Finite Group Databasebeta

← #29669596

Subgroups — #29669596

Order 15616.

736 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458061C211yesmaximal, characteristic, complemented
2#21457749C211yesmaximal, characteristic, complemented
3#21457632C211yesmaximal, characteristic, complemented
4#21445457C211yesmaximal, characteristic, complemented
5#21445378C211yesmaximal, characteristic, complemented
6#21445270C211yesmaximal, characteristic, complemented
7#21445226C211yesmaximal, characteristic, complemented
8#21445171C211yesmaximal, characteristic, complemented
9#21444998C211yesmaximal, characteristic, complemented
10#21444935C211yesmaximal, characteristic, complemented
11#21444906C211yesmaximal, characteristic, complemented
12#21444901C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15D4×D8×C61C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic, complemented
17#16840416C2211yescharacteristic, complemented
18#16840134C2211yescharacteristic, complemented
19#16840113C2211yescharacteristic, complemented
20#16840108C2211yescharacteristic, complemented
21#16840108C2211yescharacteristic, complemented
22#16840108C2211yescharacteristic, complemented
23#16840105—21no—
24#16840083—21no—
25#16840073—21no—
26#16840052C2211yescharacteristic
27#16840052C2211yescharacteristic, complemented
28#16840042C2211yescharacteristic
29#16840013—21no—
30#16839996—21no—
31#16839990—21no—
32#16839967C2211yescharacteristic, complemented
33#16839954C2211yescharacteristic, complemented
34#16839954C2211yescharacteristic, complemented
35#16839904—21no—
36#16839896—21no—
37#16839896—21no—
38C4×#11758199C2211yescharacteristic
39#16839840C2211yescharacteristic, complemented
40C8×D244C2211yescharacteristic, complemented
41D4×#92418C2211yescharacteristic, complemented
42#16839767—21no—
43#16839755—21no—
44C2×#14519698—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic, complemented
51D4⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C22×D8×C61—21no—
57D42×C61C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#466—21no—
61C61×#459C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#447—21no—
64C61×#447—21no—
65C4×D8×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758224—21no—
68C2×#11758200—21no—
69C2×#11758200—41no—
70C2×#11758199—21no—
71C2×#11758199—21no—
72C2×#11758199D411yescharacteristic
73C2×#11758194—21no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519763—41no—
79#14519761—41no—
80#14519757—21no—
81#14519751—21no—
82#14519751—21no—
83#14519748—21no—
84C4×D244C2311yescharacteristic
85#14519736—21no—
86#14519735—21no—
87C2×C4⋊Dic61—21no—
88#14519710—21no—
89#14519710D411yescharacteristic
90#14519708—21no—
91#14519708—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519702—21no—
96#14519702—21no—
97#14519700—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519698D411yescharacteristic
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×D4×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C22×D4×C61—21no—
119C2×D8×C61—21no—
120C2×D8×C61—21no—
121C2×D8×C61—21no—
122C2×D8×C61—21no—
123C2×D8×C61—44no—
124C2×D8×C61D411yescharacteristic
125C22×C8×C61—21no—
126#127×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#121×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133#120×C61—41no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C24×C61—41no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61—42no—
158C2×D4×C61—42no—
159C2×D4×C61—42no—
160C2×D4×C61—42no—
161C2×D4×C61—81no—
162C2×D4×C61—81no—
163C2×D4×C61—81no—
164C2×D4×C61—81no—
165C2×D4×C61C2×D411yescharacteristic
166C2×D4×C61C2×D411yescharacteristic
167C2×D4×C61C2×D411yescharacteristic
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C2×C22⋊D61—41no—
172C2×C22⋊D61—41no—
173C22×Dic61—41no—
174#11758224—42no—
175#11758224—81no—
176C2×D244C2×D411yescharacteristic
177C2×C4×D61—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180D8×C61—22no—
181D8×C61—42no—
182D8×C61—42no—
183D8×C61—42no—
184D8×C61—81no—
185C2×C8×C61—21no—
186C2×C8×C61—42no—
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—41no—
196C22⋊C4×C61—41no—
197C42×C61C2411yescharacteristic
198C22⋊Dic61—41no—
199#11758200—81no—
200#11758200—82no—
201#11758199—22no—
202#11758199—81no—
