Finite Group Databasebeta

← #16855849

Subgroups — #16855849

Order 3968.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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#14521583C211yesmaximal, characteristic, complemented
2#14521379C211yesmaximal, characteristic, complemented
3#14521362C211yesmaximal, characteristic, complemented
4#14521349C212yesmaximal, complemented
5#14521264C212yesmaximal, complemented
6#14521200C211yesmaximal, characteristic, complemented
7#14521183C211yesmaximal, characteristic
8#14521159C212yesmaximal, complemented
9#14521149C212yesmaximal, complemented
10#14521106C211yesmaximal, characteristic, complemented
11C2×C4×D4×C31C211yesmaximal, characteristic, complemented
12C22×D4×C31C2211yescharacteristic
13C23×C4×C31—21no—
14C4×D4×C31—21no—
15C4×D4×C31—21no—
16C4×D4×C31C2212yes—
17C4×D4×C31C2212yes—
18C2×C4⋊C4×C31C2211yescharacteristic
19C2×C22⋊C4×C31—21no—
20C2×C42×C31C2211yescharacteristic
21C2×#92480C2212yes—
22C2×#92479C2212yes—
23#11758507C2211yescharacteristic
24#11758506—21no—
25C4×C22⋊D31—21no—
26#11758501C2211yescharacteristic
27#11758453—21no—
28#11758452C2211yescharacteristic, complemented
29C4×D124C2211yescharacteristic, complemented
30#11758449—21no—
31#11758447C2211yescharacteristic, complemented
32C4×Dic62C2211yescharacteristic, complemented
33#11758420C2212yes—
34#11758420C2212yes—
35C22⋊(C31⋊C8)C2211yescharacteristic
36C22⋊(C31⋊C8)C2211yescharacteristic
37#11758398C2212yescomplemented
38#11758397C2212yescomplemented
39#11758396C2212yescomplemented
40#11758395C2212yescomplemented
41C4⋊(C31⋊C8)C2212yescomplemented
42#11758391C2212yescomplemented
43C24×C31—21no—
44C2×D4×C31—24no—
45C2×D4×C31—21no—
46C2×D4×C31C2312yes—
47C22×C4×C31—21no—
48C22×C4×C31—22no—
49C22×C4×C31—22no—
50C22×C4×C31—44no—
51C22×C4×C31D411yescharacteristic
52C22×C4×C31D411yescharacteristic
53C22×C4×C31C2311yescharacteristic
54C2×C22⋊D31—21no—
55C2×D124C2311yescharacteristic
56C2×C4×D31—21no—
57C2×Dic62C2311yescharacteristic
58C4⋊C4×C31—21no—
59C4⋊C4×C31C2312yes—
60C22⋊C4×C31—22no—
61C22⋊C4×C31—22no—
62C42×C31D411yescharacteristic
63C42×C31D411yescharacteristic
64C42×C31C2311yescharacteristic
65C42×C31C2311yescharacteristic
66C22⋊Dic31—21no—
67#92480—24no—
68#92479—24no—
69(C2×D31)⋊C4—21no—
70(C2×D31)⋊C4—21no—
71(C2×D31)⋊C4—21no—
72(C2×D31)⋊C4—21no—
73C4⋊Dic31C2311yescharacteristic
74C4⋊Dic31C2311yescharacteristic
75Dic31⋊C4—21no—
76Dic31⋊C4—21no—
77Dic31⋊C4—21no—
78Dic31⋊C4—21no—
79C4×Dic31—21no—
80C2×C31⋊C8C2312yes—
81C2×C31⋊C8C2312yes—
82C23×C31—24no—
83C23×C31—22no—
84C23×C31—42no—
85C23×C31—42no—
86C23×C31C2×D411yescharacteristic
87C22×D31—21no—
88D4×C31—24no—
89D4×C31—22no—
90D4×C31#4114yes—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—22no—
94C2×C4×C31—41no—
95C2×C4×C31—44no—
96C2×C4×C31—44no—
97C2×C4×C31—42no—
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2×D411yescharacteristic
102C2×C4×C31C2×D411yescharacteristic
103C2×C4×C31C2411yescharacteristic
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#3114—311nomaximal, Sylow
114C22×C31—21no—
115C22×C31—21no—
116C22×C31—24no—
117C22×C31—48no—
118C22×C31—42no—
119C22×C31—42no—
120C22×C31AGL(1,ℤ/8)12yes—
121C22×C31C22×D411yescharacteristic
122C2×D31—41no—
123C2×D31—41no—
124C2×D31—41no—
125C4×C31—21no—
126C4×C31—21no—
127C4×C31—21no—
128C4×C31—21no—
129C4×C31—21no—
130C4×C31—42no—
131C4×C31C2×#4112yes—
132Dic31—41no—
133Dic31—41no—
134Dic31—41no—
135#529—311no—
136C2×C4×D4D3111yescharacteristic, complemented
137#483—311no—
138#480—311no—
139#478—312no—
