Finite Group Databasebeta

← #29657137

Subgroups — #29657137

Order 15616.

1410 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456734C211yesmaximal, characteristic, complemented
2C2×#16840138C211yesmaximal, characteristic, complemented
3#21455299C211yesmaximal, characteristic, complemented
4#21444884C214yesmaximal, complemented
5#21444873C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3498C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×#11758199—22no—
12C2×#14519774C2211yescharacteristic, complemented
13#16840138—21no—
14#16840138—22no—
15#16840138C2214yescomplemented
16#16840110—21no—
17#16839962—21no—
18#16839962C2212yescomplemented
19#16839957—21no—
20#16839957C2212yescomplemented
21#16839957C2214yescomplemented
22#16839895—22no—
23#16839815C2211yescharacteristic, complemented
24#16839786C2211yescharacteristic, complemented
25#16839766—21no—
26C2×C4×#92418C2211yescharacteristic, complemented
27#16839506C2214yescomplemented
28Q8⋊#92418C2212yescomplemented
29D4⋊#92418C2212yescomplemented
30D4⋊#92418C2214yescomplemented
31C2×H(5,𝔽2)×C61—21no—
32C61×#550—21no—
33C61×#550C2212yescomplemented
34D42×C61—22no—
35D42×C61—22no—
36D42×C61—21no—
37D42×C61C2212yescomplemented
38C61×#535—22no—
39C61×#535—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×#127×C61—21no—
43C2×#127×C61C2211yescharacteristic
44C4×#41×C61C2211yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244—21no—
48#14519802—44no—
49C2×#11758201—22no—
50C2×#11758199—22no—
51C2×#11758199—28no—
52C2×#11758199—24no—
53C2×#11758199—48no—
54C2×#11758194—21no—
55C22×#92418—21no—
56#14519774—21no—
57#14519774—21no—
58#14519774—21no—
59#14519774C2312yes—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519697—22no—
64#14519697—22no—
65#14519697—24no—
66C22⋊#92418—21no—
67C4×#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70H(5,𝔽2)×C61—44no—
71H(5,𝔽2)×C61—44no—
72C2×#41×C61—22no—
73C2×#41×C61—22no—
74C2×#41×C61—21no—
75C2×#41×C61D411yescharacteristic
76C22×D4×C61—22no—
77C22×D4×C61—24no—
78C22×D4×C61—22no—
79C22×D4×C61—22no—
80C22×D4×C61—22no—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61D412yes—
84C4⋊Q8×C61D411yescharacteristic
85#127×C61—22no—
86#127×C61—22no—
87#127×C61—22no—
88#127×C61—21no—
89#127×C61—21no—
90#127×C61D411yescharacteristic
91#127×C61D412yes—
92#127×C61C2312yes—
93#124×C61—22no—
94#124×C61—21no—
95#121×C61—22no—
96#121×C61—24no—
97#121×C61—24no—
98#121×C61—21no—
99#121×C61—22no—
100#121×C61—22no—
101#121×C61—22no—
102#121×C61D412yes—
103#120×C61—24no—
104#120×C61—24no—
105#120×C61—24no—
106C4×Q8×C61D411yescharacteristic
107C4×Q8×C61C2311yescharacteristic
108C4×D4×C61—22no—
109C4×D4×C61—21no—
110C4×D4×C61—21no—
111C4×D4×C61—22no—
112C4×D4×C61—21no—
113C4×D4×C61D411yescharacteristic
114C4×D4×C61C2312yes—
115C4×D4×C61C2311yescharacteristic
116#117×C61—21no—
117#117×C61D411yescharacteristic
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120C24×C61—24no—
121C24×C61—22no—
122C23×D61—41no—
123#41×C61—22no—
124#41×C61—48no—
125#41×C61—44no—
126#41×C61—44no—
127#41×C61—42no—
128#41×C61—42no—
129#41×C61D814yescomplemented
130C2×Q8×C61—21no—
131C2×Q8×C61C2×D411yescharacteristic
132C2×D4×C61—21no—
133C2×D4×C61—24no—
134C2×D4×C61—24no—
135C2×D4×C61—22no—
136C2×D4×C61—24no—
137C2×D4×C61—24no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—22no—
141C2×D4×C61—21no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—48no—
146C2×D4×C61—44no—
147C2×D4×C61—416no—
148C2×D4×C61—44no—
149C2×D4×C61—44no—
150C2×D4×C61—42no—
151C2×D4×C61—42no—
152C2×D4×C61—82no—
153C2×D4×C61—84no—
154C2×D4×C61—84no—
155C2×D4×C61C2×D412yes—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D414yes—
158C22×C4×C61—21no—
159C22×C4×C61—24no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—22no—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—22no—
169C2×D244—44no—
170C2×D244—42no—
171C2×D244—42no—
172C2×D244—41no—
173C2×D244—41no—
174C2×D244—41no—
175C2×D244—82no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D412yes—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2411yescharacteristic
196#11758201—44no—
