Finite Group Databasebeta

← #29657575

Subgroups — #29657575

Order 15616.

1260 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic, complemented
2#21456394C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic, complemented
4#21445054C212yesmaximal, complemented
5#21444998C212yesmaximal, complemented
6#21444908C212yesmaximal, complemented
7#21444873C212yesmaximal, complemented
8#21444496C211yesmaximal, characteristic, complemented
9#21444270C211yesmaximal, characteristic, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3496C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13#16840312C2211yescharacteristic, complemented
14#16840165C2211yescharacteristic, complemented
15#16840138C2211yescharacteristic, complemented
16#16840110C2211yescharacteristic, complemented
17#16840108C2211yescharacteristic, complemented
18#16840042C2212yescomplemented
19#16840039—21no—
20#16839996—22no—
21#16839996—22no—
22#16839977—22no—
23#16839967C2212yescomplemented
24#16839957C2212yescomplemented
25#16839957C2212yescomplemented
26#16839954—21no—
27C4×#11758199—22no—
28#16839786C2211yescharacteristic, complemented
29C2×C4⋊#92418C2211yescharacteristic, complemented
30#16839761—21no—
31C2×C4×#92418C2211yescharacteristic, complemented
32#16839508C2212yescomplemented
33#16839506C2212yescomplemented
34D4⋊#92418C2212yescomplemented
35D4⋊#92418C2212yescomplemented
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C2×H(5,𝔽2)×C61—21no—
39C61×#547—21no—
40C61×#547C2212yes—
41C61×#546—22no—
42C61×#546—22no—
43D42×C61—21no—
44D42×C61C2212yes—
45C61×#537—21no—
46C61×#534—22no—
47C61×#532C2211yescharacteristic
48C61×#532C2211yescharacteristic
49C2×#121×C61—22no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758222—21no—
52#14519803—44no—
53C2×#11758200—22no—
54C2×#11758199—22no—
55C2×#11758199—22no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C22×#92418—21no—
59C22×#92418—21no—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×D244—21no—
66C4×D244—21no—
67#14519710—22no—
68#14519710—22no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2312yes—
73#14519700—22no—
74#14519698—22no—
75#14519698—22no—
76#14519697—22no—
77#14519697—22no—
78#14519697—22no—
79C4×#92418—21no—
80C4×#92418C2311yescharacteristic
81C4×#92418C2311yescharacteristic
82H(5,𝔽2)×C61—44no—
83H(5,𝔽2)×C61—44no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C22×D4×C61—24no—
89C22×D4×C61—22no—
90C22×D4×C61—22no—
91C22×D4×C61—22no—
92C22×D4×C61D411yescharacteristic
93C23×C4×C61—21no—
94C4⋊Q8×C61D411yescharacteristic
95C4⋊Q8×C61C2311yescharacteristic
96#127×C61D411yescharacteristic
97#127×C61C2311yescharacteristic
98#124×C61—21no—
99#124×C61—21no—
100#123×C61—24no—
101#123×C61—24no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—22no—
104#121×C61—24no—
105#121×C61—22no—
106#121×C61—24no—
107#121×C61—22no—
108#121×C61—21no—
109#121×C61—21no—
110#121×C61—22no—
111#121×C61—22no—
112#121×C61D412yes—
113#121×C61D412yes—
114#120×C61—24no—
115#120×C61—24no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61D412yes—
121C4×D4×C61D412yes—
122C4×D4×C61C2312yes—
123C4×D4×C61C2312yes—
124C2×C4⋊C4×C61—22no—
125C2×C4⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61—24no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C24×C61—21no—
130C24×C61—24no—
131#41×C61—44no—
132#41×C61—48no—
133#41×C61—44no—
134#41×C61—42no—
135#41×C61—42no—
136#41×C61—42no—
137#41×C61—42no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×D4×C61—24no—
141C2×D4×C61—24no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—24no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—22no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—44no—
152C2×D4×C61—48no—
153C2×D4×C61—416no—
154C2×D4×C61—44no—
155C2×D4×C61—82no—
156C2×D4×C61—82no—
157C2×D4×C61D814yescomplemented
158C2×D4×C61C2×D412yes—
159C2×D4×C61C2×D412yes—
160C2×D4×C61C2×D412yes—
161C22×C4×C61—24no—
162C22×C4×C61—21no—
163C22×C4×C61—22no—
164C22×C4×C61—22no—
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—81no—
168C22×C4×C61—81no—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C2×C22⋊D61—41no—
173#11758222—42no—
174#11758222—42no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244—21no—
178C2×C4×D61—41no—
