Finite Group Databasebeta

← #29750068

Subgroups — #29750068

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456173C211yesmaximal, characteristic, complemented
2#21455653C211yesmaximal, characteristic, complemented
3#21454647C211yesmaximal, characteristic, complemented
4#21447177C211yesmaximal, characteristic, complemented
5#21447167C211yesmaximal, characteristic, complemented
6#21447110C211yesmaximal, characteristic, complemented
7#21446832C211yesmaximal, characteristic
8#21446641C211yesmaximal, characteristic, complemented
9#21446533C211yesmaximal, characteristic, complemented
10#21446484C211yesmaximal, characteristic, complemented
11#21446480C211yesmaximal, characteristic, complemented
12#21446214C211yesmaximal, characteristic, complemented
13#21446052C211yesmaximal, characteristic, complemented
14#21445705C211yesmaximal, characteristic, complemented
15C61×#3096C211yesmaximal, characteristic, complemented
16C2×#14519802C2211yescharacteristic
17#16840390C2211yescharacteristic
18C2×#14519773C2211yescharacteristic
19#16840236C2211yescharacteristic
20#16840126—21no—
21#16840121—21no—
22#16840116—21no—
23#16840108—21no—
24#16840065—21no—
25#16840061—21no—
26#16840051—21no—
27#16840041—21no—
28#16839872—21no—
29#16839860—21no—
30C4×#11758201—21no—
31C4×#11758199—21no—
32C2×#14519713C2211yescharacteristic
33#16839805C2211yescharacteristic
34#16839802C2211yescharacteristic
35#16839788C2211yescharacteristic
36C2×#14519704C2211yescharacteristic
37#16839767C2211yescharacteristic
38#16839766C2211yescharacteristic
39#16839763C2211yescharacteristic
40#16839758C2211yescharacteristic
41C2×#14519699C2211yescharacteristic
42C2×#14519698C2211yescharacteristic
43C2×#14519697C2211yescharacteristic
44C2×#14519694C2211yescharacteristic
45C2×#14519693C2211yescharacteristic
46#16839718C2211yescharacteristic
47#16839718C2211yescharacteristic
48C2×#14519687C2211yescharacteristic
49#16839627C2211yescharacteristic
50#16839531C2211yescharacteristic
51#16839530C2211yescharacteristic
52#16839526C2211yescharacteristic
53#16839523C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839521C2211yescharacteristic
56C61×#535C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#482—21no—
59C61×#482—21no—
60C61×#481—21no—
61C61×#480—21no—
62C61×#428C2211yescharacteristic
63C61×#427C2211yescharacteristic
64C61×#418C2211yescharacteristic
65C2×#102×C61C2211yescharacteristic
66C61×#408C2211yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C22⋊D61—41no—
69#14519802—22no—
70#14519802—22no—
71C2×#11758201—21no—
72C2×#11758199—21no—
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77#14519777—21no—
78#14519776—21no—
79#14519773—22no—
80#14519773—22no—
81#14519769—21no—
82#14519768—21no—
83#14519754—41no—
84C4×D244—21no—
85C4×D244—21no—
86#14519739—21no—
87#14519739—21no—
88#14519736C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C22⋊C4×D61—41no—
91C2×C4×Dic61C2311yescharacteristic
92#14519713—22no—
93#14519710—21no—
94#14519710—21no—
95#14519708—21no—
96#14519708—21no—
97#14519706—21no—
98#14519706—21no—
99#14519704—22no—
100#14519701—21no—
101#14519701D411yescharacteristic, complemented
102#14519701D411yescharacteristic, complemented
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698—21no—
110#14519698D411yescharacteristic, complemented
111#14519698D411yescharacteristic, complemented
112#14519697—21no—
113#14519697—21no—
114#14519697—21no—
115#14519697—21no—
116#14519694—22no—
117#14519693—22no—
118#14519688—21no—
119#14519688—21no—
120#14519687—22no—
121C4×#92418—21no—
122C4×#92418—21no—
123#14519684—41no—
124C2×#41×C61C2311yescharacteristic
125C22×D4×C61C2311yescharacteristic
126C2×M16×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128#127×C61—21no—
