Finite Group Databasebeta

← #26192579

Subgroups — #26192579

Order 12032.

918 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 195 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
1#20253768C211yesmaximal, characteristic
2#20253518C212yesmaximal
3#20242609C218yesmaximal, complemented
4#20242114C211yesmaximal, characteristic
5#20241835C212yesmaximal
6C2×Q82×C47C211yesmaximal, characteristic
7#15689680C2211yescharacteristic
8#15689491—22no—
9#15689473C2214yes—
10#15689453—22no—
11#15689408C2214yes—
12#15689333C2214yes—
13#15689279—22no—
14#15689228—24no—
15#15689180C2212yes—
16#15689132—22no—
17C4×#10625329C2211yescharacteristic
18#15688872C2214yes—
19Q8⋊(C47⋊C8)C2218yes—
20Q82×C47—22no—
21Q82×C47—24no—
22Q82×C47C2214yes—
23C2×C4⋊Q8×C47—22no—
24C2×C4⋊Q8×C47—22no—
25C2×C4⋊Q8×C47C2211yescharacteristic
26C2×C4×Q8×C47—22no—
27C2×C4×Q8×C47C2212yes—
28C2×#10625337—24no—
29C2×#10625329—21no—
30#13094024C2312yes—
31#13094016—21no—
32#13094003—22no—
33C4×Dic94—22no—
34#13093983—21no—
35#13093955—24no—
36C4⋊(C47⋊C8)C2314yes—
37#13093948—28no—
38#13093948—24no—
39#13093947—24no—
40C22⋊(C47⋊C8)—22no—
41#13093935—22no—
42#13093935—21no—
43C4×C47⋊C8C2312yes—
44C22×Q8×C47—22no—
45C22×Q8×C47D412yes—
46C22×Q8×C47D412yes—
47C4⋊Q8×C47—28no—
48C4⋊Q8×C47—28no—
49C4⋊Q8×C47—28no—
50C4⋊Q8×C47—28no—
51C4⋊Q8×C47—22no—
52C4⋊Q8×C47D412yes—
53C4⋊Q8×C47C2312yes—
54C4×Q8×C47—28no—
55C4×Q8×C47—28no—
56C4×Q8×C47—22no—
57C4×Q8×C47—22no—
58C4×Q8×C47D414yes—
59C4×Q8×C47C2314yes—
60C2×C4⋊C4×C47—24no—
61C2×C4⋊C4×C47—22no—
62C2×C4⋊C4×C47—22no—
63C2×C4⋊C4×C47D412yes—
64C2×C42×C47—22no—
65C2×C42×C47—22no—
66C2×C42×C47C2311yescharacteristic
67C2×Q8×C47—28no—
68C2×Q8×C47—28no—
69C2×Q8×C47—22no—
70C2×Q8×C47—22no—
71C2×Q8×C47—44no—
72C2×Q8×C47—84no—
73C2×Q8×C47C2×D414yes—
74C2×Q8×C47C2×D414yes—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—22no—
78C22×C4×C47C2×D412yes—
79C22×C4×C47C2×D411yescharacteristic
80C2×Dic94—22no—
81C4⋊C4×C47—216no—
82C4⋊C4×C47—24no—
83C4⋊C4×C47—24no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—24no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47C2×D414yes—
88C42×C47—28no—
89C42×C47—24no—
90C42×C47—24no—
91C42×C47C2×D411yescharacteristic
92C42×C47C2×D412yes—
93C42×C47C2411yescharacteristic
94C22⋊Dic47—41no—
95#10625337—48no—
96C4⋊Dic47—22no—
97C4⋊Dic47—22no—
98Dic47⋊C4—42no—
99C4×Dic47—41no—
100#10625329—42no—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—24no—
103C23×C47#12011yescharacteristic
104Q8×C47—28no—
105Q8×C47—24no—
106Q8×C47—44no—
107Q8×C47—82no—
108Q8×C47—88no—
109Q8×C47#13718yes—
110C2×C4×C47—28no—
111C2×C4×C47—24no—
112C2×C4×C47—22no—
113C2×C4×C47—24no—
114C2×C4×C47—24no—
115C2×C4×C47—24no—
116C2×C4×C47—44no—
117C2×C4×C47—44no—
118C2×C4×C47#12011yescharacteristic
119C2×C4×C47#12012yes—
120C2×C4×C47C22×D411yescharacteristic
121C2×C4×C47C22×D412yes—
122C2×Dic47—42no—
123Dic94—44no—
124C47⋊C8—42no—
125C47⋊C8—44no—
126#18551—471nomaximal, Sylow
127C22×C47—22no—
128C22×C47#45512yes—
129C22×C47C2×#12011yescharacteristic
