Finite Group Databasebeta

← #16848993

Subgroups — #16848993

Order 3968.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×Q8×Dic31C211yesmaximal, characteristic, complemented
2C2×(C22⋊D31)⋊C4C211yesmaximal, characteristic
3#14520991C212yesmaximal, complemented
4#14520976C211yesmaximal, characteristic, complemented
5#14520974C211yesmaximal, characteristic, complemented
6#14520965C212yesmaximal
7#14520960C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic31C212yesmaximal, complemented
9#14520896C212yesmaximal, complemented
10#14520894C211yesmaximal, characteristic, complemented
11C31×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic
13C22×C22⋊D31C2211yescharacteristic
14C23×Dic31C2211yescharacteristic
15C22×C4×D31C2211yescharacteristic
16C2×C22⋊C4×C31C2212yes—
17C2×C22⋊C4×C31C2211yescharacteristic
18C2×C22⋊Dic31C2212yes—
19Q8×Dic31—24no—
20C2×(C2×D31)⋊C4C2212yes—
21C2×(C2×D31)⋊C4C2212yes—
22C2×(C2×D31)⋊C4C2211yescharacteristic
23C2×C4⋊Dic31C2212yes—
24C2×C4⋊Dic31C2211yescharacteristic
25C2×Dic31⋊C4C2211yescharacteristic
26C2×C4×Dic31C2212yes—
27C2×C4×Dic31C2212yes—
28C2×C4×Dic31C2211yescharacteristic
29(C22⋊D31)⋊C4—24no—
30C31×#95C2211yescharacteristic
31C31×#95C2212yes—
32C31⋊#95C2211yescharacteristic
33C31⋊#95C2212yes—
34C31⋊#95C2212yes—
35C31⋊#95C2212yes—
36C31⋊#95C2212yes—
37C24×C31C2311yescharacteristic
38C23×D31C2311yescharacteristic
39C2×Q8×C31—22no—
40C2×Q8×C31—24no—
41C22×C4×C31C2312yes—
42C22×C4×C31C2311yescharacteristic
43C22×C4×C31C2311yescharacteristic
44C22×C4×C31C2312yes—
45C2×C22⋊D31—22no—
46C2×C22⋊D31—22no—
47C2×C22⋊D31—22no—
48C22×Dic31—22no—
49C22×Dic31—24no—
50C22×Dic31C2312yes—
51C22×Dic31C2311yescharacteristic
52C22×Dic31C2312yes—
53C22×Dic31C2311yescharacteristic
54C22×Dic31C2311yescharacteristic
55C2×C4×D31—22no—
56C2×C4×D31—24no—
57C22⋊C4×C31—22no—
58C22⋊C4×C31—24no—
59C22⋊Dic31—24no—
60(C2×D31)⋊C4—24no—
61(C2×D31)⋊C4—22no—
62(C2×D31)⋊C4—24no—
63C4⋊Dic31—22no—
64C4⋊Dic31—24no—
65Dic31⋊C4—22no—
66C4×Dic31—24no—
67C4×Dic31—24no—
68C4×Dic31D414yes—
69C23×C31—21no—
70C23×C31—22no—
71C23×C31—22no—
72C23×C31—21no—
73C23×C31#4112yes—
74C23×C31C2411yescharacteristic
75C22×D31—21no—
76C22×D31—22no—
77C22×D31—22no—
78C22×D31—21no—
79C22×D31#4112yes—
80Q8×C31—28no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—21no—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—22no—
88C2×C4×C31—22no—
89C2×C4×C31C2×D412yes—
90C2×C4×C31#4114yes—
91C22⋊D31—28no—
92C2×Dic31—21no—
93C2×Dic31—21no—
94C2×Dic31—22no—
95C2×Dic31—21no—
96C2×Dic31—22no—
97C2×Dic31—22no—
98C2×Dic31—22no—
99C2×Dic31—22no—
100C2×Dic31—22no—
101C2×Dic31—22no—
102C2×Dic31—44no—
103C2×Dic31C2×D412yes—
104C2×Dic31C2×D412yes—
105C2×Dic31#4114yes—
106C4×D31—44no—
107#2398—311nomaximal, Sylow
108C22×C31—22no—
109C22×C31—24no—
110C22×C31—41no—
111C22×C31—41no—
112C22×C31—42no—
113C22×C31C22×D411yescharacteristic
114C22×C31C2×#4112yes—
115C22×C31C2×#4111yescharacteristic
116C22×C31C2×#4111yescharacteristic
117C22×C31C2×#4112yes—
118C2×D31—22no—
119C2×D31—24no—
120C2×D31—41no—
121C2×D31—41no—
122C2×D31—42no—
123C4×C31—22no—
124C4×C31—24no—
125C4×C31—41no—
126C4×C31—42no—
127Dic31—22no—
128Dic31—42no—
129Dic31—41no—
130Dic31—42no—
131Dic31#12414yes—
