Finite Group Databasebeta

← #30619232

Subgroups — #30619232

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462612C211yesmaximal, characteristic, complemented
2#21462545C211yesmaximal, characteristic
3#21462004C211yesmaximal, characteristic, complemented
4#21461954C211yesmaximal, characteristic
5#21461903C211yesmaximal, characteristic, complemented
6#21461902C211yesmaximal, characteristic, complemented
7#21461901C211yesmaximal, characteristic, complemented
8#21461900C211yesmaximal, characteristic, complemented
9#21461789C211yesmaximal, characteristic, complemented
10#21461788C211yesmaximal, characteristic, complemented
11#21461787C211yesmaximal, characteristic, complemented
12#21461786C211yesmaximal, characteristic, complemented
13#21461761C211yesmaximal, characteristic, complemented
14#21461535C211yesmaximal, characteristic, complemented
15#21456161C211yesmaximal, characteristic
16C2×#14519865C2211yescharacteristic
17C2×#14519864C2211yescharacteristic
18#16840814—21no—
19#16840814—21no—
20#16840813—21no—
21#16840813—21no—
22#16840794—21no—
23#16840794—21no—
24#16840791—21no—
25#16840791—21no—
26C2×#14519848C2211yescharacteristic
27C2×#14519848C2211yescharacteristic
28C2×#14519847C2211yescharacteristic
29C2×#14519847C2211yescharacteristic
30#16840769C2211yescharacteristic
31#16840769C2211yescharacteristic
32#16840767C2211yescharacteristic
33#16840766C2211yescharacteristic
34C2×#14519842C2211yescharacteristic
35C2×#14519842C2211yescharacteristic
36C2×#14519841C2211yescharacteristic
37C2×#14519841C2211yescharacteristic
38#16840737C2211yescharacteristic
39#16840736C2211yescharacteristic
40#16840727C2211yescharacteristic
41#16840727C2211yescharacteristic
42#16840721C2211yescharacteristic
43C42×#6923C2211yescharacteristic
44#16840632C2211yescharacteristic
45#16840632C2211yescharacteristic
46#16840632C2211yescharacteristic
47#16840632C2211yescharacteristic
48(C2×#6923)⋊C8C2211yescharacteristic
49(C2×#6923)⋊C8C2211yescharacteristic
50(C2×#6923)⋊C8C2211yescharacteristic
51(C2×#6923)⋊C8C2211yescharacteristic
52#16840317C2211yescharacteristic
53#16840311C2211yescharacteristic
54#16840129C411yescharacteristic, complemented
55#16840129C411yescharacteristic, complemented
56#16840125C411yescharacteristic, complemented
57#16840125C411yescharacteristic, complemented
58#16840124C411yescharacteristic, complemented
59#16840122C411yescharacteristic, complemented
60#16840121C411yescharacteristic, complemented
61#16839757C2211yescharacteristic
62#16839757C2211yescharacteristic
63#16839755C2211yescharacteristic
64#16839755C2211yescharacteristic
65#97×D61C2211yescharacteristic
66C61×#489C411yescharacteristic, complemented
67#14519865—41no—
68#14519865—41no—
69#14519864—41no—
70#14519864—41no—
71C2×#11758213—21no—
72C2×#11758213C2311yescharacteristic
73C2×#11758213C2311yescharacteristic
74C2×#11758213C2311yescharacteristic
75C2×#11758213C2311yescharacteristic
76C2×#11758217—21no—
77C2×D61⋊C8—21no—
78C2×D61⋊C8C2311yescharacteristic
79C2×D61⋊C8C2311yescharacteristic
80C22×#92420—21no—
81C2×C4×#6923—21no—