129#124×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133C4×Q8×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61—41no—
136#117×C61—21no—
137#117×C61C2311yescharacteristic
138#117×C61C2311yescharacteristic
139C2×C4⋊C4×C61—21no—
140C2×C4⋊C4×C61C2311yescharacteristic
141#107×C61—21no—
142#107×C61D411yescharacteristic, complemented
143#107×C61D411yescharacteristic, complemented
144#106×C61—21no—
145#106×C61D411yescharacteristic, complemented
146#106×C61D411yescharacteristic, complemented
147#102×C61—21no—
148#102×C61—21no—
149#102×C61—21no—
150#102×C61—21no—
151C22⋊C8×C61—21no—
152C22⋊C8×C61—21no—
153C24×C61—21no—
154C23×D61—21no—
155#41×C61—21no—
156#41×C61—21no—
157#41×C61—21no—
158#41×C61—21no—
159C2×Q8×C61C2×D411yescharacteristic
160C2×D4×C61—21no—
161C2×D4×C61—21no—
162C2×D4×C61—21no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—21no—
167C2×D4×C61—41no—
168C2×D4×C61—41no—
169C2×D4×C61C2×D411yescharacteristic
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—41no—
173C22×C4×C61C2411yescharacteristic
174C2×C22⋊D61—44no—
175C2×C22⋊D61—82no—
176C2×C22⋊D61—82no—
177C22×Dic61—21no—
178C22×Dic61—21no—
179C2×D244—21no—
180C2×D244—21no—
181C2×D244—21no—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244#4112yes—
185C2×C4×D61—41no—
186C2×C4×D61—41no—
187C2×Dic122—41no—
188M16×C61—21no—
189M16×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61C2×D411yescharacteristic
194C2×C8×C61C2×D411yescharacteristic
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—41no—
198C4⋊C4×C61C2×D411yescharacteristic
199C4⋊C4×C61C2×D411yescharacteristic
200C4⋊C4×C61C2×D411yescharacteristic
201C4⋊C4×C61C2×D411yescharacteristic
202C22⋊C4×C61—21no—
203C22⋊C4×C61—21no—
204C22⋊C4×C61—41no—
205C22⋊C4×C61—41no—
206C22⋊C4×C61—41no—
207C42×C61—21no—
208C42×C61—21no—
209C42×C61—41no—
210C22⋊Dic61—21no—
211#11758201—24no—
212#11758199—24no—
213#11758198—41no—
214#11758198—41no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—41no—
218C4⋊Dic61C2×D411yescharacteristic
219C4⋊Dic61C2×D411yescharacteristic
220#11758196—41no—
221#11758196—41no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61#4112yes—
225#11758194—21no—
226#11758194—21no—
227#11758194—21no—
228#11758194—21no—
229#11758194—21no—
230#11758194#4112yes—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418—21no—
236C2×#92418C2×D411yescharacteristic
237C2×#92418C2×D411yescharacteristic
238C23×C61—21no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61—41no—
242C23×C61—81no—
243C23×C61—81no—
244C23×C61H(5,𝔽2)11yescharacteristic
245C22×D61—22no—
246C22×D61—41no—
247C22×D61—41no—
248C22×D61—81no—
249C22×D61—81no—
250Q8×C61—21no—
251Q8×C61—21no—
252D4×C61—21no—
253D4×C61—21no—
254D4×C61—41no—
255D4×C61—41no—
256D4×C61—41no—
257D4×C61—41no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61—161no—
261D4×C61—161no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—81no—
271C2×C4×C61—161no—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C612−1+411yescharacteristic
278C22⋊D61—88no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—22no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—81no—
286D244—22no—
287D244—22no—
288D244—41no—
289D244—41no—
290C4×D61—161no—
291C4×D61—161no—
292Dic122—161no—
293C8×C61—21no—
294C8×C61—21no—
295C8×C61—41no—
296#92418—22no—
297#92418—21no—
298#92418—21no—
299#92418—41no—
300#92418—41no—
301#26855—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—82no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—161no—
312C22×C61#54211yescharacteristic
313C22×C61D4211yescharacteristic
314C22×C61#55111yescharacteristic
315C2×D61—42no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—82no—
319C2×D61—161no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—41no—
325C4×C61—81no—
