Finite Group Databasebeta

← #26147486

Subgroups — #26147486

Order 12032.

6494 conjugacy classes of subgroups; only the 675 normal subgroups are shown.

normal subgroups only

This group has 6494 conjugacy classes of subgroups — too many to draw. Only its 675 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
1C25×Dic47C211yesmaximal, characteristic
2C2×#15688995C216yesmaximal, complemented
3C2×#15688979C211yesmaximal, characteristic
4#20241610C216yesmaximal, complemented
5C2×C47×#379C211yesmaximal, characteristic
6C24×Dic47—224no—
7C24×Dic47—24no—
8C24×Dic47C2216yes—
9C22×Dic47⋊C4C2214yescomplemented
10C22×C22⋊Dic47C2216yes—
11#15688995—224no—
12#15688979C418yescomplemented
13C2×C47⋊#95C22112yescomplemented
14C26×C47C2211yescharacteristic
15C22×C22⋊C4×C47C2213yes—
16C22×C22⋊C4×C47C2213yes—
17C47×#379—24no—
18C23×Dic47—296no—
19C23×Dic47—2128no—
20C23×Dic47—212no—
21C23×Dic47—212no—
22C23×Dic47—248no—
23C23×Dic47D4124yes—
24C23×Dic47C2314yes—
25C23×Dic47C2314yes—
26C2×Dic47⋊C4—224no—
27C2×C22⋊Dic47—212no—
28C2×C22⋊Dic47—212no—
29C2×C22⋊Dic47C2×C4124yescomplemented
30C47⋊#95—224no—
31C25×C47—212no—
32C25×C47—212no—
33C25×C47C2×C414yes—
34C25×C47D414yes—
35C25×C47Q814yes—
36C25×C47C2313yes—
37C23×C4×C47C2313yes—
38C23×C4×C47C2311yescharacteristic
39C2×C22⋊C4×C47—212no—
40C2×C22⋊C4×C47—26no—
41C2×C22⋊C4×C47—26no—
42C2×C22⋊C4×C47—212no—
43C24×C47—296no—
44C24×C47—232no—
45C24×C47—248no—
46C24×C47—224no—
47C24×C47—248no—
48C24×C47—224no—
49C24×C47—212no—
50C24×C47—212no—
51C24×C47C4⋊C4116yescomplemented
52C24×C47C22×C416yes—
53C24×C47C2×D416yes—
54C24×C47C2×D4112yes—
55C24×C47C2×Q816yes—
56C24×C47#41112yes—
57C24×C47C2411yescharacteristic
58C22×C4×C47—21no—
59C22×C4×C47—23no—
60C22×C4×C47—26no—
61C22×C4×C47—26no—
62C22×C4×C47—212no—
63C22×Dic47—2256no—
64C22×Dic47—296no—
65C22×Dic47—296no—
66C22×Dic47—248no—
67C22×Dic47—212no—
68C22×Dic47—24no—
69C22×Dic47—296no—
70C22×Dic47—212no—
71C22×Dic47—212no—
72C22×Dic47C22×C418yescomplemented
73C22×Dic47C2×D4124yes—
74C22⋊C4×C47—412no—
75C22⋊C4×C47—412no—
76C22⋊Dic47—248no—
77Dic47⋊C4—416no—
78C23×C47—232no—
79C23×C47—2128no—
80C23×C47—296no—
81C23×C47—2128no—
82C23×C47—224no—
83C23×C47—224no—
84C23×C47—212no—
85C23×C47—224no—
86C23×C47—212no—
87C23×C47—224no—
88C23×C47—248no—
89C23×C47—296no—
90C23×C47C2×C4⋊C4112yes—
91C23×C47C4×D4124yes—
92C23×C47#120112yes—
93C23×C47C22⋊Q8124yes—
94C23×C47#123112yes—
95C23×C47C23×C411yescharacteristic
96C23×C47C22×D416yes—
97C23×C47C22×D411yescharacteristic
98C23×C47C22×Q811yescharacteristic
99C23×C47C2×#4116yes—
100C2×C4×C47—44no—
101C2×C4×C47—43no—
102C2×C4×C47—43no—
103C2×C4×C47—46no—
104C2×C4×C47—412no—
105C2×Dic47—2192no—
106C2×Dic47—224no—
107C2×Dic47—224no—
108C2×Dic47—224no—
109C2×Dic47—44no—
110C2×Dic47—412no—
111C2×Dic47#120116yes—
112#14693—471nomaximal, Sylow
113C22×C47—2128no—
114C22×C47—212no—
115C22×C47—248no—
116C22×C47—212no—
117C22×C47—224no—
118C22×C47—224no—
119C22×C47—248no—
120C22×C47#385124yescomplemented
121C22×C47C22×C4⋊C411yescharacteristic
122C22×C47C2×C4×D416yes—
123C22×C47C2×#12014yes—
124C22×C47C2×#12013yes—
125C22×C47C2×C22⋊Q816yes—
126C22×C47C2×#12313yes—
127C22×C47#525112yes—
128C4×C47—81no—
129C4×C47—83no—
130Dic47—216no—
131Dic47—48no—
132C25×C4—471no—
133C2×#385—476no—
134C2×#379—471no—
135C2×#379D4711yescharacteristic
136C22⋊#95—476no—
137C2×C47—224no—
138C2×C47C2×#38513yes—
