Finite Group Databasebeta

← #29654126

Subgroups — #29654126

Order 15616.

1402 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1402 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#21456625C212yesmaximal, complemented
2#21450977C211yesmaximal, characteristic, complemented
3#21444805C214yesmaximal, complemented
4#21444771C214yesmaximal, complemented
5#21444335C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C2×C61×#534C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic, complemented
9#16840097—22no—
10#16840073C2214yescomplemented
11#16840006—24no—
12#16839913—22no—
13#16839900C2214yescomplemented
14#16839726C2212yescomplemented
15#16839687C2212yescomplemented
16#16839674C2212yescomplemented
17#16839656—22no—
18#16839625—21no—
19#16839618—21no—
20C4×C22⋊Dic61C2211yescharacteristic, complemented
21#16839496C2218yescomplemented
22#16839486C2214yescomplemented
23C23×D4×C61—21no—
24C61×#534—24no—
25C61×#534—22no—
26C61×#534C2214yes—
27C2×#123×C61—22no—
28C2×#121×C61—21no—
29C2×#121×C61C2212yes—
30C2×#120×C61—22no—
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×Dic122C2311yescharacteristic, complemented
33C2×#11758200—24no—
34C2×#11758194—22no—
35#14519756—22no—
36#14519756—22no—
37C2×C4⋊Dic61C2311yescharacteristic, complemented
38C2×#11758196—21no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—22no—
43C2×C4×Dic61—21no—
44#14519710—24no—
45#14519700—24no—
46#14519698—24no—
47C22⋊#92418—22no—
48C22⋊#92418C2314yescomplemented
49#14519684—22no—
50#14519684—22no—
51#14519684C2312yescomplemented
52C25×C61—21no—
53C25×C61D411yescharacteristic
54C22×D4×C61—24no—
55C22×D4×C61—22no—
56C22×D4×C61—24no—
57C22×D4×C61—22no—
58C22×D4×C61—21no—
59C22×D4×C61—48no—
60C22×D4×C61D412yes—
61C23×C4×C61—21no—
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64#123×C61—28no—
65#123×C61—28no—
66#121×C61—24no—
67#121×C61—22no—
68#121×C61—22no—
69#121×C61—24no—
70#121×C61D414yes—
71#121×C61C2314yes—
72#120×C61—28no—
73#120×C61—28no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61D412yes—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61D412yes—
81C2×C22⋊C4×C61C2312yes—
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—24no—
85C24×C61—24no—
86C24×C61—22no—
87C24×C61—44no—
88C24×C61—42no—
89C24×C61—81no—
90C24×C61—82no—
91C24×C61—82no—
92C24×C61C2×D411yescharacteristic
93C24×C61C2×D412yes—
94C2×D4×C61—28no—
95C2×D4×C61—24no—
96C2×D4×C61—24no—
97C2×D4×C61—24no—
98C2×D4×C61—22no—
99C2×D4×C61—48no—
100C2×D4×C61—416no—
101C2×D4×C61—48no—
102C2×D4×C61—48no—
103C2×D4×C61—48no—
104C2×D4×C61—82no—
105C2×D4×C61—82no—
106C2×D4×C61C2×D414yes—
107C22×C4×C61—24no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—44no—
112C22×C4×C61—42no—
113C22×C4×C61—81no—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—22no—
121C22×Dic61—21no—
122C2×Dic122—22no—
123C2×Dic122—81no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61C2×D414yes—
127C22⋊C4×C61—28no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—24no—
130C22⋊C4×C61—24no—
131C22⋊C4×C61—24no—
132C22⋊Dic61—42no—
133C22⋊Dic61—41no—
134C22⋊Dic61—41no—
135C22⋊Dic61—44no—
136C22⋊Dic61—82no—
137#11758200—84no—
138C4⋊Dic61—22no—
139#11758196—42no—
140C4×Dic61—81no—
141#11758194—82no—
142C2×#92418—24no—
143C23×C61—24no—
144C23×C61—24no—
145C23×C61—22no—
146C23×C61—22no—
147C23×C61—22no—
148C23×C61—22no—
149C23×C61—24no—
150C23×C61—41no—
151C23×C61—42no—
152C23×C61—42no—
153C23×C61—42no—
154C23×C61—44no—
155C23×C61—48no—
156C23×C61—44no—
157C23×C61—42no—
158C23×C61—44no—
159C23×C61—81no—
160C23×C61—81no—
161C23×C61—88no—
162C23×C61—84no—
163C23×C61—84no—
164C23×C61—816no—
165C23×C61#12011yescharacteristic
166C23×C61#12011yescharacteristic
167C23×C61AGL(1,ℤ/8)14yes—
168C23×C61C22×D411yescharacteristic
169D4×C61—44no—
170D4×C61—416no—
171D4×C61—48no—
172D4×C61—48no—
173D4×C61—82no—
174D4×C61—84no—
175C2×C4×C61—22no—
176C2×C4×C61—24no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—44no—
182C2×C4×C61—48no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—81no—
186C2×C4×C61#12012yes—
187C2×C4×C61C22×D412yes—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195Dic122—44no—
196#92418—44no—
197#18317—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—24no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—44no—
207C22×C61—44no—
208C22×C61—48no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—44no—
212C22×C61—44no—
213C22×C61—44no—
214C22×C61—82no—
215C22×C61—88no—
216C22×C61—84no—
217C22×C61—816no—
218C22×C61#45512yes—
219C22×C61C2×#12011yescharacteristic
220C22×C61C2×AGL(1,ℤ/8)12yes—
221C4×C61—22no—
222C4×C61—22no—
