Finite Group Databasebeta

← #29833969

Subgroups — #29833969

Order 15616.

1088 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458953C211yesmaximal, characteristic, complemented
2C2×#16840192C211yesmaximal, characteristic, complemented
3#21455539C211yesmaximal, characteristic, complemented
4#21447797C212yesmaximal, complemented
5#21447790C212yesmaximal, complemented
6#21447783C212yesmaximal, complemented
7#21447776C212yesmaximal, complemented
8#21447568C211yesmaximal, characteristic, complemented
9C2×#16839569C211yesmaximal, characteristic, complemented
10C2×#16839546C211yesmaximal, characteristic, complemented
11C61×#3584C211yesmaximal, characteristic, complemented
12#16840560—21no—
13C2×#14519799C2211yescharacteristic, complemented
14#16840203—21no—
15#16840202C2212yescomplemented
16#16840198—21no—
17#16840197C2212yescomplemented
18#16840192—22no—
19#16840192—22no—
20#16840192—22no—
21C2×#14519723C2211yescharacteristic, complemented
22C2×#14519721C2211yescharacteristic, complemented
23C2×#14519720C2211yescharacteristic, complemented
24C2×#14519718C2211yescharacteristic, complemented
25C2×#14519717C2211yescharacteristic, complemented
26C22×#11758185C2211yescharacteristic, complemented
27#16840037C2212yescomplemented
28#16840032C2212yescomplemented
29#16840025—22no—
30C2×#14519713C2211yescharacteristic, complemented
31#16839744—21no—
32#16839578C2212yescomplemented
33#16839575C2212yescomplemented
34#16839569C2212yescomplemented
35#16839569C2212yescomplemented
36#16839546C2212yescomplemented
37#16839546C2212yescomplemented
38C61×#585—21no—
39C61×#578—21no—
40C61×#578C2212yes—
41C61×#577—22no—
42C61×#577—22no—
43C61×#576—21no—
44C61×#576C2212yes—
45C61×#575—22no—
46C61×#575—21no—
47C2×#135×C61—22no—
48C2×#135×C61C2211yescharacteristic
49C2×#135×C61C2211yescharacteristic
50C2×#131×C61C2211yescharacteristic
51C2×#11758222—21no—
52#14519808—41no—
53#14519807—22no—
54#14519803—41no—
55#14519802—22no—
56#14519799—21no—
57#14519799—21no—
58#14519799—21no—
59#14519799—21no—
60#14519797—42no—
61C2×Dic244C2311yescharacteristic
62C2×#11758190—21no—
63C2×D488C2311yescharacteristic
64C2×#11758194—21no—
65#14519723—22no—
66#14519723—22no—
67#14519721—22no—
68#14519721—22no—
69#14519720—22no—
70#14519720—22no—
71#14519718—22no—
72#14519718—22no—
73#14519717—21no—
74#14519717C2312yes—
75C2×#11758185—21no—
76C2×#11758185—21no—
77C2×#11758185—21no—
78C2×#11758185C2311yescharacteristic
79C2×#11758185C2311yescharacteristic
80#14519713—21no—
81#14519713C2312yes—
82#14519695—22no—
83#14519693—22no—
842−1+4×C61—22no—
852−1+4×C61—22no—
862−1+4×C61—41no—
87H(5,𝔽2)×C61—22no—
88H(5,𝔽2)×C61—22no—
89H(5,𝔽2)×C61—42no—
90H(5,𝔽2)×C61—41no—
91C2×#41×C61—24no—
92C2×#41×C61—21no—
93C2×#41×C61—21no—
94C2×#41×C61—42no—
95C2×#41×C61—42no—
96C2×#41×C61D411yescharacteristic
97#137×C61—22no—
98#137×C61—24no—
99#137×C61—24no—
100#137×C61—22no—
101AGL(1,ℤ/8)×C61—22no—
102AGL(1,ℤ/8)×C61—24no—
103AGL(1,ℤ/8)×C61—24no—
104AGL(1,ℤ/8)×C61—22no—
105#135×C61—24no—
106#135×C61—24no—
107#135×C61—22no—
108#135×C61—22no—
109#135×C61—21no—
110#135×C61—21no—
111#135×C61—21no—
112#135×C61—21no—
113#135×C61D412yes—
114#135×C61D412yes—
115#135×C61D412yes—
116#135×C61C2312yes—
117C2×Q16×C61—22no—
118C2×Q16×C61D411yescharacteristic
119C2×Q16×C61C2311yescharacteristic
120C2×SD16×C61—24no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×D8×C61—22no—