139C2×C47C4×#12014yes—
140C2×C47#284611yescharacteristic
141C2×C47#284814yes—
142C2×C47#284913yes—
143C26C2×D4711yescharacteristic
144C24×C4—476no—
145C24×C4—9424no—
146C24×C4—944no—
147C22×C4⋊C4—474no—
148C22×C22⋊C4—476no—
149C22×C22⋊C4C2×D4713yes—
150C22×C22⋊C4C2×D4713yes—
151#385—9424no—
152#379—24no—
153#379—478no—
154C2×#95—4712no—
155C47#1469311yescharacteristic, complemented, Sylow
156C25—212no—
157C25—212no—
158C25Dic9414yes—
159C25C4×D4714yes—
160C25C22⋊D4714yes—
161C25C22×D4713yes—
162C23×C4—474no—
163C23×C4—4724no—
164C23×C4—474no—
165C23×C4—9448no—
166C23×C4—94128no—
167C23×C4—9496no—
168C23×C4—9412no—
169C23×C4—9412no—
170C23×C4C22×D4711yescharacteristic
171C23×C4C22×D4713yes—
172C2×C4⋊C4—9424no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—26no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—4724no—
178C2×C22⋊C4—9412no—
179C2×C22⋊C4—9412no—
180#95—9424no—
181C24—232no—
182C24—296no—
183C24—248no—
184C24—224no—
185C24—248no—
186C24—224no—
187C24—212no—
188C24—212no—
189C24Dic47⋊C4116yescomplemented
190C24C2×Dic9416yes—
191C24C2×C4×D4716yes—
192C24D4×D47112yes—
193C24#10625352112yes—
194C24C2×C22⋊D4716yes—
195C24C23×D4711yescharacteristic
196C22×C4—26no—
197C22×C4—26no—
198C22×C4—21no—
199C22×C4—23no—
200C22×C4—212no—
201C22×C4—4724no—
202C22×C4—478no—
203C22×C4—9496no—
204C22×C4—94256no—
205C22×C4—9496no—
206C22×C4—9496no—
207C22×C4—9448no—
208C22×C4—9412no—
209C22×C4—944no—
210C22×C4—9412no—
211C22×C4—9412no—
212C4⋊C4—18816no—
213C22⋊C4—412no—
214C22⋊C4—412no—
215C22⋊C4—9448no—
216C23—232no—
217C23—2128no—
218C23—296no—
219C23—2128no—
220C23—296no—
221C23—224no—
222C23—224no—
223C23—212no—
224C23—224no—
225C23—212no—
226C23—224no—
227C23—248no—
228C23C2×Dic47⋊C4112yes—
229C23(C22⋊D47)⋊C4124yes—
230C23#13093992112yes—
231C23#13094001124yes—
232C23#13094005112yes—
233C23C22×C4×D4711yescharacteristic
234C23C2×D4×D4716yes—
235C23C22×C22⋊D4711yescharacteristic
236C23C22×Dic9411yescharacteristic
237C23C2×#1062535216yes—
238C2×C4—44no—
239C2×C4—43no—
240C2×C4—43no—
241C2×C4—46no—
242C2×C4—412no—
243C2×C4—4716no—
244C2×C4—94192no—
245C2×C4—9424no—
246C2×C4—9424no—
247C2×C4—9424no—
248C2×C4—1884no—
249C2×C4—18812no—
250C22—2128no—
251C22—212no—
252C22—248no—
253C22—212no—
254C22—224no—
255C22—224no—
256C22—248no—
257C22#15688995124yescomplemented
258C22C22×Dic47⋊C411yescharacteristic
259C22C2×(C22⋊D47)⋊C416yes—
260C22#120×D4714yes—
261C22C2×#1309399213yes—
262C22C2×#1309400116yes—
263C22C2×#1309400513yes—
264C22#15689629112yes—
265C4—81no—
266C4—83no—
267C4—9416no—
268C4—1888no—
269C2—224no—
270C2C2×#1568899513yes—
271C2#2024588714yes—
272C2#2025166411yescharacteristic
273C2#2025167314yes—
274C2#2025167813yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#379C2
socle, cosocleC24×C47C24
prime radicalC26×C47C22
hypercenterC2×#379D47
hypocenterC47#14693
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
2. C2×#15688995C2semidirect
4. #20241610C2semidirect
9. C22×Dic47⋊C4C22semidirect
12. #15688979C4semidirect
13. C2×C47⋊#95C22semidirect
29. C2×C22⋊Dic47C2×C4semidirect
51. C24×C47C4⋊C4semidirect
72. C22×Dic47C22×C4semidirect
120. C22×C47#385semidirect
155. C47#14693semidirect
189. C24Dic47⋊C4semidirect
257. C22#15688995semidirect

Maximal subgroups

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

subgroupindex
C25×Dic472
C2×#156889952
C2×#156889792
#202416102
C2×C47×#3792
#1469347