Finite Group Databasebeta

← #16855807

Subgroups — #16855807

Order 3968.

444 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 119 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#14521601C211yesmaximal, characteristic, complemented
2C2×#11758470C211yesmaximal, characteristic
3#14521384C211yesmaximal, characteristic, complemented
4#14521328C212yesmaximal, complemented
5#14521253C212yesmaximal, complemented
6#14521202C211yesmaximal, characteristic
7#14521199C211yesmaximal, characteristic
8#14521152C212yesmaximal, complemented
9#14521148C212yesmaximal, complemented
10#14521103C211yesmaximal, characteristic, complemented
11C31×#484C211yesmaximal, characteristic, complemented
12C22×D124C2211yescharacteristic
13C22×C4×D31—21no—
14C31×C22⋊Q8C2211yescharacteristic
15C2×C4⋊C4×C31C2211yescharacteristic
16Q8×Dic31C2211yescharacteristic, complemented
17#11758521C2211yescharacteristic, complemented
18C2×#92481C2212yes—
19C2×(C2×D31)⋊C4—21no—
20#11758501C2211yescharacteristic
21C2×C4×Dic31C2211yescharacteristic
22C2×#92470C2212yes—
23#11758470—21no—
24#11758470—21no—
25#11758470C2212yes—
26#11758470C2212yes—
27#11758466—21no—
28#11758465C2211yescharacteristic, complemented
29#11758464—21no—
30#11758464C2211yescharacteristic, complemented
31#11758455—21no—
32#11758454—21no—
33C31×#107C2212yescomplemented
34C31×#103C2212yescomplemented
35C31×C22⋊C8C2211yescharacteristic
36C22⋊(C31⋊C8)C2211yescharacteristic
37#11758409C2212yes—
38#11758403C2212yescomplemented
39Dic31⋊C8C2212yescomplemented
40#11758398C2212yescomplemented
41#11758397C2212yes—
42#11758395C2212yescomplemented
43C23×D31—21no—
44C2×Q8×C31C2311yescharacteristic
45C22×C4×C31—21no—
46C22×C4×C31C2311yescharacteristic
47C22×Dic31D411yescharacteristic
48C22×Dic31D411yescharacteristic
49C2×D124—21no—
50C2×D124—24no—
51C2×D124C2312yes—
52C2×C4×D31—22no—
53C2×C4×D31—22no—
54C2×C4×D31—44no—
55C2×Dic62—21no—
56C2×Dic62C2311yescharacteristic
57C2×C8×C31C2312yes—
58C4⋊C4×C31—21no—
59C4⋊C4×C31—21no—
60C4⋊C4×C31C2312yes—
61C4⋊C4×C31C2311yescharacteristic
62C22⋊C4×C31—21no—
63C22⋊Dic31—21no—
64#92481—24no—
65(C2×D31)⋊C4—22no—
66(C2×D31)⋊C4—22no—
67C4⋊Dic31—21no—
68C4⋊Dic31C2311yescharacteristic
69Dic31⋊C4—21no—
70Dic31⋊C4—21no—
71Dic31⋊C4—21no—
72Dic31⋊C4—21no—
73Dic31⋊C4—21no—
74C4×Dic31—21no—
75C4×Dic31—21no—
76C4×Dic31D411yescharacteristic
77C4×Dic31D411yescharacteristic
78C4×Dic31C2311yescharacteristic
79C4×Dic31C2311yescharacteristic
80C2×C31⋊C8C2312yes—
81#92470—24no—
82C23×C31C2×D411yescharacteristic
83C22×D31—24no—
84C22×D31—22no—
85C22×D31—42no—
86C22×D31—42no—
87Q8×C31—22no—
88C2×C4×C31—22no—
89C2×C4×C31—21no—
90C2×C4×C31—21no—
91C2×C4×C31—22no—
92C2×C4×C31—42no—
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2411yescharacteristic
95C2×Dic31—22no—
96C2×Dic31—22no—
97C2×Dic31—21no—
98C2×Dic31—21no—
99C2×Dic31C2×D411yescharacteristic
100C2×Dic31C2×D411yescharacteristic
101C2×Dic31C2×D411yescharacteristic
102C2×Dic31C2×D411yescharacteristic
103D124—24no—
104D124—22no—
105D124#4114yes—
106C4×D31—44no—
107C4×D31—44no—
108Dic62—22no—
109Dic62—42no—
110C8×C31—22no—
111C31⋊C8—22no—
112#3110—311nomaximal, Sylow
113C22×C31—21no—
114C22×C31—21no—
115C22×C31#13712yes—
116C22×C31C22×D411yescharacteristic
117C2×D31—24no—
118C2×D31—48no—
119C2×D31—42no—
120C2×D31—42no—
121C4×C31—21no—
122C4×C31—41no—
123C4×C31—41no—
124C4×C31—42no—
125C4×C31C2×#4112yes—
126Dic31—21no—
127Dic31—21no—
128Dic31—21no—
129Dic31—41no—
130Dic31—41no—
131Dic31AGL(1,ℤ/8)12yes—
132#533—311no—
133C2×C4×D4—311no—