223C4×C61—44no—
224C4×C61—42no—
225Dic61—82no—
226Dic61—81no—
227Dic61—81no—
228C2×#534D6111yescharacteristic
229#3193—612no—
230#2561—611no—
231#1607—614no—
232#1601—614no—
233#1507—612no—
234#1462—611no—
235C2×C61—22no—
236C2×C61—22no—
237C2×C61—41no—
238C2×C61—44no—
239C2×C61—84no—
240C2×C61#300311yescharacteristic
241C2×C61C2×#45511yescharacteristic
242C2×C61#302611yescharacteristic
243C23×D4—21no—
244#534—24no—
245#534—22no—
246#534C2×D6114yes—
247C2×#123—22no—
248C2×C22⋊Q8—611no—
249C2×#121—21no—
250C2×#121C2×D6112yes—
251C2×#120—22no—
252C22×C22⋊C4C2×D6111yescharacteristic
253#484—1222no—
254#480—614no—
255#468—1224no—
256#450—1222no—
257#448—614no—
258#407—612no—
259#397—612no—
260#393—612no—
261#388—1222no—
262#381—1221no—
263#379—1221no—
264C4×C22⋊C4—611no—
265#327—618no—
266#323—614no—
267C61#1831711yescharacteristic, complemented, Sylow
268C25—21no—
269C25C22⋊D6111yescharacteristic
270C22×Q8—611no—
271C22×D4—22no—
272C22×D4—24no—
273C22×D4—22no—
274C22×D4—24no—
275C22×D4—21no—
276C22×D4—48no—
277C22×D4C22⋊D6112yes—
278C23×C4—21no—
279C23×C4C22⋊D6111yescharacteristic
280C23×C4C22×D6111yescharacteristic
281C2×SD16—1224no—
282C2×M16—1222no—
283#123—28no—
284#123—28no—
285C22⋊Q8—1222no—
286C22⋊Q8—1222no—
287#121—24no—
288#121—24no—
289#121—22no—
290#121—22no—
291#121C22⋊D6114yes—
292#121C22×D6114yes—
293#120—28no—
294#120—28no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4C22⋊D6112yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—1222no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1222no—
307C2×C22⋊C4C22⋊D6112yes—
308C2×C22⋊C4C22×D6112yes—
309C2×C42—1221no—
310#107—1224no—
311#103—1224no—
312#102—1224no—
313C22⋊C8—614no—
314C22⋊C8—1222no—
315#95—612no—
316#95—1222no—
317#95—1222no—
318C24—21no—
319C24—21no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—44no—
324C24—42no—
325C24—81no—
326C24—82no—
327C24—82no—
328C24C2×C22⋊D6111yescharacteristic
329C24C2×C22⋊D6112yes—
330C2×Q8—1222no—
331C2×Q8—4881no—
332C2×D4—28no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—48no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4C2×C22⋊D6114yes—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—81no—
352C22×C4—1222no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4C2×C22⋊D6112yes—
356C22×C4C2×C22⋊D6112yes—
357C22×C4C2×C22⋊D6112yes—
358C22×C4C2×C22⋊D6111yescharacteristic
359C22×C4C23×D6111yescharacteristic
360SD16—4884no—
361M16—4882no—
362C2×C8—1224no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—1222no—
366C4⋊C4—2442no—
367C4⋊C4C2×C22⋊D6114yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—4882no—
374C22⋊C4—2442no—
375C22⋊C4—2441no—
376C22⋊C4—2441no—
377C22⋊C4—2444no—
378C42—4881no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—48no—
392C23—44no—
393C23—42no—
394C23—44no—
395C23—81no—
396C23—81no—
397C23—88no—
398C23—84no—
399C23—84no—
400C23—816no—
401C23#1451974611yescharacteristic
402C23#1451974611yescharacteristic
403C23#1451980314yes—
404C23C22×C22⋊D6111yescharacteristic
405Q8—2444no—
406D4—44no—
407D4—416no—
408D4—48no—
409D4—48no—
410D4—82no—
411D4—84no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4#1451974612yes—
431C2×C4C22×C22⋊D6112yes—
432C8—2444no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—24no—
440C22—21no—
441C22—44no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—82no—
450C22—88no—
451C22—84no—
452C22—816no—
453C22#1683994012yes—
454C22C2×#1451974611yescharacteristic
455C22C2×#1451980312yes—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—44no—
460C4—4882no—
461C4—4881no—
462C4—4881no—
463C2—22no—
464C2—22no—
465C2—41no—
466C2—44no—
467C2—84no—
468C2#2145504611yescharacteristic
469C2C2×#1683994011yescharacteristic
470C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#534C2
socleC22×C61C2×#120
hypercenterC2×#534D61
hypocenterC61#18317
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #21456625C2semidirect
2. #21450977C2semidirect
3. #21444805C2semidirect
4. #21444771C2semidirect
5. #21444335C2semidirect
6. #21444146C2semidirect
8. C2×#14519756C22semidirect
10. #16840073C22semidirect
13. #16839900C22semidirect
14. #16839726C22semidirect
15. #16839687C22semidirect
16. #16839674C22semidirect
20. C4×C22⋊Dic61C22semidirect
21. #16839496C22semidirect
22. #16839486C22semidirect
32. C22×Dic122C23semidirect
37. C2×C4⋊Dic61C23semidirect
48. C22⋊#92418C23semidirect
51. #14519684C23semidirect
267. C61#18317semidirect

Maximal subgroups

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

subgroupindex
#214566252
#214509772
#214448052
#214447712
#214443352
#214441462
C2×C61×#5342
#1831761