Finite Group Databasebeta

← #16855852

Subgroups — #16855852

Order 3968.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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#14521585C211yesmaximal, characteristic, complemented
2C2×#11758470C211yesmaximal, characteristic, complemented
3#14521362C211yesmaximal, characteristic, complemented
4#14521348C212yesmaximal, complemented
5#14521267C212yesmaximal, complemented
6#14521199C211yesmaximal, characteristic, complemented
7#14521185C211yesmaximal, characteristic
8#14521160C212yesmaximal, complemented
9#14521152C212yesmaximal, complemented
10#14521103C211yesmaximal, characteristic, complemented
11C31×#483C211yesmaximal, characteristic, complemented
12C22×D124C2211yescharacteristic
13C22×C4×D31—21no—
14C31×C22⋊Q8C2211yescharacteristic
15C2×C4⋊C4×C31C2211yescharacteristic
16#11758524C2211yescharacteristic, complemented
17Q8×Dic31C2211yescharacteristic, complemented
18C2×#92481C2212yes—
19#11758507C2211yescharacteristic
20C2×(C2×D31)⋊C4—21no—
21C2×C4×Dic31C2211yescharacteristic
22C2×D248C2212yes—
23#11758475—21no—
24#11758470—21no—
25#11758470—21no—
26#11758470C2212yes—
27#11758470C2212yes—
28#11758470C2211yescharacteristic, complemented
29#11758469—21no—
30#11758467C2211yescharacteristic, complemented
31#11758463—21no—
32(C22⋊D31)⋊C4—21no—
33C31×#106C2212yescomplemented
34C31×#103C2212yescomplemented
35C31×C22⋊C8C2211yescharacteristic
36C22⋊(C31⋊C8)C2211yescharacteristic
37#11758409C2212yes—
38#11758403C2212yescomplemented
39Dic31⋊C8C2212yescomplemented
40#11758397C2212yes—
41#11758397C2212yescomplemented
42#11758395C2212yescomplemented
43C23×D31—21no—
44C2×Q8×C31C2311yescharacteristic
45C22×C4×C31—21no—
46C22×C4×C31C2311yescharacteristic
47C2×C22⋊D31—21no—
48C22×Dic31D411yescharacteristic
49C22×Dic31D411yescharacteristic
50C2×D124—21no—
51C2×D124—24no—
52C2×D124C2311yescharacteristic
53C2×D124C2312yes—
54C2×C4×D31—21no—
55C2×C4×D31—22no—
56C2×C4×D31—22no—
57C2×C4×D31—44no—
58C2×C8×C31C2312yes—
59C4⋊C4×C31—21no—
60C4⋊C4×C31—21no—
61C4⋊C4×C31C2312yes—
62C4⋊C4×C31C2311yescharacteristic
63C22⋊C4×C31—21no—
64#92481—24no—
65(C2×D31)⋊C4—21no—
66(C2×D31)⋊C4—21no—
67(C2×D31)⋊C4—21no—
68(C2×D31)⋊C4—21no—
69(C2×D31)⋊C4—22no—
70(C2×D31)⋊C4—22no—
71C4⋊Dic31—21no—
72C4⋊Dic31—21no—
73C4⋊Dic31C2311yescharacteristic
74Dic31⋊C4—21no—
75C4×Dic31—21no—
76C4×Dic31D411yescharacteristic
77C4×Dic31D411yescharacteristic
78C4×Dic31C2311yescharacteristic
79C4×Dic31C2311yescharacteristic
80C2×C31⋊C8C2312yes—
81D248—24no—
82C23×C31C2×D411yescharacteristic
83C22×D31—24no—
84C22×D31—22no—
85C22×D31—21no—
86C22×D31—42no—
87C22×D31—42no—
88Q8×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31—22no—
93C2×C4×C31—42no—
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2411yescharacteristic
96C22⋊D31—42no—
97C2×Dic31—22no—
98C2×Dic31—21no—
99C2×Dic31—41no—
100C2×Dic31C2×D411yescharacteristic
101C2×Dic31C2×D411yescharacteristic
102C2×Dic31C2×D411yescharacteristic
103C2×Dic31C2×D411yescharacteristic
104D124—24no—
105D124—22no—
106D124—22no—
107D124#4114yes—
108C4×D31—42no—
109C4×D31—44no—
110C4×D31—44no—
111C8×C31—22no—
112C31⋊C8—22no—
113#3114—311nomaximal, Sylow
114C22×C31—21no—
115C22×C31—21no—
116C22×C31AGL(1,ℤ/8)12yes—
117C22×C31C22×D411yescharacteristic
118C2×D31—24no—
119C2×D31—48no—
120C2×D31—42no—
121C2×D31—42no—
122C2×D31—41no—
123C2×D31—41no—
124C2×D31—41no—
125C4×C31—21no—
126C4×C31—41no—
127C4×C31—42no—
128C4×C31—41no—
129C4×C31C2×#4112yes—
130Dic31—21no—
131Dic31—21no—
132Dic31—21no—
133Dic31—21no—
134Dic31—41no—
135#529—311no—