82C2×C4×#6923—21no—
83C2×C4×#6923C2311yescharacteristic
84C2×C4×#6923C2311yescharacteristic
85#14519848—41no—
86#14519848—41no—
87#14519847—41no—
88#14519847—41no—
89#14519844—21no—
90#14519844—21no—
91#14519844—21no—
92#14519844—21no—
93C4⋊#92420—21no—
94C4⋊#92420—21no—
95C4⋊#92420—21no—
96C4⋊#92420—21no—
97#14519842—41no—
98#14519842—41no—
99#14519841—41no—
100#14519841—41no—
101#14519834—21no—
102#14519834—21no—
103C22⋊#92420—21no—
104C22⋊#92420—21no—
105#14519832—21no—
106#14519832—21no—
107#14519832D411yescharacteristic
108#14519832D411yescharacteristic
109#14519832D411yescharacteristic
110#14519832D411yescharacteristic
111C4×#92420—21no—
112C4×#92420—21no—
113C2×#11758226C2311yescharacteristic
114C2×#11758224C2311yescharacteristic
115C2×C8×D61C2311yescharacteristic
116C2×C8×D61C2311yescharacteristic
117#14519777C2×C411yescharacteristic
118#14519775C2×C411yescharacteristic
119#14519773C2×C411yescharacteristic
120#14519768C2×C411yescharacteristic
121#14519768C2×C411yescharacteristic
122#14519765C2×C411yescharacteristic
123#14519757—21no—
124#14519753—21no—
125#14519750—21no—
126#14519750—21no—
127Q8×Dic61—21no—
128#14519742—21no—
129#14519737—21no—
130#14519737—21no—
131#14519735C2311yescharacteristic
132#14519735C2311yescharacteristic
133C42×D61C2311yescharacteristic
134#14519702C2×C411yescharacteristic
135#14519702C2×C411yescharacteristic
136#14519701C2×C411yescharacteristic
137#14519701C2×C411yescharacteristic
138#14519700C2×C411yescharacteristic
139#14519700C2×C411yescharacteristic
140#14519699C2×C411yescharacteristic
141#14519699C2×C411yescharacteristic
142#14519698C2×C411yescharacteristic
143#14519698C2×C411yescharacteristic
144#14519697C2×C411yescharacteristic
145#14519697C2×C411yescharacteristic
146#14519688C2×C411yescharacteristic
147#14519687C2×C411yescharacteristic
148#14519687C2×C411yescharacteristic
149C4⋊Q8×C61C2×C411yescharacteristic
150#124×C61C2×C411yescharacteristic
151#103×C61C2×C411yescharacteristic
152#103×C61C2×C411yescharacteristic
153#102×C61C2×C411yescharacteristic
154#102×C61C2×C411yescharacteristic
155#97×C61C2×C411yescharacteristic
156C22×#6923—21no—
157C22×#6923—21no—
158C2×Q8×C61—21no—
159C2×Q8×C61C22×C411yescharacteristic
160C2×D4×C61C22×C411yescharacteristic
161C2×C22⋊D61—21no—
162C2×C22⋊D61—21no—
163C2×C22⋊D61—21no—
164C22×Dic61—21no—
165#11758226—21no—
166#11758226—21no—
167#11758226—21no—
168#11758226—21no—
169#11758224—21no—
170#11758224—21no—
171#11758224—21no—
172#11758224—21no—
173C2×D244—21no—
174C2×D244C22×C411yescharacteristic
175C2×C4×D61—21no—
176C2×C4×D61#4111yescharacteristic
177C2×C4×D61#4111yescharacteristic
178C2×C4×D61C2411yescharacteristic
179C2×Dic122—21no—
180C2×Dic122C22×C411yescharacteristic
181#11758217—42no—
182C2×#92420—21no—
183C2×#92420—21no—
184C2×#92420—21no—
185C2×#92420—21no—
186C2×#92420—42no—
187C2×#92420C2×D411yescharacteristic
188C2×#92420C2×D411yescharacteristic
189C2×#92420C2×D411yescharacteristic
190C2×#92420C2×D411yescharacteristic
191C4×#6923—21no—
192C4×#6923—21no—
193C4×#6923—21no—
194C4×#6923—21no—
