Finite Group Databasebeta

← #26114513

Subgroups — #26114513

Order 12032.

378 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13093940C211yesmaximal, characteristic
2#20241778C211yesmaximal, characteristic
3C2×#15688850C211yesmaximal, characteristic
4#20241669C211yesmaximal, characteristic, complemented
5#20241238C211yesmaximal, characteristic, complemented
6#20241226C211yesmaximal, characteristic, complemented
7C47×#1565C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9#15689176C2211yescharacteristic
10C2×#13093940C411yescharacteristic
11C2×#13093940C411yescharacteristic
12C4×#10625329C2211yescharacteristic
13C2×C4×C47⋊C8C2211yescharacteristic
14#15688891C411yescharacteristic
15#15688891C411yescharacteristic
16#15688850—21no—
17#15688850—21no—
18#15688850—21no—
19#15688850—21no—
20#15688845—21no—
21C8×C47⋊C8—21no—
22C47×#431C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C4×M16×C47C2211yescharacteristic
25C47×#335—21no—
26C47×#334—21no—
27C22×D188C2×C411yescharacteristic
28C22×C4×D47—21no—
29C2×#10625329C2×C411yescharacteristic
30C2×#10625329C2×C411yescharacteristic
31C22×C47⋊C8—21no—
32C4×D188—21no—
33C4×D188—21no—
34C4×D188—21no—
35C4×D188—21no—
36C2×C4⋊Dic47C2×C411yescharacteristic
37C2×(C2×D47)⋊C4—21no—
38C4⋊(C47⋊C8)—21no—
39#13093940—22no—
40#13093940C814yescomplemented
41C22⋊(C47⋊C8)—21no—
42#13093935—21no—
43C4×C47⋊C8—21no—
44C4×C47⋊C8—21no—
45C4×C47⋊C8—21no—
46C4×C47⋊C8D411yescharacteristic
47C4×C47⋊C8D411yescharacteristic
48C2×M16×C47C2×C411yescharacteristic
49C2×M16×C47C2×C411yescharacteristic
50C22×C8×C47—21no—
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47—21no—
53C4⋊C8×C47—21no—
54C4⋊C8×C47D411yescharacteristic
55C4⋊C8×C47D411yescharacteristic
56C22⋊C8×C47—21no—
57#97×C47—21no—
58C4×C8×C47—21no—
59C4×C8×C47—21no—
60C23×D47—21no—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47#4111yescharacteristic
63C22×C4×C47#4111yescharacteristic
64C22×Dic47—21no—
65C2×D188—21no—
66C2×D188—21no—
67C2×D188—21no—
68C2×D188—21no—
69C2×D188—21no—
70C2×D188C2×C812yes—
71C2×C4×D47—41no—
72C2×C4×D47—41no—
73C2×C4×D47—81no—
74C2×C4×D47—81no—
75M16×C47—21no—
76M16×C47—21no—
77M16×C47—21no—
78M16×C47C2×C812yes—
79C2×C8×C47—21no—
80C2×C8×C47—21no—
81C2×C8×C47—21no—
82C2×C8×C47—21no—
83C2×C8×C47—41no—
84C2×C8×C47—81no—
85C42×C47C2×D411yescharacteristic
86C42×C47C2×D411yescharacteristic
87C42×C47#4111yescharacteristic
88C42×C47#4111yescharacteristic
89(C2×D47)⋊C4—41no—
90(C2×D47)⋊C4—41no—
91C4⋊Dic47—21no—
92C4⋊Dic47—21no—
93#10625329—21no—
94#10625329—21no—
95#10625329C2×C812yes—
96#10625329M1612yes—
97C2×C47⋊C8—21no—
98C2×C47⋊C8—21no—
99C2×C47⋊C8—21no—
100C2×C47⋊C8—21no—
101C2×C47⋊C8—42no—
102C2×C47⋊C8—41no—
103C23×C47#11711yescharacteristic
104C22×D47—22no—
105C22×D47—41no—
106C22×D47—41no—
107C22×D47—81no—
108C22×D47—81no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—41no—
112C2×C4×C47C4×D411yescharacteristic
113C2×C4×C47C4×D411yescharacteristic
114C2×C4×C47#12111yescharacteristic
115C2×C4×C47#12311yescharacteristic
116C2×C4×C47#12411yescharacteristic
117C2×C4×C47#12611yescharacteristic
118C2×C4×C47C22×C811yescharacteristic
119C2×C4×C47C2×M1611yescharacteristic
120C2×C4×C47#13111yescharacteristic
121C2×C4×C47#13111yescharacteristic
122C2×Dic47—41no—
123C2×Dic47—81no—
124D188—22no—
125D188—22no—
126D188—41no—
127D188—41no—
128C4×D47—82no—
129C4×D47—81no—
130C4×D47—81no—
131C8×C47—22no—
132C8×C47—41no—
133C8×C47—41no—
134C8×C47—41no—
135C8×C47—81no—
136C47⋊C8—22no—
137C47⋊C8—22no—
138C47⋊C8—22no—
139C47⋊C8—41no—
140C47⋊C8—81no—
141#9550—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47#38811yescharacteristic
145C22×C47#43111yescharacteristic
146C22×C47#43311yescharacteristic
147C2×D47—42no—
148C2×D47—82no—
149C2×D47—81no—
150C2×D47—81no—
151C2×D47—82no—
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47C8×D411yescharacteristic
156C4×C47C8×D411yescharacteristic
157C4×C47#43511yescharacteristic
158C4×C47#43611yescharacteristic
159C4×C47#47111yescharacteristic
160C4×C47#47211yescharacteristic
161Dic47—81no—
162C4×#100—471no—
163#1565D4711yescharacteristic, complemented