124C2×D8×C61D411yescharacteristic
125C2×D8×C61C2311yescharacteristic
126#131×C61—21no—
127#131×C61—21no—
128#131×C61D412yes—
129#131×C61C2312yes—
130C2×M16×C61—21no—
131C2×M16×C61D411yescharacteristic
132C22×C8×C61—21no—
133C22×C8×C61C2311yescharacteristic
134#41×C61—21no—
135#41×C61—21no—
136#41×C61—24no—
137#41×C61—24no—
138#41×C61—22no—
139#41×C61—22no—
140#41×C61—22no—
141#41×C61—22no—
142#41×C61—44no—
143#41×C61—44no—
144#41×C61—44no—
145#41×C61—44no—
146#41×C61—42no—
147#41×C61—42no—
148#41×C61—42no—
149#41×C61—42no—
150#41×C61—41no—
151#41×C61—41no—
152#41×C61—41no—
153#41×C61—41no—
154#41×C61D812yes—
155#41×C61C2×D412yes—
156C2×Q8×C61—24no—
157C2×Q8×C61—21no—
158C2×Q8×C61—21no—
159C2×Q8×C61—42no—
160C2×Q8×C61—42no—
161C2×Q8×C61D811yescharacteristic
162C2×D4×C61—21no—
163C2×D4×C61—24no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—44no—
167C2×D4×C61—44no—
168C2×D4×C61—42no—
169C2×D4×C61—44no—
170C2×D4×C61—41no—
171C2×D4×C61—41no—
172C2×D4×C61—42no—
173C2×D4×C61D811yescharacteristic
174C22×C4×C61—21no—
175C22×C4×C61—44no—
176C22×C4×C61—41no—
177C22×C4×C61—41no—
178C22×C4×C61C2×D411yescharacteristic
179C2×C22⋊D61—41no—
180#11758222—41no—
181#11758222—41no—
182#11758222—41no—
183#11758222—41no—
184C2×D244—21no—
185C2×C4×D61—41no—
186C2×Dic122—21no—
187Q16×C61—24no—
188Q16×C61—21no—
189Q16×C61—21no—
190Q16×C61C2×D412yes—
191Q16×C61C2×D412yes—
192SD16×C61—24no—
193SD16×C61—24no—
194SD16×C61—22no—
195SD16×C61—22no—
196SD16×C61—22no—
197SD16×C61—22no—
198D8×C61—24no—
199D8×C61—21no—
200D8×C61—21no—
201D8×C61C2×D412yes—
202D8×C61C2×D412yes—
203M16×C61—21no—
204M16×C61—22no—
205M16×C61—22no—
206M16×C61C2×D412yes—
207C2×C8×C61—21no—
208C2×C8×C61—22no—
209C2×C8×C61—22no—
210C2×C8×C61C2×D411yescharacteristic
211C2×C8×C61C2×D411yescharacteristic
212C2×C8×C61C2×D411yescharacteristic
213C2×C8×C61C2×D411yescharacteristic
214C2×C8×C61C2×D411yescharacteristic
215C2×C8×C61C2411yescharacteristic
216#11758202—42no—
217#11758201—42no—
218#11758200—42no—
219#11758199—42no—
220#11758194—22no—
221#11758194—41no—
222C2×#92418—41no—
223Dic244—21no—
224Dic244—21no—
225D488—21no—
226D488—21no—
227#11758190—41no—
228#11758190—41no—
229#11758185—21no—
230#11758185—21no—
231#11758185—22no—
232C23×C61—21no—
233C23×C61—44no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61AGL(1,ℤ/8)11yescharacteristic
237C22×D61—41no—
238Q8×C61—21no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—44no—
242Q8×C61—41no—
243Q8×C61—41no—
244Q8×C61C2×D812yes—
245D4×C61—22no—
246D4×C61—22no—
247D4×C61—22no—
248D4×C61—22no—
249D4×C61—21no—
250D4×C61—44no—
251D4×C61—44no—
252D4×C61—44no—
253D4×C61—42no—
254D4×C61—42no—
255D4×C61—42no—
256D4×C61—42no—
257D4×C61—41no—
258D4×C61—41no—
259D4×C61C2×D812yes—
260C2×C4×C61—21no—
261C2×C4×C61—22no—
262C2×C4×C61—22no—
263C2×C4×C61—44no—
264C2×C4×C61—44no—
265C2×C4×C61—44no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—42no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61#12011yescharacteristic
273C2×C4×C61#12011yescharacteristic
274C2×C4×C61C2×D811yescharacteristic
275C2×C4×C61C2×D811yescharacteristic
276C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C22⋊D61—81no—
279C22⋊D61—81no—
280C2×Dic61—41no—
281D244—41no—
282D244—41no—
283C4×D61—81no—
284C4×D61—81no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—22no—
288C8×C61#12011yescharacteristic
