Finite Group Databasebeta

← #29838163

Subgroups — #29838163

Order 15616.

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

normal subgroups only

This group has 472 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#21458957C211yesmaximal, characteristic, complemented
2#21458956C211yesmaximal, characteristic, complemented
3#21448134C211yesmaximal, characteristic, complemented
4#21448133C212yesmaximal, complemented
5#21448130C212yesmaximal, complemented
6#21448127C212yesmaximal, complemented
7#21448126C211yesmaximal, characteristic, complemented
8C2×#16839597C211yesmaximal, characteristic, complemented
9D32×D61C212yesmaximal, complemented
10#21448114C211yesmaximal, characteristic, complemented
11C61×#2263C211yesmaximal, characteristic, complemented
12#16840564—21no—
13#16840563—21no—
14#16840199—22no—
15#16840198—21no—
16#16840197—21no—
17#16840196—22no—
18#16840195C2212yescomplemented
19#16840194C2211yescharacteristic, complemented
20#16840191C2212yescomplemented
21C2×D976C2211yescharacteristic, complemented
22D16×D61C2212yescomplemented
23D16×D61C2212yescomplemented
24#16840178C2211yescharacteristic
25#16839602C2211yescharacteristic, complemented
26#16839601C2212yescomplemented
27#16839600C2211yescharacteristic, complemented
28#16839599C2211yescharacteristic, complemented
29D1952C2212yescomplemented
30#16839597C2211yescharacteristic, complemented
31#16839597C2212yescomplemented
32#16839597C2212yescomplemented
33#16839595C2212yescomplemented
34C2×#14519683C2211yescharacteristic, complemented
35C32×D61C2212yescomplemented
36C61×#508C2211yescharacteristic, complemented
37C2×D16×C61C2211yescharacteristic, complemented
38C61×SD64C2212yescomplemented
39C61×D32C2212yescomplemented
40C61×M64C2211yescharacteristic, complemented
41#14519823—41no—
42#14519822—41no—
43#14519804—42no—
44#14519803—41no—
45#14519802—41no—
46#14519801—42no—
47#14519800—22no—
48#14519799—21no—
49#14519796—22no—
50C2×D488—21no—
51D8×D61—22no—
52D8×D61—22no—
53#14519783D411yescharacteristic
54Dic488C2311yescharacteristic
55#14519722—21no—
56#14519721—22no—
57#14519720—22no—
58D976—21no—
59D976C2311yescharacteristic
60D976C2312yes—
61#14519718—22no—
62#14519718—22no—
63#14519717D411yescharacteristic
64#14519716D412yes—
65C16×D61C2312yes—
66#14519683C2311yescharacteristic
67#14519683C2311yescharacteristic
68#135×C61—21no—
69C2×D8×C61—21no—
70Q32×C61C2311yescharacteristic
71SD32×C61—21no—
72D16×C61—21no—
73D16×C61C2311yescharacteristic
74D16×C61C2312yes—
75C2×C16×C61C2311yescharacteristic
76C32×C61C2312yes—
77#41×C61—41no—
78C2×D4×C61—41no—
79C2×C22⋊D61—82no—
80#11758226—42no—
81#11758224—42no—
82#11758224—81no—
83D4×D61—42no—
84D4×D61—42no—
85D4×D61—82no—
86D4×D61—81no—
87#11758222—41no—
88#11758222—81no—
89#11758222D811yescharacteristic
90C2×D244—41no—
91C2×D244—81no—
92Q16×C61—21no—
93SD16×C61—41no—
94D8×C61—22no—
95D8×C61—21no—
96D8×C61—41no—
97C2×C8×C61C2×D411yescharacteristic
98#11758201—42no—
99#11758200—42no—
100#11758199—42no—
101#11758199—42no—
102#11758194D811yescharacteristic
103Dic244—21no—
104D488—21no—
105D488—22no—
106D488—41no—
107#11758190—41no—
108#11758189D812yes—
109C8×D61C2×D412yes—
110C16×C61C2×D411yescharacteristic
111C16×C61C2411yescharacteristic
112#11758185C2×D412yes—
113C23×C61—81no—
114C22×D61—81no—
115C22×D61—82no—
116C22×D61—82no—
117Q8×C61—41no—
118D4×C61—41no—
119D4×C61—42no—
120D4×C61—81no—
121D4×C61—81no—
122C2×C4×C61—81no—
123C2×C4×C61C2×D811yescharacteristic
124C22⋊D61—81no—
125C22⋊D61—82no—
126C22⋊D61—82no—
127C22⋊D61—82no—
128C22⋊D61D1612yes—
129C2×Dic61—82no—
130D244—41no—
131D244—42no—
132D244—82no—
133D244—81no—
134D244D1611yescharacteristic
135C4×D61—81no—
136C4×D61—82no—
137C4×D61C2×D812yes—
138Dic122—41no—
139Dic122D1611yescharacteristic
