Finite Group Databasebeta

← #29652791

Subgroups — #29652791

Order 15616.

1250 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 167 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#21459583C211yesmaximal, characteristic
2#21455941C212yesmaximal, complemented
3#21444804C214yesmaximal, complemented
4#21444770C214yesmaximal, complemented
5#21444326C212yesmaximal, complemented
6#21444160C211yesmaximal, characteristic, complemented
7C61×#2398C211yesmaximal, characteristic, complemented
8C2×#14519822—21no—
9#16840416—22no—
10#16840407—24no—
11D4×D244—22no—
12#16840343—21no—
13#16840324C2214yes—
14#16840312C2212yes—
15C2×#14519751—22no—
16#16840075C2214yescomplemented
17#16840057—22no—
18#16839962—22no—
19#16839901C2214yescomplemented
20#16839860—24no—
21#16839819C2212yescomplemented
22#16839495C2218yescomplemented
23#16839486C2214yescomplemented
24C2×C4×Q8×C61C2211yescharacteristic, complemented
25C2×C4×D4×C61C2211yescharacteristic
26C61×#400—21no—
27C61×#397C2212yescomplemented
28C61×#394—21no—
29C61×#393C2212yescomplemented
30C4×C22⋊C4×C61—22no—
31C61×#376C2211yescharacteristic, complemented
32#14519822—48no—
33C2×#11758222—22no—
34C2×#11758224—24no—
35C22×D244—22no—
36C22×C22⋊D61—24no—
37C2×D4×D61—24no—
38C2×#11758201—24no—
39#14519777D412yes—
40#14519774D412yes—
41#14519765—22no—
42#14519761—28no—
43#14519757—24no—
44#14519752—24no—
45#14519751—24no—
46#14519751—24no—
47#14519751—22no—
48#14519751—24no—
49#14519751C2314yes—
50#14519747—28no—
51C4×D244—22no—
52C4×D244D414yes—
53C2×C4⋊Dic61—22no—
54C2×#11758198—24no—
55#14519710—24no—
56C4⋊#92418—22no—
57#14519701—24no—
58#14519697—24no—
59C22⋊#92418C2314yes—
60#14519688—22no—
61C22×Q8×C61C2311yescharacteristic
62C22×D4×C61D411yescharacteristic
63C23×C4×C61D412yes—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C4×D4×C61—22no—
67C4×D4×C61—22no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C4⋊C4×C61C2311yescharacteristic
71C2×C22⋊C4×C61—22no—
72C2×C22⋊C4×C61—22no—
73C2×C22⋊C4×C61C2312yes—
74C2×C42×C61—21no—
75C2×C42×C61—21no—
76C2×C42×C61C2311yescharacteristic
77#95×C61—21no—
78#95×C61—22no—
79#95×C61—22no—
80#95×C61C2312yes—
81C24×C61C2×D412yes—
82C23×D61—24no—
83C2×Q8×C61—22no—
84C2×Q8×C61—42no—
85C2×D4×C61—24no—
86C2×D4×C61—22no—
87C22×C4×C61—21no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—44no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2411yescharacteristic
97C2×C22⋊D61—28no—
98C2×C22⋊D61—24no—
99C2×C22⋊D61—48no—
100C2×C22⋊D61—48no—
101C22×Dic61—24no—
102#11758224—416no—
103D4×D61—416no—
104#11758222—48no—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—24no—
108C2×D244—84no—
109C2×D244C2×D414yes—
110C2×C4×D61—24no—
111C2×Dic122—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—42no—
115C22⋊C4×C61—24no—
116C22⋊C4×C61—44no—
117C22⋊C4×C61—44no—
118C42×C61—42no—
119C42×C61—41no—
120C42×C61—41no—
121C42×C61C2×D412yes—
122C42×C61C2×D412yes—
123#11758201—48no—
124#11758198—28no—
125#11758198—24no—
126#11758198—24no—
127C4⋊Dic61—24no—
128C4⋊Dic61—22no—
129C4⋊Dic61C2×D414yes—
130C2×#92418—24no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61AGL(1,ℤ/8)14yes—
136C23×C61C22×D411yescharacteristic
137C22×D61—24no—
138C22×D61—48no—
139C22×D61—84no—
140C22×D61—84no—
141Q8×C61—24no—
142Q8×C61—42no—
143D4×C61—28no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—44no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—82no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12012yes—
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61C22×D412yes—
163C22⋊D61—416no—
164C22⋊D61—48no—
165C2×Dic61—24no—
166C2×Dic61—48no—
167D244—44no—
168D244—44no—
169D244—44no—
170D244—84no—
171C4×D61—48no—
172Dic122—44no—
173#92418—44no—
174#18179—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—24no—
178C22×C61—44no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—82no—
182C22×C61C2×#12011yescharacteristic
183C22×C61C2×AGL(1,ℤ/8)12yes—
184C2×D61—44no—
185C2×D61—44no—
186C2×D61—44no—
187C2×D61—168no—
188C4×C61—22no—
189C4×C61—21no—
190C4×C61—22no—
191C4×C61—42no—
192C4×C61—42no—
193C4×C61—82no—
194C4×C61—81no—
195C4×C61#45512yes—
196Dic61—44no—
197#3496—611no—
198#3112—612no—
199#2398D6111yescharacteristic, complemented
200#1607—614no—
201#1601—614no—
202#1505—612no—
203#1466—611no—
204C2×C61—22no—
205C2×C61—44no—
206C2×C61#300311yescharacteristic
207C2×C61C2×#45511yescharacteristic
