Finite Group Databasebeta

← #26374430

Subgroups — #26374430

Order 12032.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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#20256537C211yesmaximal, characteristic, complemented
2#20256525C211yesmaximal, characteristic, complemented
3#20245706C211yesmaximal, characteristic, complemented
4#20245704C212yesmaximal, complemented
5#20245702C212yesmaximal, complemented
6#20245701C211yesmaximal, characteristic, complemented
7#20245697C211yesmaximal, characteristic, complemented
8#20245696C212yesmaximal, complemented
9SD64×D47C212yesmaximal, complemented
10#20245681C211yesmaximal, characteristic, complemented
11C2×C47×SD64C211yesmaximal, characteristic, complemented
12#15689936—21no—
13#15689924—21no—
14#15689565—22no—
15#15689564—21no—
16#15689559—21no—
17#15689557—22no—
18#15689556C2212yescomplemented
19#15689555C2211yescharacteristic
20#15689555C2211yescharacteristic, complemented
21#15689552C2212yes—
22Q32×D47C2212yescomplemented
23D16×D47C2212yes—
24#15689539C2211yescharacteristic
25Dic752C2211yescharacteristic, complemented
26#15688963C2212yescomplemented
27#15688962C2211yescharacteristic, complemented
28#15688962C2211yescharacteristic, complemented
29#15688961C2212yescomplemented
30#15688960C2212yescomplemented
31D1504C2211yescharacteristic, complemented
32#15688958C2212yescomplemented
33#15688957C2211yescharacteristic, complemented
34#15688956C2212yescomplemented
35C32×D47C2212yescomplemented
36C2×Q32×C47C2211yescharacteristic, complemented
37C2×D16×C47C2211yescharacteristic
38C47×SD64C2212yescomplemented
39C47×SD64C2212yescomplemented
40C2×C32×C47C2211yescharacteristic, complemented
41#13094072—41no—
42#13094069—41no—
43#13094056—42no—
44#13094055—41no—
45#13094050—41no—
46#13094048—42no—
47#13094047—22no—
48#13094046—21no—
49#13094046—21no—
50#13094043—22no—
51Q16×D47—22no—
52D8×D47—22no—
53#13094030D411yescharacteristic
54Dic376C2311yescharacteristic
55Dic376C2311yescharacteristic
56#13093970—22no—
57#13093969—21no—
58#13093969—21no—
59#13093968—22no—
60#13093967—22no—
61D752C2311yescharacteristic
62D752C2311yescharacteristic
63#13093965—22no—
64#13093964D411yescharacteristic
65#13093963D412yes—
66C16×D47C2312yes—
67C47⋊C32C2312yes—
68C2×Q16×C47—21no—
69C2×D8×C47—21no—
70Q32×C47—21no—
71Q32×C47C2312yes—
72D16×C47—21no—
73D16×C47C2312yes—
74C2×C16×C47C2311yescharacteristic
75C32×C47C2311yescharacteristic
76C32×C47C2311yescharacteristic
77C2×Q8×C47—41no—
78C2×D4×C47—41no—
79C2×C22⋊D47—82no—
80#10625354—42no—
81#10625354—81no—
82Q8×D47—42no—
83Q8×D47—81no—
84#10625352—42no—
85#10625352—81no—
86D4×D47—42no—
87D4×D47—81no—
88#10625350—41no—
89#10625350—41no—
90#10625350—82no—
91#10625350D811yescharacteristic
92Q16×C47—22no—
93Q16×C47—41no—
94D8×C47—22no—
95D8×C47—41no—
96C2×C8×C47C2×D411yescharacteristic
97#10625337—42no—
98#10625336—42no—
99#10625335—42no—
100#10625334—42no—
101#10625329D811yescharacteristic
102Dic188—21no—
103Dic188—21no—
104D376—21no—
105D376—21no—
106#10625325—41no—
107#10625325—41no—
108#10625324D812yes—
109C8×D47C2×D412yes—
110C16×C47C2×D411yescharacteristic
111C16×C47C2411yescharacteristic
112C47⋊C16C2×D412yes—
113C23×C47—81no—
114C22×D47—82no—
115Q8×C47—42no—
116Q8×C47—81no—
117D4×C47—42no—
118D4×C47—81no—
119C2×C4×C47—81no—
120C2×C4×C47C2×D811yescharacteristic
121C22⋊D47—81no—
122C22⋊D47—81no—
123C22⋊D47—82no—
124C22⋊D47—82no—
125C22⋊D47D1612yes—
126C2×Dic47—82no—
127D188—41no—
128D188—41no—
129D188—82no—
130D188D1611yescharacteristic
131C4×D47—82no—
132C4×D47—81no—
133C4×D47—81no—
134C4×D47—82no—
135C4×D47C2×D812yes—
136Dic94—41no—
137Dic94—41no—
138Dic94—82no—