195C4×#6923—42no—
196C4×#6923—42no—
197#11758213—21no—
198#11758213—21no—
199#11758213—21no—
200#11758213—21no—
201#11758213—21no—
202#11758213—21no—
203#11758213—21no—
204#11758213—21no—
205#11758213—42no—
206D61⋊C8—21no—
207D61⋊C8—21no—
208D61⋊C8—21no—
209D61⋊C8—21no—
210D61⋊C8—42no—
211C2×C8×C61C22×C411yescharacteristic
212C2×C8×C61C22×C411yescharacteristic
213C4⋊C4×C61—21no—
214C4⋊C4×C61C22×C411yescharacteristic
215C4⋊C4×C61C22×C411yescharacteristic
216C22⋊C4×C61—21no—
217C22⋊C4×C61—21no—
218C42×C61C22×C411yescharacteristic
219#11758198—21no—
220#11758198—21no—
221#11758198—21no—
222#11758198—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61—21no—
225C4⋊Dic61—21no—
226C4⋊Dic61C22×C411yescharacteristic
227C4⋊Dic61C22×C411yescharacteristic
228#11758196—21no—
229#11758196—21no—
230#11758196—21no—
231#11758196—21no—
232C4×Dic61—21no—
233C4×Dic61—21no—
234C4×Dic61C22×C411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C2×#92418C22×C411yescharacteristic
238C2×#92418C22×C411yescharacteristic
239C8×D61—21no—
240C8×D61—21no—
241C8×D61—21no—
242C8×D61—21no—
243C23×C61—21no—
244C22×D61—21no—
245C22×D61C2×#4111yescharacteristic
246C2×#6923—22no—
247C2×#6923—22no—
248C2×#6923—41no—
249C2×#6923—41no—
250C2×#6923—41no—
251C2×#6923—41no—
252Q8×C61—21no—
253Q8×C61—21no—
254Q8×C61—81no—
255D4×C61—21no—
256D4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61C4×D411yescharacteristic
261C2×C4×C61C4×D411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C22⋊D61—41no—
264C22⋊D61—41no—
265C22⋊D61—81no—
266C22⋊D61—81no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61C4×D411yescharacteristic
273C2×Dic61C4×D411yescharacteristic
274C2×Dic61C22×D411yescharacteristic
275D244—21no—
276D244—21no—
277D244—41no—
278D244—41no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61H(5,𝔽2)11yescharacteristic
284C4×D61H(5,𝔽2)11yescharacteristic
285C4×D612−1+411yescharacteristic
286C4×D612−1+411yescharacteristic
287Dic122—21no—
288Dic122—21no—
289Dic122—81no—
290#92420—21no—
291#92420—21no—
292#92420—21no—
293#92420—21no—
294#92420—41no—
295#92420—41no—
296#92420—41no—
297#92420—41no—
298C8×C61—21no—
299C8×C61—21no—
300#92418—21no—
301#92418—21no—
302#22237—611nomaximal, Sylow
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61C2×C4×D411yescharacteristic
307C2×D61—21no—
308C2×D61—21no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61#54211yescharacteristic
313C2×D61#54211yescharacteristic
314#6923—42no—
315#6923—42no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61#52011yescharacteristic
322C4×C61#52011yescharacteristic
323Dic61—21no—
324Dic61—21no—
325Dic61—41no—
326Dic61—41no—
327Dic61—41no—
328Dic61#53511yescharacteristic
329Dic61#53611yescharacteristic
330#3135—611no—
331#2961—611no—
332#2932—611no—
333#1994—611no—
334#1962—611no—
335#1929—611no—
336#1929—611no—
337#1929—611no—
338#1929—611no—
339#1866—611no—
340#1866—611no—
341#1866—611no—
342#1866—611no—
343#1849—611no—
344#1745—611no—
345C2×C61—81no—
346C2×C61#232611yescharacteristic
347C2×C61#295611yescharacteristic
