Finite Group Databasebeta

← #29838166

Subgroups — #29838166

Order 15616.

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#21458969C211yesmaximal, characteristic, complemented
2#21458957C211yesmaximal, characteristic, complemented
3#21448138C212yesmaximal, complemented
4#21448136C211yesmaximal, characteristic, complemented
5#21448131C211yesmaximal, characteristic, complemented
6#21448130C212yesmaximal, complemented
7#21448127C212yesmaximal, complemented
8C2×#16839601C211yesmaximal, characteristic, complemented
9SD64×D61C212yesmaximal, complemented
10#21448112C211yesmaximal, characteristic, complemented
11C61×#2262C211yesmaximal, characteristic, complemented
12#16840576—21no—
13#16840564—21no—
14#16840202—21no—
15#16840201—22no—
16#16840199—22no—
17#16840197—21no—
18#16840195C2212yes—
19#16840191C2212yescomplemented
20C2×Dic488C2211yescharacteristic, complemented
21Q32×D61C2212yescomplemented
22C2×D976C2211yescharacteristic
23D16×D61C2212yes—
24#16840178C2211yescharacteristic
25#16839602C2212yescomplemented
26#16839601C2212yescomplemented
27#16839601C2212yescomplemented
28#16839600C2212yescomplemented
29#16839599C2212yescomplemented
30#16839597C2212yescomplemented
31#16839596C2211yescharacteristic, complemented
32#16839595C2212yescomplemented
33C32×D61C2212yescomplemented
34C61×#508C2211yescharacteristic
35C61×#508C2211yescharacteristic, complemented
36C61×Q64C2211yescharacteristic, complemented
37C61×SD64C2211yescharacteristic, complemented
38C61×SD64C2211yescharacteristic, complemented
39C61×D32C2211yescharacteristic, complemented
40C2×C32×C61C2211yescharacteristic, complemented
41#14519824—41no—
42#14519823—41no—
43#14519807—41no—
44#14519806—42no—
45#14519804—42no—
46#14519802—41no—
47#14519800—22no—
48#14519796—22no—
49C2×Dic244—21no—
50Q16×D61—22no—
51C2×D488—21no—
52D8×D61—22no—
53#14519783D411yescharacteristic
54Dic488—21no—
55Dic488C2312yes—
56#14519723—22no—
57#14519721—22no—
58#14519720—22no—
59D976—21no—
60D976C2312yes—
61#14519718—22no—
62#14519717D411yescharacteristic
63#14519716D412yes—
64C16×D61C2312yes—
65#14519683C2312yes—
66#135×C61—21no—
67#135×C61—21no—
68Q32×C61C2311yescharacteristic
69Q32×C61C2311yescharacteristic
70SD32×C61—21no—
71SD32×C61—21no—
72D16×C61C2311yescharacteristic
73D16×C61C2311yescharacteristic
74C2×C16×C61C2311yescharacteristic
75C32×C61C2311yescharacteristic
76C32×C61C2311yescharacteristic
77#41×C61—41no—
78#41×C61—41no—
79#11758226—42no—
80Q8×D61—42no—
81#11758224—42no—
82#11758224—82no—
83D4×D61—42no—
84D4×D61—82no—
85#11758222—81no—
86#11758222—81no—
87#11758222D811yescharacteristic
88C2×D244—41no—
89C2×D244—81no—
90C2×Dic122—41no—
91C2×Dic122—81no—
92Q16×C61—21no—
93Q16×C61—21no—
94SD16×C61—41no—
95SD16×C61—41no—
96D8×C61—21no—
97D8×C61—21no—
98C2×C8×C61C2×D411yescharacteristic
99#11758202—42no—
100#11758201—42no—
101#11758200—42no—
102#11758199—42no—