289C8×C61#12011yescharacteristic
290C8×C61C22×D411yescharacteristic
291C8×C61C22×D411yescharacteristic
292#92418—42no—
293#33538—611nomaximal, Sylow
294C22×C61—22no—
295C22×C61—41no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—81no—
299C22×C61—161no—
300C22×C61—161no—
301C22×C61—164no—
302C22×C61#44711yescharacteristic
303C22×C61#44711yescharacteristic
304C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
305C2×D61—81no—
306C2×D61—161no—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61#44711yescharacteristic
311C4×C61#44711yescharacteristic
312C4×C61C2×#12011yescharacteristic
313C4×C61C22×D811yescharacteristic
314Dic61—81no—
315#3584D6111yescharacteristic, complemented
316#3414—611no—
317C2×#508—611no—
318#3067—611no—
319#2195—612no—
320#2194—612no—
321#2193—612no—
322#2192—612no—
323#2148—611no—
324C2×#359—611no—
325C2×#350—611no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—42no—
330C2×C61—82no—
331C2×C61—82no—
332C2×C61C2×#44711yescharacteristic
333D61—161no—
334#585—21no—
335#578—21no—
336#578C2×D6112yes—
337#577—22no—
338#577—22no—
339#576—21no—
340#576C2×D6112yes—
341#575—22no—
342#575—21no—
343#575—1221no—
344C2×#135—22no—
345C2×#135—611no—
346C2×#135C2×D6111yescharacteristic
347C2×#135C2×D6111yescharacteristic
348C2×#131C2×D6111yescharacteristic
349#510—612no—
350#510—1221no—
351#509—612no—
352#509—1221no—
353#508—1222no—
354#508—1222no—
355#508—1222no—
356C2×Q32—611no—
357C2×SD32—611no—
358C2×SD32—611no—
359C2×D16—611no—
360C2×M32—611no—
361C22×C16—611no—
362#473—612no—
363#472—612no—
364#471—1222no—
365C2×#108—611no—
366#413—1221no—
367#362—612no—
368#361—612no—
369#359—612no—
370#359—612no—
371#350—612no—
372#350—612no—
373C61#3353811yescharacteristic, complemented, Sylow
3742−1+4—22no—
3752−1+4—22no—
3762−1+4—41no—
377H(5,𝔽2)—22no—
378H(5,𝔽2)—22no—
379H(5,𝔽2)—42no—
380H(5,𝔽2)—41no—
381C2×#41—24no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—42no—
385C2×#41—42no—
386C2×#41—1221no—
387C2×#41C22⋊D6111yescharacteristic
388#137—22no—
389#137—24no—
390#137—24no—
391#137—22no—
392#137—1222no—
393#137—2441no—
394AGL(1,ℤ/8)—22no—
395AGL(1,ℤ/8)—24no—
396AGL(1,ℤ/8)—24no—
397AGL(1,ℤ/8)—22no—
398AGL(1,ℤ/8)—1222no—
399AGL(1,ℤ/8)—2441no—
400#135—24no—
401#135—24no—
402#135—22no—
403#135—22no—
404#135—21no—
405#135—21no—
406#135—21no—
407#135—21no—
408#135—1221no—
409#135—1221no—
410#135—1221no—
411#135—1221no—
412#135—2442no—
413#135C22⋊D6112yes—
414#135C22⋊D6112yes—
415#135C22⋊D6112yes—
416#135C22×D6112yes—
417C2×Q16—22no—
418C2×Q16—611no—
419C2×Q16C22⋊D6111yescharacteristic
420C2×Q16C22×D6111yescharacteristic
421C2×SD16—24no—
422C2×SD16—21no—
423C2×SD16—21no—
424C2×SD16—1221no—
425C2×D8—22no—
426C2×D8—611no—
427C2×D8C22⋊D6111yescharacteristic
428C2×D8C22×D6111yescharacteristic
429#131—21no—
430#131—21no—
431#131C22⋊D6112yes—
432#131C22×D6112yes—
433C2×M16—21no—
434C2×M16—1221no—
435C2×M16C22⋊D6111yescharacteristic
436C22×C8—21no—
437C22×C8C22×D6111yescharacteristic
438Q32—1222no—
439Q32—1222no—
440SD32—1222no—
441SD32—1222no—
442SD32—1222no—
443SD32—1222no—
444D16—1222no—
445D16—1222no—
446M32—612no—
447M32—1221no—
448C2×C16—611no—
449C2×C16—611no—
450C2×C16—1221no—
451C2×C16—1221no—
452C2×C16—1221no—
453#108—612no—
454#108—1221no—
455#101—1222no—
456#100—1222no—
457#41—21no—
458#41—21no—
459#41—24no—
460#41—24no—
461#41—22no—
462#41—22no—
463#41—22no—
464#41—22no—