348C2×C61#295611yescharacteristic
349D61—21no—
350D61—21no—
351D61—81no—
352C2×#131—611no—
353C2×#131—611no—
354#533—611no—
355#532—611no—
356#491—611no—
357#491—611no—
358#490—611no—
359#490—611no—
360#489—611no—
361#489—611no—
362#489—611no—
363#489#692311yescharacteristic, complemented
364#446—1221no—
365#446—1221no—
366#446—1221no—
367#446—1221no—
368#435—1221no—
369#435—1221no—
370#435—1221no—
371#435—1221no—
372C2×#108—611no—
373C2×#108—611no—
374C2×#108—611no—
375C2×#108—611no—
376#424—611no—
377#424—611no—
378#423—611no—
379#423—611no—
380C2×#104—611no—
381C2×#104—611no—
382C2×#104—611no—
383C2×#104—611no—
384#416—611no—
385#416—611no—
386#416—611no—
387#416—611no—
388#408—611no—
389#408—611no—
390#405—611no—
391#405—611no—
392C2×#97—611no—
393C2×#97—611no—
394C43—611no—
395#339—611no—
396#339—611no—
397#339—611no—
398#339—611no—
399#336—611no—
400#336—611no—
401#336—611no—
402#336—611no—
403C61#2223711yescharacteristic, complemented, Sylow
404C2×#41—611no—
405C2×#41—611no—
406#131—2441no—
407#131—2441no—
408#131—2441no—
409#131—2441no—
410C2×M16—611no—
411C2×M16—611no—
412C2×M16—611no—
413C2×M16—611no—
414C2×M16—1221no—
415C2×M16—1221no—
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8—1221no—
421C22×C8—1221no—
422C4⋊Q8—611no—
423C4⋊Q8—611no—
424C4⋊Q8C2×#692311yescharacteristic
425#127—611no—
426#125—611no—
427#125—611no—
428#124—611no—
429#124C2×#692311yescharacteristic
430C22⋊Q8—1221no—
431C22⋊Q8—1221no—
432#121—1221no—
433#121—1221no—
434C4×Q8—1221no—
435C4×Q8—1221no—
436C4×D4—1221no—
437C4×D4—1221no—
438#117—611no—
439#117—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—1221no—
444C2×C42—1221no—
445#108—2441no—
446#108—2441no—
447#108—2441no—
448#108—2441no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C4⋊C8—1221no—
452C4⋊C8—1221no—
453C4⋊C8—1221no—
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456C4⋊C8—1221no—
457#104—2441no—
458#104—2441no—
459#104—2441no—
460#104—2441no—
461#103—611no—
462#103—611no—
463#103—611no—
464#103—611no—
465#103—611no—
466#103—611no—
467#103C2×#692311yescharacteristic
468#103C2×#692311yescharacteristic
469#102—611no—
470#102—611no—
471#102—611no—
472#102—611no—
473#102—611no—
474#102—611no—
475#102C2×#692311yescharacteristic
476#102C2×#692311yescharacteristic
477C22⋊C8—1221no—
478C22⋊C8—1221no—
479C22⋊C8—1221no—
480C22⋊C8—1221no—
481#97—611no—
482#97—611no—
483#97—611no—
484#97—611no—
485#97—611no—
486#97—611no—
487#97—611no—
488#97—1221no—
489#97—1221no—
490#97C2×#692311yescharacteristic
491C4×C8—1221no—
492C4×C8—1221no—
493#41—1221no—
494#41—1221no—
495#41—1221no—
496#41—1221no—
497#41—1221no—
498#41—1221no—
499#41—1221no—
500#41—1221no—
501C2×Q8—21no—
502C2×Q8—611no—
503C2×Q8—1221no—
504C2×Q8C22×#692311yescharacteristic
505C2×D4—611no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4—1221no—
509C2×D4—1221no—
510C2×D4C22×#692311yescharacteristic
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518M16—1221no—
519M16—1221no—
520M16—1221no—
521M16—1221no—
522M16—1221no—
523M16—1221no—
524M16—1221no—
