Finite Group Databasebeta

← #26125671

Subgroups — #26125671

Order 12032.

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

normal subgroups only

This group has 510 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#20245972C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241321C214yesmaximal, complemented
4C47×#1464C211yesmaximal, characteristic
5C23×Dic94C411yescharacteristic
6C2×#13093983C411yescharacteristic
7#15689180—21no—
8#15689029—21no—
9#15689029—21no—
10#15689028C412yes—
11#15689028C2212yes—
12#15688984—21no—
13#15688984—21no—
14#15688849C2212yes—
15#15688847—21no—
16C2×#117×C47C2211yescharacteristic
17C47×#432—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C22×Dic94—21no—
21C22×Dic94—24no—
22C22×Dic94—42no—
23C22×Dic94—41no—
24C22×Dic94—41no—
25C22×Dic94C2×C412yes—
26C23×Dic47—21no—
27#13093983—42no—
28C2×C4⋊Dic47—21no—
29C2×C22⋊Dic47—21no—
30C2×C4×Dic47C2×C412yes—
31C4⋊(C47⋊C8)—41no—
32C22⋊(C47⋊C8)—22no—
33#13093935—41no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C23×C4×C47C2×C411yescharacteristic
39#117×C47—41no—
40#117×C47—41no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2311yescharacteristic
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C4⋊C8×C47—41no—
48C22⋊C8×C47—22no—
49#97×C47—41no—
50C24×C47C22⋊C411yescharacteristic
51C22×C4×C47—22no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—24no—
62C22×Dic47—22no—
63C22×Dic47—22no—
64C22×Dic47—42no—
65C22×Dic47—42no—
66C2×Dic94—24no—
67C2×Dic94—22no—
68C2×Dic94—42no—
69C2×Dic94—48no—
70C2×Dic94—48no—
71C2×Dic94—41no—
72C2×Dic94—41no—
73C2×Dic94—42no—
74C2×Dic94—42no—
75C2×Dic94M1614yes—
76C2×C8×C47—42no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C22⋊Dic47—42no—
86C4⋊Dic47—42no—
87C4×Dic47—42no—
88C2×C47⋊C8—42no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47#10012yes—
94C23×C47C2×C22⋊C411yescharacteristic
95C2×C4×C47—24no—
96C2×C4×C47—24no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47#9911yescharacteristic
110C2×C4×C47#9911yescharacteristic
111C2×C4×C47C2×M1612yes—
112C2×Dic47—24no—
113C2×Dic47—48no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119Dic94—44no—
120Dic94—44no—
121Dic94—44no—
122Dic94—44no—
123C8×C47—82no—
124C47⋊C8—82no—
125#11510—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—22no—
129C22×C47—42no—
130C22×C47—42no—
131C22×C47—41no—
132C22×C47—41no—
133C22×C47#40711yescharacteristic
134C22×C47C2×#9911yescharacteristic
135C22×C47C2×#10011yescharacteristic
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—22no—
139C4×C47—41no—
140C4×C47—41no—
141C4×C47—41no—
142C4×C47—41no—
143Dic47—44no—
144Dic47—82no—
145#2310—471no—
146#1464—471no—
147#1464D4711yescharacteristic
148#1319—474no—
149C2×C47—21no—
150C2×C47—21no—
151C2×C47—42no—
152C2×C47#146211yescharacteristic
153C2×C47#212911yescharacteristic
154C2×C47#213511yescharacteristic
155C23×Q8—471no—
156C2×#117—471no—
157C2×#117C2×D4711yescharacteristic
158#432—21no—
159#432—941no—
160#391—472no—
161#391—472no—
162#391—941no—
163#391—941no—
164#381—941no—
165#381—941no—
166#324—472no—
167#324C2×D4712yes—
168#323—21no—
169#323—941no—
170C47#1151011yescharacteristic, complemented, Sylow
171C22×Q8—472no—
172C22×Q8—944no—
173C22×Q8—941no—
174C22×Q8—1881no—
175C22×Q8—1881no—
176C22×Q8—1882no—
177C23×C4—941no—
178C23×C4C4×D4711yescharacteristic
179#117—41no—
180#117—41no—
181#117—1882no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4C4×D4711yescharacteristic
184C2×C4⋊C4C22×D4711yescharacteristic
185C2×C22⋊C4—941no—
186C2×C22⋊C4D18811yescharacteristic
187C2×C22⋊C4C22⋊D4711yescharacteristic
188C2×C42—472no—
189C2×C42D18811yescharacteristic
190C2×C42C22⋊D4711yescharacteristic
191C4⋊C8—41no—
192C4⋊C8—1881no—
193C22⋊C8—22no—
194C22⋊C8—942no—
195#97—41no—
196#97—1881no—
197#95—942no—
198#95—942no—
199#95—942no—
200#95—942no—
201C24(C2×D47)⋊C411yescharacteristic
202C2×Q8—474no—
203C2×Q8—944no—
204C2×Q8—942no—
205C2×Q8—1888no—
206C2×Q8—1888no—
207C2×Q8—1882no—
208C2×Q8—1882no—
209C2×Q8—1882no—
210C2×Q8—1881no—
211C2×Q8—1881no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—944no—
217C22×C4—942no—
218C22×C4—942no—
219C22×C4—1882no—
220C22×C4—1882no—
221C22×C4(C2×D47)⋊C411yescharacteristic
222C22×C4(C2×D47)⋊C411yescharacteristic
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4C2×C4×D4711yescharacteristic
225C22×C4C2×D18811yescharacteristic
226C22×C4C2×C22⋊D4711yescharacteristic
227C2×C8—42no—
228C2×C8—1882no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—1882no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—1882no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—1882no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23#1309394012yes—
245C23C2×(C2×D47)⋊C411yescharacteristic
246Q8—1884no—
247Q8—1884no—
248Q8—1884no—
249Q8—1884no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—944no—
265C2×C4—1882no—
266C2×C4—1882no—
267C2×C4—1882no—
268C2×C4—1888no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4#1309393811yescharacteristic
272C2×C4#1309393811yescharacteristic
273C2×C4M16×D4712yes—
274C8—82no—
275C8—3762no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22#1568908611yescharacteristic
284C22C2×#1309393811yescharacteristic
285C22C2×#1309394011yescharacteristic
286C4—21no—
287C4—21no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—3762no—
294C4—1884no—
295C2—21no—
296C2—21no—
297C2—42no—
298C2#2024171411yescharacteristic
299C2#2024506211yescharacteristic
300C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1464C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1464D47
hypocenterC47#11510
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #20241321C2semidirect
170. C47#11510semidirect

Maximal subgroups

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

subgroupindex
#202459722
#202417222
#202413212
C47×#14642
#1151047