130C4×C47—28no—
131C4×C47—22no—
132C4×C47—21no—
133C4×C47—21no—
134C4×C47—44no—
135C4×C47—44no—
136C4×C47C2×#13714yes—
137Dic47—82no—
138C2×Q82D4711yescharacteristic
139#3139—471no—
140#3113—472no—
141#1645—478no—
142#1558—471no—
143#1491—472no—
144C2×C47—22no—
145C2×C47#300412yes—
146C2×C47C2×#45511yescharacteristic
147Q82—22no—
148Q82—24no—
149Q82C2×D4714yes—
150#533—471no—
151C2×C4⋊Q8—22no—
152C2×C4⋊Q8—22no—
153C2×C4⋊Q8C2×D4711yescharacteristic
154C2×C4×Q8—22no—
155C2×C4×Q8C2×D4712yes—
156#491—942no—
157#487—474no—
158#483—942no—
159#475—474no—
160#462—474no—
161#451—942no—
162#441—944no—
163#432—472no—
164#419—942no—
165C4×M16—471no—
166#333—474no—
167Q8⋊C8—478no—
168C47#1855111yescharacteristic, complemented, Sylow
169C22×Q8—22no—
170C22×Q8C22⋊D4712yes—
171C22×Q8C22⋊D4712yes—
172C2×Q16—944no—
173C2×M16—941no—
174C4⋊Q8—28no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—22no—
179C4⋊Q8—472no—
180C4⋊Q8C22⋊D4712yes—
181C4⋊Q8C22×D4712yes—
182#125—941no—
183C22⋊Q8—942no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—942no—
189C4×Q8C22⋊D4714yes—
190C4×Q8C22×D4714yes—
191#117—941no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C22⋊D4712yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C22×D4711yescharacteristic
199#106—944no—
200C4⋊C8—474no—
201#103—948no—
202#103—944no—
203#103—944no—
204C22⋊C8—942no—
205#97—941no—
206#97—942no—
207C4×C8—472no—
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—84no—
214C2×Q8—942no—
215C2×Q8C2×C22⋊D4714yes—
216C2×Q8C2×C22⋊D4714yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4C2×C22⋊D4712yes—
221C22×C4C2×C22⋊D4711yescharacteristic
222Q16—1888no—
223M16—1882no—
224C2×C8—942no—
225C2×C8—944no—
226C4⋊C4—216no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—942no—
233C4⋊C4—942no—
234C4⋊C4—1882no—
235C4⋊C4C2×C22⋊D4714yes—
236C22⋊C4—1881no—
237C42—28no—
238C42—24no—
239C42—24no—
240C42—1881no—
241C42C2×C22⋊D4711yescharacteristic
242C42C2×C22⋊D4712yes—
243C42C23×D4711yescharacteristic
244C23#1309399311yescharacteristic
245Q8—28no—
246Q8—24no—
247Q8—44no—
248Q8—82no—
249Q8—88no—
250Q8—1884no—
251Q8#1309405518yes—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—1882no—
261C2×C4#1309399312yes—
262C2×C4#1309399311yescharacteristic
263C2×C4C22×C22⋊D4711yescharacteristic
264C2×C4C22×C22⋊D4712yes—
265C8—1882no—
266C8—1884no—
267C22—22no—
268C22#1568930112yes—
269C22C2×#1309399311yescharacteristic
270C4—21no—
271C4—21no—
272C4—28no—
273C4—22no—
274C4—44no—
275C4—44no—
276C4—3762no—
277C4C2×#1309405514yes—
278C2—22no—
279C2#2025262212yes—
280C2C2×#1568930111yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×Q82×C47C2
socleC22×C47C2×#120
prime radicalC23×C47#120
hypercenterC2×Q82D47
hypocenterC47#18551
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #20242609C2semidirect
168. C47#18551semidirect

Maximal subgroups

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

subgroupindex
#202537682
#202535182
#202426092
#202421142
#202418352
C2×Q82×C472
#1855147