Finite Group Databasebeta

← #26374431

Subgroups — #26374431

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#20256538C211yesmaximal, characteristic, complemented
2#20256526C211yesmaximal, characteristic, complemented
3#20245707C212yesmaximal, complemented
4#20245705C211yesmaximal, characteristic, complemented
5#20245700C211yesmaximal, characteristic, complemented
6#20245699C212yesmaximal, complemented
7#20245696C212yesmaximal, complemented
8C2×#15688962C211yesmaximal, characteristic, complemented
9SD64×D47C212yesmaximal, complemented
10#20245681C211yesmaximal, characteristic, complemented
11C47×#2262C211yesmaximal, characteristic, complemented
12#15689937—21no—
13#15689925—21no—
14#15689563—21no—
15#15689562—22no—
16#15689560—22no—
17#15689558—21no—
18#15689556C2212yes—
19#15689552C2212yescomplemented
20C2×Dic376C2211yescharacteristic, complemented
21Q32×D47C2212yescomplemented
22C2×D752C2211yescharacteristic
23D16×D47C2212yes—
24#15689539C2211yescharacteristic
25#15688963C2212yescomplemented
26#15688962C2212yescomplemented
27#15688962C2212yescomplemented
28#15688961C2212yescomplemented
29#15688960C2212yescomplemented
30#15688958C2212yescomplemented
31#15688957C2211yescharacteristic, complemented
32#15688956C2212yescomplemented
33C32×D47C2212yescomplemented
34C47×#508C2211yescharacteristic
35C47×#508C2211yescharacteristic, complemented
36C47×Q64C2211yescharacteristic, complemented
37C47×SD64C2211yescharacteristic, complemented
38C47×SD64C2211yescharacteristic, complemented
39C47×D32C2211yescharacteristic, complemented
40C2×C32×C47C2211yescharacteristic, complemented
41#13094071—41no—
42#13094070—41no—
43#13094054—41no—
44#13094053—42no—
45#13094051—42no—
46#13094049—41no—
47#13094047—22no—
48#13094043—22no—
49C2×Dic188—21no—
50Q16×D47—22no—
51C2×D376—21no—
52D8×D47—22no—
53#13094030D411yescharacteristic
54Dic376—21no—
55Dic376C2312yes—
56#13093970—22no—
57#13093968—22no—
58#13093967—22no—
59D752—21no—
60D752C2312yes—
61#13093965—22no—
62#13093964D411yescharacteristic
63#13093963D412yes—
64C16×D47C2312yes—
65C47⋊C32C2312yes—
66#135×C47—21no—
67#135×C47—21no—
68Q32×C47C2311yescharacteristic
69Q32×C47C2311yescharacteristic
70SD32×C47—21no—
71SD32×C47—21no—
72D16×C47C2311yescharacteristic
73D16×C47C2311yescharacteristic
74C2×C16×C47C2311yescharacteristic
75C32×C47C2311yescharacteristic
76C32×C47C2311yescharacteristic
77#41×C47—41no—
78#41×C47—41no—
79#10625354—42no—
80Q8×D47—42no—
81#10625352—42no—
82#10625352—82no—
83D4×D47—42no—
84D4×D47—82no—
85#10625350—81no—
86#10625350—81no—
87#10625350D811yescharacteristic
88C2×D188—41no—
89C2×D188—81no—
90C2×Dic94—41no—
91C2×Dic94—81no—
92Q16×C47—21no—
93Q16×C47—21no—
94SD16×C47—41no—
95SD16×C47—41no—
96D8×C47—21no—
97D8×C47—21no—
98C2×C8×C47C2×D411yescharacteristic
99#10625337—42no—
100#10625336—42no—
101#10625335—42no—
102#10625334—42no—
103#10625329D811yescharacteristic
104Dic188—22no—
105Dic188—41no—
106D376—22no—
107D376—41no—
108#10625324D812yes—
109C8×D47C2×D412yes—
110C16×C47C2×D411yescharacteristic
111C16×C47C2411yescharacteristic
112C47⋊C16C2×D412yes—
113C22×D47—81no—
114C22×D47—82no—
115Q8×C47—41no—
116Q8×C47—41no—
117D4×C47—41no—
118D4×C47—41no—
119D4×C47—81no—
120D4×C47—81no—
121C2×C4×C47—81no—
122C2×C4×C47—81no—
123C2×C4×C47C2×D811yescharacteristic
124C22⋊D47—82no—
125C22⋊D47—82no—
126C22⋊D47D1612yes—
127C2×Dic47—81no—
128C2×Dic47—82no—
129D188—42no—
130D188—82no—
131D188—81no—
132D188D1611yescharacteristic
133C4×D47—82no—
134C4×D47—82no—
135C4×D47C2×D812yes—
136Dic94—42no—
137Dic94—81no—
138Dic94—82no—
139Dic94D1611yescharacteristic
140C8×C47C2×D811yescharacteristic
141C8×C47C22×D411yescharacteristic
142C47⋊C8C2×D812yes—
143#33983—471nomaximal, Sylow
144C22×C47—81no—
145C22×C47—81no—
146C22×C47C2×D1611yescharacteristic
147C2×D47—82no—
148C2×D47—322no—
149C2×D47—321no—
150C2×D47C2×D1612yes—
151C4×C47—81no—
152C4×C47—81no—
153C4×C47C2×D1611yescharacteristic
154C4×C47C22×D811yescharacteristic
155Dic47—82no—
156Dic47C2×D1612yes—
157#3417—471no—
158#3415—471no—
159#2264—471no—
160#2264—472no—
161#2263—471no—
162#2263—472no—
