Finite Group Databasebeta

← #26213735

Subgroups — #26213735

Order 12032.

1996 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689890C211yesmaximal, characteristic
2#20256591C211yesmaximal, characteristic
3C2×#15688894C2112yesmaximal, complemented
4C2×C47×#568C211yesmaximal, characteristic
5#15689890—24no—
6#15689890C418yescomplemented
7C2×#13094031—24no—
8C22×#10625329C2216yes—
9#15689606—24no—
10#15689601C412yes—
11#15689598C416yes—
12#41×Dic47—24no—
13C2×#13093983C2216yes—
14#15688894C22116yes—
15C22×#41×C47C2211yescharacteristic
16C47×#568—24no—
17C47×#568—24no—
18C2×#131×C47C418yescomplemented
19C22×M16×C47C2216yes—
20#13094031—232no—
21C2×#10625329—26no—
22C2×#10625329—212no—
23C2×#10625329C2×C4124yescomplemented
24C2×#10625329C2314yes—
25C22×C47⋊C8—26no—
26Q8×Dic47—28no—
27D4×Dic47—224no—
28#13093983—26no—
29#13093983—26no—
30#13093983C2×C4112yes—
31#13093983C2314yes—
32C2×C4⋊Dic47—26no—
33C2×C22⋊Dic47C2×C4112yes—
34C2×C4×Dic47—26no—
35C2×#41×C47C2×C414yes—
36C2×#41×C47C2×C414yes—
37C2×#41×C47D414yes—
38C2×#41×C47D414yes—
39C2×#41×C47D414yes—
40C2×#41×C47Q814yes—
41C22×Q8×C47C2×C411yescharacteristic
42C22×D4×C47C2×C413yes—
43C23×C4×C47C2313yes—
44#131×C47—232no—
45C2×M16×C47—212no—
46C2×M16×C47—212no—
47C2×M16×C47C2×C4112yescomplemented
48C2×M16×C47C2314yes—
49C22×C8×C47C2×C4112yescomplemented
50C24×C47C22×C413yes—
51#41×C47C22⋊C4116yes—
52#41×C47C22⋊C4116yes—
53#41×C47C4⋊C4116yes—
54#41×C47C4⋊C4116yes—
55C2×Q8×C47C4214yes—
56C2×Q8×C47C22⋊C414yes—
57C2×Q8×C47C4⋊C414yes—
58C2×D4×C47C42112yes—
59C2×D4×C47C22⋊C4112yes—
60C2×D4×C47C4⋊C4112yes—
61C22×C4×C47C22×C413yes—
62C22×C4×C47C22×C416yes—
63C22×C4×C47C22×C416yes—
64C22×C4×C47C2×D416yes—
65C22×C4×C47C2×D416yes—
66C22×C4×C47C2×D416yes—
67C22×C4×C47C2×Q816yes—
68C22×C4×C47C2411yescharacteristic
69C22×Dic47—212no—
70M16×C47—224no—
71M16×C47—24no—
72M16×C47—24no—
73C2×C8×C47—224no—
74C2×C8×C47C22×C418yes—
75C22⋊Dic47—212no—
76C22⋊Dic47C22×C418yes—
77C4⋊Dic47—212no—
78C4⋊Dic47—24no—
79C4×Dic47—212no—
80C4×Dic47—24no—
81#10625329—224no—
82#10625329—24no—
83#10625329C22×C418yes—
84C2×C47⋊C8—224no—
85C2×C47⋊C8—24no—
86C23×C47—83no—
87C23×C47C2×C4216yes—
88C23×C47C2×C22⋊C416yes—
89C23×C47C2×C4⋊C416yes—
90C23×C47C23×C411yescharacteristic
91Q8×C47#95116yes—
92D4×C47#95148yes—
93C2×C4×C47C2×C4216yes—
94C2×C4×C47C2×C22⋊C4112yes—
95C2×C4×C47C2×C22⋊C416yes—
96C2×C4×C47C2×C22⋊C4112yes—
97C2×C4×C47C2×C4⋊C4112yes—
98C2×C4×C47C2×C4⋊C416yes—
99C2×C4×C47C2×C4⋊C4112yes—
100C2×C4×C47C23×C411yescharacteristic
101C2×C4×C47C23×C411yescharacteristic
102C2×C4×C47C22×D411yescharacteristic
103C2×C4×C47C22×D411yescharacteristic
104C2×C4×C47C22×D411yescharacteristic
105C2×C4×C47C22×Q811yescharacteristic
106C2×Dic47—28no—
107C2×Dic47—812no—
108C8×C47—28no—
109C47⋊C8—28no—
110#20360—471nomaximal, Sylow
111C22×C47—46no—
112C22×C47—41no—
113C22×C47—46no—
114C22×C47—46no—
115C22×C47C2×#95112yes—
116C22×C47C22×C4211yescharacteristic
117C22×C47C22×C22⋊C411yescharacteristic
118C22×C47C22×C4⋊C411yescharacteristic
119C4×C47C2×#95112yes—
120C4×C47C22×C22⋊C411yescharacteristic
121C4×C47C22×C22⋊C411yescharacteristic
122C4×C47C22×C4⋊C411yescharacteristic
123C4×C47C22×C4⋊C411yescharacteristic
124Dic47—48no—
125C2×#568—471no—
126C2×#568D4711yescharacteristic
127#3428—471no—
128C2×#344—4712no—
129C2×C47—21no—
130C2×C47—21no—
131C2×C47—21no—
132C2×C47—212no—
133C2×C47C22×#9511yescharacteristic
134C22×#41C2×D4711yescharacteristic