136C2×C4×D4—311no—
137#483D3111yescharacteristic, complemented
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#52511yescharacteristic
148C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C31#57511yescharacteristic
150D31—44no—
151D31—81no—
152C22×D4—311no—
153C23×C4—621no—
154C2×SD16—312no—
155C2×D8—312no—
156#126—621no—
157#125—311no—
158#124—311no—
159#123—621no—
160C22⋊Q8C2×D3111yescharacteristic
161#121—311no—
162C4×Q8—311no—
163C4×D4—312no—
164C4×D4—312no—
165C4×D4—311no—
166C4×D4—621no—
167C4×D4—621no—
168C4×D4—621no—
169#117—621no—
170C2×C4⋊C4C2×D3111yescharacteristic
171C2×C22⋊C4—621no—
172C2×C42—311no—
173#107—312no—
174#106C2×D3112yescomplemented
175C4⋊C8—312no—
176#103C2×D3112yescomplemented
177#102—312no—
178#102—312no—
179#102—312no—
180C22⋊C8—311no—
181C22⋊C8C2×D3111yescharacteristic
182#97—312no—
183C31#311411yescharacteristic, complemented, Sylow
184C24—621no—
185C2×Q8C22×D3111yescharacteristic
186C2×D4—312no—
187C2×D4—311no—
188C2×D4—624no—
189C2×D4—621no—
190C2×D4—621no—
191C22×C4—21no—
192C22×C4—311no—
193C22×C4—311no—
194C22×C4—621no—
195C22×C4—622no—
196C22×C4—622no—
197C22×C4—1244no—
198C22×C4C22×D3111yescharacteristic
199SD16—624no—
200D8—624no—
201C2×C8—312no—
202C2×C8C22×D3112yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—311no—
206C4⋊C4—621no—
207C4⋊C4—621no—
208C4⋊C4—621no—
209C4⋊C4C22×D3111yescharacteristic
210C4⋊C4C22×D3112yes—
211C22⋊C4—21no—
212C22⋊C4—622no—
213C22⋊C4—622no—
214C22⋊C4—621no—
215C22⋊C4—621no—
216C22⋊C4—621no—
217C22⋊C4—621no—
218C42—311no—
219C42—311no—
220C42—311no—
221C42—311no—
222C42—621no—
223C23—624no—
224C23—622no—
225C23—621no—
226C23—1242no—
227C23—1242no—
228C23D4×D3111yescharacteristic
229Q8—22no—
230D4—314no—
231D4—624no—
232D4—622no—
233D4—622no—
234D4—1242no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—311no—
241C2×C4—311no—
242C2×C4—311no—
243C2×C4—311no—
244C2×C4—622no—
245C2×C4—621no—
246C2×C4—1242no—
247C2×C4—1244no—
248C2×C4—1244no—
249C2×C4—1241no—
250C2×C4D4×D3111yescharacteristic
251C2×C4C23×D3111yescharacteristic
252C8—22no—
253C8—622no—
254C22—21no—
255C22—21no—
256C22—624no—
257C22—1248no—
258C22—1242no—
259C22—1242no—
260C22—1241no—
261C22—1241no—
262C22—1241no—
263C22#1175849112yes—
264C22C2×D4×D3111yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—621no—
270C4—621no—
271C4—621no—
272C4—621no—
273C4—1241no—
274C4#41×D3112yes—
275C2—22no—
276C2—1244no—
277C2—2481no—
278C2#1452155711yescharacteristic
279C2C2×#1175849111yescharacteristic
280C2#1452184811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#483C2
socleC22×C31C22×D4
prime radical#14521185C2
hypercenter#483D31
hypocenterC31#3114
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #14521585C2semidirect
2. C2×#11758470C2semidirect
3. #14521362C2semidirect
4. #14521348C2semidirect
5. #14521267C2semidirect
6. #14521199C2semidirect
8. #14521160C2semidirect
9. #14521152C2semidirect
10. #14521103C2semidirect
11. C31×#483C2semidirect
16. #11758524C22semidirect
17. Q8×Dic31C22semidirect
28. #11758470C22semidirect
30. #11758467C22semidirect
33. C31×#106C22semidirect
34. C31×#103C22semidirect
38. #11758403C22semidirect
39. Dic31⋊C8C22semidirect
41. #11758397C22semidirect
42. #11758395C22semidirect
137. #483D31semidirect
174. #106C2×D31semidirect
176. #103C2×D31semidirect
183. C31#3114semidirect

Maximal subgroups

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

subgroupindex
#145215852
C2×#117584702
#145213622
#145213482
#145212672
#145211992
#145211852
#145211602
#145211522
#145211032
C31×#4832
#311431