Finite Group Databasebeta

← #26108923

Subgroups — #26108923

Order 12032.

778 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 147 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#20243468C212yesmaximal
2#20241718C212yesmaximal
3#20241718C212yesmaximal
4C22×#125×C47C211yesmaximal, characteristic
5#15689087C2214yes—
6#15688892—24no—
7#15688849—24no—
8#15688849—24no—
9C2×#125×C47—24no—
10C2×#125×C47—24no—
11C2×#125×C47—24no—
12C22×C4⋊C4×C47C411yescharacteristic
13C22×C4⋊C4×C47C411yescharacteristic
14C22×C4⋊C4×C47C411yescharacteristic
15C22×C4⋊C4×C47C2211yescharacteristic
16C22×C4⋊C4×C47C2211yescharacteristic
17C22×C4⋊C4×C47C2211yescharacteristic
18C22×C42×C47C411yescharacteristic
19C2×#10625329—24no—
20C22⋊(C47⋊C8)—24no—
21C22⋊(C47⋊C8)—24no—
22C23×C4×C47C2×C411yescharacteristic
23C23×C4×C47C2×C411yescharacteristic
24C23×C4×C47C2×C411yescharacteristic
25C23×C4×C47C2×C411yescharacteristic
26C23×C4×C47C2×C411yescharacteristic
27C23×C4×C47C2×C411yescharacteristic
28C23×C4×C47C2311yescharacteristic
29#125×C47—232no—
30C2×C4⋊C4×C47—22no—
31C2×C4⋊C4×C47—22no—
32C2×C4⋊C4×C47—22no—
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47D414yes—
46C2×C4⋊C4×C47D414yes—
47C2×C4⋊C4×C47D414yes—
48C2×C42×C47—22no—
49C2×C42×C47—22no—
50C2×C42×C47—22no—
51C24×C47C22×C411yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—24no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47—42no—
69C22×C4×C47—42no—
70C22×C4×C47—42no—
71C22×C4×C47C22⋊C412yes—
72C22×C4×C47C22⋊C412yes—
73C22×C4×C47C22⋊C412yes—
74C22×C4×C47C22⋊C412yes—
75C22×C4×C47C22⋊C412yes—
76C22×C4×C47C22⋊C412yes—
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D412yes—
79C22×C4×C47C2×D412yes—
80C4⋊C4×C47—28no—
81C4⋊C4×C47—28no—
82C4⋊C4×C47—28no—
83C4⋊C4×C47—28no—
84C4⋊C4×C47—28no—
85C4⋊C4×C47—28no—
86C42×C47—28no—
87#10625329—48no—
88C2×C47⋊C8—44no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47C2×C22⋊C411yescharacteristic
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47C2×C22⋊C411yescharacteristic
95C2×C4×C47—24no—
96C2×C4×C47—24no—
97C2×C4×C47—22no—
98C2×C4×C47—24no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—24no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47#10114yes—
123C2×C4×C47#12014yes—
124C47⋊C8—84no—
125#8554—471nomaximal, Sylow
126C22×C47—28no—
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47#37911yescharacteristic
134C22×C47#41012yes—
135C22×C47C2×#10112yes—
136C22×C47#41312yes—
137C4×C47—24no—
138C4×C47—24no—
139C4×C47—42no—
140C4×C47—42no—
141C4×C47—42no—
142C4×C47—42no—
143C4×C47—42no—
144C22×#125D4711yescharacteristic
145#1821—472no—
146#1463—472no—
147#1463—472no—
148C2×C47—22no—
149C2×C47—22no—
150C2×C47—22no—
151C2×C47#178211yescharacteristic
152C2×C47#178411yescharacteristic
153C2×C47#178511yescharacteristic
154C2×#125—24no—
155C2×#125—24no—
156C2×#125—24no—
157C22×C4⋊C4Dic4711yescharacteristic
158C22×C4⋊C4Dic4711yescharacteristic
159C22×C4⋊C4Dic4711yescharacteristic
160C22×C4⋊C4C2×D4711yescharacteristic
161C22×C4⋊C4C2×D4711yescharacteristic
162C22×C4⋊C4C2×D4711yescharacteristic
163C22×C42Dic4711yescharacteristic
164#407—474no—
165#343—944no—
166#324—944no—
167#324—944no—
168C47#855411yescharacteristic, complemented, Sylow
169C23×C4C2×Dic4711yescharacteristic
170C23×C4C2×Dic4711yescharacteristic
171C23×C4C2×Dic4711yescharacteristic
172C23×C4C2×Dic4711yescharacteristic
173C23×C4C2×Dic4711yescharacteristic
174C23×C4C2×Dic4711yescharacteristic
175C23×C4C22×D4711yescharacteristic
176C2×M16—944no—
177#125—232no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C22⋊D4714yes—
194C2×C4⋊C4C22⋊D4714yes—
195C2×C4⋊C4C22⋊D4714yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C22⋊C8—944no—
200C22⋊C8—944no—
201C24C22×Dic4711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4C22⋊Dic4712yes—
222C22×C4C22⋊Dic4712yes—
223C22×C4C22⋊Dic4712yes—
224C22×C4C22⋊Dic4712yes—
225C22×C4C22⋊Dic4712yes—
226C22×C4C22⋊Dic4712yes—
227C22×C4C2×C22⋊D4712yes—
228C22×C4C2×C22⋊D4712yes—
229C22×C4C2×C22⋊D4712yes—
230M16—1888no—
231C2×C8—1884no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C42—28no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23C2×C22⋊Dic4711yescharacteristic
243C23C2×C22⋊Dic4711yescharacteristic
244C23C2×C22⋊Dic4711yescharacteristic
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#1309394314yes—
273C2×C4#1309399314yes—
274C8—3764no—
275C22—28no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1568897911yescharacteristic
283C22#1568909712yes—
284C22C2×#1309394312yes—
285C22#1568910712yes—
286C4—24no—
287C4—24no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—42no—
292C4—42no—
293C2—22no—
294C2—22no—
295C2—22no—
296C2#2024330611yescharacteristic
297C2#2024331211yescharacteristic
298C2#2024331611yescharacteristic
subgroupG/H
centerC22#15688979
commutator, prime radicalC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC22×#125×C47C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenterC22×#125D47
hypocenterC47#8554
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
168. C47#8554semidirect

Maximal subgroups

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

subgroupindex
#202434682
#202417182
C22×#125×C472
#855447