Finite Group Databasebeta

← #29838168

Subgroups — #29838168

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#21458957C211yesmaximal, characteristic, complemented
3#21448138C212yesmaximal, complemented
4#21448134C211yesmaximal, characteristic, complemented
5#21448133C212yesmaximal, complemented
6#21448129C212yesmaximal, complemented
7#21448126C211yesmaximal, characteristic, complemented
8C2×#16839599C211yesmaximal, characteristic, complemented
9SD64×D61C212yesmaximal, complemented
10#21448114C211yesmaximal, characteristic, complemented
11C61×#2264C211yesmaximal, characteristic, complemented
12#16840575—21no—
13#16840564—21no—
14#16840204—22no—
15#16840203—21no—
16#16840199—22no—
17#16840197—21no—
18#16840195C2212yes—
19#16840195C2212yescomplemented
20#16840194C2211yescharacteristic, complemented
21Q32×D61C2212yescomplemented
22C2×D976C2211yescharacteristic
23D16×D61C2212yes—
24#16840178C2211yescharacteristic
25#16839602C2211yescharacteristic, complemented
26#16839601C2212yescomplemented
27#16839600C2211yescharacteristic, complemented
28#16839599C2212yescomplemented
29#16839599C2212yescomplemented
30#16839599C2211yescharacteristic, complemented
31D1952C2212yescomplemented
32#16839597C2211yescharacteristic, complemented
33#16839595C2212yescomplemented
34C2×#14519683C2211yescharacteristic, complemented
35C32×D61C2212yescomplemented
36C61×#508C2211yescharacteristic
37C2×Q32×C61C2211yescharacteristic, complemented
38C61×Q64C2212yescomplemented
39C61×SD64C2212yescomplemented
40C61×M64C2211yescharacteristic, complemented
41#14519825—41no—
42#14519823—41no—
43#14519809—42no—
44#14519808—41no—
45#14519804—42no—
46#14519802—41no—
47#14519800—22no—
48#14519800—22no—
49#14519799—21no—
50Q16×D61—22no—
51C2×D488—21no—
52D8×D61—22no—
53#14519783D411yescharacteristic
54Dic488C2311yescharacteristic
55#14519723—22no—
56#14519722—21no—
57#14519720—22no—
58#14519720—22no—
59D976—21no—
60D976C2311yescharacteristic
61D976C2312yes—
62#14519718—22no—
63#14519717D411yescharacteristic
64#14519716D412yes—
65C16×D61C2312yes—
66#14519683C2311yescharacteristic
67#14519683C2311yescharacteristic
68#135×C61—21no—
69C2×Q16×C61—21no—
70Q32×C61—21no—
71Q32×C61C2311yescharacteristic
72Q32×C61C2312yes—
73SD32×C61—21no—
74D16×C61C2311yescharacteristic
75C2×C16×C61C2311yescharacteristic
76C32×C61C2312yes—
77#41×C61—41no—
78C2×Q8×C61—41no—
79#11758226—42no—
80#11758226—42no—
81#11758226—81no—
82Q8×D61—42no—
83Q8×D61—81no—
84D4×D61—42no—
85D4×D61—82no—
86#11758222—41no—
87#11758222—82no—
88#11758222—81no—
89#11758222D811yescharacteristic
90C2×D244—41no—
91C2×D244—81no—
92Q16×C61—21no—
93Q16×C61—22no—
94Q16×C61—41no—
95SD16×C61—41no—
96D8×C61—21no—
97C2×C8×C61C2×D411yescharacteristic
98#11758202—42no—
99#11758201—42no—
100#11758201—42no—
101#11758199—42no—
102#11758194D811yescharacteristic
103Dic244—21no—
104D488—22no—
105D488—21no—
106D488—41no—
107#11758190—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—42no—
117Q8×C61—81no—
118D4×C61—41no—
119D4×C61—81no—
120C2×C4×C61—81no—
121C2×C4×C61—81no—
122C2×C4×C61C2×D811yescharacteristic
123C22⋊D61—81no—
124C22⋊D61—82no—
125C22⋊D61D1612yes—
126D244—41no—
127D244—42no—
128D244—82no—
129D244—82no—
130D244—81no—
131D244D1611yescharacteristic
132C4×D61—82no—
133C4×D61—81no—
134C4×D61—82no—
135C4×D61—82no—
136C4×D61C2×D812yes—
137Dic122—41no—
138Dic122—82no—
139Dic122D1611yescharacteristic
140C8×C61C2×D811yescharacteristic
141C8×C61C22×D411yescharacteristic
142#92418C2×D812yes—
143#33983—611nomaximal, Sylow
144C22×C61—81no—
145C22×C61C2×D1611yescharacteristic
146C2×D61—81no—
147C2×D61—82no—
148C2×D61—322no—
149C2×D61—321no—
150C2×D61C2×D1612yes—
151C4×C61—82no—
152C4×C61—81no—
153C4×C61C2×D1611yescharacteristic
154C4×C61C22×D811yescharacteristic
155Dic61—81no—
156Dic61C2×D1612yes—
157#3417—611no—
158#3415—611no—
159#2264—612no—
160#2264D6111yescharacteristic, complemented
161#2263—612no—
162#2263—611no—
163#2262—611no—
164#2262—612no—
165C2×SD64—611no—
166C2×SD64—612no—
167#2258—611no—
168C2×C61—21no—
169C2×C61—161no—
170C2×C61C22×D1611yescharacteristic
171D61—22no—
172D61—161no—
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#508—611no—
183#508C2×D6111yescharacteristic
184C2×Q32—612no—
185C2×Q32C2×D6111yescharacteristic, complemented
186C2×D16—612no—
187C2×D16—611no—
188#504—611no—
189Q64—611no—
190Q64C2×D6112yescomplemented
191SD64—612no—
192SD64—612no—
193SD64—611no—
194SD64—611no—
195SD64—612no—
196SD64C2×D6112yescomplemented
197D32—611no—
198D32—612no—
199M64—612no—
200M64C2×D6111yescharacteristic, complemented
201C2×C32—611no—
202C2×C32—612no—
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—1222no—
212#135—1222no—
213#135—1221no—
214C2×Q16—21no—
215C2×Q16—1222no—
216C2×D8—1222no—
217C2×D8—1221no—
218#131—611no—
219Q32—21no—
220Q32—611no—
221Q32—1222no—
222Q32C22×D6112yes—
223Q32C22×D6111yescharacteristic
224SD32—21no—
225SD32—1222no—
226SD32—1222no—
227SD32—1221no—
228D16—611no—
229D16—612no—
230D16—1222no—
231D16—1221no—
232D16C22×D6111yescharacteristic
233M32—611no—
234M32—612no—
235C2×C16—612no—
236C2×C16C22×D6111yescharacteristic
237C32—611no—
238C32—611no—
239C32C22×D6112yes—
240#41—41no—
241#41—611no—
242#41—4882no—
243#41—4881no—
244#41—4881no—
245#41—2442no—
246#41—2442no—
247#41—2441no—
248C2×Q8—41no—
249C2×Q8—4881no—
250C2×Q8—2442no—
251C2×D4—4881no—
252C2×D4—4882no—
253C2×D4—2442no—
254C2×D4—2441no—
255Q16—21no—
256Q16—22no—
257Q16—41no—
258Q16—1221no—
259Q16—2442no—
260SD16—41no—
261SD16—2442no—
262SD16—2442no—
263SD16—2441no—
264D8—21no—
265D8—1222no—
266D8—1221no—
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—42no—
280Q8—81no—
281Q8—611no—
282Q8—4882no—
283Q8—2441no—
284D4—41no—
285D4—81no—
286D4—612no—
287D4—611no—
288D4—4882no—
289D4—4882no—
290D4—4882no—
291D4—4881no—
292D4—4881no—
293D4—2442no—
294D4—2441no—
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—612no—
308C22—19521no—
309C22—19522no—
310C22—4881no—
311C22—4882no—
312C22D16×D6111yescharacteristic
313C4—81no—
314C4—82no—
315C4—612no—
316C4—4881no—
317C4D16×D6111yescharacteristic
318C4C2×D8×D6111yescharacteristic
319C2—21no—
320C2—161no—
321C2—1222no—
322C2—9762no—
323C2—9761no—
324C2C2×D16×D6111yescharacteristic
subgroupG/H
centerC2C2×D16×D61
commutator, cosocleC16×C61C24
FrattiniC16C23×D61
FittingC61×#2264C2
socleC2×C61C22×D16
hypercenter#2264D61
hypocenterC61#33983
LCSG ▷ C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2264

