Finite Group Databasebeta

← #29838165

Subgroups — #29838165

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#21458968C211yesmaximal, characteristic, complemented
2#21458956C211yesmaximal, characteristic, complemented
3#21448137C211yesmaximal, characteristic, complemented
4#21448135C212yesmaximal, complemented
5#21448133C212yesmaximal, complemented
6#21448132C211yesmaximal, characteristic, complemented
7#21448128C211yesmaximal, characteristic, complemented
8#21448127C212yesmaximal, complemented
9SD64×D61C212yesmaximal, complemented
10#21448112C211yesmaximal, characteristic, complemented
11C2×C61×SD64C211yesmaximal, characteristic, complemented
12#16840575—21no—
13#16840563—21no—
14#16840204—22no—
15#16840203—21no—
16#16840198—21no—
17#16840196—22no—
18#16840195C2212yescomplemented
19#16840194C2211yescharacteristic
20#16840194C2211yescharacteristic, complemented
21#16840191C2212yes—
22Q32×D61C2212yescomplemented
23D16×D61C2212yes—
24#16840178C2211yescharacteristic
25Dic976C2211yescharacteristic, complemented
26#16839602C2212yescomplemented
27#16839601C2211yescharacteristic, complemented
28#16839601C2211yescharacteristic, complemented
29#16839600C2212yescomplemented
30#16839599C2212yescomplemented
31D1952C2211yescharacteristic, complemented
32#16839597C2212yescomplemented
33#16839596C2211yescharacteristic, complemented
34#16839595C2212yescomplemented
35C32×D61C2212yescomplemented
36C2×Q32×C61C2211yescharacteristic, complemented
37C2×D16×C61C2211yescharacteristic
38C61×SD64C2212yescomplemented
39C61×SD64C2212yescomplemented
40C2×C32×C61C2211yescharacteristic, complemented
41#14519825—41no—
42#14519822—41no—
43#14519809—42no—
44#14519808—41no—
45#14519803—41no—
46#14519801—42no—
47#14519800—22no—
48#14519799—21no—
49#14519799—21no—
50#14519796—22no—
51Q16×D61—22no—
52D8×D61—22no—
53#14519783D411yescharacteristic
54Dic488C2311yescharacteristic
55Dic488C2311yescharacteristic
56#14519723—22no—
57#14519722—21no—
58#14519722—21no—
59#14519721—22no—
60#14519720—22no—
61D976C2311yescharacteristic
62D976C2311yescharacteristic
63#14519718—22no—
64#14519717D411yescharacteristic
65#14519716D412yes—
66C16×D61C2312yes—
67#14519683C2312yes—
68C2×Q16×C61—21no—
69C2×D8×C61—21no—
70Q32×C61—21no—
71Q32×C61C2312yes—
72D16×C61—21no—
73D16×C61C2312yes—
74C2×C16×C61C2311yescharacteristic
75C32×C61C2311yescharacteristic
76C32×C61C2311yescharacteristic
77C2×Q8×C61—41no—
78C2×D4×C61—41no—
79C2×C22⋊D61—82no—
80#11758226—42no—
81#11758226—81no—
82Q8×D61—42no—
83Q8×D61—81no—
84#11758224—42no—
85#11758224—81no—
86D4×D61—42no—
87D4×D61—81no—
88#11758222—41no—
89#11758222—41no—
90#11758222—82no—
91#11758222D811yescharacteristic
92Q16×C61—22no—
93Q16×C61—41no—
94D8×C61—22no—
95D8×C61—41no—
96C2×C8×C61C2×D411yescharacteristic
97#11758202—42no—
98#11758201—42no—
99#11758200—42no—
100#11758199—42no—
101#11758194D811yescharacteristic
102Dic244—21no—
103Dic244—21no—
104D488—21no—
105D488—21no—
106#11758190—41no—
107#11758190—41no—
108#11758189D812yes—
109C8×D61C2×D412yes—
110C16×C61C2×D411yescharacteristic
111C16×C61C2411yescharacteristic
112#11758185C2×D412yes—
113C23×C61—81no—
114C22×D61—82no—
115Q8×C61—42no—
116Q8×C61—81no—
117D4×C61—42no—
118D4×C61—81no—
119C2×C4×C61—81no—
120C2×C4×C61C2×D811yescharacteristic
121C22⋊D61—81no—
122C22⋊D61—81no—
123C22⋊D61—82no—
124C22⋊D61—82no—
125C22⋊D61D1612yes—
126C2×Dic61—82no—
127D244—41no—
128D244—41no—
129D244—82no—
130D244D1611yescharacteristic
131C4×D61—82no—
132C4×D61—81no—
133C4×D61—81no—
134C4×D61—82no—
135C4×D61C2×D812yes—
136Dic122—41no—
137Dic122—41no—
138Dic122—82no—
139Dic122D1611yescharacteristic
140C8×C61C2×D811yescharacteristic
141C8×C61C22×D411yescharacteristic
142#92418C2×D812yes—
143#33983—611nomaximal, Sylow
144C22×C61—82no—
145C22×C61—321no—
146C22×C61C2×D1611yescharacteristic
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—322no—
150C2×D61C2×D1612yes—
151C4×C61—82no—
152C4×C61C2×D1611yescharacteristic
153C4×C61C22×D811yescharacteristic
154Dic61—81no—
155Dic61—81no—
156Dic61C2×D1612yes—