197#11758199—416no—
198#11758199—44no—
199#11758199—84no—
200#11758194—42no—
201C2×#92418—24no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C23×C61—21no—
206C23×C61—22no—
207C23×C61—24no—
208C23×C61—48no—
209C23×C61—44no—
210C23×C61—82no—
211C23×C61—82no—
212C23×C61—84no—
213C23×C61—84no—
214C23×C61#12011yescharacteristic
215C22×D61—42no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—82no—
219Q8×C61—21no—
220Q8×C61—42no—
221Q8×C61C2×D812yescomplemented
222D4×C61—24no—
223D4×C61—21no—
224D4×C61—24no—
225D4×C61—44no—
226D4×C61—48no—
227D4×C61—48no—
228D4×C61—44no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—44no—
232D4×C61—42no—
233D4×C61—44no—
234D4×C61—44no—
235D4×C61—42no—
236D4×C61—82no—
237D4×C61—84no—
238D4×C61—84no—
239D4×C61C2×D812yescomplemented
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—48no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61C2×D812yes—
256C2×C4×C61AGL(1,ℤ/8)12yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260D244—44no—
261D244—42no—
262D244—42no—
263D244—44no—
264D244—82no—
265D244—82no—
266D244—82no—
267#92418—24no—
268#92418—42no—
269#92418—42no—
270#18620—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—44no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—44no—
280C22×C61—44no—
281C22×C61—42no—
282C22×C61—164no—
283C22×C61—168no—
284C22×C61#44712yes—
285C22×C61C2×#12011yescharacteristic
286C2×D61—82no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—164no—
290C4×C61—22no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61#44712yes—
297C4×C61#45312yes—
298C4×C61C22×D811yescharacteristic
299C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C61#57612yes—
301#3498D6111yescharacteristic, complemented
302#3205—611no—
303C2×#493—611no—
304#3038—611no—
305#1621—614no—
306#1619—614no—
307#1545—611no—
308#1492—611no—
309#1476—611no—
310C2×C61—22no—
311C2×C61—42no—
312C2×C61—82no—
313C2×C61—84no—
314C2×C61C2×#44711yescharacteristic
315C2×C61#300811yescharacteristic
316C2×C61C2×#45311yescharacteristic
317D61—162no—
318C2×H(5,𝔽2)—21no—
319C2×AGL(1,ℤ/8)—1221no—
320C22×D8—1222no—
321#550—21no—
322#550C2×D6112yescomplemented
323D42—22no—
324D42—22no—
325D42—21no—
326D42C2×D6112yescomplemented
327#535—22no—
328#535—21no—
329#532—21no—
330#532C2×D6111yescharacteristic
331C2×#127—21no—
332C2×#127—611no—
333C2×#127C2×D6111yescharacteristic
334C4×#41C2×D6111yescharacteristic
335#493—614no—
336#493—1222no—
337#493—1221no—
338#486—1221no—
339#460—612no—
340#460—1221no—
341#459—612no—
342#459—614no—
343#459—1221no—
344#447—1222no—
345#431—611no—
346#423—611no—
347#418—1221no—
348C2×C4×C8—611no—
349#331—614no—
350Q8⋊C8—612no—
351D4⋊C8—612no—
352D4⋊C8—614no—
353C61#1862011yescharacteristic, complemented, Sylow
354H(5,𝔽2)—44no—
355H(5,𝔽2)—44no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—21no—
359C2×#41C22⋊D6111yescharacteristic
360C22×D4—22no—
361C22×D4—24no—
362C22×D4—22no—
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—21no—
366C22×D4—21no—
367C22×D4—1221no—
368C22×D4—1221no—
369C22×D4—1221no—
370C22×D4C22⋊D6112yes—
371AGL(1,ℤ/8)—2444no—
372C2×SD16—1222no—
373C2×D8—1224no—
374C2×D8—1228no—
375C2×D8—1222no—
376C2×D8—2448no—
377C2×M16—1221no—
378C22×C8—1221no—
379C4⋊Q8C22⋊D6111yescharacteristic
380#127—22no—
381#127—22no—
382#127—22no—
383#127—21no—
384#127—21no—
385#127—612no—
386#127—1221no—
387#127—1221no—
388#127—1221no—
389#127C22⋊D6111yescharacteristic
390#127C22⋊D6112yes—
391#127C22×D6112yes—
392#124—22no—
393#124—21no—
394#121—22no—
395#121—24no—
396#121—24no—
397#121—21no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121C22⋊D6112yes—
402#120—24no—
403#120—24no—
404#120—24no—
405C4×Q8C22⋊D6111yescharacteristic
406C4×Q8C22×D6111yescharacteristic
407C4×D4—22no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—22no—
411C4×D4—21no—
412C4×D4C22⋊D6111yescharacteristic
413C4×D4C22×D6111yescharacteristic
414C4×D4C22×D6112yes—
415#117—21no—
416#117C22⋊D6111yescharacteristic
417C2×C42—21no—
418C2×C42C22×D6111yescharacteristic
419C4⋊C8—612no—
420C4⋊C8—612no—
421C4⋊C8—1221no—
422#102—1222no—
423#102—1222no—