139Dic94D1611yescharacteristic
140C8×C47C2×D811yescharacteristic
141C8×C47C22×D411yescharacteristic
142C47⋊C8C2×D812yes—
143#33983—471nomaximal, Sylow
144C22×C47—82no—
145C22×C47—321no—
146C22×C47C2×D1611yescharacteristic
147C2×D47—81no—
148C2×D47—81no—
149C2×D47—322no—
150C2×D47C2×D1612yes—
151C4×C47—82no—
152C4×C47C2×D1611yescharacteristic
153C4×C47C22×D811yescharacteristic
154Dic47—81no—
155Dic47—81no—
156Dic47C2×D1612yes—
157#3417—471no—
158#3415—471no—
159#2264—471no—
160#2264—472no—
161#2263—471no—
162#2263—472no—
163#2262—472no—
164#2262—471no—
165C2×SD64—472no—
166C2×SD64D4711yescharacteristic, complemented
167#2258—471no—
168C2×C47—21no—
169C2×C47—162no—
170C2×C47C22×D1611yescharacteristic
171D47—22no—
172D47—161no—
173D47—161no—
174#578—941no—
175#576—941no—
176#510—942no—
177#510—941no—
178#509—942no—
179#509—941no—
180#508—472no—
181#508—472no—
182#508—471no—
183#508—471no—
184C2×Q32—472no—
185C2×Q32C2×D4711yescharacteristic, complemented
186C2×D16—472no—
187C2×D16C2×D4711yescharacteristic
188#504—471no—
189Q64—472no—
190Q64—471no—
191SD64—472no—
192SD64—472no—
193SD64—471no—
194SD64—471no—
195SD64C2×D4712yescomplemented
196SD64C2×D4712yescomplemented
197D32—472no—
198D32—471no—
199M64—471no—
200M64—472no—
201C2×C32—472no—
202C2×C32C2×D4711yescharacteristic, complemented
203C47#3398311yescharacteristic, complemented, Sylow
2042−1+4—1881no—
205H(5,𝔽2)—1881no—
206#137—1882no—
207#137—1881no—
208AGL(1,ℤ/8)—1882no—
209AGL(1,ℤ/8)—1881no—
210#135—942no—
211#135—942no—
212#135—941no—
213#135—941no—
214C2×Q16—21no—
215C2×Q16—942no—
216C2×D8—21no—
217C2×D8—942no—
218#131—471no—
219Q32—21no—
220Q32—471no—
221Q32—471no—
222Q32—942no—
223Q32C22×D4712yes—
224SD32—942no—
225SD32—942no—
226SD32—941no—
227SD32—941no—
228D16—21no—
229D16—471no—
230D16—471no—
231D16—942no—
232D16C22×D4712yes—
233M32—471no—
234M32—472no—
235C2×C16—472no—
236C2×C16C22×D4711yescharacteristic
237C32—472no—
238C32C22×D4711yescharacteristic
239C32C22×D4711yescharacteristic
240#41—471no—
241#41—3762no—
242#41—3761no—
243#41—3761no—
244#41—1882no—
245#41—1882no—
246#41—1881no—
247#41—1881no—
248C2×Q8—41no—
249C2×Q8—3761no—
250C2×Q8—1882no—
251C2×D4—41no—
252C2×D4—3761no—
253C2×D4—3762no—
254C2×D4—1882no—
255Q16—22no—
256Q16—41no—
257Q16—941no—
258Q16—941no—
259Q16—1882no—
260SD16—1882no—
261SD16—1882no—
262SD16—1881no—
263SD16—1881no—
264D8—22no—
265D8—41no—
266D8—941no—
267D8—941no—
268D8—1882no—
269M16—471no—
270M16—472no—
271C2×C8—472no—
272C2×C8D4×D4711yescharacteristic
273C16—472no—
274C16D4×D4711yescharacteristic
275C16C23×D4711yescharacteristic
276C23—81no—
277C23—3762no—
278Q8—42no—
279Q8—81no—
280Q8—471no—
281Q8—3762no—
282Q8—1881no—
283Q8—1881no—
284D4—42no—
285D4—81no—
286D4—472no—
287D4—471no—
288D4—3762no—
289D4—3762no—
290D4—3762no—
291D4—3761no—
292D4—3761no—
293D4—1881no—
294D4—1881no—
295C2×C4—81no—
296C2×C4—472no—
297C2×C4—3762no—
298C2×C4—3762no—
299C2×C4—3762no—
300C2×C4—3761no—
301C2×C4—3761no—
302C2×C4D8×D4711yescharacteristic
303C8—472no—
304C8D8×D4711yescharacteristic
305C8C2×D4×D4711yescharacteristic
306C22—82no—
307C22—321no—
308C22—472no—
309C22—3761no—
310C22—3761no—
311C22—15042no—
312C22D16×D4711yescharacteristic
313C4—82no—
314C4—472no—
315C4—3761no—
316C4—3761no—
317C4D16×D4711yescharacteristic
318C4C2×D8×D4711yescharacteristic
319C2—21no—
320C2—162no—
321C2—942no—
322C2—7521no—
323C2—7521no—
324C2C2×D16×D4711yescharacteristic
subgroupG/H
centerC2C2×D16×D47
commutator, cosocleC16×C47C24
FrattiniC16C23×D47
FittingC2×C47×SD64C2
socleC2×C47C22×D16
hypercenterC2×SD64D47
hypocenterC47#33983
LCSG ▷ C16×C47 ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×SD64