163#2262—472no—
164#2262D4711yescharacteristic, complemented
165C2×SD64—472no—
166C2×SD64—471no—
167#2258—471no—
168C2×C47—21no—
169C2×C47—161no—
170C2×C47—161no—
171C2×C47C22×D1611yescharacteristic
172D47—22no—
173D47—162no—
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#508C2×D4711yescharacteristic
183#508C2×D4711yescharacteristic, complemented
184C2×Q32—472no—
185C2×Q32—471no—
186C2×D16—472no—
187C2×D16—471no—
188#504—471no—
189Q64—472no—
190Q64C2×D4711yescharacteristic, complemented
191SD64—472no—
192SD64—472no—
193SD64—472no—
194SD64—472no—
195SD64C2×D4711yescharacteristic, complemented
196SD64C2×D4711yescharacteristic, complemented
197D32—472no—
198D32C2×D4711yescharacteristic, complemented
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—21no—
211#135—21no—
212#135—942no—
213#135—942no—
214C2×Q16—942no—
215C2×Q16—941no—
216C2×D8—942no—
217C2×D8—941no—
218#131—471no—
219Q32—472no—
220Q32—942no—
221Q32—941no—
222Q32C22×D4711yescharacteristic
223Q32C22×D4711yescharacteristic
224SD32—21no—
225SD32—21no—
226SD32—942no—
227SD32—942no—
228D16—472no—
229D16—942no—
230D16—941no—
231D16C22×D4711yescharacteristic
232D16C22×D4711yescharacteristic
233M32—471no—
234M32—472no—
235C2×C16—472no—
236C2×C16C22×D4711yescharacteristic
237C32—472no—
238C32C22×D4711yescharacteristic
239C32C22×D4711yescharacteristic
240#41—41no—
241#41—41no—
242#41—471no—
243#41—3761no—
244#41—3761no—
245#41—3762no—
246#41—1882no—
247#41—1882no—
248C2×Q8—3761no—
249C2×Q8—1882no—
250C2×Q8—1881no—
251C2×D4—3761no—
252C2×D4—3762no—
253C2×D4—1882no—
254C2×D4—1881no—
255Q16—21no—
256Q16—21no—
257Q16—942no—
258Q16—1882no—
259Q16—1881no—
260SD16—41no—
261SD16—41no—
262SD16—1882no—
263SD16—1882no—
264D8—21no—
265D8—21no—
266D8—942no—
267D8—1882no—
268D8—1881no—
269M16—471no—
270M16—472no—
271C2×C8—472no—
272C2×C8D4×D4711yescharacteristic
273C16—472no—
274C16D4×D4711yescharacteristic
275C16C23×D4711yescharacteristic
276C23—3762no—
277C23—3761no—
278Q8—41no—
279Q8—41no—
280Q8—471no—
281Q8—3762no—
282Q8—3761no—
283Q8—1882no—
284D4—41no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—472no—
289D4—471no—
290D4—3762no—
291D4—3762no—
292D4—3762no—
293D4—3761no—
294D4—1882no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—472no—
298C2×C4—3762no—
299C2×C4—3762no—
300C2×C4—3762no—
301C2×C4—3761no—
302C2×C4D8×D4711yescharacteristic
303C8—472no—
304C8D8×D4711yescharacteristic
305C8C2×D4×D4711yescharacteristic
306C22—81no—
307C22—81no—
308C22—472no—
309C22—3762no—
310C22—15041no—
311C22—15042no—
312C22D16×D4711yescharacteristic
313C4—81no—
314C4—81no—
315C4—472no—
316C4—3762no—
317C4D16×D4711yescharacteristic
318C4C2×D8×D4711yescharacteristic
319C2—21no—
320C2—161no—
321C2—161no—
322C2—942no—
323C2—7522no—
324C2C2×D16×D4711yescharacteristic
subgroupG/H
centerC2C2×D16×D47
commutator, cosocleC16×C47C24
FrattiniC16C23×D47
FittingC47×#2262C2
socleC2×C47C22×D16
hypercenter#2262D47
hypocenterC47#33983
LCSG ▷ C16×C47 ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #20256538C2semidirect
2. #20256526C2semidirect
3. #20245707C2semidirect
4. #20245705C2semidirect
5. #20245700C2semidirect
6. #20245699C2semidirect
7. #20245696C2semidirect
8. C2×#15688962C2semidirect
9. SD64×D47C2semidirect
10. #20245681C2semidirect
11. C47×#2262C2semidirect
19. #15689552C22semidirect
20. C2×Dic376C22semidirect
21. Q32×D47C22semidirect
25. #15688963C22semidirect
26. #15688962C22semidirect
27. #15688962C22semidirect
28. #15688961C22semidirect
29. #15688960C22semidirect
30. #15688958C22semidirect
31. #15688957C22semidirect
32. #15688956C22semidirect
33. C32×D47C22semidirect
35. C47×#508C22semidirect
36. C47×Q64C22semidirect
37. C47×SD64C22semidirect
38. C47×SD64C22semidirect
39. C47×D32C22semidirect
40. C2×C32×C47C22semidirect
164. #2262D47semidirect
183. #508C2×D47semidirect
190. Q64C2×D47semidirect
195. SD64C2×D47semidirect
196. SD64C2×D47semidirect
198. D32C2×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
#202565382
#202565262
#202457072
#202457052
#202457002
#202456992
#202456962
C2×#156889622
SD64×D472
#202456812
C47×#22622
#3398347