Finite Group Databasebeta

← #26125896

Subgroups — #26125896

Order 12032.

444 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 87 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#20247696C211yesmaximal, characteristic
2#20241709C211yesmaximal, characteristic
3#20241317C214yesmaximal, complemented
4C47×#1465C211yesmaximal, characteristic
5C2×#13094005C412yes—
6C22×C4⋊Dic47C411yescharacteristic
7#15689176—21no—
8#15689053—21no—
9#15689048—21no—
10#15689021—21no—
11#15688984C2212yes—
12#15688983—21no—
13#15688983C411yescharacteristic
14#15688847—21no—
15#15688847C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#407—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C23×Dic47C2×C412yes—
21#13094005—44no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47—41no—
27C2×Dic47⋊C4—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47C2×C412yes—
30C4⋊(C47⋊C8)—41no—
31C22⋊(C47⋊C8)—22no—
32C4×C47⋊C8—41no—
33C47⋊#95—21no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C22×D4×C47C2×C411yescharacteristic
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C2×M16×C47—41no—
40C4×D4×C47—41no—
41C4×D4×C47—41no—
42C2×C4⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47C2311yescharacteristic
45C2×C42×C47D411yescharacteristic
46C22⋊C8×C47—22no—
47C22⋊C8×C47—41no—
48C24×C47C22⋊C411yescharacteristic
49C24×C47C22×C411yescharacteristic
50C2×D4×C47—22no—
51C2×D4×C47—22no—
52C2×D4×C47—41no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—42no—
58C22×C4×C47—42no—
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×Dic47—22no—
65C22×Dic47—22no—
66C22×Dic47—24no—
67C22×Dic47—42no—
68C22×Dic47—42no—
69M16×C47—82no—
70C2×C8×C47—42no—
71C4⋊C4×C47—41no—
72C22⋊C4×C47—41no—
73C22⋊C4×C47—41no—
74C42×C47—21no—
75C42×C47—21no—
76C22⋊Dic47—44no—
77C22⋊Dic47—42no—
78C4⋊Dic47—42no—
79C4⋊Dic47—42no—
80C4⋊Dic47—41no—
81C4⋊Dic47—41no—
82Dic47⋊C4—44no—
83C2×C47⋊C8—42no—
84C23×C47—21no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47#13112yes—
95D4×C47—44no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—81no—
109C2×C4×C47#9911yescharacteristic
110C2×C4×C47#9911yescharacteristic
111C2×C4×C47#10012yes—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—84no—
119C8×C47—82no—
120C47⋊C8—82no—
121#11562—471nomaximal, Sylow
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—22no—
125C22×C47—42no—
126C22×C47—42no—
127C22×C47—42no—
128C22×C47—41no—
129C22×C47—41no—
130C22×C47—41no—
131C22×C47—41no—
132C22×C47—41no—
133C22×C47—81no—
134C22×C47#40811yescharacteristic
135C22×C47C2×#9911yescharacteristic
136C22×C47C2×#10011yescharacteristic
137C4×C47—21no—
138C4×C47—22no—
139C4×C47—41no—
140C4×C47—41no—
141C4×C47—41no—
142C4×C47#35212yes—
143Dic47—82no—
144Dic47—82no—
145#2486—471no—
146#1465D4711yescharacteristic
147#1460—471no—
148#1317—474no—
149C2×C47—21no—
150C2×C47—21no—
151C2×C47—42no—
152C2×C47—81no—
153C2×C47#147111yescharacteristic
154C2×C47C2×#35211yescharacteristic
155C2×C47#212311yescharacteristic
156C2×#123—472no—
157C2×C4×D4C2×D4711yescharacteristic
158C22×C4⋊C4—471no—
159#431—941no—
160#407—21no—
161#398—941no—
162#397—941no—
163#389—941no—
164#381—472no—
165#380—471no—
166#380—941no—
167#324—21no—
168#323—472no—
169#323—941no—
170#323C2×D4712yes—
171C47#1156211yescharacteristic, complemented, Sylow
172C22×D4C4×D4711yescharacteristic
173C23×C4—472no—
174C23×C4C4×D4711yescharacteristic
175C23×C4C22⋊D4711yescharacteristic
176C2×M16—41no—
177#123—1884no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—941no—
181C2×C4⋊C4—942no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4—941no—
184C2×C4⋊C4—1881no—
185C2×C4⋊C4—1881no—
186C2×C4⋊C4C22⋊D4711yescharacteristic
187C2×C22⋊C4—472no—
188C2×C22⋊C4—942no—
189C2×C22⋊C4D18811yescharacteristic
190C2×C22⋊C4C22×D4711yescharacteristic
191C2×C42D18811yescharacteristic
192C4⋊C8—1881no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—942no—
196C4×C8—1881no—
197#95—941no—
198#95—942no—
199#95—942no—
200C24(C2×D47)⋊C411yescharacteristic
201C24C2×C4×D4711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—944no—
212C22×C4—942no—
213C22×C4—942no—
214C22×C4—1882no—
215C22×C4—1882no—
216C22×C4(C2×D47)⋊C411yescharacteristic
217C22×C4(C2×D47)⋊C411yescharacteristic
218C22×C4(C2×D47)⋊C411yescharacteristic
219C22×C4C2×D18811yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221M16—82no—
222C2×C8—42no—
223C2×C8—1882no—
224C4⋊C4—41no—
225C4⋊C4—1882no—
226C4⋊C4—1882no—
227C4⋊C4—1881no—
228C4⋊C4—1881no—
229C4⋊C4—1884no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—1884no—
233C22⋊C4—1882no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23C2×(C2×D47)⋊C411yescharacteristic
246C23#1309403212yes—
247D4—44no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—81no—
261C2×C4—3764no—
262C2×C4—1882no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4—1882no—
267C2×C4—1882no—
268C2×C4#1309393811yescharacteristic
269C2×C4#1309393811yescharacteristic
270C2×C4#1309394012yes—
271C8—82no—
272C8—3762no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#1568909111yescharacteristic
286C22C2×#1309393811yescharacteristic
287C22C2×#1309394011yescharacteristic
288C4—21no—
289C4—22no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—3762no—
294C4—3762no—
295C4#1568891212yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—81no—
300C2#2024174911yescharacteristic
301C2C2×#1568891211yescharacteristic
302C2#2024504311yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC24×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1465C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C22⋊C4×C47C23
hypercenter#1465D47
hypocenterC47#11562
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20241317C2semidirect
171. C47#11562semidirect

Maximal subgroups

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

subgroupindex
#202476962
#202417092
#202413172
C47×#14652
#1156247