424#102—1224no—
425C22⋊C8—1221no—
426C4×C8—612no—
427C4×C8—1221no—
428C4×C8—1221no—
429C24—24no—
430C24—22no—
431C24—2441no—
432#41—22no—
433#41—48no—
434#41—44no—
435#41—44no—
436#41—42no—
437#41—42no—
438#41#1175819914yescomplemented
439C2×Q8—21no—
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4—21no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—48no—
455C2×D4—44no—
456C2×D4—416no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—82no—
462C2×D4—84no—
463C2×D4—84no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—4882no—
468C2×D4—2444no—
469C2×D4—2441no—
470C2×D4—2441no—
471C2×D4—2441no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4C2×C22⋊D6112yes—
475C2×D4C2×C22⋊D6111yescharacteristic
476C2×D4C2×C22⋊D6114yes—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485SD16—2444no—
486D8—4884no—
487D8—24416no—
488D8—2444no—
489M16—2442no—
490C2×C8—1224no—
491C2×C8—1222no—
492C2×C8—1222no—
493C2×C8—1222no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4C2×C22⋊D6111yescharacteristic
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C4⋊C4C2×C22⋊D6112yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42C2×C22⋊D6111yescharacteristic
511C42C2×C22⋊D6111yescharacteristic
512C42C2×C22⋊D6111yescharacteristic
513C42C23×D6111yescharacteristic
514C23—21no—
515C23—24no—
516C23—22no—
517C23—48no—
518C23—44no—
519C23—82no—
520C23—82no—
521C23—84no—
522C23—84no—
523C23—4882no—
524C23—4882no—
525C23—4882no—
526C23—2442no—
527C23#1451974611yescharacteristic
528Q8—21no—
529Q8—42no—
530Q8C2×#1175819912yescomplemented
531D4—24no—
532D4—21no—
533D4—24no—
534D4—44no—
535D4—48no—
536D4—48no—
537D4—44no—
538D4—42no—
539D4—42no—
540D4—44no—
541D4—42no—
542D4—44no—
543D4—44no—
544D4—42no—
545D4—82no—
546D4—84no—
547D4—84no—
548D4—4882no—
549D4—4882no—
550D4—4882no—
551D4—2444no—
552D4—2442no—
553D4—2442no—
554D4—2444no—
555D4C2×#1175819912yescomplemented
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4#1451974611yescharacteristic
569C2×C4#1451974611yescharacteristic
570C2×C4#1451974611yescharacteristic
571C2×C4C2×#1175819912yes—
572C2×C4#1451980312yes—
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4C22×C22⋊D6111yescharacteristic
576C8—1224no—
577C8—2442no—
578C8—2442no—
579C22—21no—
580C22—21no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—164no—
591C22—168no—
592C22—9764no—
593C22—4882no—
594C22—4882no—
595C22—4882no—
596C22#1683989612yes—
597C22C2×#1451974611yescharacteristic
598C4—21no—
599C4—22no—
600C4—21no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4#1683989612yes—
605C4#1683993012yes—
606C4C22×#1175819911yescharacteristic
607C4C2×#1451980311yescharacteristic
608C4#1684056512yes—
609C2—22no—
610C2—42no—
611C2—84no—
612C2—82no—
613C2—9762no—
614C2C2×#1683989611yescharacteristic
615C2#2145508711yescharacteristic
616C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3498C2
socleC22×C61C2×#120
hypercenter#3498D61
hypocenterC61#18620
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3498

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. C2×#16840138C2semidirect
3. #21455299C2semidirect
4. #21444884C2semidirect
5. #21444873C2semidirect
6. #21444487C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
9. C61×#3498C2semidirect
12. C2×#14519774C22semidirect
15. #16840138C22semidirect
18. #16839962C22semidirect
20. #16839957C22semidirect
21. #16839957C22semidirect
23. #16839815C22semidirect
24. #16839786C22semidirect
26. C2×C4×#92418C22semidirect
27. #16839506C22semidirect
28. Q8⋊#92418C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
33. C61×#550C22semidirect
37. D42×C61C22semidirect
129. #41×C61D8semidirect
221. Q8×C61C2×D8semidirect
239. D4×C61C2×D8semidirect
301. #3498D61semidirect
322. #550C2×D61semidirect
326. D42C2×D61semidirect
353. C61#18620semidirect
438. #41#11758199semidirect
530. Q8C2×#11758199semidirect
555. D4C2×#11758199semidirect

Maximal subgroups

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

subgroupindex
#214567342
C2×#168401382
#214552992
#214448842
#214448732
#214444872
#214442702
#214442012
C61×#34982
#1862061