Finite Group Databasebeta

← #29671287

Subgroups — #29671287

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457634C211yesmaximal, characteristic, complemented
2#21457419C211yesmaximal, characteristic, complemented
3#21457415C211yesmaximal, characteristic, complemented
4#21445410C211yesmaximal, characteristic, complemented
5#21445286C211yesmaximal, characteristic, complemented
6#21445282C211yesmaximal, characteristic, complemented
7#21445256C211yesmaximal, characteristic, complemented
8#21445253C211yesmaximal, characteristic, complemented
9#21445054C211yesmaximal, characteristic, complemented
10#21445053C211yesmaximal, characteristic, complemented
11#21444873C211yesmaximal, characteristic, complemented
12#21444872C211yesmaximal, characteristic, complemented
13#21444704C211yesmaximal, characteristic, complemented
14C8×#11758199C211yesmaximal, characteristic, complemented
15C61×#3294C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17C2×#14519802—21no—
18#16840417C2211yescharacteristic
19D4×D244C2211yescharacteristic, complemented
20#16840165C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840136C2211yescharacteristic, complemented
23#16840134C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic, complemented
25#16840077—21no—
26#16840042C2211yescharacteristic, complemented
27#16840010—21no—
28#16840006—21no—
29#16839997—21no—
30#16839967C2211yescharacteristic, complemented
31#16839964C2211yescharacteristic, complemented
32#16839957C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839954C2211yescharacteristic
35#16839913—21no—
36#16839912—21no—
37#16839907—21no—
38#16839905—21no—
39#16839894—21no—
40C4×#11758199C2211yescharacteristic
41#16839840C2211yescharacteristic, complemented
42#16839840C2211yescharacteristic, complemented
43C8×D244C2211yescharacteristic, complemented
44#102×D61—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic, complemented
51D4⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#547C2211yescharacteristic
57D42×C61C2211yescharacteristic, complemented
58C61×#486C2211yescharacteristic, complemented
59C61×#480—21no—
60C61×#474C2211yescharacteristic, complemented
61C61×#466—21no—
62C61×#459C2211yescharacteristic
63C61×#447—21no—
64C4×D8×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic, complemented
66C2×#102×C61—21no—
67C2×#11758222—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519803—44no—
71#14519802—44no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758199—21no—
81C2×#11758199—41no—
82C2×#11758199D411yescharacteristic
83C2×#11758194—21no—
84C2×#11758194—21no—
85C2×C8×D61—21no—
86#14519777C2311yescharacteristic
87#14519774C2311yescharacteristic
88#14519759—41no—
89#14519756—21no—
90#14519752—21no—
91#14519751—21no—
92#14519748—21no—
93#14519747—41no—
94C4×D244C2311yescharacteristic
95C4⋊C4×D61—21no—
96#14519710—21no—
97#14519710—41no—
98#14519710D411yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—21no—
103#14519700—21no—
104#14519699—21no—
105#14519699—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697—21no—
111#14519697D411yescharacteristic
112C22⋊#92418—21no—
113C22⋊#92418—21no—
114#14519689—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C2×#41×C61—21no—
118C22×D4×C61—21no—
119C22×D4×C61—21no—
120C2×D8×C61—21no—
121C2×D8×C61D411yescharacteristic
122C22×C8×C61—21no—
123C4⋊Q8×C61C2311yescharacteristic
124#127×C61C2311yescharacteristic
125#123×C61—41no—
126C22⋊Q8×C61—21no—
127#121×C61—21no—
128#121×C61—21no—
129#121×C61—21no—
130#120×C61—41no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134C2×C4⋊C4×C61—21no—
135#107×C61—21no—
136#107×C61D411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—41no—
147C23×D61—41no—
148#41×C61—42no—
149#41×C61—81no—
150C2×Q8×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—42no—
157C2×D4×C61—42no—
158C2×D4×C61—41no—
159C2×D4×C61—81no—
160C2×D4×C61—81no—
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C2×D4×C61C2×D411yescharacteristic
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—41no—
168C2×C22⋊D61—41no—
169C2×C22⋊D61—41no—
170D4×D61—42no—
171D4×D61—81no—
172#11758222—42no—
173#11758222—81no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244—42no—
178C2×D244—41no—
179C2×D244—81no—
180C2×D244C2×D411yescharacteristic
181C2×C4×D61—21no—
182C2×C4×D61—41no—
183C2×Dic122—21no—
