Finite Group Databasebeta

← #16848294

Subgroups — #16848294

Order 3968.

784 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×Q8×D31C211yesmaximal, characteristic
2C2×#11758469C211yesmaximal, characteristic
3C2×#11758449C211yesmaximal, characteristic
4#14520956C211yesmaximal, characteristic, complemented
5#14520955C211yesmaximal, characteristic, complemented
6#14520954C211yesmaximal, characteristic, complemented
7#14520953C214yesmaximal, complemented
8#14520911C214yesmaximal
9C31×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C2×Q8×D31—24no—
12C2×Q8×D31—24no—
13C2×Q8×D31C418yescomplemented
14C22×C4×D31C2212yes—
15C22×C4×D31C2211yescharacteristic
16C22×Dic62C2212yes—
17C22×Dic62C2211yescharacteristic
18C2×C4⋊C4×C31C2211yescharacteristic
19C2×C42×C31C2211yescharacteristic
20C2×(C2×D31)⋊C4C2212yes—
21C2×(C2×D31)⋊C4C2212yes—
22C2×C4⋊Dic31C2211yescharacteristic
23C2×Dic31⋊C4C2212yes—
24C2×C4×Dic31C2212yes—
25C2×C4×Dic31C2211yescharacteristic
26#11758469—24no—
27#11758449—24no—
28C31×#95C2214yes—
29C31⋊#95C2214yes—
30C31⋊#95C2214yes—
31C31⋊#95C2214yes—
32C23×D31C2311yescharacteristic
33C2×Q8×C31—22no—
34C2×Q8×C31—22no—
35C2×Q8×C31C2×C414yes—
36C22×C4×C31C2312yes—
37C22×C4×C31C2312yes—
38C22×C4×C31C2312yes—
39C22×C4×C31C2311yescharacteristic
40C22×Dic31C2312yes—
41C22×Dic31C2312yes—
42C22×Dic31C2312yes—
43C22×Dic31C2311yescharacteristic
44Q8×D31—232no—
45C2×C4×D31—24no—
46C2×C4×D31—24no—
47C2×C4×D31C2×C418yes—
48C2×C4×D31C2×C414yes—
49C2×C4×D31D414yes—
50C2×C4×D31D414yes—
51C2×Dic62—22no—
52C2×Dic62—22no—
53C2×Dic62—24no—
54C2×Dic62—24no—
55C2×Dic62C2×C414yes—
56C2×Dic62C2×C418yes—
57C4⋊C4×C31—22no—
58C42×C31—22no—
59(C2×D31)⋊C4—24no—
60(C2×D31)⋊C4—24no—
61C4⋊Dic31—22no—
62Dic31⋊C4—24no—
63C4×Dic31—22no—
64C4×Dic31—24no—
65C23×C31C2411yescharacteristic
66C22×D31—22no—
67C22×D31—22no—
68C22×D31C22×C412yes—
69C22×D31C2×D412yes—
70C22×D31C2×D412yes—
71Q8×C31—28no—
72C2×C4×C31—22no—
73C2×C4×C31—22no—
74C2×C4×C31—22no—
75C2×C4×C31—22no—
76C2×C4×C31—22no—
77C2×C4×C31—24no—
78C2×C4×C31—22no—
79C2×C4×C31C22×C414yes—
80C2×C4×C31C22×C412yes—
81C2×C4×C31C2×D412yes—
82C2×C4×C31C2×D412yes—
83C2×Dic31—22no—
84C2×Dic31—22no—
85C2×Dic31—22no—
86C2×Dic31—22no—
87C2×Dic31—22no—
88C2×Dic31—24no—
89C2×Dic31—22no—
90C2×Dic31C22×C414yes—
91C2×Dic31C22×C412yes—
92C2×Dic31C2×D412yes—
93C2×Dic31C2×D412yes—
94C4×D31—216no—
95C4×D31C22⋊C4116yes—
96Dic62—28no—
97Dic62—216no—
98#2310—311nomaximal, Sylow
99C22×C31C23×C411yescharacteristic
100C22×C31C22×D411yescharacteristic
101C22×C31C22×D411yescharacteristic
102C22×C312−1+412yes—
103C22×C312−1+412yes—
104C2×D31—24no—
105C2×D31—22no—
106C2×D31—24no—
107C2×D31—22no—
108C2×D31C2×C22⋊C414yes—
109C4×C31—24no—
110C4×C31—42no—
111C4×C31—42no—
112C4×C31C2×C22⋊C414yes—
113Dic31—24no—
114Dic31—42no—
115Dic31—42no—
116Dic31C2×C22⋊C414yes—
117C23×Q8—311no—
118C2×#117—311no—
119C2×#117—311no—