179C2×Dic122—21no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D412yes—
185C4⋊C4×C61C2×D412yes—
186C4⋊C4×C61C2×D412yes—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—24no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2411yescharacteristic
199#11758200—44no—
200#11758199—44no—
201#11758199—44no—
202#11758199—82no—
203#11758198—41no—
204#11758198—41no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207#11758194—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—42no—
213C2×#92418—81no—
214C23×C61—22no—
215C23×C61—24no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—48no—
219C23×C61—81no—
220C23×C61—81no—
221C23×C61—84no—
222C23×C61—84no—
223C23×C61#12011yescharacteristic
224C22×D61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—42no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—24no—
232D4×C61—42no—
233D4×C61—48no—
234D4×C61—48no—
235D4×C61—44no—
236D4×C61—44no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—42no—
242D4×C61—81no—
243D4×C61—84no—
244D4×C61C2×D814yes—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—48no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—44no—
257C2×C4×C61—82no—
258C2×C4×C61—81no—
259C2×C4×C61#12011yescharacteristic
260C2×C4×C61#12011yescharacteristic
261C2×C4×C61#12011yescharacteristic
262C2×C4×C61C2×D812yes—
263C2×C4×C61AGL(1,ℤ/8)12yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C22⋊D61—82no—
268C2×Dic61—41no—
269D244—42no—
270D244—41no—
271D244—41no—
272D244—41no—
273D244—41no—
274C4×D61—82no—
275Dic122—41no—
276Dic122—41no—
277#92418—22no—
278#92418—41no—
279#92418—42no—
280#92418—42no—
281#18662—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—24no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—84no—
294C22×C61—168no—
295C22×C61#44712yes—
296C22×C61C2×#12011yescharacteristic
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61#44712yes—
307C4×C61C22×D811yescharacteristic
308C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C4×C61#57611yescharacteristic
310C4×C61#57711yescharacteristic
311Dic61—81no—
312#3496D6111yescharacteristic, complemented
313#3245—611no—
314#3169—611no—
315#3036—611no—
316#1647—612no—
317#1639—612no—
318#1625—612no—
319#1619—612no—
320#1547—611no—
321#1492—611no—
322C2×D4⋊C8—611no—
323C2×C61—22no—
324C2×C61—44no—
325C2×C61—82no—
326C2×C61—82no—
327C2×C61C2×#44711yescharacteristic
328C2×C61#301211yescharacteristic
329C2×C61#301811yescharacteristic
330D61—161no—
331C2×H(5,𝔽2)—21no—
332C2×AGL(1,ℤ/8)—1221no—
333#547—21no—
334#547C2×D6112yes—
335#546—22no—
336#546—22no—
337D42—21no—
338D42C2×D6112yes—
339#537—21no—
340#534—22no—
341#532—611no—
342#532C2×D6111yescharacteristic
343#532C2×D6111yescharacteristic
344C2×#121—22no—
345C2×C4×D4C2×D6111yescharacteristic
346#500—611no—
347#493—611no—
348#486—611no—
349#486—611no—
350#474—612no—
351#474—1221no—
352#466—1222no—
353#466—1222no—
354#463—1222no—
355#461—612no—
356#459—612no—
357#459—612no—
358#459—1221no—
359C4×D8—1222no—
360#423—611no—
361C2×C4⋊C8—611no—
362#417—1221no—
363C2×C4×C8—611no—
364#332—612no—
365#331—612no—
366D4⋊C8—612no—
367D4⋊C8—612no—
368D4⋊C8—612no—
369D4⋊C8—612no—
370C61#1866211yescharacteristic, complemented, Sylow
371H(5,𝔽2)—44no—
372H(5,𝔽2)—44no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—1221no—
378C22×D4—24no—
379C22×D4—22no—
380C22×D4—22no—
381C22×D4—22no—
382C22×D4C22⋊D6111yescharacteristic
383C23×C4—21no—
384AGL(1,ℤ/8)—2444no—
385C2×SD16—1222no—
386C2×D8—1222no—
387C2×D8—1222no—
388C2×D8—1222no—
389C2×M16—1221no—
390C22×C8—1221no—
391C22×C8—1221no—
392C4⋊Q8—611no—
393C4⋊Q8C22⋊D6111yescharacteristic
394C4⋊Q8C22×D6111yescharacteristic
395#127—611no—
396#127C22⋊D6111yescharacteristic
397#127C22×D6111yescharacteristic
398#124—21no—
399#124—21no—
400#124—1221no—
401#123—24no—
402#123—24no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405#121—24no—
406#121—24no—
407#121—22no—
408#121—22no—
409#121—21no—
410#121—21no—
411#121—22no—
412#121—22no—
413#121—1221no—
414#121—1221no—
415#121C22⋊D6112yes—
416#121C22⋊D6112yes—
417#120—24no—
418#120—24no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—1221no—
424C4×D4—1221no—
425C4×D4C22⋊D6112yes—
426C4×D4C22⋊D6112yes—
427C4×D4C22×D6112yes—
428C4×D4C22×D6112yes—
