Finite Group Databasebeta

← #26213652

Subgroups — #26213652

Order 12032.

2036 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689890C211yesmaximal, characteristic
2#20256591C211yesmaximal, characteristic
3C2×#15688879C2112yesmaximal, complemented
4C47×#3428C211yesmaximal, characteristic
5#15689890—24no—
6#15689890—24no—
7C2×#13094031C418yescomplemented
8C22×#10625329C2216yes—
9#15689606—24no—
10#15689601—21no—
11#15689598—23no—
12#41×Dic47C418yescomplemented
13C2×#13093983C2216yes—
14#15688879C22116yes—
15C22×#41×C47C2211yescharacteristic
16C47×#520—24no—
17C47×#519—21no—
18C47×#518—23no—
19C4×#41×C47C418yescomplemented
20C2×#117×C47C2216yes—
21#13094031—232no—
22C2×#10625329—212no—
23C2×#10625329—212no—
24C2×#10625329C2×C4112yescomplemented
25C2×#10625329C2314yes—
26C22×C47⋊C8C2×C4112yescomplemented
27Q8×Dic47—28no—
28D4×Dic47—224no—
29#13093983—26no—
30#13093983—26no—
31#13093983C2×C4112yescomplemented
32#13093983C2314yes—
33C2×C4⋊Dic47—26no—
34C2×C22⋊Dic47—26no—
35C2×C4×Dic47C2×C4112yescomplemented
36C2×#41×C47C2×C414yes—
37C2×#41×C47C2×C414yes—
38C2×#41×C47C2×C414yes—
39C2×#41×C47D414yes—
40C2×#41×C47D414yes—
41C2×#41×C47Q814yes—
42C22×Q8×C47D411yescharacteristic
43C22×D4×C47D413yes—
44C23×C4×C47C2313yes—
45C4×Q8×C47—28no—
46C4×D4×C47—224no—
47#117×C47—26no—
48#117×C47—26no—
49#117×C47C2×C4112yescomplemented
50#117×C47C2314yes—
51C2×C4⋊C4×C47—26no—
52C2×C22⋊C4×C47—26no—
53C2×C42×C47C2×C4112yescomplemented
54C24×C47C2×D413yes—
55#41×C47C42116yes—
56#41×C47C22⋊C4116yes—
57#41×C47C4⋊C4116yes—
58#41×C47C4⋊C4116yes—
59C2×Q8×C47C22⋊C414yes—
60C2×Q8×C47C22⋊C414yes—
61C2×Q8×C47C4⋊C414yes—
62C2×D4×C47C22⋊C4112yes—
63C2×D4×C47C22⋊C4112yes—
64C2×D4×C47C4⋊C4112yes—
65C22×C4×C47—212no—
66C22×C4×C47C22×C416yes—
67C22×C4×C47C22×C416yes—
68C22×C4×C47C22×C416yes—
69C22×C4×C47C2×D413yes—
70C22×C4×C47C2×D416yes—
71C22×C4×C47C2×D416yes—
72C22×C4×C47C2×Q816yes—
73C22×C4×C47C2411yescharacteristic
74C22×Dic47—212no—
75C4⋊C4×C47—212no—
76C4⋊C4×C47—24no—
77C22⋊C4×C47—212no—
78C22⋊C4×C47—24no—
79C42×C47—212no—
80C42×C47C22×C418yes—
81C22⋊Dic47—212no—
82C22⋊Dic47—24no—
83C4⋊Dic47—212no—
84C4⋊Dic47—24no—
85C4×Dic47—212no—
86C4×Dic47C22×C418yes—
87#10625329—224no—
88#10625329—24no—
89#10625329—24no—
90C2×C47⋊C8—224no—
91C2×C47⋊C8C22×C418yes—
92C23×C47—83no—
93C23×C47C2×C22⋊C416yes—
94C23×C47C2×C22⋊C416yes—
95C23×C47C2×C4⋊C416yes—
96C23×C47C22×D411yescharacteristic
97Q8×C47#95116yes—
98D4×C47#95148yes—