135#568—24no—
136#568—24no—
137#568—478no—
138#568—944no—
139C2×#131—944no—
140C2×#131Dic4718yescomplemented
141C22×M16—476no—
142C22×M16C2×D4716yes—
143#520—944no—
144#519—472no—
145#518—476no—
146C4×#41—944no—
147C2×#117—476no—
148#344—4716no—
149C47#2036011yescharacteristic, complemented, Sylow
150C2×#41Dic9414yes—
151C2×#41C4×D4714yes—
152C2×#41D18814yes—
153C2×#41C2×Dic4714yes—
154C2×#41C22⋊D4714yes—
155C2×#41C22⋊D4714yes—
156C22×Q8C4×D4711yescharacteristic
157C22×D4C4×D4713yes—
158C23×C4C22×D4713yes—
159#131—232no—
160#131—9432no—
161C2×M16—212no—
162C2×M16—212no—
163C2×M16—4724no—
164C2×M16—474no—
165C2×M16—9412no—
166C2×M16—946no—
167C2×M16C2×Dic47112yescomplemented
168C2×M16C22×D4714yes—
169C22×C8—946no—
170C22×C8C2×Dic47112yescomplemented
171C4×Q8—948no—
172C4×D4—9424no—
173#117—4712no—
174#117—474no—
175#117—946no—
176#117—946no—
177C2×C4⋊C4—946no—
178C2×C22⋊C4—4712no—
179C2×C42—946no—
180C24C2×C4×D4713yes—
181#41Dic47⋊C4116yes—
182#41C4⋊Dic47116yes—
183#41(C2×D47)⋊C4116yes—
184#41C22⋊Dic47116yes—
185C2×Q8C4×Dic4714yes—
186C2×Q8Dic47⋊C414yes—
187C2×Q8(C2×D47)⋊C414yes—
188C2×D4C4×Dic47112yes—
189C2×D4Dic47⋊C4112yes—
190C2×D4(C2×D47)⋊C4112yes—
191C22×C4—9412no—
192C22×C4C2×Dic9416yes—
193C22×C4C2×C4×D4713yes—
194C22×C4C2×C4×D4716yes—
195C22×C4C2×D18816yes—
196C22×C4C22×Dic4716yes—
197C22×C4C2×C22⋊D4716yes—
198C22×C4C2×C22⋊D4716yes—
199C22×C4C23×D4711yescharacteristic
200M16—224no—
201M16—24no—
202M16—24no—
203M16—478no—
204M16—9424no—
205M16—944no—
206C2×C8—224no—
207C2×C8—9424no—
208C2×C8—944no—
209C2×C8C22×Dic4718yes—
210C4⋊C4—9412no—
211C4⋊C4—944no—
212C22⋊C4—478no—
213C22⋊C4—9412no—
214C42—9412no—
215C42—944no—
216C23—83no—
217C23C2×C4×Dic4716yes—
218C23C2×(C2×D47)⋊C416yes—
219C23C2×Dic47⋊C416yes—
220C23C22×C4×D4711yescharacteristic
221Q8C47⋊#95116yes—
222D4C47⋊#95148yes—
223C2×C4—948no—
224C2×C4—37612no—
225C2×C4C2×C4×Dic4716yes—
226C2×C4C2×(C2×D47)⋊C416yes—
227C2×C4C2×(C2×D47)⋊C4112yes—
228C2×C4C2×C22⋊Dic47112yes—
229C2×C4C2×Dic47⋊C416yes—
230C2×C4C2×Dic47⋊C4112yes—
231C2×C4C2×C4⋊Dic47112yes—
232C2×C4C22×C4×D4711yescharacteristic
233C2×C4C23×Dic4711yescharacteristic
234C2×C4C22×C22⋊D4711yescharacteristic
235C2×C4C22×C22⋊D4711yescharacteristic
236C2×C4C22×D18811yescharacteristic
237C2×C4C22×Dic9411yescharacteristic
238C8—28no—
239C8—948no—
240C22—46no—
241C22—41no—
242C22—46no—
243C22—46no—
244C22C2×C47⋊#95112yes—
245C22C22×C4×Dic4711yescharacteristic
246C22C22×(C2×D47)⋊C411yescharacteristic
247C22C22×Dic47⋊C411yescharacteristic
248C4—1888no—
249C4C2×C47⋊#95112yes—
250C4C22×(C2×D47)⋊C411yescharacteristic
251C4C22×C22⋊Dic4711yescharacteristic
252C4C22×Dic47⋊C411yescharacteristic
253C4C22×C4⋊Dic4711yescharacteristic
254C2—21no—
255C2—21no—
256C2—21no—
257C2—212no—
258C2C22×C47⋊#9511yescharacteristic
subgroupG/H
centerC4C22×C22⋊Dic47
commutatorC22×C47C22×C42
FrattiniC22×C4C23×D47
FittingC2×C47×#568C2
socleC2×C47C22×#95
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#568D47
hypocenterC47#20360
LCSG ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
3. C2×#15688894C2semidirect
6. #15689890C4semidirect
18. C2×#131×C47C4semidirect
23. C2×#10625329C2×C4semidirect
47. C2×M16×C47C2×C4semidirect
49. C22×C8×C47C2×C4semidirect
140. C2×#131Dic47semidirect
149. C47#20360semidirect
167. C2×M16C2×Dic47semidirect
170. C22×C8C2×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×#156888942
C2×C47×#5682
#2036047