465#41—44no—
466#41—44no—
467#41—44no—
468#41—44no—
469#41—42no—
470#41—42no—
471#41—42no—
472#41—42no—
473#41—41no—
474#41—41no—
475#41—41no—
476#41—41no—
477#41—2441no—
478#41—2441no—
479#41—2441no—
480#41—2441no—
481#41#1175819912yes—
482#41C2×C22⋊D6112yes—
483C2×Q8—24no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—42no—
487C2×Q8—42no—
488C2×Q8—1221no—
489C2×Q8#1175819911yescharacteristic
490C2×D4—21no—
491C2×D4—24no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—44no—
495C2×D4—44no—
496C2×D4—42no—
497C2×D4—44no—
498C2×D4—41no—
499C2×D4—41no—
500C2×D4—42no—
501C2×D4—1221no—
502C2×D4—2441no—
503C2×D4#1175819911yescharacteristic
504C22×C4—21no—
505C22×C4—44no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—2441no—
509C22×C4C2×C22⋊D6111yescharacteristic
510Q16—24no—
511Q16—21no—
512Q16—21no—
513Q16—1221no—
514Q16—1221no—
515Q16—2442no—
516Q16C2×C22⋊D6112yes—
517Q16C2×C22⋊D6112yes—
518SD16—24no—
519SD16—24no—
520SD16—22no—
521SD16—22no—
522SD16—22no—
523SD16—22no—
524SD16—2442no—
525SD16—2442no—
526SD16—2441no—
527SD16—2441no—
528D8—24no—
529D8—21no—
530D8—21no—
531D8—1221no—
532D8—1221no—
533D8—2442no—
534D8C2×C22⋊D6112yes—
535D8C2×C22⋊D6112yes—
536M16—21no—
537M16—22no—
538M16—22no—
539M16—1222no—
540M16—2441no—
541M16C2×C22⋊D6112yes—
542C2×C8—21no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—2441no—
546C2×C8C2×C22⋊D6111yescharacteristic
547C2×C8C2×C22⋊D6111yescharacteristic
548C2×C8C2×C22⋊D6111yescharacteristic
549C2×C8C2×C22⋊D6111yescharacteristic
550C2×C8C2×C22⋊D6111yescharacteristic
551C2×C8C23×D6111yescharacteristic
552C16—1221no—
553C16—1221no—
554C16—1222no—
555C23—21no—
556C23—44no—
557C23—41no—
558C23—41no—
559C23—2441no—
560C23#1451980311yescharacteristic
561Q8—21no—
562Q8—22no—
563Q8—22no—
564Q8—44no—
565Q8—41no—
566Q8—41no—
567Q8—2441no—
568Q8—2441no—
569Q8C2×#1175819912yes—
570D4—22no—
571D4—22no—
572D4—22no—
573D4—22no—
574D4—21no—
575D4—44no—
576D4—44no—
577D4—44no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—41no—
583D4—41no—
584D4—4881no—
585D4—4881no—
586D4—2441no—
587D4—2441no—
588D4C2×#1175819912yes—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—4881no—
602C2×C4—4881no—
603C2×C4—2441no—
604C2×C4#1451974611yescharacteristic
605C2×C4#1451974611yescharacteristic
606C2×C4C2×#1175819911yescharacteristic
607C2×C4C2×#1175819911yescharacteristic
608C2×C4#1451980311yescharacteristic
609C2×C4C22×C22⋊D6111yescharacteristic
610C8—22no—
611C8—2442no—
612C8#1451974611yescharacteristic
613C8#1451974611yescharacteristic
614C8C22×C22⋊D6111yescharacteristic
615C8C22×C22⋊D6111yescharacteristic
616C22—22no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—161no—
622C22—161no—
623C22—164no—
624C22—9761no—
625C22—4881no—
626C22#1683989611yescharacteristic
627C22#1683989611yescharacteristic
628C22C2×#1451980311yescharacteristic
629C4—22no—
630C4—42no—
631C4—42no—
632C4—4881no—
633C4#1683989611yescharacteristic
634C4#1683989611yescharacteristic
635C4C2×#1451974611yescharacteristic
636C4C22×#1175819911yescharacteristic
637C2—21no—
638C2—21no—
639C2—21no—
640C2—42no—
641C2—82no—
642C2—82no—
643C2—9761no—
644C2C2×#1683989611yescharacteristic
subgroupG/H
centerC4#16839896
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3584C2
socleC2×C61C2×#447
hypercenter#3584D61
hypocenterC61#33538
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3584

