Finite Group Databasebeta

← #29652792

Subgroups — #29652792

Order 15616.

1250 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455937C212yesmaximal, complemented
2#21449103C211yesmaximal, characteristic, complemented
3#21444805C214yesmaximal, complemented
4#21444771C214yesmaximal, complemented
5#21444327C212yesmaximal, complemented
6#21444160C211yesmaximal, characteristic, complemented
7C61×#3496C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic, complemented
9#16840073C2214yescomplemented
10#16840054—22no—
11#16839959—22no—
12#16839900C2214yescomplemented
13#16839862—24no—
14#16839819C2212yescomplemented
15#16839700—21no—
16#16839687C2212yescomplemented
17#16839676—21no—
18#16839674C2212yescomplemented
19C4×C22⋊Dic61—22no—
20#16839608C2211yescharacteristic, complemented
21#16839496C2218yescomplemented
22#16839486C2214yescomplemented
23C2×H(5,𝔽2)×C61—21no—
24C61×#547—22no—
25C61×#546—24no—
26D42×C61—22no—
27C61×#537—21no—
28C61×#534C2214yes—
29C61×#532C2212yes—
30C2×#121×C61—22no—
31C2×C4×D4×C61C2211yescharacteristic
32C22×Dic122C2311yescharacteristic
33C2×#11758200—24no—
34C4×Dic122—22no—
35C4×Dic122—22no—
36C2×C4⋊Dic61C2311yescharacteristic
37C2×#11758196—21no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61—21no—
42#14519710—24no—
43C4⋊#92418—22no—
44#14519700—24no—
45#14519698—24no—
46C22⋊#92418C2314yes—
47#14519688—22no—
48#14519684—22no—
49#14519684—21no—
50#14519684—22no—
51#14519684C2312yes—
52H(5,𝔽2)×C61—48no—
53C2×#41×C61—24no—
54C2×#41×C61—22no—
55C22×D4×C61—24no—
56C22×D4×C61—24no—
57C22×D4×C61—22no—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61D412yes—
60C4⋊Q8×C61D412yes—
61#127×C61D412yes—
62#124×C61—22no—
63#123×C61—28no—
64C22⋊Q8×C61—24no—
65#121×C61—24no—
66#121×C61—24no—
67#121×C61—24no—
68#121×C61—22no—
69#121×C61—24no—
70#121×C61C2314yes—
71#120×C61—28no—
72C4×D4×C61—22no—
73C4×D4×C61—22no—
74C4×D4×C61—22no—
75C4×D4×C61D414yes—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C22⋊C4×C61—24no—
79C2×C22⋊C4×C61C2312yes—
80C2×C42×C61C2311yescharacteristic
81C24×C61—24no—
82C24×C61C2×D412yes—
83#41×C61—48no—
84#41×C61—416no—
85C2×Q8×C61—22no—
86C2×D4×C61—24no—
87C2×D4×C61—28no—
88C2×D4×C61—24no—
89C2×D4×C61—24no—
90C2×D4×C61—22no—
91C2×D4×C61—22no—
92C2×D4×C61—22no—
93C2×D4×C61—416no—
94C2×D4×C61—48no—
95C2×D4×C61—48no—
96C2×D4×C61—84no—
97C2×D4×C61C2×D414yes—
98C22×C4×C61—24no—
99C22×C4×C61—22no—
100C22×C4×C61—24no—
101C22×C4×C61—22no—
102C22×C4×C61—44no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—22no—
109C22×Dic61—21no—
110C2×Dic122—22no—
111C2×Dic122—42no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61C2×D414yes—
116C22⋊C4×C61—28no—
117C22⋊C4×C61—24no—
118C22⋊C4×C61—24no—
119C22⋊C4×C61—24no—
120C42×C61C2×D412yes—
121C42×C61C2×D412yes—
122C22⋊Dic61—44no—
123C22⋊Dic61—44no—
124#11758200—48no—
125C4⋊Dic61—22no—
126#11758196—42no—
127C4×Dic61—42no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130C2×#92418—24no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—24no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61—48no—
137C23×C61—84no—
138C23×C61—84no—
139C23×C61AGL(1,ℤ/8)14yes—
140C23×C61C22×D411yescharacteristic
141Q8×C61—44no—
142D4×C61—28no—
143D4×C61—44no—
144D4×C61—44no—
145D4×C61—416no—
146D4×C61—48no—
147D4×C61—44no—
148D4×C61—84no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—24no—
153C2×C4×C61—41no—
154C2×C4×C61—48no—
155C2×C4×C61—48no—
156C2×C4×C61—44no—
157C2×C4×C61—42no—
158C2×C4×C61—82no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12012yes—
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61C22×D412yes—
163C2×Dic61—22no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—41no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171Dic122—24no—
172Dic122—42no—
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—44no—
182C22×C61—44no—
183C22×C61—44no—
184C22×C61—82no—
185C22×C61—168no—
186C22×C61C2×#12011yescharacteristic
187C22×C61C2×AGL(1,ℤ/8)12yes—
188C4×C61—22no—
189C4×C61—21no—
190C4×C61—22no—
191C4×C61—44no—
192C4×C61—42no—
193C4×C61#45512yes—
194Dic61—42no—
195Dic61—82no—
196Dic61—81no—
197#3496D6111yescharacteristic
198#3112—612no—
199#2398—611no—
200#1607—614no—
201#1601—614no—
202#1505—612no—
203#1466—611no—
204C2×C61—22no—
205C2×C61—44no—
