Finite Group Databasebeta

← #26213727

Subgroups — #26213727

Order 12032.

2046 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689890C211yesmaximal, characteristic
2C2×#15689606C211yesmaximal, characteristic
3#20243126C216yesmaximal, complemented
4C2×#15688891C216yesmaximal, complemented
5C2×C47×#568C211yesmaximal, characteristic
6#15689890—24no—
7#15689890C418yescomplemented
8C2×#13094031—24no—
9C22×#10625329C2216yes—
10#15689606—24no—
11#15689606—24no—
12C2×Q8×Dic47C412yes—
13C2×D4×Dic47C416yes—
14C2×#13093983C2213yes—
15C22×C4⋊Dic47C2213yes—
16#15688891C22112yes—
17#15688891C2214yes—
18C22×#41×C47C2211yescharacteristic
19C47×#568—24no—
20C47×#568C418yescomplemented
21C2×#131×C47—24no—
22C22×M16×C47C2216yes—
23C23×Dic47C2×C416yes—
24#13094031—232no—
25C2×#10625329—26no—
26C2×#10625329—212no—
27C2×#10625329C2×C4124yescomplemented
28C2×#10625329C2314yes—
29C22×C47⋊C8—26no—
30Q8×Dic47—28no—
31D4×Dic47—224no—
32#13093983—26no—
33#13093983—26no—
34C2×C4⋊Dic47—26no—
35C2×C4⋊Dic47—26no—
36C2×C4⋊Dic47C2×C416yes—
37C2×C4⋊Dic47C2313yes—
38C2×C4⋊Dic47C2311yescharacteristic
39C2×C22⋊Dic47C2×C416yes—
40C2×C4×Dic47C2×C416yes—
41C2×#41×C47C2×C414yes—
42C2×#41×C47C2×C414yes—
43C2×#41×C47D414yes—
44C2×#41×C47D414yes—
45C2×#41×C47D414yes—
46C2×#41×C47Q814yes—
47C22×Q8×C47C2×C411yescharacteristic
48C22×D4×C47C2×C413yes—
49C23×C4×C47C2313yes—
50#131×C47—232no—
51C2×M16×C47—26no—
52C2×M16×C47—212no—
53C2×M16×C47C2×C4124yescomplemented
54C2×M16×C47C2314yes—
55C22×C8×C47—26no—
56C24×C47C22×C413yes—
57#41×C47C22⋊C4116yes—
58#41×C47C22⋊C4116yes—
59#41×C47C4⋊C4116yes—
60#41×C47C4⋊C4116yes—
61C2×Q8×C47C4214yes—
62C2×Q8×C47C22⋊C414yes—
63C2×Q8×C47C4⋊C414yes—
64C2×D4×C47C42112yes—
65C2×D4×C47C22⋊C4112yes—
66C2×D4×C47C4⋊C4112yes—
67C22×C4×C47C22×C413yes—
68C22×C4×C47C22×C416yes—
69C22×C4×C47C22×C416yes—
70C22×C4×C47C2×D416yes—
71C22×C4×C47C2×D416yes—
72C22×C4×C47C2×D416yes—
73C22×C4×C47C2×Q816yes—
74C22×C4×C47C2411yescharacteristic
75C22×Dic47—212no—
76C22×Dic47—86no—
77C22×Dic47C22×C412yes—
78C22×Dic47C22×C416yes—
79M16×C47—224no—
80M16×C47—24no—
81M16×C47C22×C418yes—
82C2×C8×C47—224no—
83C2×C8×C47—24no—
84C22⋊Dic47—212no—
85C4⋊Dic47—212no—
86C4⋊Dic47—23no—
87C4⋊Dic47—23no—
88C4⋊Dic47—21no—
89C4⋊Dic47—21no—
90C4×Dic47—212no—
91#10625329—224no—
92#10625329—24no—
93#10625329C22×C418yes—
94C2×C47⋊C8—224no—
95C2×C47⋊C8—24no—
96C23×C47—43no—
97C23×C47—43no—
98C23×C47C2×C4216yes—
99C23×C47C2×C22⋊C416yes—
100C23×C47C2×C4⋊C416yes—
101C23×C47C23×C411yescharacteristic
102Q8×C47#95116yes—