164#1478—471no—
165C2×D4⋊C8—471no—
166C8×M16—471no—
167#1283—471no—
168#1278—471no—
169C2×C47—21no—
170C2×C47—21no—
171C2×C47#192311yescharacteristic
172C2×C47#193111yescharacteristic
173C2×C47#193211yescharacteristic
174D47—82no—
175C2×C4×D4—471no—
176#431—471no—
177#431C2×D4711yescharacteristic
178C2×C4⋊C8C2×D4711yescharacteristic
179C2×#100—471no—
180C2×#100—471no—
181C4×M16—471no—
182C4×M16C2×D4711yescharacteristic
183C2×C4×C8—471no—
184#343—471no—
185#343—471no—
186#335—21no—
187#334—21no—
188D4⋊C8—941no—
189D4⋊C8—941no—
190D4⋊C8—941no—
191D4⋊C8—941no—
192#322—941no—
193C82—941no—
194C47#955011yescharacteristic, complemented, Sylow
195C22×D4—471no—
196C23×C4—941no—
197C2×M16—471no—
198C2×M16—471no—
199C2×M16C4×D4711yescharacteristic
200C2×M16C4×D4711yescharacteristic
201C22×C8—21no—
202C22×C8—941no—
203C4×D4—941no—
204C4×D4—941no—
205C4×D4—941no—
206C4×D4—941no—
207C2×C4⋊C4—471no—
208C2×C22⋊C4—941no—
209C2×C42C22×D4711yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—941no—
213C4⋊C8C22⋊D4711yescharacteristic
214C4⋊C8C22⋊D4711yescharacteristic
215#100—474no—
216#100—942no—
217C22⋊C8—21no—
218C22⋊C8—941no—
219#97—21no—
220#97—941no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—471no—
224C4×C8—471no—
225C4×C8—941no—
226C4×C8—941no—
227C4×C8—941no—
228C24—941no—
229C2×D4—472no—
230C2×D4—941no—
231C2×D4—941no—
232C2×D4—941no—
233C2×D4—941no—
234C2×D4—941no—
235C22×C4—941no—
236C22×C4—3761no—
237C22×C4—3761no—
238C22×C4—1881no—
239C22×C4—1881no—
240C22×C4C2×C4×D4711yescharacteristic
241C22×C4#1062535211yescharacteristic
242C22×C4#1062535411yescharacteristic
243M16—21no—
244M16—21no—
245M16—21no—
246M16—472no—
247M16—472no—
248M16—941no—
249M16—941no—
250M16C8×D4712yes—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—81no—
257C2×C8—941no—
258C2×C8—941no—
259C2×C8—941no—
260C2×C8—941no—
261C2×C8—1882no—
262C2×C8—1881no—
263C4⋊C4—941no—
264C4⋊C4—941no—
265C22⋊C4—1881no—
266C22⋊C4—1881no—
267C42#1062535011yescharacteristic
268C42D4×D4711yescharacteristic
269C42#1062535411yescharacteristic
270C42C2×C22⋊D4711yescharacteristic
271C23—942no—
272C23—3761no—
273C23—3761no—
274C23—1881no—
275C23—1881no—
276C23#1309398211yescharacteristic
277D4—942no—
278D4—942no—
279D4—1881no—
280D4—1881no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—3762no—
285C2×C4—3761no—
286C2×C4—3761no—
287C2×C4—3761no—
288C2×C4—1881no—
289C2×C4C4×C22⋊D4711yescharacteristic
290C2×C4#1309398611yescharacteristic
291C2×C4#1309399611yescharacteristic
292C2×C4#1309400911yescharacteristic
293C2×C4#1309401311yescharacteristic
294C2×C4#1309401911yescharacteristic
295C2×C4C2×C8×D4711yescharacteristic
296C2×C4M16×D4711yescharacteristic
297C2×C4#1309403011yescharacteristic
298C2×C4#1309403211yescharacteristic
299C8—22no—
300C8—41no—
301C8—41no—
302C8—41no—
303C8—81no—
304C8—942no—
305C8—942no—
306C8—942no—
307C8—3761no—
308C8—1881no—
309C22—21no—
310C22—21no—
311C22—3762no—
312C22—3762no—
313C22—3761no—
314C22—3761no—
315C22—1882no—
316C22#1568901311yescharacteristic
317C22#1568917511yescharacteristic
318C22#1568918311yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—3761no—
323C4C8×C22⋊D4711yescharacteristic
324C4#1568918811yescharacteristic
325C4#1568919311yescharacteristic
326C4#1568919911yescharacteristic
327C4#1568938811yescharacteristic
328C4#1568939111yescharacteristic
329C2—21no—
330C2—21no—
331C2—3762no—
332C2#2024399311yescharacteristic
333C2#2024404911yescharacteristic
334C2#2024405611yescharacteristic
subgroupG/H
centerC2×C4#13093996
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1565C2
socleC22×C47#388
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1565D47
hypocenterC47#9550
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
4. #20241669C2semidirect
5. #20241238C2semidirect
6. #20241226C2semidirect
7. C47×#1565C2semidirect
40. #13093940C8semidirect
163. #1565D47semidirect
194. C47#9550semidirect

Maximal subgroups

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

subgroupindex
C4×#130939402
#202417782
C2×#156888502
#202416692
#202412382
#202412262
C47×#15652
#955047