Complements

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

NH ≅ G/Ntype
1. #21458953C2semidirect
2. C2×#16840192C2semidirect
3. #21455539C2semidirect
4. #21447797C2semidirect
5. #21447790C2semidirect
6. #21447783C2semidirect
7. #21447776C2semidirect
8. #21447568C2semidirect
9. C2×#16839569C2semidirect
10. C2×#16839546C2semidirect
11. C61×#3584C2semidirect
13. C2×#14519799C22semidirect
15. #16840202C22semidirect
17. #16840197C22semidirect
21. C2×#14519723C22semidirect
22. C2×#14519721C22semidirect
23. C2×#14519720C22semidirect
24. C2×#14519718C22semidirect
25. C2×#14519717C22semidirect
26. C22×#11758185C22semidirect
27. #16840037C22semidirect
28. #16840032C22semidirect
30. C2×#14519713C22semidirect
32. #16839578C22semidirect
33. #16839575C22semidirect
34. #16839569C22semidirect
35. #16839569C22semidirect
36. #16839546C22semidirect
37. #16839546C22semidirect
315. #3584D61semidirect
373. C61#33538semidirect

Maximal subgroups

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

subgroupindex
#214589532
C2×#168401922
#214555392
#214477972
#214477902
#214477832
#214477762
#214475682
C2×#168395692
C2×#168395462
C61×#35842
#3353861