Finite Group Databasebeta

← #26126303

Subgroups — #26126303

Order 12032.

430 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C4×D8×D47C211yesmaximal, characteristic
2#20245185C211yesmaximal, characteristic, complemented
3#20242145C211yesmaximal, characteristic, complemented
4#20241350C211yesmaximal, characteristic, complemented
5#20241349C211yesmaximal, characteristic, complemented
6#20241348C211yesmaximal, characteristic, complemented
7C47×#1331C211yesmaximal, characteristic, complemented
8C2×D8×D47C411yescharacteristic
9C4×D4×D47—21no—
10C4×D376C2211yescharacteristic
11D8×Dic47C411yescharacteristic
12#15689205C411yescharacteristic
13C4×#10625334—21no—
14#15689203—21no—
15#107×D47C411yescharacteristic
16#15689145—21no—
17#102×D47—21no—
18C4×C8×D47C2211yescharacteristic
19Dic47⋊C16—21no—
20(C2×D47)⋊C16—21no—
21C4×C47⋊C16C2211yescharacteristic, complemented
22#15688876C2211yescharacteristic, complemented
23#15688875—21no—
24C4×D8×C47C2211yescharacteristic
25C4×C16×C47C2211yescharacteristic, complemented
26C47×#335C2211yescharacteristic, complemented
27C2×D4×D47—21no—
28C22×C4×D47—41no—
29C2×D376C2×C411yescharacteristic
30C2×#10625334—21no—
31D8×D47—22no—
32D8×D47—22no—
33C2×#10625324—41no—
34C2×C8×D47C2×C411yescharacteristic
35C2×C8×D47D411yescharacteristic
36C4×D188—21no—
37D4×Dic47—21no—
38#13093986—21no—
39C4×C22⋊D47—41no—
40(C22⋊D47)⋊C4—41no—
41C4⋊C4×D47—21no—
42C22⋊C4×D47—41no—
43C42×D47D411yescharacteristic
44C2×C47⋊C16—21no—
45#13093958C2×C411yescharacteristic
46#13093957—21no—
47C4⋊(C47⋊C8)—21no—
48Dic47⋊C8—41no—
49#13093946—21no—
50#13093945—21no—
51#13093944—21no—
52C4×C47⋊C8D411yescharacteristic
53C8×Dic47C2×C411yescharacteristic
54C8×Dic47D411yescharacteristic
55C2×D8×C47C2×C411yescharacteristic
56C4×D4×C47—21no—
57C2×C16×C47—21no—
58#107×C47C2×C411yescharacteristic
59C4⋊C8×C47—21no—
60#102×C47—21no—
61C4×C8×C47C2311yescharacteristic
62C23×D47—41no—
63C2×D4×C47—21no—
64C22×C4×C47—41no—
65C2×C22⋊D47—41no—
66C22×Dic47—41no—
67D4×D47—24no—
68D4×D47—42no—
69C2×D188—21no—
70C2×C4×D47—41no—
71C2×C4×D47—41no—
72C2×C4×D47—41no—
73C2×C4×D47—82no—
74C2×C4×D47—82no—
75C2×C4×D47C22⋊C411yescharacteristic
76C2×C4×D47D811yescharacteristic
77C2×C4×D47SD1611yescharacteristic
78D8×C47—21no—
79D8×C47M1612yescomplemented
80C2×C8×C47—41no—
81C2×C8×C47C22×C411yescharacteristic
82C2×C8×C47C2×D411yescharacteristic
83C4⋊C4×C47—21no—
84C22⋊C4×C47—41no—
85C42×C47C2×D411yescharacteristic
86C22⋊Dic47—41no—
87#10625334—24no—
88(C2×D47)⋊C4—41no—
89(C2×D47)⋊C4—41no—
90C4⋊Dic47—21no—
91Dic47⋊C4—41no—
92C4×Dic47C22⋊C411yescharacteristic
93C4×Dic47D811yescharacteristic
94C4×Dic47SD1611yescharacteristic
95C2×C47⋊C8—41no—
96C2×C47⋊C8C22⋊C411yescharacteristic
97C2×C47⋊C8C22⋊C411yescharacteristic
98D376—21no—
99D376M1612yescomplemented
100#10625324—82no—