206C2×C61—84no—
207C2×C61#300311yescharacteristic
208C2×C61C2×#45511yescharacteristic
209C2×C61#302511yescharacteristic
210C2×H(5,𝔽2)—21no—
211#547—22no—
212#546—24no—
213D42—22no—
214#537—21no—
215#534C2×D6114yes—
216#532C2×D6112yes—
217C2×#121—22no—
218C2×C4×Q8—611no—
219C2×C4×D4C2×D6111yescharacteristic
220#480—614no—
221#477—1222no—
222#460—1222no—
223#448—614no—
224#440—1224no—
225#432—612no—
226#400—1221no—
227#397—612no—
228#394—1221no—
229#393—612no—
230C4×C22⋊C4—1222no—
231#376—611no—
232#327—618no—
233#323—614no—
234C61#1817911yescharacteristic, complemented, Sylow
235H(5,𝔽2)—48no—
236C2×#41—24no—
237C2×#41—22no—
238C22×Q8—611no—
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—22no—
242C22×D4C22⋊D6111yescharacteristic
243C23×C4C22⋊D6112yes—
244C2×SD16—1224no—
245C4⋊Q8C22⋊D6112yes—
246#127C22⋊D6112yes—
247#124—22no—
248#123—28no—
249C22⋊Q8—24no—
250#121—24no—
251#121—24no—
252#121—24no—
253#121—22no—
254#121—24no—
255#121C22×D6114yes—
256#120—28no—
257C4×Q8—1222no—
258C4×Q8—1222no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4C22⋊D6114yes—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—611no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4C22⋊D6111yescharacteristic
267C2×C22⋊C4—24no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1222no—
270C2×C22⋊C4C22×D6112yes—
271C2×C42—1221no—
272C2×C42—1221no—
273C2×C42C22×D6111yescharacteristic
274#107—1224no—
275C4⋊C8—1222no—
276#103—1224no—
277#102—1224no—
278C22⋊C8—614no—
279#97—1222no—
280#95—612no—
281#95—1221no—
282#95—1222no—
283#95—1222no—
284C24—24no—
285C24C2×C22⋊D6112yes—
286#41—48no—
287#41—416no—
288C2×Q8—22no—
289C2×Q8—1222no—
290C2×Q8—2442no—
291C2×D4—24no—
292C2×D4—28no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—416no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—84no—
302C2×D4C2×C22⋊D6114yes—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—44no—
308C22×C4—1221no—
309C22×C4—1222no—
310C22×C4—1221no—
311C22×C4C2×C22⋊D6111yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313C22×C4C2×C22⋊D6112yes—
314C22×C4C23×D6111yescharacteristic
315SD16—2448no—
316C2×C8—1224no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—1222no—
321C4⋊C4—2442no—
322C4⋊C4C2×C22⋊D6114yes—
323C22⋊C4—28no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—2444no—
328C22⋊C4—2444no—
329C42—2442no—
330C42—2441no—
331C42—2441no—
332C42C2×C22⋊D6112yes—
333C42C2×C22⋊D6112yes—
334C23—22no—
335C23—22no—
336C23—24no—
337C23—42no—
338C23—42no—
339C23—48no—
340C23—84no—
341C23—84no—
342C23#1451980314yes—
343C23C22×C22⋊D6111yescharacteristic
344Q8—44no—
345Q8—1224no—
346Q8—2442no—
347D4—28no—
348D4—44no—
349D4—44no—
350D4—416no—
351D4—48no—
352D4—44no—
353D4—84no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—41no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4—1222no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2441no—
368C2×C4—2442no—
369C2×C4—2442no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4#1451974611yescharacteristic
373C2×C4#1451974612yes—
374C2×C4#1451974611yescharacteristic
375C2×C4C22×C22⋊D6112yes—
376C8—2444no—
377C22—21no—
378C22—21no—
379C22—24no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22—44no—
384C22—44no—
385C22—44no—
386C22—82no—
387C22—168no—
388C22C2×#1451974611yescharacteristic
389C22C2×#1451980312yes—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—42no—
394C4—44no—
395C4—4882no—
396C4—4881no—
397C4—2442no—
398C4#1683994012yes—
399C2—22no—
400C2—44no—
401C2—84no—
402C2#2145504611yescharacteristic
403C2C2×#1683994011yescharacteristic
404C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18179
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21455937C2semidirect
2. #21449103C2semidirect
3. #21444805C2semidirect
4. #21444771C2semidirect
5. #21444327C2semidirect
6. #21444160C2semidirect
8. C2×C4×Dic122C22semidirect
9. #16840073C22semidirect
12. #16839900C22semidirect
14. #16839819C22semidirect
16. #16839687C22semidirect
18. #16839674C22semidirect
20. #16839608C22semidirect
21. #16839496C22semidirect
22. #16839486C22semidirect
234. C61#18179semidirect

Maximal subgroups

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

subgroupindex
#214559372
#214491032
#214448052
#214447712
#214443272
#214441602
C61×#34962
#1817961