120#391—311no—
121#391—311no—
122#391—311no—
123#391—314no—
124#391D3111yescharacteristic, complemented
125#381—314no—
126C2×C31C22×C22⋊C411yescharacteristic
127C2×C31#51912yes—
128C2×C31#53611yescharacteristic
129C2×C31#53612yes—
130C2×C31#53611yescharacteristic
131D31—24no—
132C22×Q8—318no—
133C22×Q8—312no—
134C22×Q8—311no—
135C22×Q8—624no—
136C22×Q8—624no—
137C22×Q8C2×D3111yescharacteristic
138C23×C4—311no—
139C23×C4—312no—
140#117—624no—
141#117—624no—
142C2×C4⋊C4—312no—
143C2×C4⋊C4—311no—
144C2×C4⋊C4C2×D3111yescharacteristic
145C2×C22⋊C4—312no—
146C2×C22⋊C4—312no—
147C2×C42—312no—
148C2×C42—311no—
149C2×C42C2×D3111yescharacteristic
150#95—314no—
151#95—314no—
152#95—314no—
153#95C2×D3114yes—
154C31#231011yescharacteristic, complemented, Sylow
155C24—311no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—318no—
159C2×Q8—314no—
160C2×Q8—624no—
161C2×Q8—624no—
162C2×Q8—6232no—
163C2×Q8—622no—
164C2×Q8—622no—
165C2×Q8C4×D3114yes—
166C22×C4—311no—
167C22×C4—314no—
168C22×C4—314no—
169C22×C4—314no—
170C22×C4—312no—
171C22×C4—318no—
172C22×C4—312no—
173C22×C4—312no—
174C22×C4—624no—
175C22×C4—624no—
176C22×C4C22×D3112yes—
177C22×C4C22×D3112yes—
178C22×C4C22×D3112yes—
179C22×C4C22×D3111yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—624no—
182C4⋊C4—622no—
183C22⋊C4—624no—
184C22⋊C4—624no—
185C42—22no—
186C42—624no—
187C42—622no—
188C23—312no—
189C23—312no—
190C23—312no—
191C23—622no—
192C23—622no—
193C23C23×D3111yescharacteristic
194Q8—28no—
195Q8—6216no—
196Q8—628no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—312no—
205C2×C4—312no—
206C2×C4—312no—
207C2×C4—3116no—
208C2×C4—314no—
209C2×C4—6216no—
210C2×C4—622no—
211C2×C4—622no—
212C2×C4—622no—
213C2×C4—622no—
214C2×C4—622no—
215C2×C4—624no—
216C2×C4—622no—
217C2×C4C2×C4×D3114yes—
218C2×C4C2×C4×D3112yes—
219C2×C4D4×D3112yes—
220C2×C4D4×D3112yes—
221C22—314no—
222C22—622no—
223C22—624no—
224C22—622no—
225C22—624no—
226C22C22×C4×D3111yescharacteristic
227C22C2×D4×D3111yescharacteristic
228C22C2×D4×D3111yescharacteristic
229C22#1175856412yes—
230C22#1175856712yes—
231C4—24no—
232C4—42no—
233C4—42no—
234C4—314no—
235C4—624no—
236C4—1242no—
237C4—1242no—
238C4C22⋊C4×D3114yes—
239C2—624no—
240C2C2×C22⋊C4×D3111yescharacteristic
241C2#1452152512yes—
242C2#1452161911yescharacteristic
243C2#1452162212yes—
244C2#1452162411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#391C2
socle, cosocleC23×C31C24
prime radicalC23×D31C23
hypercenter#391D31
hypocenterC31#2310
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #14520956C2semidirect
5. #14520955C2semidirect
6. #14520954C2semidirect
7. #14520953C2semidirect
9. C31×#391C2semidirect
13. C2×Q8×D31C4semidirect
124. #391D31semidirect
154. C31#2310semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D312
C2×#117584692
C2×#117584492
#145209562
#145209552
#145209542
#145209532
#145209112
C31×#3912
#231031