132C2×C4×Q8—311no—
133C2×C4×D4—311no—
134#400—312no—
135#397—311no—
136#397—311no—
137#394—312no—
138#393—311no—
139#393D3111yescharacteristic, complemented
140C4×C22⋊C4—312no—
141C4×C22⋊C4—312no—
142#376—311no—
143C2×C31—42no—
144C2×C31#52511yescharacteristic
145C2×C31#52511yescharacteristic
146C2×C31C2×#12411yescharacteristic
147C2×C31#52912yes—
148C2×C31#52912yes—
149D31—42no—
150C22×Q8C2×D3111yescharacteristic
151C22×D4—311no—
152C23×C4—311no—
153C23×C4—311no—
154C4×Q8—624no—
155C4×D4—624no—
156C2×C4⋊C4—311no—
157C2×C4⋊C4—312no—
158C2×C4⋊C4—311no—
159C2×C22⋊C4—312no—
160C2×C22⋊C4—311no—
161C2×C22⋊C4—312no—
162C2×C22⋊C4—312no—
163C2×C22⋊C4C2×D3111yescharacteristic
164C2×C22⋊C4C2×D3112yes—
165C2×C42—311no—
166C2×C42—312no—
167C2×C42—312no—
168#95—311no—
169#95—312no—
170#95—312no—
171#95—312no—
172#95—312no—
173#95C2×D3111yescharacteristic
174#95C2×D3112yes—
175C31#239811yescharacteristic, complemented, Sylow
176C24—311no—
177C24C22×D3111yescharacteristic
178C2×Q8—22no—
179C2×Q8—24no—
180C2×D4—622no—
181C2×D4—622no—
182C2×D4—622no—
183C22×C4—311no—
184C22×C4—311no—
185C22×C4—312no—
186C22×C4—311no—
187C22×C4—312no—
188C22×C4—622no—
189C22×C4—624no—
190C22×C4—622no—
191C22×C4—624no—
192C22×C4C22×D3112yes—
193C22×C4C22×D3111yescharacteristic
194C22×C4C22×D3111yescharacteristic
195C22×C4C22×D3112yes—
196C4⋊C4—622no—
197C4⋊C4—624no—
198C4⋊C4—622no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—624no—
202C22⋊C4—624no—
203C22⋊C4—622no—
204C22⋊C4—624no—
205C42—314no—
206C42—624no—
207C42—624no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—312no—
213C23—621no—
214C23—622no—
215C23—621no—
216C23—622no—
217C23#9249912yes—
218C23C23×D3111yescharacteristic
219Q8—28no—
220D4—628no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—312no—
230C2×C4—314no—
231C2×C4—312no—
232C2×C4—621no—
233C2×C4—621no—
234C2×C4—622no—
235C2×C4—621no—
236C2×C4—622no—
237C2×C4—622no—
238C2×C4—622no—
239C2×C4—622no—
240C2×C4—622no—
241C2×C4—622no—
242C2×C4—1244no—
243C2×C4—1244no—
244C2×C4D4×D3112yes—
245C2×C4#9249714yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—622no—
252C22—624no—
253C22—1241no—
254C22—1241no—
255C22—1242no—
256C22C2×D4×D3111yescharacteristic
257C22C2×#9249712yes—
258C22C2×#9249911yescharacteristic
259C22#41×D3111yescharacteristic
260C22#41×D3112yes—
261C4—22no—
262C4—24no—
263C4—41no—
264C4—42no—
265C4—314no—
266C4—622no—
267C4—1242no—
268C4—1241no—
269C4—1242no—
270C2—42no—
271C2—1242no—
272C2#1452155511yescharacteristic
273C2#1452156211yescharacteristic
274C2#124×D3111yescharacteristic
275C2#1452157812yes—
276C2#1452158512yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#393C2
prime radicalC22×C22⋊D31C22
hypercenter#393D31
hypocenterC31#2398
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic31C2semidirect
3. #14520991C2semidirect
4. #14520976C2semidirect
5. #14520974C2semidirect
7. #14520960C2semidirect
8. C22⋊C4×Dic31C2semidirect
9. #14520896C2semidirect
10. #14520894C2semidirect
11. C31×#393C2semidirect
139. #393D31semidirect
175. C31#2398semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic312
C2×(C22⋊D31)⋊C42
#145209912
#145209762
#145209742
#145209652
#145209602
C22⋊C4×Dic312
#145208962
#145208942
C31×#3932
#239831