134#484—311no—
135#484D3111yescharacteristic, complemented
136#474—312no—
137#461—312no—
138#450—311no—
139#450—311no—
140#440—312no—
141#440—312no—
142#432—311no—
143C2×C31—22no—
144C2×C31#52511yescharacteristic
145C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C31C2×#13711yescharacteristic
147D31—44no—
148C22×D4—311no—
149C23×C4—621no—
150C2×SD16—312no—
151C2×SD16—312no—
152C4⋊Q8—311no—
153C4⋊Q8—311no—
154#125—621no—
155C22⋊Q8—311no—
156C22⋊Q8—621no—
157C22⋊Q8C2×D3111yescharacteristic
158C4×Q8—311no—
159C4×Q8—311no—
160C4×Q8—621no—
161C4×D4—312no—
162C4×D4—312no—
163C4×D4—621no—
164C4×D4—621no—
165#117—621no—
166C2×C4⋊C4C2×D3111yescharacteristic
167C2×C22⋊C4—621no—
168C2×C42—311no—
169#107—312no—
170#107C2×D3112yescomplemented
171C4⋊C8—312no—
172#103—312no—
173#103C2×D3112yescomplemented
174#102—312no—
175#102—312no—
176C22⋊C8—311no—
177C22⋊C8C2×D3111yescharacteristic
178#97—312no—
179C31#311011yescharacteristic, complemented, Sylow
180C24—621no—
181C2×Q8—311no—
182C2×Q8—621no—
183C2×Q8C22×D3111yescharacteristic
184C2×D4—312no—
185C2×D4—624no—
186C2×D4—621no—
187C22×C4—21no—
188C22×C4—311no—
189C22×C4—311no—
190C22×C4—622no—
191C22×C4—622no—
192C22×C4—1244no—
193C22×C4C22×D3111yescharacteristic
194SD16—624no—
195SD16—624no—
196C2×C8—312no—
197C2×C8C22×D3112yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—311no—
201C4⋊C4—621no—
202C4⋊C4—621no—
203C4⋊C4—621no—
204C4⋊C4—621no—
205C4⋊C4—621no—
206C4⋊C4—621no—
207C4⋊C4C22×D3111yescharacteristic
208C4⋊C4C22×D3112yes—
209C22⋊C4—21no—
210C22⋊C4—622no—
211C22⋊C4—622no—
212C22⋊C4—621no—
213C42—311no—
214C42—311no—
215C42—311no—
216C42—311no—
217C42—621no—
218C42—621no—
219C23—624no—
220C23—622no—
221C23—1242no—
222C23—1242no—
223C23D4×D3111yescharacteristic
224Q8—22no—
225Q8—622no—
226Q8—1242no—
227D4—314no—
228D4—624no—
229D4—622no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—311no—
236C2×C4—311no—
237C2×C4—311no—
238C2×C4—311no—
239C2×C4—622no—
240C2×C4—622no—
241C2×C4—621no—
242C2×C4—621no—
243C2×C4—1244no—
244C2×C4—1244no—
245C2×C4D4×D3111yescharacteristic
246C2×C4C23×D3111yescharacteristic
247C8—22no—
248C8—622no—
249C22—21no—
250C22—21no—
251C22—624no—
252C22—1248no—
253C22—1242no—
254C22—1242no—
255C22#1175849512yes—
256C22C2×D4×D3111yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—312no—
262C4—621no—
263C4—621no—
264C4—621no—
265C4—1241no—
266C4—1241no—
267C4#41×D3112yes—
268C2—22no—
269C2—1244no—
270C2#1452155711yescharacteristic
271C2AGL(1,ℤ/8)×D3111yescharacteristic
272C2C2×#1175849511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#484C2
socleC22×C31C22×D4
prime radicalC22×D124C22
hypercenter#484D31
hypocenterC31#3110
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #14521601C2semidirect
3. #14521384C2semidirect
4. #14521328C2semidirect
5. #14521253C2semidirect
8. #14521152C2semidirect
9. #14521148C2semidirect
10. #14521103C2semidirect
11. C31×#484C2semidirect
16. Q8×Dic31C22semidirect
17. #11758521C22semidirect
28. #11758465C22semidirect
30. #11758464C22semidirect
33. C31×#107C22semidirect
34. C31×#103C22semidirect
38. #11758403C22semidirect
39. Dic31⋊C8C22semidirect
40. #11758398C22semidirect
42. #11758395C22semidirect
135. #484D31semidirect
170. #107C2×D31semidirect
173. #103C2×D31semidirect
179. C31#3110semidirect

Maximal subgroups

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

subgroupindex
#145216012
C2×#117584702
#145213842
#145213282
#145212532
#145212022
#145211992
#145211522
#145211482
#145211032
C31×#4842
#311031