103D4×C47#95148yes—
104C2×C4×C47C2×C4216yes—
105C2×C4×C47C2×C22⋊C4112yes—
106C2×C4×C47C2×C22⋊C416yes—
107C2×C4×C47C2×C22⋊C4112yes—
108C2×C4×C47C2×C4⋊C4112yes—
109C2×C4×C47C2×C4⋊C416yes—
110C2×C4×C47C2×C4⋊C4112yes—
111C2×C4×C47C23×C411yescharacteristic
112C2×C4×C47C23×C411yescharacteristic
113C2×C4×C47C22×D411yescharacteristic
114C2×C4×C47C22×D411yescharacteristic
115C2×C4×C47C22×D411yescharacteristic
116C2×C4×C47C22×Q811yescharacteristic
117C2×Dic47—22no—
118C2×Dic47—26no—
119C2×Dic47—46no—
120C2×Dic47—42no—
121C2×Dic47—812no—
122C8×C47—28no—
123C47⋊C8—28no—
124#20358—471nomaximal, Sylow
125C22×C47—21no—
126C22×C47—26no—
127C22×C47—21no—
128C22×C47—26no—
129C22×C47—46no—
130C22×C47—46no—
131C22×C47C2×#95112yes—
132C22×C47C22×C4211yescharacteristic
133C22×C47C22×C22⋊C411yescharacteristic
134C22×C47C22×C4⋊C411yescharacteristic
135C4×C47C2×#95112yes—
136C4×C47C22×C22⋊C411yescharacteristic
137C4×C47C22×C22⋊C411yescharacteristic
138C4×C47C22×C4⋊C411yescharacteristic
139C4×C47C22×C4⋊C411yescharacteristic
140Dic47—46no—
141Dic47—42no—
142C2×#568—471no—
143C2×#568D4711yescharacteristic
144C2×#520—471no—
145#1742—476no—
146C2×#343—476no—
147C2×C47—21no—
148C2×C47—21no—
149C2×C47—212no—
150C2×C47C22×#9511yescharacteristic
151C2×C47#288811yescharacteristic
152C2×C47#288911yescharacteristic
153C22×#41C2×D4711yescharacteristic
154#568—24no—
155#568—478no—
156#568—944no—
157#568Dic4718yescomplemented
158C2×#131—24no—
159C2×#131—944no—
160C22×M16—476no—
161C22×M16C2×D4716yes—
162#520—944no—
163#520—944no—
164C2×C4×Q8—472no—
165C2×C4×D4—476no—
166C2×#117—473no—
167C22×C4⋊C4—473no—
168#343—4712no—
169#343—474no—
170C47#2035811yescharacteristic, complemented, Sylow
171C2×#41Dic9414yes—
172C2×#41C4×D4714yes—
173C2×#41D18814yes—
174C2×#41C2×Dic4714yes—
175C2×#41C22⋊D4714yes—
176C2×#41C22⋊D4714yes—
177C22×Q8C4×D4711yescharacteristic
178C22×D4C4×D4713yes—
179C23×C4—476no—
180C23×C4C22×D4713yes—
181#131—232no—
182#131—9432no—
183C2×M16—26no—
184C2×M16—212no—
185C2×M16—4724no—
186C2×M16—474no—
187C2×M16—946no—
188C2×M16—9412no—
189C2×M16C2×Dic47124yescomplemented
190C2×M16C22×D4714yes—
191C22×C8—26no—
192C22×C8—946no—
193C4×Q8—948no—
194C4×D4—9424no—
195#117—946no—
196#117—946no—
197C2×C4⋊C4—476no—
198C2×C4⋊C4—473no—
199C2×C4⋊C4—471no—
200C2×C4⋊C4—946no—
201C2×C4⋊C4—946no—
202C2×C22⋊C4—476no—
203C2×C42—476no—
204C24C2×C4×D4713yes—
205#41Dic47⋊C4116yes—
206#41C4⋊Dic47116yes—
207#41(C2×D47)⋊C4116yes—
208#41C22⋊Dic47116yes—
209C2×Q8C4×Dic4714yes—
210C2×Q8Dic47⋊C414yes—
