Finite Group Databasebeta

← #16854014

Subgroups — #16854014

Order 3968.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×#11758524C211yesmaximal, characteristic, complemented
2C2×#11758507C211yesmaximal, characteristic
3C22×C4×Dic31C211yesmaximal, characteristic, complemented
4#14520956C211yesmaximal, characteristic, complemented
5#14520950C211yesmaximal, characteristic, complemented
6#14520939C214yesmaximal, complemented
7#14520926C214yesmaximal, complemented
8#14520914C211yesmaximal, characteristic, complemented
9C2×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C23×C4×C31C2211yescharacteristic
12C22×C22⋊D31C2212yes—
13C23×Dic31C2212yes—
14C22×D124C2211yescharacteristic
15C31×C22⋊Q8—24no—
16C2×C4⋊C4×C31C2211yescharacteristic
17C2×C4⋊C4×C31C2212yes—
18C2×C22⋊C4×C31C2212yes—
19#11758524—24no—
20#11758507—24no—
21C2×(C2×D31)⋊C4C2214yes—
22C2×(C2×D31)⋊C4C2214yes—
23C2×(C2×D31)⋊C4C2212yes—
24C2×C4⋊Dic31C2211yescharacteristic
25C2×C4×Dic31—24no—
26C2×C4×Dic31C2212yes—
27C2×C4×Dic31C2211yescharacteristic
28C2×C4×Dic31C2211yescharacteristic
29C31⋊#95C2214yes—
30C31⋊#95C2214yes—
31C24×C31C2311yescharacteristic
32C23×D31C2312yes—
33C2×Q8×C31—22no—
34C2×Q8×C31—22no—
35C2×Q8×C31—22no—
36C22×C4×C31—22no—
37C22×C4×C31—22no—
38C22×C4×C31D414yes—
39C22×C4×C31C2312yes—
40C22×C4×C31C2312yes—
41C22×C4×C31C2311yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C2×C22⋊D31—24no—
44C2×C22⋊D31—28no—
45C22×Dic31—24no—
46C22×Dic31—24no—
47C22×Dic31—24no—
48C22×Dic31C2312yes—
49C22×Dic31C2312yes—
50C22×Dic31C2312yes—
51C2×D124—22no—
52C2×D124—22no—
53C2×D124—22no—
54C4⋊C4×C31—22no—
55C4⋊C4×C31—24no—
56C22⋊C4×C31—24no—
57(C2×D31)⋊C4—24no—
58(C2×D31)⋊C4—28no—
59(C2×D31)⋊C4—28no—
60C4⋊Dic31—22no—
61C4×Dic31—22no—
62C4×Dic31—24no—
63C4×Dic31D414yescomplemented
64C23×C31—21no—
65C23×C31—22no—
66C23×C31—21no—
67C23×C31C2×D412yes—
68C23×C31#4114yes—
69C23×C31C2411yescharacteristic
70C22×D31—22no—
71C22×D31—22no—
72C22×D31—24no—
73C22×D31—22no—
74C22×D31—24no—
75Q8×C31—44no—
76C2×C4×C31—21no—
77C2×C4×C31—21no—
78C2×C4×C31—28no—
79C2×C4×C31—22no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—24no—
83C2×C4×C31—22no—
84C2×C4×C31C2×D412yes—
85C2×C4×C31C2×D412yes—
86C2×C4×C31C2×D412yes—
87C2×C4×C31#4112yes—
88C22⋊D31—48no—
89C2×Dic31—216no—
90C2×Dic31—22no—
91C2×Dic31—22no—
92C2×Dic31—22no—
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—24no—
96C2×Dic31C2×D414yes—
97C2×Dic31#4114yes—
98D124—44no—
99#2850—311nomaximal, Sylow
100C22×C31—22no—
101C22×C31—24no—
102C22×C31—24no—
103C22×C31—22no—
104C22×C31#12414yescomplemented
105C22×C31C22×D411yescharacteristic
106C22×C31C22×D411yescharacteristic
107C22×C31C2×#4111yescharacteristic
108C22×C31C2×#4112yes—
109C22×C31C2×#4112yes—
110C2×D31—42no—
111C2×D31—44no—
112C2×D31—42no—
113C2×D31—44no—
114C2×D31—42no—
115C4×C31—22no—
116C4×C31—42no—
117C4×C31—42no—
118C4×C31#12114yes—
119Dic31—24no—
120Dic31—24no—
121Dic31—42no—
122C2×#124—311no—
123C2×C22⋊Q8D3111yescharacteristic, complemented
124C2×#121—311no—
125C22×C42—311no—
126#391—311no—
127#390—311no—
128#388—314no—
129#386—314no—
