12411yescharacteristic
231C23C2×D12411yescharacteristic
232Q8—622no—
233D4—622no—
234D4—1242no—
235D4—1242no—
236D4—1242no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—622no—
243C2×C4—622no—
244C2×C4—621no—
245C2×C4—621no—
246C2×C4—621no—
247C2×C4—1242no—
248C2×C4—1242no—
249C2×C4—1242no—
250C2×C4C2×D12411yescharacteristic
251C2×C4C2×D12411yescharacteristic
252C2×C4C2×D12411yescharacteristic
253C2×C4#9249714yes—
254C2×C4C23×D3111yescharacteristic
255C8—22no—
256C8—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—1241no—
266C22—1241no—
267C22—1241no—
268C22#1175846312yes—
269C22#1175848112yes—
270C22C22×D12411yescharacteristic
271C4—21no—
272C4—21no—
273C4—1241no—
274C4—1242no—
275C4—1241no—
276C4—1242no—
277C4—1241no—
278C4#1175846312yes—
279C4C2×#9249712yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—2481no—
284C2C2×#1175846311yescharacteristic
285C2C2×#1175848111yescharacteristic
286C2#1452184911yescharacteristic
subgroupG/H
centerC2×C4C2×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×C22⋊C8C2
socleC22×C31C22×D4
prime radical#14521560C2
hypercenterC2×C22⋊C8D31
hypocenterC31#3091
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#11758503C2semidirect
3. #14521391C2semidirect
4. #14521385C2semidirect
5. #14521380C2semidirect
6. #14521374C2semidirect
7. #14521368C2semidirect
8. #14521363C2semidirect
9. #14521087C2semidirect
10. #14521046C2semidirect
11. C2×C31×C22⋊C8C2semidirect
35. #11758409C22semidirect
36. #11758409C22semidirect
37. #11758406C22semidirect
38. #11758406C22semidirect
39. #11758405C22semidirect
40. #11758405C22semidirect
41. #11758404C22semidirect
42. #11758404C22semidirect
147. C2×C22⋊C8D31semidirect
186. C31#3091semidirect

Maximal subgroups

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

subgroupindex
#145215602
C2×#117585032
#145213912
#145213852
#145213802
#145213742
#145213682
#145213632
#145210872
#145210462
C2×C31×C22⋊C82
#309131