203#11758199—81no—
204#11758198—21no—
205#11758198—41no—
206#11758198—41no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—41no—
212C4⋊Dic61C2×D411yescharacteristic
213C4×Dic61—41no—
214#11758194—81no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—81no—
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×D411yescharacteristic
222C8×D61—81no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—82no—
229C23×C61—82no—
230C23×C61—82no—
231C23×C61—82no—
232C23×C61—161no—
233C23×C61—161no—
234C22×D61—21no—
235D4×C61—21no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—81no—
248D4×C61—81no—
249D4×C61—82no—
250D4×C61—82no—
251D4×C61—82no—
252D4×C61—82no—
253D4×C61#13512yes—
254D4×C61AGL(1,ℤ/8)12yes—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C22⋊D61—82no—
268C22⋊D61—161no—
269C2×Dic61—21no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—82no—
273D244—21no—
274D244—21no—
275C4×D61—41no—
276C4×D61—41no—
277Dic122—41no—
278Dic122—41no—
279Dic122—81no—
280C8×C61—21no—
281C8×C61—21no—
282C8×C61—41no—
283#92418—21no—
284#92418—21no—
285#92418—41no—
286#92418—41no—
287#92418—41no—
288#92418—41no—
289#19771—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—164no—
306C22×C61—164no—
307C22×C61D4211yescharacteristic
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C4×C61—21no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61C2×#13511yescharacteristic
316C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
317C4×C61#57611yescharacteristic
318C4×C61#57611yescharacteristic
319Dic61—41no—
320Dic61—81no—
321Dic61—81no—
322#3325—611no—
323#3303—611no—
324#3294—611no—
325D4×D8D6111yescharacteristic, complemented
326#1717—611no—
327#1701—611no—
328#1683—611no—
329#1677—611no—
330#1667—611no—
331#1639—611no—
332#1629—611no—
333#1625—611no—
334#1624—611no—
335#1581—611no—
336#1580—611no—
337C2×C61—42no—
338C2×C61—42no—
339C2×C61—81no—
340C2×C61—161no—
341C2×C61—161no—
342C2×C61#326411yescharacteristic
343C2×C61#328011yescharacteristic
344C2×C61#328811yescharacteristic
345D61—81no—
346C22×D8—21no—
347#555—611no—
348#547—611no—
349D42C2×D6111yescharacteristic
350D42C2×D6111yescharacteristic
351#493C2×D6111yescharacteristic
352#492—611no—
353#487—611no—
354#486—611no—
355#486—611no—
356#486—611no—
357#485—1221no—
358#482—1221no—
359#480—1221no—
360#476—611no—
361#476—611no—
362#474—611no—
363#469—1221no—
364#466—21no—
365#466—1221no—
366#465—1221no—
367#461—611no—
368#459—611no—
369#459—611no—
370#459C2×D6111yescharacteristic
371#459C2×D6111yescharacteristic
372#449—1221no—
373#447—21no—
374#447—21no—
375#447—1221no—
376#447—1221no—
377C4×D8—611no—
378C4×D8C2×D6111yescharacteristic
379#436—611no—
380C8×D4—611no—
381C8×D4—611no—
382C8×D4C2×D6111yescharacteristic
383#418—1221no—
384#416—1221no—
385C2×#102—1221no—
386#335—611no—
387#334—611no—
388#333—611no—
389#332—611no—
390#331—611no—
391#331—611no—
392D4⋊C8—611no—
393D4⋊C8—611no—
394D4⋊C8—611no—
395D4⋊C8—611no—
396#322—611no—
397C61#1977111yescharacteristic, complemented, Sylow
398C2×#41—1221no—
399C22×D4—21no—
400C22×D4—21no—
401C22×D4—21no—
402C22×D4—21no—
403C2×SD16—1221no—
404C2×SD16—2441no—
405C2×D8—21no—
406C2×D8—21no—
407C2×D8—21no—
408C2×D8—21no—
409C2×D8—44no—
410C2×D8—611no—
411C2×D8—1221no—
412C2×D8—1221no—
413C2×D8C22⋊D6111yescharacteristic
414C2×M16—1221no—
415C22×C8—21no—
416C22×C8—1221no—
417C22×C8—1221no—
418C4⋊Q8—611no—
419C4⋊Q8—611no—
420#127C22×D6111yescharacteristic
421#127C22×D6111yescharacteristic
422#124—2441no—
423#123—2441no—
424C22⋊Q8—1221no—
425#121—21no—
426#121—21no—
427#121—21no—
428#121—21no—
429#121—1221no—
430#121—1221no—
431#121—1221no—
432#120—41no—
433#120—41no—
434C4×D4—611no—
435C4×D4C22×D6111yescharacteristic
436C4×D4C22×D6111yescharacteristic
437C4×D4C22×D6111yescharacteristic
438#117—1221no—
439#117—1221no—
440C2×C4⋊C4—1221no—
441#107—611no—
442#107—1221no—
443#107C22⋊D6111yescharacteristic
444#106—1221no—
445#106—1221no—
446C4⋊C8—611no—
447C4⋊C8—611no—
448C4⋊C8—611no—
449C4⋊C8C22×D6111yescharacteristic
450#103—1221no—
451#103—1221no—
452#103—1221no—
453#102—21no—
454#102—21no—
455#102—611no—
456#102—611no—
457#102—1221no—
458#102—1221no—
459#102—1221no—
460#102—1221no—
461#102—1221no—
462#102—1221no—
463#102—1221no—
464#102—1221no—