157#3417—611no—
158#3415—611no—
159#2264—611no—
160#2264—612no—
161#2263—611no—
162#2263—612no—
163#2262—612no—
164#2262—611no—
165C2×SD64—612no—
166C2×SD64D6111yescharacteristic, complemented
167#2258—611no—
168C2×C61—21no—
169C2×C61—162no—
170C2×C61C22×D1611yescharacteristic
171D61—22no—
172D61—161no—
173D61—161no—
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#508—611no—
183#508—611no—
184C2×Q32—612no—
185C2×Q32C2×D6111yescharacteristic, complemented
186C2×D16—612no—
187C2×D16C2×D6111yescharacteristic
188#504—611no—
189Q64—612no—
190Q64—611no—
191SD64—612no—
192SD64—612no—
193SD64—611no—
194SD64—611no—
195SD64C2×D6112yescomplemented
196SD64C2×D6112yescomplemented
197D32—612no—
198D32—611no—
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—1222no—
211#135—1222no—
212#135—1221no—
213#135—1221no—
214C2×Q16—21no—
215C2×Q16—1222no—
216C2×D8—21no—
217C2×D8—1222no—
218#131—611no—
219Q32—21no—
220Q32—611no—
221Q32—611no—
222Q32—1222no—
223Q32C22×D6112yes—
224SD32—1222no—
225SD32—1222no—
226SD32—1221no—
227SD32—1221no—
228D16—21no—
229D16—611no—
230D16—611no—
231D16—1222no—
232D16C22×D6112yes—
233M32—611no—
234M32—612no—
235C2×C16—612no—
236C2×C16C22×D6111yescharacteristic
237C32—612no—
238C32C22×D6111yescharacteristic
239C32C22×D6111yescharacteristic
240#41—611no—
241#41—4882no—
242#41—4881no—
243#41—4881no—
244#41—2442no—
245#41—2442no—
246#41—2441no—
247#41—2441no—
248C2×Q8—41no—
249C2×Q8—4881no—
250C2×Q8—2442no—
251C2×D4—41no—
252C2×D4—4881no—
253C2×D4—4882no—
254C2×D4—2442no—
255Q16—22no—
256Q16—41no—
257Q16—1221no—
258Q16—1221no—
259Q16—2442no—
260SD16—2442no—
261SD16—2442no—
262SD16—2441no—
263SD16—2441no—
264D8—22no—
265D8—41no—
266D8—1221no—
267D8—1221no—
268D8—2442no—
269M16—611no—
270M16—612no—
271C2×C8—612no—
272C2×C8D4×D6111yescharacteristic
273C16—612no—
274C16D4×D6111yescharacteristic
275C16C23×D6111yescharacteristic
276C23—81no—
277C23—4882no—
278Q8—42no—
279Q8—81no—
280Q8—611no—
281Q8—4882no—
282Q8—2441no—
283Q8—2441no—
284D4—42no—
285D4—81no—
286D4—612no—
287D4—611no—
288D4—4882no—
289D4—4882no—
290D4—4882no—
291D4—4881no—
292D4—4881no—
293D4—2441no—
294D4—2441no—
295C2×C4—81no—
296C2×C4—612no—
297C2×C4—4882no—
298C2×C4—4882no—
299C2×C4—4882no—
300C2×C4—4881no—
301C2×C4—4881no—
302C2×C4D8×D6111yescharacteristic
303C8—612no—
304C8D8×D6111yescharacteristic
305C8C2×D4×D6111yescharacteristic
306C22—82no—
307C22—321no—
308C22—612no—
309C22—19522no—
310C22—4881no—
311C22—4881no—
312C22D16×D6111yescharacteristic
313C4—82no—
314C4—612no—
315C4—4881no—
316C4—4881no—
317C4D16×D6111yescharacteristic
318C4C2×D8×D6111yescharacteristic
319C2—21no—
320C2—162no—
321C2—1222no—
322C2—9761no—
323C2—9761no—
324C2C2×D16×D6111yescharacteristic
subgroupG/H
centerC2C2×D16×D61
commutator, cosocleC16×C61C24
FrattiniC16C23×D61
FittingC2×C61×SD64C2
socleC2×C61C22×D16
hypercenterC2×SD64D61
hypocenterC61#33983
LCSG ▷ C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
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. #21458968C2semidirect
2. #21458956C2semidirect
3. #21448137C2semidirect
4. #21448135C2semidirect
5. #21448133C2semidirect
6. #21448132C2semidirect
7. #21448128C2semidirect
8. #21448127C2semidirect
9. SD64×D61C2semidirect
10. #21448112C2semidirect
11. C2×C61×SD64C2semidirect
18. #16840195C22semidirect
20. #16840194C22semidirect
22. Q32×D61C22semidirect
25. Dic976C22semidirect
26. #16839602C22semidirect
27. #16839601C22semidirect
28. #16839601C22semidirect
29. #16839600C22semidirect
30. #16839599C22semidirect
31. D1952C22semidirect
32. #16839597C22semidirect
33. #16839596C22semidirect
34. #16839595C22semidirect
35. C32×D61C22semidirect
36. C2×Q32×C61C22semidirect
38. C61×SD64C22semidirect
39. C61×SD64C22semidirect
40. C2×C32×C61C22semidirect
166. C2×SD64D61semidirect
185. C2×Q32C2×D61semidirect
195. SD64C2×D61semidirect
196. SD64C2×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
#214589682
#214589562
#214481372
#214481352
#214481332
#214481322
#214481282
#214481272
SD64×D612
#214481122
C2×C61×SD642
#3398361