Finite Group Databasebeta

← #29661607

Subgroups — #29661607

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic, complemented
2#21456366C211yesmaximal, characteristic, complemented
3C2×#16840121C211yesmaximal, characteristic, complemented
4#21445153C214yesmaximal, complemented
5#21445136C214yesmaximal, complemented
6#21444434C211yesmaximal, characteristic, complemented
7#21444417C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3561C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172C2214yescomplemented
12#16840165—21no—
13#16840158—21no—
14#16840157C2212yescomplemented
15#16840132C2212yescomplemented
16#16840121—22no—
17#16840121C2214yes—
18#16839972—21no—
19#16839967—21no—
20#16839917—21no—
21#16839900—21no—
22#16839819C2211yescharacteristic
23#16839805—22no—
24C2×#14519701—21no—
25C2×#14519698—21no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2214yescomplemented
29#16839504C2212yescomplemented
30#16839502C2214yes—
31#16839502C2212yescomplemented
32C61×#563C2212yes—
33C61×#562C2212yes—
34C61×#556—21no—
35C61×#551—22no—
36D4×Q8×C61—21no—
37C61×#548—21no—
38C61×#546—21no—
39C61×#543—21no—
40C61×#542—21no—
41C61×#542—22no—
42C61×#538—21no—
43C61×#536—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#124×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758202—22no—
49C2×#11758200—22no—
50C2×#11758194—21no—
51C22×#92418—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58#14519710—24no—
59#14519708—24no—
60C4⋊#92418—21no—
61#14519701—24no—
62#14519700—22no—
63#14519700—22no—
64#14519698—24no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688C2312yes—
68#14519688C2312yes—
69#14519688C2312yes—
70C4×#92418—21no—
71C2×#41×C61—21no—
72C22×Q8×C61—21no—
73C22×Q8×C61D411yescharacteristic
74C22×D4×C61D411yescharacteristic
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61D411yescharacteristic
78C4⋊Q8×C61D411yescharacteristic
79#126×C61—22no—
80#126×C61—22no—
81#126×C61—22no—
82#126×C61—21no—
83#125×C61C2312yes—
84#125×C61C2311yescharacteristic
85#124×C61—21no—
86#124×C61—22no—
87#124×C61—22no—
88#124×C61—21no—
89#124×C61—21no—
90#124×C61D412yes—
91#124×C61C2311yescharacteristic
92#124×C61C2312yes—
93#123×C61—22no—
94#123×C61—22no—
95#123×C61—21no—
96#123×C61—22no—
97#123×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61D411yescharacteristic
109#121×C61—21no—
110#121×C61—22no—
111#121×C61—22no—
112#121×C61D411yescharacteristic
113#120×C61—22no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61—21no—
117C4×Q8×C61D411yescharacteristic
118C4×D4×C61—22no—
119C4×D4×C61—21no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61D411yescharacteristic
123#117×C61—22no—
124#117×C61—21no—
125#117×C61D412yes—
126C2×C22⋊C4×C61—22no—
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C24×C61—21no—
131#41×C61—82no—
132C2×Q8×C61—22no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—42no—
137C2×Q8×C61—82no—
138C2×Q8×C61C2×D411yescharacteristic
139C2×Q8×C61C2×D412yes—
140C2×D4×C61—21no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—21no—
144C2×D4×C61—42no—
145C2×D4×C61C2×D411yescharacteristic
146C2×D4×C61C2×D412yes—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×Dic61—41no—
156C2×Dic122—22no—
157C2×Dic122—41no—
158C2×Dic122—82no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61C2×D412yes—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61—21no—
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2411yescharacteristic
197#11758202—82no—
198#11758200—82no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203#11758194—24no—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—42no—
208C23×C61—22no—
209C23×C61—21no—
210C23×C61—42no—
211C23×C61—81no—
212C23×C61—81no—
213C23×C61#12011yescharacteristic
214Q8×C61—41no—
215Q8×C61—42no—
216Q8×C61—42no—
217Q8×C61—42no—
218Q8×C61—81no—
219Q8×C61—82no—
220D4×C61—41no—
221D4×C61—42no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—82no—
225D4×C61—82no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—81no—
237C2×C4×C61—82no—
238C2×C4×C61—82no—
239C2×C4×C61—82no—
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×Dic61—42no—
247C2×Dic61—82no—
248Dic122—42no—
249Dic122—42no—
250Dic122—82no—
251#92418—24no—
252#92418—42no—
253#92418—42no—
254#19032—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—41no—
258C22×C61—42no—
259C22×C61—41no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—41no—
263C22×C61—162no—
264C22×C61#45612yes—
265C22×C61C2×#12011yescharacteristic
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—41no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—42no—
277C4×C61#45512yes—
278Dic61—82no—
279#3561D6111yescharacteristic
280#3218—611no—
281#3165—611no—
282C2×#489—611no—
283#1664—614no—
284#1661—614no—
285#1531—611no—
286#1527—611no—
287#1524—611no—
288C2×C61—22no—
289C2×C61—81no—
290C2×C61—82no—
291C2×C61C2×#45511yescharacteristic