326C4×C61#52511yescharacteristic
327C4×C61#53211yescharacteristic
328C4×C61#54711yescharacteristic
329C4×C61#54811yescharacteristic
330Dic61—42no—
331Dic61—41no—
332Dic61—41no—
333Dic61—81no—
334#3136—611no—
335#3096D6111yescharacteristic, complemented
336#3081—611no—
337#2946—611no—
338#2063—611no—
339#2061—611no—
340#2051—611no—
341#2002—611no—
342#1961—611no—
343#1938—611no—
344#1931—611no—
345#1931—611no—
346#1887—611no—
347#1860—611no—
348#1773—611no—
349C2×C61—21no—
350C2×C61—21no—
351C2×C61—81no—
352C2×C61—161no—
353C2×C61#246111yescharacteristic
354C2×C61#328811yescharacteristic
355C2×C61#328811yescharacteristic
356D61—82no—
357C2×AGL(1,ℤ/8)—611no—
358#544—611no—
359#535C2×D6111yescharacteristic
360C2×#127—611no—
361#520C2×D6111yescharacteristic
362#518—611no—
363#490—1221no—
364#489—1221no—
365#488—1221no—
366#486—1221no—
367#482—21no—
368#482—21no—
369#481—21no—
370#480—21no—
371#478—1221no—
372#478—1221no—
373#476—1221no—
374#474—1221no—
375#442—1221no—
376#440—1221no—
377C4×SD16—1221no—
378C4×D8—1221no—
379C2×#108—611no—
380#428—611no—
381#428—611no—
382#428C2×D6111yescharacteristic
383#427C2×D6111yescharacteristic
384#424—611no—
385C2×#104—611no—
386#418—611no—
387#418—611no—
388#418C2×D6111yescharacteristic
389#417—611no—
390#416—611no—
391C2×#102—611no—
392C2×#102—611no—
393C2×#102—611no—
394C2×#102C2×D6111yescharacteristic
395C2×#100—611no—
396C2×#100—611no—
397#408C2×D6111yescharacteristic
398#405—611no—
399#405—611no—
400C2×#97—611no—
401#382—611no—
402#343—611no—
403#343—611no—
404#341—611no—
405#339—611no—
406#339—611no—
407#339—611no—
408C61#2685511yescharacteristic, complemented, Sylow
409C2×#41C22×D6111yescharacteristic
410C22×D4—611no—
411C22×D4—2441no—
412C22×D4C22×D6111yescharacteristic
413AGL(1,ℤ/8)—1222no—
414AGL(1,ℤ/8)—1222no—
415C2×SD16—1221no—
416C2×D8—1221no—
417C2×M16—611no—
418C2×M16—611no—
419C2×M16—611no—
420C2×M16C22×D6111yescharacteristic
421C22×C8—611no—
422C22×C8C22×D6111yescharacteristic
423C4⋊Q8—1221no—
424C4⋊Q8—1221no—
425#127—21no—
426#127—1222no—
427#127—1222no—
428#125—1221no—
429#125—1221no—
430#124—21no—
431C22⋊Q8—2441no—
432#121—21no—
433#121—21no—
434#120—41no—
435C4×Q8—21no—
436C4×D4—21no—
437C4×D4—41no—
438C4×D4—1221no—
439C4×D4—1221no—
440C4×D4—1221no—
441C4×D4—1221no—
442#117—21no—
443#117—611no—
444#117C22×D6111yescharacteristic
445#117C22×D6111yescharacteristic
446C2×C4⋊C4—21no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C22⋊C4—2441no—
450C2×C42—611no—
451#108—1222no—
452#107—21no—
453#107—1221no—
454#107—1221no—
455#107C22⋊D6111yescharacteristic, complemented
456#107C22⋊D6111yescharacteristic, complemented
457#106—21no—
458#106—1221no—
459#106—1221no—
460#106C22⋊D6111yescharacteristic, complemented
461#106C22⋊D6111yescharacteristic, complemented
462C4⋊C8—1221no—
463C4⋊C8—1221no—
464#104—1222no—
465#103—611no—
466#103—611no—
467#103—1221no—
468#102—21no—
469#102—21no—
470#102—21no—
471#102—21no—
472#102—611no—
473#102—611no—
474#102—1221no—
475#102—1221no—
476#102—1221no—
477#102—1221no—
478#102—1221no—
479#102—1221no—
480#102—1221no—
481#102—1221no—
482#102—1221no—
483#102—1221no—
484#102—1221no—
485#100—1222no—
486#100—1222no—
487C22⋊C8—21no—
488C22⋊C8—21no—
489#97—1221no—
490#97—1221no—
491#97—1222no—
492C4×C8—1221no—
493C4×C8—1221no—
494#95—2441no—
495C24—21no—
496C24—1221no—
497#41—21no—
498#41—21no—
499#41—21no—
500#41—21no—
501C2×Q8—2441no—
502C2×Q8D4×D6111yescharacteristic
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—41no—
511C2×D4—41no—
512C2×D4—612no—
513C2×D4—1221no—
514C2×D4—1221no—
515C2×D4—1221no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4—4882no—
519C2×D4—4882no—
520C2×D4—2444no—
521C2×D4D4×D6111yescharacteristic
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—41no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—2441no—