140C8×C61C2×D811yescharacteristic
141C8×C61C22×D411yescharacteristic
142#92418C2×D812yes—
143#33982—611nomaximal, Sylow
144C22×C61—82no—
145C22×C61—81no—
146C22×C61—321no—
147C22×C61C2×D1611yescharacteristic
148C2×D61—82no—
149C2×D61—81no—
150C2×D61—322no—
151C2×D61—322no—
152C2×D61—321no—
153C2×D61C2×D1612yes—
154C4×C61—81no—
155C4×C61C2×D1611yescharacteristic
156C4×C61C22×D811yescharacteristic
157Dic61—81no—
158Dic61C2×D1612yes—
159#3415—611no—
160#3415—611no—
161#2263—612no—
162#2263—612no—
163#2263—611no—
164#2263D6111yescharacteristic, complemented
165#2262—611no—
166#2262—612no—
167C2×D32—611no—
168C2×D32—612no—
169#2258—611no—
170C2×C61—21no—
171C2×C61—161no—
172C2×C61—162no—
173C2×C61C22×D1611yescharacteristic
174D61—22no—
175D61—162no—
176D61—161no—
177#576—1221no—
178#576—1221no—
179#509—1222no—
180#509—1222no—
181#509—1221no—
182#509—1221no—
183#508—612no—
184#508—612no—
185#508—611no—
186#508C2×D6111yescharacteristic, complemented
187C2×D16—612no—
188C2×D16—612no—
189C2×D16—611no—
190C2×D16C2×D6111yescharacteristic, complemented
191#504—611no—
192Q64—611no—
193SD64—611no—
194SD64—611no—
195SD64—612no—
196SD64C2×D6112yescomplemented
197D32—612no—
198D32—611no—
199D32—612no—
200D32—612no—
201D32C2×D6112yescomplemented
202M64—612no—
203M64C2×D6111yescharacteristic, complemented
204C2×C32—611no—
205C2×C32—612no—
206C61#3398211yescharacteristic, complemented, Sylow
207H(5,𝔽2)—2441no—
208H(5,𝔽2)—2441no—
209AGL(1,ℤ/8)—2442no—
210AGL(1,ℤ/8)—2442no—
211AGL(1,ℤ/8)—2441no—
212AGL(1,ℤ/8)—2441no—
213#135—21no—
214#135—1222no—
215#135—1222no—
216#135—1221no—
217C2×D8—21no—
218C2×D8—1222no—
219C2×D8—1222no—
220C2×D8—1221no—
221#131—611no—
222Q32—611no—
223Q32C22×D6111yescharacteristic
224SD32—21no—
225SD32—1222no—
226SD32—1222no—
227SD32—1221no—
228D16—21no—
229D16—612no—
230D16—611no—
231D16—1222no—
232D16—1222no—
233D16—1221no—
234D16C22×D6112yes—
235D16C22×D6111yescharacteristic
236M32—611no—
237M32—612no—
238C2×C16—612no—
239C2×C16C22×D6111yescharacteristic
240C32—611no—
241C32—611no—
242C32C22×D6112yes—
243#41—41no—
244#41—611no—
245#41—4881no—
246#41—4881no—
247#41—2442no—
248#41—2442no—
249#41—2441no—
250C2×D4—41no—
251C2×D4—4881no—
252C2×D4—4882no—
253C2×D4—4881no—
254C2×D4—4882no—
255C2×D4—2442no—
256C2×D4—2442no—
257C2×D4—2441no—
258Q16—21no—
259Q16—1221no—
260SD16—41no—
261SD16—2442no—
262SD16—2442no—
263SD16—2441no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—1221no—
268D8—1222no—
269D8—2442no—
270D8—2442no—
271D8—2441no—
272M16—611no—
273M16—612no—
274C2×C8—612no—
275C2×C8D4×D6111yescharacteristic
276C16—612no—
277C16D4×D6111yescharacteristic
278C16C23×D6111yescharacteristic
279C23—81no—
280C23—4882no—
281C23—4882no—
282C23—4881no—
283Q8—41no—
284Q8—611no—
285Q8—2441no—
286D4—42no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—612no—
291D4—611no—
292D4—4882no—
293D4—4882no—
294D4—4882no—
295D4—4882no—
296D4—4881no—
297D4—4881no—
298D4—2441no—
299D4—2442no—
300C2×C4—81no—
301C2×C4—612no—
302C2×C4—4882no—
303C2×C4—4882no—
304C2×C4—4881no—
305C2×C4D8×D6111yescharacteristic
306C8—612no—
307C8D8×D6111yescharacteristic
308C8C2×D4×D6111yescharacteristic
309C22—81no—
310C22—82no—
311C22—321no—
312C22—612no—
313C22—19521no—
314C22—19522no—
315C22—19522no—
316C22—4882no—
317C22—4881no—
318C22D16×D6111yescharacteristic
319C4—81no—
320C4—612no—
321C4—4881no—
322C4D16×D6111yescharacteristic
323C4C2×D8×D6111yescharacteristic
324C2—21no—
325C2—162no—
326C2—161no—
327C2—1222no—
328C2—9762no—
329C2—9761no—
330C2C2×D16×D6111yescharacteristic
subgroupG/H
centerC2C2×D16×D61
commutator, cosocleC16×C61C24
FrattiniC16C23×D61
FittingC61×#2263C2
socleC2×C61C22×D16
hypercenter#2263D61
hypocenterC61#33982
LCSG ▷ C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2263