103#11758194D811yescharacteristic
104Dic244—22no—
105Dic244—41no—
106D488—22no—
107D488—41no—
108#11758189D812yes—
109C8×D61C2×D412yes—
110C16×C61C2×D411yescharacteristic
111C16×C61C2411yescharacteristic
112#11758185C2×D412yes—
113C22×D61—81no—
114C22×D61—82no—
115Q8×C61—41no—
116Q8×C61—41no—
117D4×C61—41no—
118D4×C61—41no—
119D4×C61—81no—
120D4×C61—81no—
121C2×C4×C61—81no—
122C2×C4×C61—81no—
123C2×C4×C61C2×D811yescharacteristic
124C22⋊D61—82no—
125C22⋊D61—82no—
126C22⋊D61D1612yes—
127C2×Dic61—81no—
128C2×Dic61—82no—
129D244—42no—
130D244—82no—
131D244—81no—
132D244D1611yescharacteristic
133C4×D61—82no—
134C4×D61—82no—
135C4×D61C2×D812yes—
136Dic122—42no—
137Dic122—81no—
138Dic122—82no—
139Dic122D1611yescharacteristic
140C8×C61C2×D811yescharacteristic
141C8×C61C22×D411yescharacteristic
142#92418C2×D812yes—
143#33983—611nomaximal, Sylow
144C22×C61—81no—
145C22×C61—81no—
146C22×C61C2×D1611yescharacteristic
147C2×D61—82no—
148C2×D61—322no—
149C2×D61—321no—
150C2×D61C2×D1612yes—
151C4×C61—81no—
152C4×C61—81no—
153C4×C61C2×D1611yescharacteristic
154C4×C61C22×D811yescharacteristic
155Dic61—82no—
156Dic61C2×D1612yes—
157#3417—611no—
158#3415—611no—
159#2264—611no—
160#2264—612no—
161#2263—611no—
162#2263—612no—
163#2262—612no—
164#2262D6111yescharacteristic, complemented
165C2×SD64—612no—
166C2×SD64—611no—
167#2258—611no—
168C2×C61—21no—
169C2×C61—161no—
170C2×C61—161no—
171C2×C61C22×D1611yescharacteristic
172D61—22no—
173D61—162no—
174#578—1221no—
175#576—1221no—
176#510—1222no—
177#510—1221no—
178#509—1222no—
179#509—1221no—
180#508—612no—
181#508—612no—
182#508C2×D6111yescharacteristic
183#508C2×D6111yescharacteristic, complemented
184C2×Q32—612no—
185C2×Q32—611no—
186C2×D16—612no—
187C2×D16—611no—
188#504—611no—
189Q64—612no—
190Q64C2×D6111yescharacteristic, complemented
191SD64—612no—
192SD64—612no—
193SD64—612no—
194SD64—612no—
195SD64C2×D6111yescharacteristic, complemented
196SD64C2×D6111yescharacteristic, complemented
197D32—612no—
198D32C2×D6111yescharacteristic, complemented
199M64—611no—
200M64—612no—
201C2×C32—612no—
202C2×C32C2×D6111yescharacteristic, complemented
203C61#3398311yescharacteristic, complemented, Sylow
2042−1+4—2441no—
205H(5,𝔽2)—2441no—
206#137—2442no—
207#137—2441no—
208AGL(1,ℤ/8)—2442no—
209AGL(1,ℤ/8)—2441no—
210#135—21no—
211#135—21no—
212#135—1222no—
213#135—1222no—
214C2×Q16—1222no—
215C2×Q16—1221no—
216C2×D8—1222no—
217C2×D8—1221no—
218#131—611no—
219Q32—612no—
220Q32—1222no—
221Q32—1221no—
222Q32C22×D6111yescharacteristic
223Q32C22×D6111yescharacteristic
224SD32—21no—
225SD32—21no—
226SD32—1222no—
227SD32—1222no—
228D16—612no—
229D16—1222no—
230D16—1221no—
231D16C22×D6111yescharacteristic
232D16C22×D6111yescharacteristic
233M32—611no—
234M32—612no—
235C2×C16—612no—
236C2×C16C22×D6111yescharacteristic
237C32—612no—
238C32C22×D6111yescharacteristic
