Finite Group Databasebeta

← #26110794

Subgroups — #26110794

Order 12032.

356 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252490C211yesmaximal, characteristic
2#20244366C211yesmaximal, characteristic
3#20242177C211yesmaximal, characteristic
4#20242177C211yesmaximal, characteristic
5#20241220C211yesmaximal, characteristic
6#20241220C211yesmaximal, characteristic
7C47×#1544C211yesmaximal, characteristic
8#15689254—21no—
9#15689254—21no—
10Q8×C47⋊C8—21no—
11Q8×C47⋊C8—21no—
12#15689184—22no—
13#15689176C2211yescharacteristic
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16C42×Dic47C2211yescharacteristic
17#15688882—22no—
18#15688876—21no—
19#15688876—21no—
20#15688874—21no—
21#15688874—21no—
22C47×#533C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#332C411yescharacteristic, complemented
26C47×#332C411yescharacteristic, complemented
27C47×#332C411yescharacteristic, complemented
28C47×#332C411yescharacteristic, complemented
29C2×#10625329—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47—21no—
33C4⋊(C47⋊C8)—22no—
34C4⋊(C47⋊C8)—22no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)D411yescharacteristic
38C4⋊(C47⋊C8)D411yescharacteristic
39C4⋊(C47⋊C8)D411yescharacteristic
40C4⋊(C47⋊C8)D411yescharacteristic
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43#13093935—22no—
44C4×C47⋊C8—22no—
45C4×C47⋊C8—21no—
46C4×C47⋊C8—21no—
47C4⋊Q8×C47C2×C411yescharacteristic
48C4⋊Q8×C47C2×C411yescharacteristic
49#125×C47Q812yes—
50C22⋊Q8×C47—21no—
51C22⋊Q8×C47—21no—
52C4×Q8×C47—21no—
53C4×Q8×C47—21no—
54#117×C47D412yes—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47C2×C411yescharacteristic
57C4⋊C8×C47C2×C411yescharacteristic
58C4⋊C8×C47C2×C411yescharacteristic
59C4⋊C8×C47C2×C411yescharacteristic
60C22⋊C8×C47—21no—
61C22⋊C8×C47—21no—
62C4×C8×C47—21no—
63C4×C8×C47—21no—
64C2×Q8×C47—21no—
65C2×Q8×C47—21no—
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47#4111yescharacteristic
68C22×C4×C47#4111yescharacteristic
69C22×Dic47—41no—
70C2×C8×C47—21no—
71C2×C8×C47—21no—
72C2×C8×C47—21no—
73C2×C8×C47—21no—
74C4⋊C4×C47—21no—
75C4⋊C4×C47—21no—
76C4⋊C4×C47—22no—
77C4⋊C4×C47—22no—
78C4⋊C4×C47C4⋊C412yes—
79C4⋊C4×C47C4⋊C412yes—
80C22⋊C4×C47—22no—
81C42×C47—22no—
82C42×C47C22×C411yescharacteristic
83C42×C47C2×D411yescharacteristic
84C42×C47C2×D411yescharacteristic
85C42×C47C2×Q811yescharacteristic
86C4×Dic47—22no—
87C4×Dic47—22no—
88C4×Dic47—22no—
89C4×Dic47—41no—
90C4×Dic47—41no—
91C4×Dic47—41no—
92#10625329—42no—
93C2×C47⋊C8—21no—
94C2×C47⋊C8—21no—
95C2×C47⋊C8—21no—
96C2×C47⋊C8—21no—
97C2×C47⋊C8—41no—
98C23×C47#12311yescharacteristic
99Q8×C47—41no—
100Q8×C47—41no—
101Q8×C47—41no—
102Q8×C47—41no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47C2×C4⋊C411yescharacteristic
112C2×C4×C47C4×D411yescharacteristic
113C2×C4×C47C4×D411yescharacteristic
114C2×C4×C47#12011yescharacteristic
115C2×C4×C47C22⋊Q811yescharacteristic
116C2×C4×C47C22⋊Q811yescharacteristic
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C8×C47—41no—
121C8×C47—41no—
122C8×C47—41no—
123C8×C47—41no—
124C47⋊C8—41no—
125C47⋊C8—41no—
126C47⋊C8—41no—
127C47⋊C8—41no—
128C47⋊C8—81no—
129#8936—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47#38511yescharacteristic
134C4×C47—21no—
135C4×C47—21no—
136C4×C47—21no—
137C4×C47—42no—
138C4×C47—42no—
139C4×C47#44311yescharacteristic
140C4×C47#44311yescharacteristic
141C4×C47#44411yescharacteristic
142C4×C47#44411yescharacteristic
143C4×C47#45411yescharacteristic
144C4×C47#45611yescharacteristic
145Dic47—44no—
146#2986—471no—
147#1994—471no—
148#1574—471no—