101C8×D47—21no—
102C8×D47—21no—
103C8×D47—41no—
104C16×C47—41no—
105C47⋊C16—41no—
106C23×C47—41no—
107C22×D47—44no—
108C22×D47—41no—
109C22×D47—82no—
110C22×D47—82no—
111C22×D47#10211yescharacteristic
112D4×C47—22no—
113D4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—81no—
116C2×C4×C47—81no—
117C2×C4×C47C2×C22⋊C411yescharacteristic
118C2×C4×C47C2×D811yescharacteristic
119C2×C4×C47C2×SD1611yescharacteristic
120C22⋊D47—44no—
121C2×Dic47—41no—
122C2×Dic47—82no—
123C2×Dic47#10211yescharacteristic
124C2×Dic47#10211yescharacteristic
125C2×Dic47#10211yescharacteristic
126D188—22no—
127D188—41no—
128C4×D47—21no—
129C4×D47—21no—
130C4×D47—41no—
131C4×D47—82no—
132C4×D47—81no—
133C4×D47—81no—
134C4×D47—82no—
135C4×D47D1612yes—
136C4×D47SD3212yes—
137C8×C47—21no—
138C8×C47—81no—
139C8×C47C2×M1611yescharacteristic
140C8×C47C2×M1611yescharacteristic
141C47⋊C8—21no—
142C47⋊C8—21no—
143C47⋊C8—81no—
144#11657—471nomaximal, Sylow
145C22×C47—42no—
146C22×C47—81no—
147C22×C47—81no—
148C22×C47C2×#10211yescharacteristic
149C2×D47—21no—
150C2×D47—21no—
151C2×D47—42no—
152C2×D47—88no—
153C2×D47—81no—
154C2×D47—81no—
155C2×D47#35712yes—
156C4×C47—21no—
157C4×C47—81no—
158C4×C47#40711yescharacteristic
159C4×C47#42111yescharacteristic
160C4×C47C2×D1611yescharacteristic
161C4×C47C2×SD3211yescharacteristic
162Dic47—21no—
163Dic47—21no—
164Dic47—81no—
165Dic47#35712yes—
166C2×C4×D8—471no—
167#2162—471no—
168#1567—471no—
169#1331—471no—
170#1331—471no—
171#1331—471no—
172#1331D4711yescharacteristic, complemented
173C2×C47—82no—
174C2×C47#148611yescharacteristic
175C2×C47C2×#35711yescharacteristic
176C2×C47#214311yescharacteristic
177D47—22no—
178D47—82no—
179C22×D8—471no—
180C2×C4×D4—941no—
181C4×D8—471no—
182C4×D8—471no—
183C4×D8—471no—
184C4×D8—941no—
185C4×D8—941no—
186C4×D8C2×D4711yescharacteristic
187C2×#107—471no—
188#423—941no—
189C2×#102—941no—
190C2×C4×C8—471no—
191C4⋊C16—941no—
192C22⋊C16—941no—
193C4×C16—471no—
194C4×C16C2×D4711yescharacteristic, complemented
195#335—471no—
196#335—941no—
197#335C2×D4711yescharacteristic, complemented
198C47#1165711yescharacteristic, complemented, Sylow
199C22×D4—941no—
200C23×C4—1881no—
201C2×D8—471no—
202C2×D8—941no—
203C2×D8—942no—
204C2×D8—942no—
205C2×D8C4×D4711yescharacteristic
206C2×M16—1881no—
207C22×C8—471no—
208C22×C8—471no—
209C4×D4—21no—
210C4×D4—941no—
211C4×D4—941no—
212C4×D4—941no—
213C4×D4—1881no—
214C4×D4—1881no—
215C2×C4⋊C4—941no—
216C2×C22⋊C4—1881no—
217C2×C42—471no—
218C2×C16—21no—
219C2×C16—941no—
220#107—471no—
221#107—941no—
222#107C4×D4711yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—941no—
225C4⋊C8—1881no—
226#102—21no—
227#102—941no—
228#102—941no—
229#102—941no—
230C4×C8—471no—
231C4×C8—471no—
232C4×C8—471no—
233C4×C8C22×D4711yescharacteristic
234C24—1881no—
235C2×D4—21no—
236C2×D4—944no—