Complements

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

NH ≅ G/Ntype
1. #20256537C2semidirect
2. #20256525C2semidirect
3. #20245706C2semidirect
4. #20245704C2semidirect
5. #20245702C2semidirect
6. #20245701C2semidirect
7. #20245697C2semidirect
8. #20245696C2semidirect
9. SD64×D47C2semidirect
10. #20245681C2semidirect
11. C2×C47×SD64C2semidirect
18. #15689556C22semidirect
20. #15689555C22semidirect
22. Q32×D47C22semidirect
25. Dic752C22semidirect
26. #15688963C22semidirect
27. #15688962C22semidirect
28. #15688962C22semidirect
29. #15688961C22semidirect
30. #15688960C22semidirect
31. D1504C22semidirect
32. #15688958C22semidirect
33. #15688957C22semidirect
34. #15688956C22semidirect
35. C32×D47C22semidirect
36. C2×Q32×C47C22semidirect
38. C47×SD64C22semidirect
39. C47×SD64C22semidirect
40. C2×C32×C47C22semidirect
166. C2×SD64D47semidirect
185. C2×Q32C2×D47semidirect
195. SD64C2×D47semidirect
196. SD64C2×D47semidirect
202. C2×C32C2×D47semidirect
203. C47#33983semidirect

Maximal subgroups

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

subgroupindex
#202565372
#202565252
#202457062
#202457042
#202457022
#202457012
#202456972
#202456962
SD64×D472
#202456812
C2×C47×SD642
#3398347