149#1574—471no—
150#1544D4711yescharacteristic
151#1276—471no—
152#1276—471no—
153C2×C47—41no—
154C2×C47#185711yescharacteristic
155C2×C47#186611yescharacteristic
156C2×C47#186611yescharacteristic
157#533C2×D4711yescharacteristic
158#446—941no—
159#446—941no—
160C8×Q8—941no—
161C8×Q8—941no—
162#433—942no—
163#431—471no—
164#431—471no—
165#431C2×D4711yescharacteristic
166#431C2×D4711yescharacteristic
167#423—471no—
168C43—471no—
169#339—942no—
170#335—941no—
171#335—941no—
172#334—941no—
173#334—941no—
174#332Dic4711yescharacteristic, complemented
175#332Dic4711yescharacteristic, complemented
176#332Dic4711yescharacteristic, complemented
177#332Dic4711yescharacteristic, complemented
178C47#893611yescharacteristic, complemented, Sylow
179C2×M16—941no—
180C4⋊Q8C2×Dic4711yescharacteristic
181C4⋊Q8C2×Dic4711yescharacteristic
182#125Dic9412yes—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187#117D18812yes—
188C2×C42—941no—
189C2×C42—941no—
190C2×C42—941no—
191C2×C42C22×D4711yescharacteristic
192C4⋊C8—471no—
193C4⋊C8—471no—
194C4⋊C8—471no—
195C4⋊C8—471no—
196C4⋊C8—942no—
197C4⋊C8—942no—
198C4⋊C8—941no—
199C4⋊C8—941no—
200C4⋊C8C2×Dic4711yescharacteristic
201C4⋊C8C2×Dic4711yescharacteristic
202C4⋊C8C2×Dic4711yescharacteristic
203C4⋊C8C2×Dic4711yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—941no—
207C22⋊C8—941no—
208#97—942no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—942no—
212C4×C8—941no—
213C4×C8—941no—
214C2×Q8—21no—
215C2×Q8—21no—
216C22×C4—1881no—
217C22×C4C2×D18811yescharacteristic
218C22×C4#1062535211yescharacteristic
219C22×C4#1062535211yescharacteristic
220M16—1882no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—941no—
226C2×C8—941no—
227C2×C8—941no—
228C2×C8—941no—
229C2×C8—1881no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4C4⋊Dic4712yes—
235C4⋊C4C4⋊Dic4712yes—
236C22⋊C4—22no—
237C42—22no—
238C42—942no—
239C42—942no—
240C42—942no—
241C42—1881no—
242C42—1881no—
243C42—1881no—
244C42C2×Dic9411yescharacteristic
245C42D4×D4711yescharacteristic
246C42D4×D4711yescharacteristic
247C42C22×Dic4711yescharacteristic
248C23#1309400811yescharacteristic
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—1882no—
262C2×C4—1882no—
263C2×C4—1882no—
264C2×C4C2×C4⋊Dic4711yescharacteristic
265C2×C4D4×Dic4711yescharacteristic
266C2×C4D4×Dic4711yescharacteristic
267C2×C4#1309399411yescharacteristic
268C2×C4#1309400111yescharacteristic
269C2×C4#1309400111yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—3761no—
275C8—1881no—
276C8—1881no—
277C8—1881no—
278C8—1881no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1568899811yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—1884no—
289C4#1568924011yescharacteristic
290C4#1568924011yescharacteristic
291C4#1568924511yescharacteristic
292C4#1568924511yescharacteristic
293C4#1568929811yescharacteristic
294C4#1568930811yescharacteristic
295C2—41no—
296C2#2024360511yescharacteristic
297C2#2024366511yescharacteristic
298C2#2024366511yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1544C2
socleC22×C47#385
prime radicalC23×C47#123
cosocleC2×C42×C47C23
hypercenter#1544D47
hypocenterC47#8936
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
25. C47×#332C4semidirect
26. C47×#332C4semidirect
27. C47×#332C4semidirect
28. C47×#332C4semidirect
174. #332Dic47semidirect
175. #332Dic47semidirect
176. #332Dic47semidirect
177. #332Dic47semidirect
178. C47#8936semidirect

Maximal subgroups

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

subgroupindex
#202524902
#202443662
#202421772
#202412202
C47×#15442
#893647