465#102—1221no—
466#102—2441no—
467#102C22⋊D6111yescharacteristic
468#102C22⋊D6111yescharacteristic
469C22⋊C8—21no—
470C22⋊C8—1221no—
471C22⋊C8—1221no—
472C22⋊C8—1221no—
473C4×C8—611no—
474C4×C8—611no—
475C4×C8C22×D6111yescharacteristic
476C24—41no—
477C24—41no—
478#41—4881no—
479#41—2442no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—41no—
489C2×D4—41no—
490C2×D4—41no—
491C2×D4—41no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—81no—
497C2×D4—81no—
498C2×D4—81no—
499C2×D4—81no—
500C2×D4—611no—
501C2×D4—2441no—
502C2×D4—2441no—
503C2×D4D4×D6111yescharacteristic
504C2×D4C2×C22⋊D6111yescharacteristic
505C2×D4C2×C22⋊D6111yescharacteristic
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—1221no—
510C22×C4—2441no—
511SD16—4881no—
512SD16—4882no—
513D8—22no—
514D8—42no—
515D8—42no—
516D8—42no—
517D8—81no—
518D8—1222no—
519D8—4881no—
520D8—4881no—
521M16—4881no—
522C2×C8—21no—
523C2×C8—42no—
524C2×C8—611no—
525C2×C8—611no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—4881no—
531C2×C8—4881no—
532C2×C8C2×C22⋊D6111yescharacteristic
533C2×C8C2×C22⋊D6111yescharacteristic
534C4⋊C4—611no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—2441no—
540C4⋊C4D4×D6111yescharacteristic
541C4⋊C4C2×C22⋊D6111yescharacteristic
542C4⋊C4C2×C22⋊D6111yescharacteristic
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—1221no—
549C22⋊C4—2441no—
550C22⋊C4—2441no—
551C22⋊C4—2441no—
552C42—2441no—
553C42C23×D6111yescharacteristic
554C23—21no—
555C23—21no—
556C23—21no—
557C23—41no—
558C23—41no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—161no—
564C23—161no—
565C23—1221no—
566Q8—4881no—
567Q8—2441no—
568Q8—2441no—
569D4—21no—
570D4—21no—
571D4—21no—
572D4—21no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—81no—
582D4—81no—
583D4—82no—
584D4—82no—
585D4—82no—
586D4—82no—
587D4—1221no—
588D4—1221no—
589D4—9761no—
590D4—4882no—
591D4#1451979612yes—
592D4#1451980312yes—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—1221no—
603C2×C4—4882no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C22×C22⋊D6111yescharacteristic
610C2×C4#1451982211yescharacteristic
611C8—21no—
612C8—21no—
613C8—41no—
614C8—1221no—
615C8—1221no—
616C8—2441no—
617C8—2441no—
618C8—2441no—
619C8—2441no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—164no—
636C22—164no—
637C22—2441no—
638C22—2441no—
639C22—2441no—
640C22D4×C22⋊D6111yescharacteristic
641C4—21no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—4881no—
646C4—4881no—
647C4—2441no—
648C4C2×#1451979611yescharacteristic
649C4C2×#1451980311yescharacteristic
650C4#1684056311yescharacteristic
651C4#1684056511yescharacteristic
652C2—42no—
653C2—42no—
654C2—81no—
655C2—161no—
656C2—161no—
657C2—4881no—
658C2#2145729511yescharacteristic
659C2#2145750411yescharacteristic
660C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×D8×C61C2
socleC22×C61D42
hypercenterD4×D8D61
hypocenterC61#19771
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. #21458061C2semidirect
2. #21457749C2semidirect
3. #21457632C2semidirect
4. #21445457C2semidirect
5. #21445378C2semidirect
6. #21445270C2semidirect
7. #21445226C2semidirect
8. #21445171C2semidirect
9. #21444998C2semidirect
10. #21444935C2semidirect
11. #21444906C2semidirect
12. #21444901C2semidirect
13. #21444649C2semidirect
14. #21444641C2semidirect
15. D4×D8×C61C2semidirect
16. #16840477C22semidirect
17. #16840416C22semidirect
18. #16840134C22semidirect
19. #16840113C22semidirect
20. #16840108C22semidirect
21. #16840108C22semidirect
22. #16840108C22semidirect
27. #16840052C22semidirect
32. #16839967C22semidirect
33. #16839954C22semidirect
34. #16839954C22semidirect
39. #16839840C22semidirect
40. C8×D244C22semidirect
41. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839509C22semidirect
49. #16839506C22semidirect
50. #16839506C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
54. #16839489C22semidirect
55. #16839484C22semidirect
325. D4×D8D61semidirect
397. C61#19771semidirect

Maximal subgroups

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

subgroupindex
#214580612
#214577492
#214576322
#214454572
#214453782
#214452702
#214452262
#214451712
#214449982
#214449352
#214449062
#214449012
#214446492
#214446412
D4×D8×C612
#1977161