239C32C22×D6111yescharacteristic
240#41—41no—
241#41—41no—
242#41—611no—
243#41—4881no—
244#41—4881no—
245#41—4882no—
246#41—2442no—
247#41—2442no—
248C2×Q8—4881no—
249C2×Q8—2442no—
250C2×Q8—2441no—
251C2×D4—4881no—
252C2×D4—4882no—
253C2×D4—2442no—
254C2×D4—2441no—
255Q16—21no—
256Q16—21no—
257Q16—1222no—
258Q16—2442no—
259Q16—2441no—
260SD16—41no—
261SD16—41no—
262SD16—2442no—
263SD16—2442no—
264D8—21no—
265D8—21no—
266D8—1222no—
267D8—2442no—
268D8—2441no—
269M16—611no—
270M16—612no—
271C2×C8—612no—
272C2×C8D4×D6111yescharacteristic
273C16—612no—
274C16D4×D6111yescharacteristic
275C16C23×D6111yescharacteristic
276C23—4882no—
277C23—4881no—
278Q8—41no—
279Q8—41no—
280Q8—611no—
281Q8—4882no—
282Q8—4881no—
283Q8—2442no—
284D4—41no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—612no—
289D4—611no—
290D4—4882no—
291D4—4882no—
292D4—4882no—
293D4—4881no—
294D4—2442no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—612no—
298C2×C4—4882no—
299C2×C4—4882no—
300C2×C4—4882no—
301C2×C4—4881no—
302C2×C4D8×D6111yescharacteristic
303C8—612no—
304C8D8×D6111yescharacteristic
305C8C2×D4×D6111yescharacteristic
306C22—81no—
307C22—81no—
308C22—612no—
309C22—19521no—
310C22—19522no—
311C22—4882no—
312C22D16×D6111yescharacteristic
313C4—81no—
314C4—81no—
315C4—612no—
316C4—4882no—
317C4D16×D6111yescharacteristic
318C4C2×D8×D6111yescharacteristic
319C2—21no—
320C2—161no—
321C2—161no—
322C2—1222no—
323C2—9762no—
324C2C2×D16×D6111yescharacteristic
subgroupG/H
centerC2C2×D16×D61
commutator, cosocleC16×C61C24
FrattiniC16C23×D61
FittingC61×#2262C2
socleC2×C61C22×D16
hypercenter#2262D61
hypocenterC61#33983
LCSG ▷ C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #21458969C2semidirect
2. #21458957C2semidirect
3. #21448138C2semidirect
4. #21448136C2semidirect
5. #21448131C2semidirect
6. #21448130C2semidirect
7. #21448127C2semidirect
8. C2×#16839601C2semidirect
9. SD64×D61C2semidirect
10. #21448112C2semidirect
11. C61×#2262C2semidirect
19. #16840191C22semidirect
20. C2×Dic488C22semidirect
21. Q32×D61C22semidirect
25. #16839602C22semidirect
26. #16839601C22semidirect
27. #16839601C22semidirect
28. #16839600C22semidirect
29. #16839599C22semidirect
30. #16839597C22semidirect
31. #16839596C22semidirect
32. #16839595C22semidirect
33. C32×D61C22semidirect
35. C61×#508C22semidirect
36. C61×Q64C22semidirect
37. C61×SD64C22semidirect
38. C61×SD64C22semidirect
39. C61×D32C22semidirect
40. C2×C32×C61C22semidirect
164. #2262D61semidirect
183. #508C2×D61semidirect
190. Q64C2×D61semidirect
195. SD64C2×D61semidirect
196. SD64C2×D61semidirect
198. D32C2×D61semidirect
202. C2×C32C2×D61semidirect
203. C61#33983semidirect

Maximal subgroups

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

subgroupindex
#214589692
#214589572
#214481382
#214481362
#214481312
#214481302
#214481272
C2×#168396012
SD64×D612
#214481122
C61×#22622
#3398361