Finite Group Databasebeta

← #16855624

Subgroups — #16855624

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
2#14521390C212yesmaximal, complemented
3#14521384C211yesmaximal, characteristic, complemented
4#14521379C211yesmaximal, characteristic, complemented
5#14521373C212yesmaximal, complemented
6#14521367C211yesmaximal, characteristic, complemented
7#14521362C211yesmaximal, characteristic, complemented
8#14521085C212yesmaximal, complemented
9#14521043C212yesmaximal, complemented
10C2×C22⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
11C2×C31×#117C211yesmaximal, characteristic, complemented
12C23×C4×C31C2211yescharacteristic
13C2×#92495C2211yescharacteristic
14C31×#117—21no—
15C31×#117—21no—
16C31×#117C2212yes—
17C31×#117C2212yes—
18C2×C4⋊C4×C31C2211yescharacteristic
19C2×C4⋊C4×C31C2211yescharacteristic
20C2×C22⋊C4×C31—21no—
21C2×C42×C31—21no—
22#11758530—21no—
23#11758507C2211yescharacteristic
24#11758507C2211yescharacteristic
25#11758506—21no—
26C4×C22⋊D31—21no—
27C4×C22⋊D31—21no—
28#11758503C2211yescharacteristic
29#11758501C2211yescharacteristic
30#11758501C2211yescharacteristic
31C22×C31⋊C8C2212yes—
32C22⋊(C31⋊C8)C2211yescharacteristic
33C22⋊(C31⋊C8)C2211yescharacteristic
34C22⋊(C31⋊C8)C2212yes—
35#11758398C2212yescomplemented
36#11758398C2212yescomplemented
37#11758397C2212yescomplemented
38#11758397C2212yescomplemented
39#11758396C2212yescomplemented
40#11758396C2212yescomplemented
41#11758395C2212yescomplemented
42#11758395C2212yescomplemented
43C24×C31D411yescharacteristic
44C22×C4×C31—24no—
45C22×C4×C31—21no—
46C22×C4×C31—21no—
47C22×C4×C31D411yescharacteristic
48C22×C4×C31D411yescharacteristic
49C22×C4×C31D411yescharacteristic
50C22×C4×C31C2311yescharacteristic
51C22×C4×C31C2311yescharacteristic
52C22×C4×C31C2311yescharacteristic
53C2×C22⋊D31—21no—
54C2×C22⋊D31—21no—
55C2×C22⋊D31—21no—
56#92495—24no—
57C2×D124C2311yescharacteristic
58C2×C4×D31—21no—
59C2×Dic62C2311yescharacteristic
60C4⋊C4×C31—21no—
61C4⋊C4×C31—21no—
62C4⋊C4×C31C2312yes—
63C4⋊C4×C31C2312yes—
64C22⋊C4×C31—22no—
65C22⋊C4×C31—22no—
66C42×C31—22no—
67C42×C31—22no—
68C22⋊Dic31—21no—
69C22⋊Dic31—21no—
70C22⋊Dic31—21no—
71(C2×D31)⋊C4—21no—
72(C2×D31)⋊C4—21no—
73C4⋊Dic31C2311yescharacteristic
74C4⋊Dic31C2311yescharacteristic
75Dic31⋊C4—21no—
76Dic31⋊C4—21no—
77C4×Dic31—21no—
78C2×C31⋊C8—24no—
79C2×C31⋊C8C2312yes—
80C2×C31⋊C8C2312yes—
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×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31#4114yes—
101C2×C4×C31C2411yescharacteristic
102C22⋊D31—42no—
103C22⋊D31—42no—
104C22⋊D31—42no—
105C2×Dic31—21no—
106C2×Dic31—21no—
107C2×Dic31—21no—
108D124—22no—
109C4×D31—42no—
110Dic62—22no—
111C31⋊C8—22no—
112C31⋊C8—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—42no—
131C4×C31—42no—
132C4×C31#12312yes—
133C4×C31C2×#4112yes—
134Dic31—41no—
135Dic31—41no—
136Dic31—41no—
137#525—311no—
138C2×#117D3111yescharacteristic, complemented
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⋊C8—311no—
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×C8—312no—
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—21no—
167#117—21no—
168#117—311no—
169#117C2×D3112yes—
170#117C2×D3112yes—
171C2×C4⋊C4C2×D3111yescharacteristic
172C2×C4⋊C4C2×D3111yescharacteristic
173C2×C22⋊C4—21no—
174C2×C42—21no—
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⋊C8—311no—
184C22⋊C8—311no—
185C22⋊C8—312no—
186C31#309111yescharacteristic, complemented, Sylow
187C24C22⋊D3111yescharacteristic
188#41—624no—
189C2×Q8—311no—
190C2×D4—311no—
191C2×D4—621no—
192C2×D4—621no—
193C2×D4—621no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—621no—
198C22×C4C22⋊D3111yescharacteristic
199C22×C4C22⋊D3111yescharacteristic
200C22×C4C22⋊D3111yescharacteristic
201C22×C4C22×D3111yescharacteristic
202C22×C4C22×D3111yescharacteristic
203C22×C4C22×D3111yescharacteristic
204C2×C8—312no—
205C2×C8—312no—
206C2×C8—624no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—311no—
210C4⋊C4—311no—
211C4⋊C4—621no—
212C4⋊C4—621no—
213C4⋊C4C22×D3112yes—
214C4⋊C4C22×D3112yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—621no—
218C22⋊C4—621no—
219C22⋊C4—621no—
220C22⋊C4—621no—
221C22⋊C4—621no—
222C42—22no—
223C42—22no—
224C42—621no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—621no—
229C23C2×C22⋊D3111yescharacteristic
230C23C2×C22⋊D3111yescharacteristic
231C23C2×C22⋊D3111yescharacteristic
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—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—621no—
247C2×C4—621no—
248C2×C4—621no—
249C2×C4—1242no—
250C2×C4#9249514yes—
251C2×C4C2×C22⋊D3111yescharacteristic
252C2×C4C2×C22⋊D3111yescharacteristic
253C2×C4C2×C22⋊D3111yescharacteristic
254C2×C4C23×D3111yescharacteristic
255C8—622no—
256C8—622no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—1241no—
266C22—1241no—
267C22—1241no—
268C22#1175850612yes—
269C22#1175852712yes—
270C22C22×C22⋊D3111yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—1241no—
276C4—1241no—
277C4—1241no—
278C4#1175850612yes—
279C4C2×#9249512yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—2481no—
284C2C2×#1175850611yescharacteristic
285C2C2×#1175852711yescharacteristic
286C2#1452184711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#117C2
socleC22×C31C22×D4
prime radical#14521560C2
hypercenterC2×#117D31
hypocenterC31#3091
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #14521390C2semidirect
3. #14521384C2semidirect
4. #14521379C2semidirect
5. #14521373C2semidirect
6. #14521367C2semidirect
7. #14521362C2semidirect
8. #14521085C2semidirect
9. #14521043C2semidirect
10. C2×C22⋊(C31⋊C8)C2semidirect
11. C2×C31×#117C2semidirect
35. #11758398C22semidirect
36. #11758398C22semidirect
37. #11758397C22semidirect
38. #11758397C22semidirect
39. #11758396C22semidirect
40. #11758396C22semidirect
41. #11758395C22semidirect
42. #11758395C22semidirect
138. C2×#117D31semidirect
186. C31#3091semidirect

Maximal subgroups

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

subgroupindex
#145215602
#145213902
#145213842
#145213792
#145213732
#145213672
#145213622
#145210852
#145210432
C2×C22⋊(C31⋊C8)2
C2×C31×#1172
#309131