Complements

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

NH ≅ G/Ntype
1. #21458957C2semidirect
2. #21458956C2semidirect
3. #21448134C2semidirect
4. #21448133C2semidirect
5. #21448130C2semidirect
6. #21448127C2semidirect
7. #21448126C2semidirect
8. C2×#16839597C2semidirect
9. D32×D61C2semidirect
10. #21448114C2semidirect
11. C61×#2263C2semidirect
18. #16840195C22semidirect
19. #16840194C22semidirect
20. #16840191C22semidirect
21. C2×D976C22semidirect
22. D16×D61C22semidirect
23. D16×D61C22semidirect
25. #16839602C22semidirect
26. #16839601C22semidirect
27. #16839600C22semidirect
28. #16839599C22semidirect
29. D1952C22semidirect
30. #16839597C22semidirect
31. #16839597C22semidirect
32. #16839597C22semidirect
33. #16839595C22semidirect
34. C2×#14519683C22semidirect
35. C32×D61C22semidirect
36. C61×#508C22semidirect
37. C2×D16×C61C22semidirect
38. C61×SD64C22semidirect
39. C61×D32C22semidirect
40. C61×M64C22semidirect
164. #2263D61semidirect
186. #508C2×D61semidirect
190. C2×D16C2×D61semidirect
196. SD64C2×D61semidirect
201. D32C2×D61semidirect
203. M64C2×D61semidirect
206. C61#33982semidirect

Maximal subgroups

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

subgroupindex
#214589572
#214589562
#214481342
#214481332
#214481302
#214481272
#214481262
C2×#168395972
D32×D612
#214481142
C61×#22632
#3398261