130#382—311no—
131C2×C31—24no—
132C2×C31C2×#12111yescharacteristic
133C2×C31#52512yes—
134C2×C31C2×#12411yescharacteristic
135C2×C31#52912yes—
136C2×C31#53211yescharacteristic
137D31—82no—
138C22×Q8C2×D3111yescharacteristic
139C22×D4—312no—
140C22×D4—311no—
141C23×C4—312no—
142C23×C4C2×D3111yescharacteristic
143#124—624no—
144C22⋊Q8—24no—
145#121—624no—
146C2×C4⋊C4—311no—
147C2×C4⋊C4C2×D3111yescharacteristic
148C2×C4⋊C4C2×D3112yes—
149C2×C22⋊C4—312no—
150C2×C22⋊C4—314no—
151C2×C22⋊C4—314no—
152C2×C22⋊C4C2×D3112yes—
153C2×C42—311no—
154C2×C42—312no—
155C2×C42—311no—
156C2×C42—624no—
157#95—314no—
158#95—314no—
159C31#285011yescharacteristic, complemented, Sylow
160C24—312no—
161C24C22×D3111yescharacteristic
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—628no—
166C2×D4—624no—
167C2×D4—622no—
168C2×D4—622no—
169C2×D4—622no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—312no—
173C22×C4—312no—
174C22×C4—312no—
175C22×C4—624no—
176C22×C4—624no—
177C22×C4—624no—
178C22×C4C22⋊D3114yes—
179C22×C4C22×D3112yes—
180C22×C4C22×D3112yes—
181C22×C4C22×D3111yescharacteristic
182C22×C4C22×D3111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—622no—
186C22⋊C4—24no—
187C22⋊C4—628no—
188C22⋊C4—628no—
189C22⋊C4—624no—
190C42—314no—
191C42—624no—
192C42—622no—
193C23—21no—
194C23—21no—
195C23—22no—
196C23—622no—
197C23—622no—
198C23—624no—
199C23—624no—
200C23—622no—
201C23#9249914yes—
202C23C2×C22⋊D3112yes—
203C23C23×D3111yescharacteristic
204Q8—44no—
205D4—1248no—
206D4—1244no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—314no—
216C2×C4—314no—
217C2×C4—6216no—
218C2×C4—622no—
219C2×C4—622no—
220C2×C4—622no—
221C2×C4—622no—
222C2×C4—624no—
223C2×C4—622no—
224C2×C4D4×D3112yes—
225C2×C4#9249712yes—
226C2×C4C2×C22⋊D3112yes—
227C2×C4C2×C22⋊D3112yes—
228C22—22no—
229C22—24no—
230C22—24no—
231C22—22no—
232C22—1242no—
233C22—1244no—
234C22—1242no—
235C22—1244no—
236C22—1242no—
237C22#1175852414yescomplemented
238C22C2×D4×D3111yescharacteristic
239C22C2×#9249711yescharacteristic
240C22C2×#9249912yes—
241C22#41×D3112yes—
242C22C22×C22⋊D3111yescharacteristic
243C4—22no—
244C4—42no—
245C4—42no—
246C4—624no—
247C4—624no—
248C4—1242no—
249C4#1175851614yes—
250C2—24no—
251C2—2482no—
252C2C2×#1175851611yescharacteristic
253C2#1452156212yes—
254C2C2×#1175852411yescharacteristic
255C2#1452158512yes—
256C2#1452159511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC2×C31×C22⋊Q8C2
prime radicalC2×#11758507C2
hypercenterC2×C22⋊Q8D31
hypocenterC31#2850
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11758524C2semidirect
3. C22×C4×Dic31C2semidirect
4. #14520956C2semidirect
5. #14520950C2semidirect
6. #14520939C2semidirect
7. #14520926C2semidirect
8. #14520914C2semidirect
9. C2×C31×C22⋊Q8C2semidirect
63. C4×Dic31D4semidirect
104. C22×C31#124semidirect
123. C2×C22⋊Q8D31semidirect
159. C31#2850semidirect
237. C22#11758524semidirect

Maximal subgroups

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

subgroupindex
C2×#117585242
C2×#117585072
C22×C4×Dic312
#145209562
#145209502
#145209392
#145209262
#145209142
C2×C31×C22⋊Q82
#285031