99C2×C4×C47—28no—
100C2×C4×C47—812no—
101C2×C4×C47C2×C42112yes—
102C2×C4×C47C2×C22⋊C4112yes—
103C2×C4×C47C2×C22⋊C416yes—
104C2×C4×C47C2×C22⋊C416yes—
105C2×C4×C47C2×C4⋊C4112yes—
106C2×C4×C47C2×C4⋊C416yes—
107C2×C4×C47C2×C4⋊C4112yes—
108C2×C4×C47C23×C411yescharacteristic
109C2×C4×C47C23×C411yescharacteristic
110C2×C4×C47C23×C411yescharacteristic
111C2×C4×C47C22×D411yescharacteristic
112C2×C4×C47C22×D411yescharacteristic
113C2×C4×C47C22×Q811yescharacteristic
114C2×Dic47—28no—
115C2×Dic47—812no—
116C47⋊C8—28no—
117#20338—471nomaximal, Sylow
118C22×C47—46no—
119C22×C47—46no—
120C22×C47—41no—
121C22×C47—46no—
122C22×C47C2×#95112yes—
123C22×C47C22×C22⋊C411yescharacteristic
124C22×C47C22×C22⋊C411yescharacteristic
125C22×C47C22×C4⋊C411yescharacteristic
126C4×C47—48no—
127C4×C47C2×#95112yes—
128C4×C47C22×C4211yescharacteristic
129C4×C47C22×C22⋊C411yescharacteristic
130C4×C47C22×C4⋊C411yescharacteristic
131C4×C47C22×C4⋊C411yescharacteristic
132Dic47—48no—
133C2×#568—471no—
134#3428—471no—
135#3428D4711yescharacteristic
136C2×H(3,ℤ/4)—4712no—
137C2×C47—21no—
138C2×C47—21no—
139C2×C47—21no—
140C2×C47—212no—
141C2×C47C22×#9511yescharacteristic
142C22×#41C2×D4711yescharacteristic
143#568—944no—
144#568—944no—
145C2×#131—478no—
146C22×M16—476no—
147#520—24no—
148#520—944no—
149#519—21no—
150#519—941no—
151#518—23no—
152#518—943no—
153C4×#41—478no—
154C4×#41Dic4718yescomplemented
155C2×#117—476no—
156C2×#117C2×D4716yes—
157H(3,ℤ/4)—4716no—
158C47#2033811yescharacteristic, complemented, Sylow
159C2×#41Dic9414yes—
160C2×#41C4×D4714yes—
161C2×#41C4×D4714yes—
162C2×#41D18814yes—
163C2×#41C2×Dic4714yes—
164C2×#41C22⋊D4714yes—
165C22×Q8C22⋊D4711yescharacteristic
166C22×D4C22⋊D4713yes—
167C23×C4C22×D4713yes—
168#131—9432no—
169C2×M16—4712no—
170C2×M16—474no—
171C2×M16—9412no—
172C2×M16—9412no—
173C22×C8—4712no—
174C4×Q8—28no—
175C4×Q8—948no—
176C4×D4—224no—
177C4×D4—9424no—
178#117—26no—
179#117—26no—
180#117—4712no—
181#117—474no—
182#117—946no—
183#117—946no—
184#117C2×Dic47112yescomplemented
185#117C22×D4714yes—
186C2×C4⋊C4—26no—
187C2×C4⋊C4—946no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—946no—
190C2×C42—4712no—
191C2×C42C2×Dic47112yescomplemented
192C24C2×C22⋊D4713yes—
193#41C4×Dic47116yes—
194#41Dic47⋊C4116yes—
195#41C4⋊Dic47116yes—
196#41(C2×D47)⋊C4116yes—
197C2×Q8Dic47⋊C414yes—
198C2×Q8(C2×D47)⋊C414yes—
199C2×Q8C22⋊Dic4714yes—
200C2×D4Dic47⋊C4112yes—
201C2×D4(C2×D47)⋊C4112yes—
202C2×D4C22⋊Dic47112yes—