237C2×D4—941no—
238C2×D4—1881no—
239C2×D4—1882no—
240C22×C4—41no—
241C22×C4—471no—
242C22×C4—471no—
243C22×C4—471no—
244C22×C4—3762no—
245C22×C4—3762no—
246C22×C4—1881no—
247C22×C4—1881no—
248C22×C4—1881no—
249C22×C4—1881no—
250D8—21no—
251D8—472no—
252D8—944no—
253D8—941no—
254D8#1062532412yescomplemented
255M16—3762no—
256C2×C8—41no—
257C2×C8—471no—
258C2×C8—471no—
259C2×C8—941no—
260C2×C8—941no—
261C2×C8—1881no—
262C2×C8—1881no—
263C2×C8C2×C4×D4711yescharacteristic
264C2×C8D4×D4711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—941no—
267C4⋊C4—1881no—
268C22⋊C4—41no—
269C22⋊C4—1881no—
270C22⋊C4—1881no—
271C22⋊C4—1881no—
272C42—471no—
273C42—471no—
274C42—471no—
275C42D4×D4711yescharacteristic
276C16—41no—
277C16—1881no—
278C23—41no—
279C23—471no—
280C23—3762no—
281C23—3762no—
282C23—1884no—
283C23—1881no—
284D4—22no—
285D4—41no—
286D4—942no—
287D4—1884no—
288D4—1881no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—471no—
293C2×C4—472no—
294C2×C4—472no—
295C2×C4—471no—
296C2×C4—471no—
297C2×C4—941no—
298C2×C4—941no—
299C2×C4—3762no—
300C2×C4—3762no—
301C2×C4—3762no—
302C2×C4—3761no—
303C2×C4—3761no—
304C2×C4—1881no—
305C2×C4—1881no—
306C2×C4C22⋊C4×D4711yescharacteristic
307C2×C4D8×D4711yescharacteristic
308C2×C4SD16×D4711yescharacteristic
309C8—21no—
310C8—81no—
311C8—941no—
312C8—941no—
313C8—3761no—
314C8M16×D4711yescharacteristic
315C8C2×#1062532411yescharacteristic
316C22—42no—
317C22—81no—
318C22—81no—
319C22—472no—
320C22—941no—
321C22—941no—
322C22—3768no—
323C22—3761no—
324C22—3761no—
325C22—1882no—
326C22#102×D4711yescharacteristic
327C4—21no—
328C4—81no—
329C4—472no—
330C4—941no—
331C4—941no—
332C4—3761no—
333C4#1568908511yescharacteristic
334C4#1568913811yescharacteristic
335C4D16×D4711yescharacteristic
336C4SD32×D4711yescharacteristic
337C2—82no—
338C2—942no—
339C2—3762no—
340C2#2024181111yescharacteristic
341C2#357×D4711yescharacteristic
342C2#2024511411yescharacteristic
subgroupG/H
centerC2×C4D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#1331C2
socleC22×C47C2×#102
prime radicalC2×D8×D47C4
cosocleC4×C8×C47C23
hypercenter#1331D47
hypocenterC47#11657
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #20245185C2semidirect
3. #20242145C2semidirect
4. #20241350C2semidirect
5. #20241349C2semidirect
6. #20241348C2semidirect
7. C47×#1331C2semidirect
21. C4×C47⋊C16C22semidirect
22. #15688876C22semidirect
25. C4×C16×C47C22semidirect
26. C47×#335C22semidirect
79. D8×C47M16semidirect
99. D376M16semidirect
172. #1331D47semidirect
194. C4×C16C2×D47semidirect
197. #335C2×D47semidirect
198. C47#11657semidirect
254. D8#10625324semidirect

Maximal subgroups

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

subgroupindex
C4×D8×D472
#202451852
#202421452
#202413502
#202413492
#202413482
C47×#13312
#1165747