140#463—312no—
141#450—311no—
142#447—311no—
143#442—312no—
144#440—312no—
145#432—311no—
146C2×C31—22no—
147C2×C31—44no—
148C2×C31#52511yescharacteristic
149C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C31#57511yescharacteristic
151D31—81no—
152C22×D4C2×D3111yescharacteristic
153C23×C4—21no—
154C2×SD16—312no—
155C2×D8—312no—
156#126—621no—
157#125—311no—
158#124—311no—
159#123—621no—
160C22⋊Q8—311no—
161#121—311no—
162C4×Q8—311no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—311no—
166C4×D4—621no—
167C4×D4C2×D3112yes—
168C4×D4C2×D3112yes—
169#117—621no—
170C2×C4⋊C4C2×D3111yescharacteristic
171C2×C22⋊C4—21no—
172C2×C42C2×D3111yescharacteristic
173#107—312no—
174#106—312no—
175C4⋊C8—312no—
176#103—312no—
177#102—312no—
178#102—312no—
179#102—312no—
180C22⋊C8—311no—
181C22⋊C8—311no—
182#97—312no—
183C31#311411yescharacteristic, complemented, Sylow
184C24—21no—
185C2×Q8—311no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—311no—
189C2×D4—621no—
190C2×D4C22×D3112yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—621no—
196C22×C4C22⋊D3111yescharacteristic
197C22×C4C22⋊D3111yescharacteristic
198C22×C4C22×D3111yescharacteristic
199SD16—624no—
200D8—624no—
201C2×C8—312no—
202C2×C8—312no—
203C4⋊C4—21no—
204C4⋊C4—311no—
205C4⋊C4—311no—
206C4⋊C4—621no—
207C4⋊C4—621no—
208C4⋊C4—621no—
209C4⋊C4—621no—
210C4⋊C4C22×D3112yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—621no—
214C22⋊C4—621no—
215C22⋊C4—621no—
216C22⋊C4—621no—
217C22⋊C4—621no—
218C42—621no—
219C42C22⋊D3111yescharacteristic
220C42C22⋊D3111yescharacteristic
221C42C22×D3111yescharacteristic
222C42C22×D3111yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—621no—
228C23C2×C22⋊D3111yescharacteristic
229Q8—622no—
230D4—24no—
231D4—22no—
232D4—622no—
233D4—1242no—
234D4#9249514yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—621no—
243C2×C4—621no—
244C2×C4—621no—
245C2×C4—1242no—
246C2×C4C2×C22⋊D3111yescharacteristic
247C2×C4C2×C22⋊D3111yescharacteristic
248C2×C4C2×C22⋊D3111yescharacteristic
249C2×C4C2×C22⋊D3111yescharacteristic
250C2×C4C2×C22⋊D3111yescharacteristic
251C2×C4C23×D3111yescharacteristic
252C8—622no—
253C8—622no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—1241no—
261C22—1241no—
262C22—1241no—
263C22#1175850912yes—
264C22C22×C22⋊D3111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—1241no—
272C4—1241no—
273C4—1241no—
274C4C2×#9249512yes—
275C2—22no—
276C2—44no—
277C2—2481no—
278C2#1452156011yescharacteristic
279C2C2×#1175850911yescharacteristic
280C2#1452184711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C4×D4×C31C2
socleC22×C31C22×D4
prime radical#14521183C2
hypercenterC2×C4×D4D31
hypocenterC31#3114
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #14521583C2semidirect
2. #14521379C2semidirect
3. #14521362C2semidirect
4. #14521349C2semidirect
5. #14521264C2semidirect
6. #14521200C2semidirect
8. #14521159C2semidirect
9. #14521149C2semidirect
10. #14521106C2semidirect
11. C2×C4×D4×C31C2semidirect
28. #11758452C22semidirect
29. C4×D124C22semidirect
31. #11758447C22semidirect
32. C4×Dic62C22semidirect
37. #11758398C22semidirect
38. #11758397C22semidirect
39. #11758396C22semidirect
40. #11758395C22semidirect
41. C4⋊(C31⋊C8)C22semidirect
42. #11758391C22semidirect
136. C2×C4×D4D31semidirect
183. C31#3114semidirect

Maximal subgroups

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

subgroupindex
#145215832
#145213792
#145213622
#145213492
#145212642
#145212002
#145211832
#145211592
#145211492
#145211062
C2×C4×D4×C312
#311431