203C22×C4—212no—
204C22×C4—9412no—
205C22×C4C2×Dic9416yes—
206C22×C4C2×C4×D4716yes—
207C22×C4C2×C4×D4716yes—
208C22×C4C2×D18816yes—
209C22×C4C22×Dic4716yes—
210C22×C4C2×C22⋊D4713yes—
211C22×C4C2×C22⋊D4716yes—
212C22×C4C23×D4711yescharacteristic
213M16—9424no—
214M16—944no—
215M16—944no—
216C2×C8—478no—
217C2×C8—9424no—
218C4⋊C4—212no—
219C4⋊C4—24no—
220C4⋊C4—9412no—
221C4⋊C4—944no—
222C22⋊C4—212no—
223C22⋊C4—24no—
224C22⋊C4—9412no—
225C22⋊C4—944no—
226C42—212no—
227C42—478no—
228C42—9412no—
229C42C22×Dic4718yes—
230C23—83no—
231C23C2×(C2×D47)⋊C416yes—
232C23C2×C22⋊Dic4716yes—
233C23C2×Dic47⋊C416yes—
234C23C22×C22⋊D4711yescharacteristic
235Q8C47⋊#95116yes—
236D4C47⋊#95148yes—
237C2×C4—28no—
238C2×C4—812no—
239C2×C4—948no—
240C2×C4—37612no—
241C2×C4C2×C4×Dic47112yes—
242C2×C4C2×(C2×D47)⋊C416yes—
243C2×C4C2×(C2×D47)⋊C4112yes—
244C2×C4C2×C22⋊Dic4716yes—
245C2×C4C2×Dic47⋊C416yes—
246C2×C4C2×Dic47⋊C4112yes—
247C2×C4C2×C4⋊Dic47112yes—
248C2×C4C22×C4×D4711yescharacteristic
249C2×C4C22×C4×D4711yescharacteristic
250C2×C4C23×Dic4711yescharacteristic
251C2×C4C22×C22⋊D4711yescharacteristic
252C2×C4C22×D18811yescharacteristic
253C2×C4C22×Dic9411yescharacteristic
254C8—948no—
255C22—46no—
256C22—46no—
257C22—41no—
258C22—46no—
259C22C2×C47⋊#95112yes—
260C22C22×(C2×D47)⋊C411yescharacteristic
261C22C22×C22⋊Dic4711yescharacteristic
262C22C22×Dic47⋊C411yescharacteristic
263C4—48no—
264C4—1888no—
265C4C2×C47⋊#95112yes—
266C4C22×C4×Dic4711yescharacteristic
267C4C22×(C2×D47)⋊C411yescharacteristic
268C4C22×Dic47⋊C411yescharacteristic
269C4C22×C4⋊Dic4711yescharacteristic
270C2—21no—
271C2—21no—
272C2—21no—
273C2—212no—
274C2C22×C47⋊#9511yescharacteristic
subgroupG/H
centerC4C22×C4×Dic47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC47×#3428C2
socleC2×C47C22×#95
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenter#3428D47
hypocenterC47#20338
LCSG ▷ C4×C47 ▷ C47
UCSC1 ◁ C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
3. C2×#15688879C2semidirect
7. C2×#13094031C4semidirect
12. #41×Dic47C4semidirect
19. C4×#41×C47C4semidirect
24. C2×#10625329C2×C4semidirect
26. C22×C47⋊C8C2×C4semidirect
31. #13093983C2×C4semidirect
35. C2×C4×Dic47C2×C4semidirect
49. #117×C47C2×C4semidirect
53. C2×C42×C47C2×C4semidirect
154. C4×#41Dic47semidirect
158. C47#20338semidirect
184. #117C2×Dic47semidirect
191. C2×C42C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156898902
#202565912
C2×#156888792
C47×#34282
#2033847