Complements

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

NH ≅ G/Ntype
1. #21458968C2semidirect
2. #21458957C2semidirect
3. #21448138C2semidirect
4. #21448134C2semidirect
5. #21448133C2semidirect
6. #21448129C2semidirect
7. #21448126C2semidirect
8. C2×#16839599C2semidirect
9. SD64×D61C2semidirect
10. #21448114C2semidirect
11. C61×#2264C2semidirect
19. #16840195C22semidirect
20. #16840194C22semidirect
21. Q32×D61C22semidirect
25. #16839602C22semidirect
26. #16839601C22semidirect
27. #16839600C22semidirect
28. #16839599C22semidirect
29. #16839599C22semidirect
30. #16839599C22semidirect
31. D1952C22semidirect
32. #16839597C22semidirect
33. #16839595C22semidirect
34. C2×#14519683C22semidirect
35. C32×D61C22semidirect
37. C2×Q32×C61C22semidirect
38. C61×Q64C22semidirect
39. C61×SD64C22semidirect
40. C61×M64C22semidirect
160. #2264D61semidirect
185. C2×Q32C2×D61semidirect
190. Q64C2×D61semidirect
196. SD64C2×D61semidirect
200. M64C2×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
#214589572
#214481382
#214481342
#214481332
#214481292
#214481262
C2×#168395992
SD64×D612
#214481142
C61×#22642
#3398361