528C22×C4—2441no—
529C22×C4C23×D6111yescharacteristic
530SD16—1224no—
531D8—1224no—
532M16—21no—
533M16—21no—
534M16—612no—
535M16—1221no—
536M16—1221no—
537M16—1221no—
538M16—1221no—
539M16—1221no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—611no—
544C2×C8—611no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8C2×C22⋊D6111yescharacteristic
551C2×C8C2×C22⋊D6111yescharacteristic
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—41no—
555C4⋊C4—611no—
556C4⋊C4—611no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—2441no—
560C4⋊C4—2441no—
561C4⋊C4—2441no—
562C4⋊C4C2×C22⋊D6111yescharacteristic
563C4⋊C4C2×C22⋊D6111yescharacteristic
564C4⋊C4C2×C22⋊D6111yescharacteristic
565C4⋊C4C2×C22⋊D6111yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—41no—
571C22⋊C4—1221no—
572C22⋊C4—2441no—
573C22⋊C4—2441no—
574C42—21no—
575C42—21no—
576C42—41no—
577C42—612no—
578C42—1221no—
579C42—1221no—
580C23—21no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—81no—
585C23—81no—
586C23—1222no—
587C23—4881no—
588C23—4881no—
589C23—2441no—
590C23—2441no—
591C23#1451982211yescharacteristic
592Q8—21no—
593Q8—21no—
594Q8—9761no—
595D4—21no—
596D4—21no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—41no—
603D4—161no—
604D4—161no—
605D4—1222no—
606D4—1222no—
607D4—4888no—
608D4—2441no—
609D4—2441no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—81no—
619C2×C4—161no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1222no—
623C2×C4—9761no—
624C2×C4—9761no—
625C2×C4—4881no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C22×C22⋊D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4#1451982411yescharacteristic
635C8—21no—
636C8—21no—
637C8—41no—
638C8—1222no—
639C8—1221no—
640C8—1221no—
641C8—2441no—
642C8—2441no—
643C22—21no—
644C22—21no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—82no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—161no—
653C22—9761no—
654C22—4882no—
655C22—4881no—
656C22—4881no—
657C22—2442no—
658C22#1684038011yescharacteristic
659C22D4×C22⋊D6111yescharacteristic
660C22#1684044011yescharacteristic
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—81no—
667C4—4881no—
668C4—2442no—
669C4—2441no—
670C4—2441no—
671C4#1684027011yescharacteristic
672C4#1684030811yescharacteristic
673C4#1684041011yescharacteristic
674C4#1684041911yescharacteristic
675C2—21no—
676C2—21no—
677C2—81no—
678C2—161no—
679C2—4882no—
680C2#2144978511yescharacteristic
681C2#2145758111yescharacteristic
682C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3096C2
socleC22×C61D42
prime radical#21446832C2
hypercenter#3096D61
hypocenterC61#26855
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3096

Complements

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

NH ≅ G/Ntype
1. #21456173C2semidirect
2. #21455653C2semidirect
3. #21454647C2semidirect
4. #21447177C2semidirect
5. #21447167C2semidirect
6. #21447110C2semidirect
8. #21446641C2semidirect
9. #21446533C2semidirect
10. #21446484C2semidirect
11. #21446480C2semidirect
12. #21446214C2semidirect
13. #21446052C2semidirect
14. #21445705C2semidirect
15. C61×#3096C2semidirect
101. #14519701D4semidirect
102. #14519701D4semidirect
110. #14519698D4semidirect
111. #14519698D4semidirect
142. #107×C61D4semidirect
143. #107×C61D4semidirect
145. #106×C61D4semidirect
146. #106×C61D4semidirect
335. #3096D61semidirect
408. C61#26855semidirect
455. #107C22⋊D61semidirect
456. #107C22⋊D61semidirect
460. #106C22⋊D61semidirect
461. #106C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214561732
#214556532
#214546472
#214471772
#214471672
#214471102
#214468322
#214466412
#214465332
#214464842
#214464802
#214462142
#214460522
#214457052
C61×#30962
#2685561