184D8×C61—41no—
185D8×C61—81no—
186C2×C8×C61—21no—
187C2×C8×C61—81no—
188C2×C8×C61C2×D411yescharacteristic
189C2×C8×C61C2×D411yescharacteristic
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—41no—
193C4⋊C4×C61C2×D411yescharacteristic
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D411yescharacteristic
196C22⋊C4×C61—21no—
197C22⋊C4×C61—21no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—41no—
200C22⋊C4×C61—41no—
201C42×C61C2411yescharacteristic
202C22⋊Dic61—41no—
203#11758201—42no—
204#11758201—42no—
205#11758201—81no—
206#11758200—42no—
207#11758200—42no—
208#11758200—81no—
209#11758199—42no—
210#11758199—42no—
211#11758199—82no—
212#11758199—81no—
213#11758199D814yescomplemented
214#11758198—21no—
215#11758198—41no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61C2×D411yescharacteristic
219#11758196—41no—
220#11758194—42no—
221#11758194—42no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418C2×D411yescharacteristic
227C2×#92418C2×D411yescharacteristic
228C8×D61—81no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—41no—
233C23×C61—82no—
234C23×C61—82no—
235C23×C61—161no—
236C22×D61—21no—
237C22×D61—41no—
238C22×D61—82no—
239C22×D61—82no—
240C22×D61—161no—
241Q8×C61—41no—
242Q8×C61—41no—
243D4×C61—21no—
244D4×C61—21no—
245D4×C61—21no—
246D4×C61—21no—
247D4×C61—21no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—41no—
251D4×C61—41no—
252D4×C61—81no—
253D4×C61—82no—
254D4×C61—82no—
255D4×C61—82no—
256D4×C61C2×D812yes—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—82no—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C22⋊D61—82no—
272C22⋊D61—82no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275D244—21no—
276D244—41no—
277D244—41no—
278D244—41no—
279D244—41no—
280D244—81no—
281D244—82no—
282D244C2×D812yes—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—82no—
286Dic122—41no—
287Dic122—41no—
288C8×C61—21no—
289C8×C61—21no—
290C8×C61—41no—
291#92418—22no—
292#92418—21no—
293#92418—41no—
294#92418—41no—
295#92418—41no—
296#92418C2×D812yes—
297#19884—611nomaximal, Sylow
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—164no—
311C22×C61D4211yescharacteristic
312C2×D61—22no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—164no—
319C4×C61—21no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—81no—
324C4×C61C22×D811yescharacteristic
325C4×C61#57511yescharacteristic
326C4×C61#57611yescharacteristic
327C4×C61#57611yescharacteristic
328Dic61—41no—
329Dic61—81no—
330#3294—611no—
331#3294D6111yescharacteristic, complemented
332#3272—611no—
333#3272—611no—
334#1706—611no—
335#1685—611no—
336#1685—611no—
337#1681—611no—
338#1681—611no—
339#1647—611no—
340#1647—611no—
341#1619—611no—
342#1619—611no—
343#1589—611no—
344C8×D8—611no—
345C2×C61—42no—
346C2×C61—81no—
347C2×C61—81no—
348C2×C61—161no—
349C2×C61#327311yescharacteristic
350C2×C61D4×D811yescharacteristic
351C2×C61#329111yescharacteristic
352D61—42no—
353D61—161no—
354C2×AGL(1,ℤ/8)—1221no—
355C2×AGL(1,ℤ/8)—1221no—
356#547—611no—
357#547C2×D6111yescharacteristic
358D42—611no—
359D42C2×D6111yescharacteristic, complemented
360#500—611no—
361#493—611no—
362#492—611no—
363#492—611no—
364#486—611no—
365#486C2×D6111yescharacteristic, complemented
366#480—21no—
367#480—1221no—
368#474—611no—
369#474C2×D6111yescharacteristic, complemented
370#468—1221no—
371#468—1221no—
372#466—21no—
373#466—1221no—
374#461—611no—
375#461—611no—
376#459—611no—
377#459—611no—
378#459—611no—
379#459C2×D6111yescharacteristic
380#450—1221no—
381#450—1221no—
382#449—1221no—
383#449—1221no—
384#447—21no—
385#447—1221no—
386C4×D8—611no—
387C4×D8C2×D6111yescharacteristic
388#436—611no—
389#436—611no—
390C8×D4—611no—
391C8×D4C2×D6111yescharacteristic, complemented
392C2×#102—21no—
393C2×#102—1221no—
394#335—611no—
395#335—611no—
396#332—611no—
397#332—611no—
398#331—611no—
399#331—611no—
400D4⋊C8—611no—
401D4⋊C8—611no—
402D4⋊C8—611no—
403D4⋊C8—611no—
404C82—611no—
405C61#1988411yescharacteristic, complemented, Sylow
406C2×#41—21no—
407C2×#41—1221no—
408C22×D4—21no—
409C22×D4—21no—
410C22×D4—1221no—
411C22×D4—1221no—
412AGL(1,ℤ/8)—2444no—
413AGL(1,ℤ/8)—2444no—
414C2×SD16—1221no—
415C2×SD16—1221no—
416C2×SD16—1221no—
417C2×SD16—1221no—
418C2×SD16—1221no—
419C2×SD16—1221no—
420C2×D8—21no—
421C2×D8—611no—
422C2×D8—1221no—
423C2×D8—1221no—
