Finite Group Databasebeta

← #29669405

Subgroups — #29669405

Order 15616.

736 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458162C211yesmaximal, characteristic, complemented
2#21457626C211yesmaximal, characteristic, complemented
3#21457420C211yesmaximal, characteristic, complemented
4#21445491C211yesmaximal, characteristic, complemented
5#21445305C211yesmaximal, characteristic, complemented
6#21445258C211yesmaximal, characteristic, complemented
7#21445206C211yesmaximal, characteristic, complemented
8#21445160C211yesmaximal, characteristic, complemented
9#21445054C211yesmaximal, characteristic, complemented
10#21445052C211yesmaximal, characteristic, complemented
11#21444996C211yesmaximal, characteristic, complemented
12#21444908C211yesmaximal, characteristic, complemented
13#21444648C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3283C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17#16840416C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840165C2211yescharacteristic, complemented
21#16840135C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840075—21no—
25#16840075—21no—
26#16840073—21no—
27#16840042C2211yescharacteristic, complemented
28#16840039C2211yescharacteristic
29#16840039C2211yescharacteristic
30#16840009—21no—
31#16839996—21no—
32#16839996—21no—
33#16839967C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839914—21no—
37#16839908—21no—
38#16839895—21no—
39C4×#11758199C2211yescharacteristic
40#16839841C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43C2×#14519710—21no—
44C2×#14519697—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51D4⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C2×#135×C61—21no—
57C61×#547C2211yescharacteristic
58C61×#547C2211yescharacteristic, complemented
59C61×#494C2211yescharacteristic, complemented
60C61×#467—21no—
61C61×#461C2211yescharacteristic
62C61×#461C2211yescharacteristic, complemented
63C61×#452—21no—
64C61×#452—21no—
65C4×D8×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519801—44no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199D411yescharacteristic
77C2×#11758189—21no—
78C22×#92418—21no—
79C22×#92418—21no—
80#14519777C2311yescharacteristic
81#14519774C2311yescharacteristic
82#14519761—41no—
83#14519757—21no—
84#14519752—21no—
85#14519751—21no—
86#14519751—21no—
87#14519747—41no—
88C4×D244C2311yescharacteristic
89C2×C4⋊Dic61—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519710—41no—
96#14519710D411yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519700—21no—
101#14519698—21no—
102#14519698D411yescharacteristic
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×#41×C61—21no—
118#135×C61—44no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×D8×C61D411yescharacteristic
124C22×C8×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126C4⋊Q8×C61C2311yescharacteristic
127#123×C61—41no—
128#123×C61—41no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—21no—
131#121×C61—21no—
132#121×C61—21no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137C2×C4⋊C4×C61—21no—
138#107×C61D411yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147#41×C61—42no—
148#41×C61—42no—
149#41×C61—81no—
150#41×C61—81no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C2×D4×C61C2×D411yescharacteristic
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C22×C4×C61—41no—
163C2×C22⋊D61—41no—
164C2×C22⋊D61—41no—
165C22×Dic61—41no—
166#11758224—42no—
167#11758224—81no—
168D4×D61—42no—
169D4×D61—81no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—42no—
174C2×D244—41no—
175C2×D244—81no—
176C2×D244C2×D411yescharacteristic
177C2×C4×D61—21no—
178C2×Dic122—21no—
179Q16×C61—42no—
180Q16×C61—81no—
181SD16×C61—42no—
182SD16×C61—42no—
183D8×C61D814yescomplemented
184C2×C8×C61—21no—
185C2×C8×C61—42no—
186C2×C8×C61C2×D411yescharacteristic
187C2×C8×C61C2×D411yescharacteristic
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—41no—
193C4⋊C4×C61—41no—
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D411yescharacteristic
196C4⋊C4×C61C2×D411yescharacteristic
197C22⋊C4×C61—21no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61—41no—
201C22⋊C4×C61—41no—
202C42×C61C2411yescharacteristic
203#11758200—42no—
204#11758199—41no—
205#11758199—42no—
206#11758199—81no—
207#11758198—21no—
208#11758198—41no—