208C2×C61#302511yescharacteristic
209D61—84no—
210C2×H(5,𝔽2)—1221no—
211#547—1222no—
212#546—1224no—
213D42—1222no—
214#537—1221no—
215#534—614no—
216#532—612no—
217C2×#121—1222no—
218C2×C4×Q8C2×D6111yescharacteristic, complemented
219C2×C4×D4C2×D6111yescharacteristic
220#480—614no—
221#477—1222no—
222#460—1222no—
223#448—614no—
224#440—1224no—
225#432—612no—
226#400—21no—
227#397C2×D6112yescomplemented
228#394—21no—
229#393C2×D6112yescomplemented
230C4×C22⋊C4—22no—
231#376C2×D6111yescharacteristic, complemented
232#327—618no—
233#323—614no—
234C61#1817911yescharacteristic, complemented, Sylow
235H(5,𝔽2)—2448no—
236C2×#41—1224no—
237C2×#41—1222no—
238C22×Q8C22×D6111yescharacteristic
239C22×D4—1222no—
240C22×D4—1224no—
241C22×D4—1224no—
242C22×D4C22⋊D6111yescharacteristic
243C23×C4C22⋊D6112yes—
244C2×SD16—1224no—
245C4⋊Q8—612no—
246#127—612no—
247#124—1222no—
248#123—1228no—
249C22⋊Q8—1224no—
250#121—614no—
251#121—1224no—
252#121—1224no—
253#121—1224no—
254#121—1222no—
255#121—1224no—
256#120—1228no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—614no—
262C4×D4—1222no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—1222no—
265C2×C4⋊C4C22⋊D6111yescharacteristic
266C2×C4⋊C4C22×D6111yescharacteristic
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—1224no—
270C2×C22⋊C4C22×D6112yes—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42C22×D6111yescharacteristic
274#107—1224no—
275C4⋊C8—1222no—
276#103—1224no—
277#102—1224no—
278C22⋊C8—614no—
279#97—1222no—
280#95—21no—
281#95—22no—
282#95—22no—
283#95C22×D6112yes—
284C24—1224no—
285C24C2×C22⋊D6112yes—
286#41—2448no—
287#41—24416no—
288C2×Q8—22no—
289C2×Q8—42no—
290C2×Q8—1222no—
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—614no—
294C2×D4—1224no—
295C2×D4—1228no—
296C2×D4—1224no—
297C2×D4—1222no—
298C2×D4—1222no—
299C2×D4—4884no—
300C2×D4—24416no—
301C2×D4—2448no—
302C2×D4—2448no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—44no—
309C22×C4—1224no—
310C22×C4—1224no—
311C22×C4C2×C22⋊D6111yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313C22×C4C2×C22⋊D6112yes—
314C22×C4C23×D6111yescharacteristic
315SD16—2448no—
316C2×C8—1224no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—614no—
321C4⋊C4—1224no—
322C4⋊C4—1222no—
323C22⋊C4—24no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—1228no—
327C22⋊C4—1224no—
328C22⋊C4—1224no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42D4×D6112yes—
333C42D4×D6112yes—
334C23—22no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—1224no—
339C23—4884no—
340C23—4884no—
341C23—2448no—
342C23#1451980214yes—
343C23C22×C22⋊D6111yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—2444no—
347D4—28no—
348D4—4884no—
349D4—2444no—
350D4—2444no—
351D4—24416no—
352D4—2448no—
353D4—2444no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—1224no—
370C2×C4—2448no—
371C2×C4—2448no—
372C2×C4#1451974511yescharacteristic
373C2×C4#1451974512yes—
374C2×C4#1451974511yescharacteristic
375C2×C4C2×D4×D6112yes—
376C8—2444no—
377C22—21no—
378C22—21no—
379C22—24no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22—9768no—
385C22—2444no—
386C22—2444no—
387C22—2444no—
388C22C2×#1451974511yescharacteristic
389C22C2×#1451980212yes—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—42no—
394C4—42no—
395C4—82no—
396C4—81no—
397C4—2444no—
398C4#1683993912yes—
399C2—22no—
400C2—44no—
401C2—4884no—
402C2#2145504511yescharacteristic
403C2C2×#1683993911yescharacteristic
404C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2398C2
socleC22×C61C2×#120
prime radical#21459583C2
hypercenter#2398D61
hypocenterC61#18179
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
2. #21455941C2semidirect
3. #21444804C2semidirect
4. #21444770C2semidirect
5. #21444326C2semidirect
6. #21444160C2semidirect
7. C61×#2398C2semidirect
16. #16840075C22semidirect
19. #16839901C22semidirect
21. #16839819C22semidirect
22. #16839495C22semidirect
23. #16839486C22semidirect
24. C2×C4×Q8×C61C22semidirect
27. C61×#397C22semidirect
29. C61×#393C22semidirect
31. C61×#376C22semidirect
199. #2398D61semidirect
218. C2×C4×Q8C2×D61semidirect
227. #397C2×D61semidirect
229. #393C2×D61semidirect
231. #376C2×D61semidirect
234. C61#18179semidirect

Maximal subgroups

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

subgroupindex
#214595832
#214559412
#214448042
#214447702
#214443262
#214441602
C61×#23982
#1817961