211C2×Q8(C2×D47)⋊C414yes—
212C2×D4C4×Dic47112yes—
213C2×D4Dic47⋊C4112yes—
214C2×D4(C2×D47)⋊C4112yes—
215C22×C4—472no—
216C22×C4—476no—
217C22×C4—9412no—
218C22×C4—3766no—
219C22×C4C2×Dic9416yes—
220C22×C4C2×C4×D4713yes—
221C22×C4C2×C4×D4716yes—
222C22×C4C2×D18816yes—
223C22×C4C22×Dic4716yes—
224C22×C4C2×C22⋊D4716yes—
225C22×C4C2×C22⋊D4716yes—
226C22×C4C23×D4711yescharacteristic
227M16—224no—
228M16—24no—
229M16—478no—
230M16—9424no—
231M16—944no—
232M16C22×Dic4718yes—
233C2×C8—224no—
234C2×C8—24no—
235C2×C8—9424no—
236C2×C8—944no—
237C4⋊C4—9412no—
238C4⋊C4—943no—
239C4⋊C4—941no—
240C4⋊C4—943no—
241C4⋊C4—941no—
242C22⋊C4—9412no—
243C42—9412no—
244C23—43no—
245C23—43no—
246C23C2×C4×Dic4716yes—
247C23C2×(C2×D47)⋊C416yes—
248C23C2×Dic47⋊C416yes—
249C23C22×C4×D4711yescharacteristic
250Q8C47⋊#95116yes—
251D4C47⋊#95148yes—
252C2×C4—942no—
253C2×C4—946no—
254C2×C4—37612no—
255C2×C4—1882no—
256C2×C4—1886no—
257C2×C4C2×C4×Dic4716yes—
258C2×C4C2×(C2×D47)⋊C416yes—
259C2×C4C2×(C2×D47)⋊C4112yes—
260C2×C4C2×C22⋊Dic47112yes—
261C2×C4C2×Dic47⋊C416yes—
262C2×C4C2×Dic47⋊C4112yes—
263C2×C4C2×C4⋊Dic47112yes—
264C2×C4C22×C4×D4711yescharacteristic
265C2×C4C23×Dic4711yescharacteristic
266C2×C4C22×C22⋊D4711yescharacteristic
267C2×C4C22×C22⋊D4711yescharacteristic
268C2×C4C22×D18811yescharacteristic
269C2×C4C22×Dic9411yescharacteristic
270C8—28no—
271C8—948no—
272C22—26no—
273C22—21no—
274C22—21no—
275C22—26no—
276C22—46no—
277C22—46no—
278C22C2×C47⋊#95112yes—
279C22C22×C4×Dic4711yescharacteristic
280C22C22×(C2×D47)⋊C411yescharacteristic
281C22C22×Dic47⋊C411yescharacteristic
282C4—1886no—
283C4—1882no—
284C4C2×C47⋊#95112yes—
285C4C22×(C2×D47)⋊C411yescharacteristic
286C4C22×C22⋊Dic4711yescharacteristic
287C4C22×Dic47⋊C411yescharacteristic
288C4C22×C4⋊Dic4711yescharacteristic
289C2—21no—
290C2—21no—
291C2—212no—
292C2C22×C47⋊#9511yescharacteristic
293C2#2025186611yescharacteristic
294C2#2025187011yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC22×C47C22×C42
FrattiniC22×C4C23×D47
FittingC2×C47×#568C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#568D47
hypocenterC47#20358
LCSG ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
3. #20243126C2semidirect
4. C2×#15688891C2semidirect
7. #15689890C4semidirect
20. C47×#568C4semidirect
27. C2×#10625329C2×C4semidirect
53. C2×M16×C47C2×C4semidirect
157. #568Dic47semidirect
170. C47#20358semidirect
189. C2×M16C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156898902
C2×#156896062
#202431262
C2×#156888912
C2×C47×#5682
#2035847