209#11758198—41no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—41no—
213C4⋊Dic61C2×D411yescharacteristic
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—42no—
219C2×#92418—81no—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222#11758190—42no—
223#11758190—81no—
224#11758189—42no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C22×D61—21no—
229C22×D61—41no—
230C22×D61—82no—
231C22×D61—82no—
232C22×D61—161no—
233Q8×C61—41no—
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61—41no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61—21no—
240D4×C61—21no—
241D4×C61—82no—
242D4×C61—82no—
243D4×C61C2×D812yes—
244D4×C61C2×D812yes—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—82no—
257C2×C4×C61—82no—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C22⋊D61—82no—
262C22⋊D61—82no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265C2×Dic61—82no—
266D244—21no—
267D244—21no—
268D244—41no—
269D244—41no—
270D244—41no—
271D244—41no—
272D244—81no—
273D244—82no—
274C4×D61—41no—
275C4×D61—41no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—21no—
279C8×C61—41no—
280C8×C61C2×D812yes—
281#92418—21no—
282#92418—21no—
283#92418—41no—
284#92418—41no—
285#92418—41no—
286#92418—41no—
287#19758—611nomaximal, Sylow
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61D4211yescharacteristic
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—164no—
305C4×C61—21no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—81no—
310C4×C61—81no—
311C4×C61C22×D811yescharacteristic
312C4×C61#57511yescharacteristic
313C4×C61#57611yescharacteristic
314C4×C61#57711yescharacteristic
315Dic61—41no—
316Dic61—81no—
317#3334—611no—
318#3294—611no—
319#3283D6111yescharacteristic, complemented
320#3272—611no—
321#1722—611no—
322#1689—611no—
323#1681—611no—
324#1673—611no—
325#1665—611no—
326#1647—611no—
327#1647—611no—
328#1639—611no—
329#1625—611no—
330#1581—611no—
331#1580—611no—
332C2×C61—42no—
333C2×C61—42no—
334C2×C61—81no—
335C2×C61#326711yescharacteristic
336C2×C61D4×D811yescharacteristic
337C2×C61#328911yescharacteristic
338D61—81no—
339D61—161no—
340C2×AGL(1,ℤ/8)—1221no—
341C2×#135—21no—
342#547—611no—
343#547C2×D6111yescharacteristic
344#547C2×D6111yescharacteristic, complemented
345D42—611no—
346#500—611no—
347#500—611no—
348#494C2×D6111yescharacteristic, complemented
349#492—611no—
350#486—611no—
351#486—611no—
352#480—1221no—
353#480—1221no—
354#480—1221no—
355#474—611no—
356#474—611no—
357#474—611no—
358#468—1221no—
359#467—21no—
360#466—1221no—
361#466—1221no—
362#461—611no—
363#461C2×D6111yescharacteristic
364#461C2×D6111yescharacteristic, complemented
365#459—611no—
366#459—611no—
367#452—21no—
368#452—21no—
369#450—1221no—
370#449—1221no—
371#447—1221no—
372C4×D8—611no—
373C4×D8C2×D6111yescharacteristic
374#436—611no—
375C8×D4—611no—
376C8×D4—611no—
377C8×D4C2×D6111yescharacteristic, complemented
378C2×#107—1221no—
379C2×#102—1221no—
380#335—611no—
381#334—611no—
382#332—611no—
383#332—611no—
384#332—611no—
385#331—611no—
386D4⋊C8—611no—
387D4⋊C8—611no—
388D4⋊C8—611no—
389D4⋊C8—611no—
390#322—611no—
391C61#1975811yescharacteristic, complemented, Sylow
392C2×#41—21no—
393C2×#41—21no—
394C2×#41—1221no—
395C22×D4—1221no—
396C22×D4—1221no—
397AGL(1,ℤ/8)—2444no—
398#135—44no—
399C2×Q16—21no—
400C2×Q16—21no—
401C2×SD16—21no—
402C2×SD16—21no—
403C2×SD16—1221no—
404C2×SD16—1221no—
405C2×SD16—1221no—
406C2×D8—611no—
407C2×D8—1221no—
408C2×D8—1221no—
409C2×D8C22⋊D6111yescharacteristic
410C2×M16—1221no—
411C22×C8—21no—
412C22×C8—1221no—
413C22×C8—1221no—
414C4⋊Q8—611no—
415C4⋊Q8C22×D6111yescharacteristic
416C4⋊Q8C22×D6111yescharacteristic
417#127—611no—
418#123—41no—
419#123—41no—
420#123—2441no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—1221no—
424#121—21no—
425#121—21no—
426#121—1221no—
427#121—1221no—
428#121—1221no—
429#120—2441no—
430C4×D4—611no—
431C4×D4C22×D6111yescharacteristic
432C4×D4C22×D6111yescharacteristic
433C4×D4C22×D6111yescharacteristic
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—1221no—
437#107—611no—
438#107—1221no—
439#107—1221no—
440#107—1221no—
441#107—1221no—
442#107—1221no—
443#107—2441no—
444#107C22⋊D6111yescharacteristic
445C4⋊C8—611no—
446C4⋊C8—611no—
447C4⋊C8—611no—
448C4⋊C8C22×D6111yescharacteristic
449#103—21no—
450#103—21no—
451#103—1221no—
452#102—611no—