424C2×D8—1221no—
425C2×D8—2441no—
426C2×D8C22⋊D6111yescharacteristic
427C2×M16—1221no—
428C2×M16—1221no—
429C22×C8—21no—
430C22×C8—1221no—
431C4⋊Q8—611no—
432C4⋊Q8C22×D6111yescharacteristic
433#127—611no—
434#127C22×D6111yescharacteristic
435#123—41no—
436#123—2441no—
437C22⋊Q8—21no—
438C22⋊Q8—1221no—
439#121—21no—
440#121—21no—
441#121—21no—
442#121—1221no—
443#121—1221no—
444#121—1221no—
445#120—41no—
446#120—2441no—
447C4×D4—611no—
448C4×D4C22×D6111yescharacteristic
449C4×D4C22×D6111yescharacteristic
450C4×D4C22×D6111yescharacteristic
451C2×C4⋊C4—21no—
452C2×C4⋊C4—1221no—
453#107—21no—
454#107—611no—
455#107—1221no—
456#107—2441no—
457#107C22⋊D6111yescharacteristic
458C4⋊C8—611no—
459C4⋊C8—611no—
460C4⋊C8—611no—
461C4⋊C8C22×D6111yescharacteristic
462#103—1221no—
463#103—1221no—
464#102—21no—
465#102—21no—
466#102—21no—
467#102—21no—
468#102—611no—
469#102—611no—
470#102—1221no—
471#102—1221no—
472#102—1221no—
473#102—1221no—
474#102—1221no—
475#102—1221no—
476#102C22⋊D6111yescharacteristic
477#102C22⋊D6111yescharacteristic
478C22⋊C8—21no—
479C22⋊C8—1221no—
480C22⋊C8—1221no—
481C22⋊C8—1221no—
482C4×C8—611no—
483C4×C8—611no—
484C4×C8C22×D6111yescharacteristic
485C24—41no—
486C24—2441no—
487#41—42no—
488#41—81no—
489#41—4881no—
490#41—2442no—
491C2×Q8—21no—
492C2×Q8—1221no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—41no—
497C2×D4—41no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—41no—
501C2×D4—81no—
502C2×D4—81no—
503C2×D4—611no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—4881no—
508C2×D4—4881no—
509C2×D4—2441no—
510C2×D4—2442no—
511C2×D4—2441no—
512C2×D4—2442no—
513C2×D4—2441no—
514C2×D4D4×D6111yescharacteristic
515C2×D4C2×C22⋊D6111yescharacteristic
516C2×D4C2×C22⋊D6111yescharacteristic
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—41no—
521C22×C4—1221no—
522C22×C4—2441no—
523SD16—4881no—
524SD16—4881no—
525SD16—2442no—
526SD16—2442no—
527SD16—2442no—
528SD16—2442no—
529D8—41no—
530D8—81no—
531D8—614no—
532D8—4882no—
533D8—4881no—
534D8—2442no—
535D8—2442no—
536M16—2442no—
537M16—2442no—
538C2×C8—21no—
539C2×C8—81no—
540C2×C8—611no—
541C2×C8—611no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—4881no—
547C2×C8C2×C22⋊D6111yescharacteristic
548C2×C8C2×C22⋊D6111yescharacteristic
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—41no—
552C4⋊C4—611no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—2441no—
556C4⋊C4D4×D6111yescharacteristic
557C4⋊C4C2×C22⋊D6111yescharacteristic
558C4⋊C4C2×C22⋊D6111yescharacteristic
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—1221no—
565C22⋊C4—2441no—
566C22⋊C4—2441no—
567C42C23×D6111yescharacteristic
568C23—21no—
569C23—21no—
570C23—21no—
571C23—41no—
572C23—82no—
573C23—82no—
574C23—161no—
575C23—1221no—
576C23—9761no—
577C23—4882no—
578C23—4882no—
579C23—2441no—
580Q8—41no—
581Q8—41no—
582Q8—2441no—
583Q8—2441no—
584D4—21no—
585D4—21no—
586D4—21no—
587D4—21no—
588D4—21no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—81no—
594D4—82no—
595D4—82no—
596D4—82no—
597D4—612no—
598D4—1221no—
599D4—4881no—
600D4—4882no—
601D4—4882no—
602D4—4882no—
603D4—2441no—
604D4—2441no—
605D4—2441no—
606D4—2441no—
607D4D8×D6112yes—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—82no—
619C2×C4—1221no—
620C2×C4—4882no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4C2×D4×D6111yescharacteristic
625C2×C4C22×C22⋊D6111yescharacteristic
626C2×C4#1451982211yescharacteristic
627C8—21no—
628C8—21no—
629C8—41no—
630C8—612no—
631C8—1222no—
632C8—1221no—
633C8—2441no—
634C8—2441no—
635C8—2441no—
636C22—22no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—164no—
649C22—1222no—
650C22—9764no—
651C22—4881no—
652C22—4881no—
653C22—4881no—
654C22—2441no—
655C22—2441no—
656C22D4×C22⋊D6111yescharacteristic
657C4—21no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—81no—
662C4—4881no—
663C4—2441no—
664C4C2×D8×D6111yescharacteristic
665C4#1684056011yescharacteristic
666C4#1684056311yescharacteristic
667C4#1684056511yescharacteristic
668C2—42no—
669C2—81no—
670C2—81no—
671C2—161no—
672C2—9761no—
673C2—2442no—
674C2#2145742611yescharacteristic
675C2D8×C22⋊D6111yescharacteristic
676C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3294C2
socleC22×C61D42
hypercenter#3294D61
hypocenterC61#19884
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3294