525M16—1221no—
526M16—2442no—
527M16—2442no—
528C2×C8—611no—
529C2×C8—611no—
530C2×C8—611no—
531C2×C8—611no—
532C2×C8—611no—
533C2×C8—611no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—2442no—
547C2×C8—2442no—
548C2×C8C22×#692311yescharacteristic
549C2×C8C22×#692311yescharacteristic
550C4⋊C4—21no—
551C4⋊C4—611no—
552C4⋊C4—611no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4C22×#692311yescharacteristic
561C4⋊C4C22×#692311yescharacteristic
562C22⋊C4—21no—
563C22⋊C4—21no—
564C22⋊C4—1221no—
565C22⋊C4—1221no—
566C22⋊C4—1221no—
567C22⋊C4—1221no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—1221no—
572C42—1221no—
573C42—1221no—
574C42—1221no—
575C42—1221no—
576C42—1221no—
577C42—2442no—
578C42—2442no—
579C42C22×#692311yescharacteristic
580C23—21no—
581C23—611no—
582C23—1221no—
583Q8—21no—
584Q8—21no—
585Q8—81no—
586Q8—1221no—
587Q8—1221no—
588Q8—4881no—
589D4—21no—
590D4—21no—
591D4—1221no—
592D4—1221no—
593D4—4881no—
594D4—4881no—
595D4—2441no—
596D4—2441no—
597D4—2441no—
598D4—2441no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—611no—
606C2×C4—611no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4D4×#692311yescharacteristic
625C2×C4D4×#692311yescharacteristic
626C2×C4C23×#692311yescharacteristic
627C8—21no—
628C8—21no—
629C8—1221no—
630C8—1221no—
631C8—1221no—
632C8—1221no—
633C8—1221no—
634C8—1221no—
635C8—2441no—
636C8—2441no—
637C8—2441no—
638C8—2441no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—611no—
643C22—611no—
644C22—1221no—
645C22—1221no—
646C22—2441no—
647C22—2441no—
648C22—2441no—
649C22C2×D4×#692311yescharacteristic
650C4—21no—
651C4—21no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—611no—
656C4—611no—
657C4—1221no—
658C4—1221no—
659C4—2442no—
660C4—2442no—
661C4—2441no—
662C4—2441no—
663C4—2441no—
664C4#1684083111yescharacteristic
665C4#1684083111yescharacteristic
666C2—81no—
667C2—1221no—
668C2—1221no—
669C2—4881no—
670C2#2146230911yescharacteristic
671C2#2146259311yescharacteristic
672C2#2146259411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#489C4
socleC22×C61C2×C4×D4
prime radical#16840311C22
cosocleC2×C4×D61C24
hypercenter#489#6923
hypocenterC61#22237
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #21462612C2semidirect
3. #21462004C2semidirect
5. #21461903C2semidirect
6. #21461902C2semidirect
7. #21461901C2semidirect
8. #21461900C2semidirect
9. #21461789C2semidirect
10. #21461788C2semidirect
11. #21461787C2semidirect
12. #21461786C2semidirect
13. #21461761C2semidirect
14. #21461535C2semidirect
54. #16840129C4semidirect
55. #16840129C4semidirect
56. #16840125C4semidirect
57. #16840125C4semidirect
58. #16840124C4semidirect
59. #16840122C4semidirect
60. #16840121C4semidirect
66. C61×#489C4semidirect
363. #489#6923semidirect
403. C61#22237semidirect

Maximal subgroups

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

subgroupindex
#214626122
#214625452
#214620042
#214619542
#214619032
#214619022
#214619012
#214619002
#214617892
#214617882
#214617872
#214617862
#214617612
#214615352
#214561612
#2223761