429C2×C4⋊C4—22no—
430C2×C4⋊C4C22⋊D6111yescharacteristic
431C2×C22⋊C4—24no—
432C2×C22⋊C4—21no—
433C2×C42C22×D6111yescharacteristic
434#107—1222no—
435#107—1222no—
436C4⋊C8—612no—
437C4⋊C8—612no—
438C4⋊C8—1221no—
439C4⋊C8—1221no—
440#103—1222no—
441#102—1222no—
442#102—1222no—
443#102—1222no—
444#102—1222no—
445#102—1222no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—1221no—
449C24—21no—
450C24—24no—
451#41—44no—
452#41—48no—
453#41—44no—
454#41—42no—
455#41—42no—
456#41—42no—
457#41—42no—
458#41—2442no—
459#41—2442no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—1221no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—24no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—44no—
475C2×D4—48no—
476C2×D4—416no—
477C2×D4—44no—
478C2×D4—82no—
479C2×D4—82no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—2441no—
484C2×D4#1175819914yescomplemented
485C2×D4C2×C22⋊D6112yes—
486C2×D4C2×C22⋊D6112yes—
487C2×D4C2×C22⋊D6112yes—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—2441no—
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500SD16—2444no—
501D8—4882no—
502D8—2444no—
503D8—2444no—
504M16—2442no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—4881no—
510C2×C8—2442no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4C2×C22⋊D6112yes—
518C4⋊C4C2×C22⋊D6112yes—
519C4⋊C4C2×C22⋊D6112yes—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—2441no—
529C22⋊C4—2441no—
530C42C2×C22⋊D6111yescharacteristic
531C42C2×C22⋊D6111yescharacteristic
532C42C2×C22⋊D6111yescharacteristic
533C42C23×D6111yescharacteristic
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—48no—
539C23—81no—
540C23—81no—
541C23—84no—
542C23—84no—
543C23—2441no—
544C23#1451974611yescharacteristic
545Q8—41no—
546Q8—41no—
547Q8—42no—
548Q8—2441no—
549Q8—2441no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—24no—
554D4—42no—
555D4—48no—
556D4—48no—
557D4—44no—
558D4—44no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—42no—
564D4—81no—
565D4—84no—
566D4—4882no—
567D4—2442no—
568D4—2441no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4C2×#1175819914yes—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—4882no—
588C2×C4—2441no—
589C2×C4#1451974611yescharacteristic
590C2×C4#1451974611yescharacteristic
591C2×C4#1451974611yescharacteristic
592C2×C4C2×#1175819912yes—
593C2×C4#1451980312yes—
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4C22×C22⋊D6111yescharacteristic
597C8—1222no—
598C8—2441no—
599C8—2442no—
600C8—2442no—
601C22—21no—
602C22—21no—
603C22—24no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—84no—
613C22—168no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22#1683989612yes—
618C22C2×#1451974611yescharacteristic
619C4—21no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—4881no—
626C4#1683989612yes—
627C4C22×#1175819911yescharacteristic
628C4C2×#1451980311yescharacteristic
629C4#1684056511yescharacteristic
630C4#1684057311yescharacteristic
631C2—22no—
632C2—44no—
633C2—82no—
634C2—82no—
635C2—9761no—
636C2C2×#1683989611yescharacteristic
637C2#2145512311yescharacteristic
638C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18662
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21457126C2semidirect
2. #21456394C2semidirect
3. #21455283C2semidirect
4. #21445054C2semidirect
5. #21444998C2semidirect
6. #21444908C2semidirect
7. #21444873C2semidirect
8. #21444496C2semidirect
9. #21444270C2semidirect
10. C2×D4⋊#92418C2semidirect
11. C61×#3496C2semidirect
13. #16840312C22semidirect
14. #16840165C22semidirect
15. #16840138C22semidirect
16. #16840110C22semidirect
17. #16840108C22semidirect
18. #16840042C22semidirect
23. #16839967C22semidirect
24. #16839957C22semidirect
25. #16839957C22semidirect
28. #16839786C22semidirect
29. C2×C4⋊#92418C22semidirect
31. C2×C4×#92418C22semidirect
32. #16839508C22semidirect
33. #16839506C22semidirect
34. D4⋊#92418C22semidirect
35. D4⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
157. C2×D4×C61D8semidirect
312. #3496D61semidirect
370. C61#18662semidirect
484. C2×D4#11758199semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214563942
#214552832
#214450542
#214449982
#214449082
#214448732
#214444962
#214442702
C2×D4⋊#924182
C61×#34962
#1866261