Complements

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

NH ≅ G/Ntype
1. #21457634C2semidirect
2. #21457419C2semidirect
3. #21457415C2semidirect
4. #21445410C2semidirect
5. #21445286C2semidirect
6. #21445282C2semidirect
7. #21445256C2semidirect
8. #21445253C2semidirect
9. #21445054C2semidirect
10. #21445053C2semidirect
11. #21444873C2semidirect
12. #21444872C2semidirect
13. #21444704C2semidirect
14. C8×#11758199C2semidirect
15. C61×#3294C2semidirect
19. D4×D244C22semidirect
20. #16840165C22semidirect
22. #16840136C22semidirect
23. #16840134C22semidirect
24. #16840109C22semidirect
26. #16840042C22semidirect
30. #16839967C22semidirect
31. #16839964C22semidirect
41. #16839840C22semidirect
42. #16839840C22semidirect
43. C8×D244C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
50. #16839505C22semidirect
51. D4⋊#92418C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
57. D42×C61C22semidirect
58. C61×#486C22semidirect
60. C61×#474C22semidirect
65. C8×D4×C61C22semidirect
213. #11758199D8semidirect
331. #3294D61semidirect
359. D42C2×D61semidirect
365. #486C2×D61semidirect
369. #474C2×D61semidirect
391. C8×D4C2×D61semidirect
405. C61#19884semidirect

Maximal subgroups

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

subgroupindex
#214576342
#214574192
#214574152
#214454102
#214452862
#214452822
#214452562
#214452532
#214450542
#214450532
#214448732
#214448722
#214447042
C8×#117581992
C61×#32942
#1988461