453#102—611no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102—1221no—
458#102—1221no—
459#102—1221no—
460#102—1221no—
461#102—2441no—
462#102C22⋊D6111yescharacteristic
463#102C22⋊D6111yescharacteristic
464C22⋊C8—21no—
465C22⋊C8—1221no—
466C22⋊C8—1221no—
467C22⋊C8—1221no—
468C4×C8—611no—
469C4×C8—611no—
470C4×C8C22×D6111yescharacteristic
471C24—2441no—
472#41—42no—
473#41—42no—
474#41—81no—
475#41—81no—
476#41—4881no—
477#41—2442no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—1221no—
481C2×D4—41no—
482C2×D4—41no—
483C2×D4—611no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—4881no—
488C2×D4—4881no—
489C2×D4—2441no—
490C2×D4—2442no—
491C2×D4—2441no—
492C2×D4—2441no—
493C2×D4—2442no—
494C2×D4D4×D6111yescharacteristic
495C2×D4C2×C22⋊D6111yescharacteristic
496C2×D4C2×C22⋊D6111yescharacteristic
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—1221no—
503C22×C4—2441no—
504Q16—42no—
505Q16—81no—
506SD16—42no—
507SD16—42no—
508SD16—4881no—
509SD16—2442no—
510SD16—2442no—
511D8—4881no—
512D8—2441no—
513D8—2442no—
514D8#1175819914yescomplemented
515M16—2442no—
516C2×C8—21no—
517C2×C8—42no—
518C2×C8—611no—
519C2×C8—611no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—4881no—
525C2×C8—2442no—
526C2×C8C2×C22⋊D6111yescharacteristic
527C2×C8C2×C22⋊D6111yescharacteristic
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—611no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—2441no—
538C4⋊C4D4×D6111yescharacteristic
539C4⋊C4C2×C22⋊D6111yescharacteristic
540C4⋊C4C2×C22⋊D6111yescharacteristic
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—1221no—
547C22⋊C4—2441no—
548C22⋊C4—2441no—
549C42C23×D6111yescharacteristic
550C23—21no—
551C23—21no—
552C23—21no—
553C23—1221no—
554C23—9761no—
555C23—4882no—
556C23—4882no—
557C23—2441no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—2441no—
563Q8—2441no—
564D4—21no—
565D4—21no—
566D4—21no—
567D4—21no—
568D4—82no—
569D4—82no—
570D4—1221no—
571D4—1221no—
572D4—4881no—
573D4—4882no—
574D4—4882no—
575D4—4882no—
576D4—2441no—
577D4—2441no—
578D4—2441no—
579D4—2441no—
580D4C2×#1175819912yes—
581D4C2×#1175819912yes—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—1221no—
596C2×C4—4882no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C2×C4#1451982211yescharacteristic
603C8—21no—
604C8—41no—
605C8—1221no—
606C8—1221no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C8—2441no—
611C8C2×#1175819912yes—
612C22—22no—
613C22—22no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—9764no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—2441no—
626C22—2441no—
627C22—2441no—
628C22D4×C22⋊D6111yescharacteristic
629C4—21no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—4881no—
636C4—2441no—
637C4C22×#1175819911yescharacteristic
638C4#1684055911yescharacteristic
639C4#1684056511yescharacteristic
640C4#1684056911yescharacteristic
641C2—42no—
642C2—42no—
643C2—81no—
644C2—9761no—
645C2—4881no—
646C2#2145734411yescharacteristic
647C2D4×#1175819911yescharacteristic
648C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3283C2
socleC22×C61D42
hypercenter#3283D61
hypocenterC61#19758
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #21458162C2semidirect
2. #21457626C2semidirect
3. #21457420C2semidirect
4. #21445491C2semidirect
5. #21445305C2semidirect
6. #21445258C2semidirect
7. #21445206C2semidirect
8. #21445160C2semidirect
9. #21445054C2semidirect
10. #21445052C2semidirect
11. #21444996C2semidirect
12. #21444908C2semidirect
13. #21444648C2semidirect
14. #21444643C2semidirect
15. C61×#3283C2semidirect
19. #16840165C22semidirect
20. #16840165C22semidirect
27. #16840042C22semidirect
42. D4×#92418C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839508C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839483C22semidirect
58. C61×#547C22semidirect
59. C61×#494C22semidirect
62. C61×#461C22semidirect
66. C8×D4×C61C22semidirect
183. D8×C61D8semidirect
319. #3283D61semidirect
344. #547C2×D61semidirect
348. #494C2×D61semidirect
364. #461C2×D61semidirect
377. C8×D4C2×D61semidirect
391. C61#19758semidirect
514. D8#11758199semidirect

Maximal subgroups

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

subgroupindex
#214581622
#214576262
#214574202
#214454912
#214453052
#214452582
#214452062
#214451602
#214450542
#214450522
#214449962
#214449082
#214446482
#214446432
C61×#32832
#1975861