292C2×C61C2×#45611yescharacteristic
293C2×C61#302111yescharacteristic
294#563C2×D6112yes—
295#562C2×D6112yes—
296#556—21no—
297#551—22no—
298D4×Q8—21no—
299#548—21no—
300#546—21no—
301#543—21no—
302#542—21no—
303#542—22no—
304#538—21no—
305#536—21no—
306#533C2×D6111yescharacteristic
307C2×C4⋊Q8—611no—
308#529C2×D6111yescharacteristic
309C2×#124C2×D6111yescharacteristic
310#501—614no—
311#500—1221no—
312#498—1221no—
313#497—612no—
314#491—612no—
315#489—614no—
316#489—1222no—
317#462—1221no—
318#461—1221no—
319#451—1221no—
320#448—1221no—
321#432—611no—
322#428—1222no—
323C2×#103—1221no—
324C2×#102—1221no—
325C4×M16—611no—
326C2×#97—611no—
327#330—614no—
328#330—612no—
329#329—614no—
330#329—612no—
331C61#1903211yescharacteristic, complemented, Sylow
332C2×#41—21no—
333C22×Q8—21no—
334C22×Q8—1221no—
335C22×Q8C22⋊D6111yescharacteristic
336C22×D4C22⋊D6111yescharacteristic
337C2×Q16—1222no—
338C2×SD16—1222no—
339C2×M16—1221no—
340C22×C8—1221no—
341C4⋊Q8—21no—
342C4⋊Q8—22no—
343C4⋊Q8—612no—
344C4⋊Q8—1221no—
345C4⋊Q8—1221no—
346C4⋊Q8—1221no—
347C4⋊Q8C22⋊D6111yescharacteristic
348C4⋊Q8C22⋊D6111yescharacteristic
349#126—22no—
350#126—22no—
351#126—22no—
352#126—21no—
353#125C22×D6111yescharacteristic
354#125C22×D6112yes—
355#124—21no—
356#124—22no—
357#124—22no—
358#124—21no—
359#124—21no—
360#124C22⋊D6112yes—
361#124C22×D6112yes—
362#124C22×D6111yescharacteristic
363#123—22no—
364#123—22no—
365#123—21no—
366#123—22no—
367#123—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8C22⋊D6111yescharacteristic
379#121—21no—
380#121—22no—
381#121—22no—
382#121C22⋊D6111yescharacteristic
383#120—22no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8C22⋊D6111yescharacteristic
388C4×D4—22no—
389C4×D4—21no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4C22⋊D6111yescharacteristic
393#117—22no—
394#117—21no—
395#117C22⋊D6112yes—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C42C22×D6111yescharacteristic
402#107—1224no—
403#106—1224no—
404C4⋊C8—1221no—
405#103—1224no—
406#103—1222no—
407#103—1222no—
408#102—1224no—
409C22⋊C8—1221no—
410#97—612no—
411#97—612no—
412#97—612no—
413#97—1221no—
414C4×C8—1221no—
415C24—21no—
416#41—82no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—42no—
422C2×Q8—82no—
423C2×Q8—1222no—
424C2×Q8—4882no—
425C2×Q8—2441no—
426C2×Q8C2×C22⋊D6112yes—
427C2×Q8C2×C22⋊D6111yescharacteristic
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—42no—
433C2×D4C2×C22⋊D6112yes—
434C2×D4C2×C22⋊D6111yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—2441no—
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444Q16—4882no—
445SD16—4882no—
446M16—1224no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—1222no—
450C2×C8—2442no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4C2×C22⋊D6111yescharacteristic
468C4⋊C4C2×C22⋊D6111yescharacteristic
469C4⋊C4C2×C22⋊D6112yes—
470C4⋊C4C2×C22⋊D6112yes—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C42—22no—
487C42—22no—
488C42—21no—
489C42C2×C22⋊D6111yescharacteristic
490C42C2×C22⋊D6111yescharacteristic
491C42C2×C22⋊D6111yescharacteristic
492C42C23×D6111yescharacteristic
493C23—22no—
494C23—21no—
495C23—42no—
496C23—81no—
497C23—81no—
498C23#1451974611yescharacteristic
499Q8—41no—
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—81no—
504Q8—82no—
505Q8—4882no—
506Q8—2442no—
507Q8—2442no—
508D4—41no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—82no—
513D4—82no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—4882no—
529C2×C4—2442no—
530C2×C4#1451974611yescharacteristic
531C2×C4#1451974611yescharacteristic
532C2×C4#1451974611yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C22×C22⋊D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C8—1224no—
537C8—2442no—
538C8—2442no—
539C22—21no—
540C22—21no—
541C22—41no—
542C22—42no—
543C22—41no—
544C22—42no—
545C22—42no—
546C22—41no—
547C22—162no—
548C22#1683994512yes—
549C22C2×#1451974611yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—4882no—
562C4#1683994012yes—
563C2—22no—
564C2—82no—
565C2—81no—
566C2C2×#1683994011yescharacteristic
567C2C2×#1683994511yescharacteristic
568C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3561C2
socleC22×C61C2×#120
hypercenter#3561D61
hypocenterC61#19032
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3561

Complements

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

NH ≅ G/Ntype
1. #21456843C2semidirect
2. #21456366C2semidirect
3. C2×#16840121C2semidirect
4. #21445153C2semidirect
5. #21445136C2semidirect
6. #21444434C2semidirect
7. #21444417C2semidirect
8. #21444407C2semidirect
11. #16840172C22semidirect
14. #16840157C22semidirect
15. #16840132C22semidirect
28. #16839504C22semidirect
29. #16839504C22semidirect
31. #16839502C22semidirect
331. C61#19032semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214563662
C2×#168401212
#214451532
#214451362
#214444342
#214444172
#214444072
C61×#35612
#1903261