Finite Group Databasebeta

← #30620605

Subgroups — #30620605

Order 15616.

1006 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462329C211yesmaximal, characteristic, complemented
2#21462322C211yesmaximal, characteristic
3#21461908C212yesmaximal, complemented
4#21461898C212yesmaximal, complemented
5#21461784C212yesmaximal, complemented
6#21461724C211yesmaximal, characteristic, complemented
7#21461695C211yesmaximal, characteristic, complemented
8#21461659C211yesmaximal, characteristic
9#102×#6923C212yesmaximal, complemented
10C8×C22⋊#6923C211yesmaximal, characteristic, complemented
11#416×D61C211yesmaximal, characteristic
12C2×D4×#6923C2212yes—
13C2×#14519845C2212yes—
14C2×#14519840C2212yes—
15C2×#14519839C2212yes—
16D61⋊(C22⋊C8)C2211yescharacteristic
17C2×C8×#6923C2212yes—
18(C2×#6923)⋊Q8C2212yes—
19#16840713C2212yes—
20#16840712—22no—
21#16840710—22no—
22#16840706—21no—
23#16840705—21no—
24#16840704C2211yescharacteristic
25#16840704C2211yescharacteristic
26C4⋊C4×#6923C2212yes—
27C4×C22⋊#6923C2211yescharacteristic
28#16840698—21no—
29C2×(C2×#6923)⋊C4—21no—
30#16840636C2212yes—
31#16840635C2212yes—
32#16840635C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C2×#41×D61C2211yescharacteristic
35C22×C8×D61C2211yescharacteristic
36#117×D61C2211yescharacteristic
37#16839759C411yescharacteristic, complemented
38#16839758C411yescharacteristic, complemented
39#16839757C411yescharacteristic, complemented
40#16839757C411yescharacteristic, complemented
41#16839756C411yescharacteristic, complemented
42#16839755C411yescharacteristic, complemented
43#16839755C411yescharacteristic, complemented
44#103×D61C2211yescharacteristic
45#103×D61C2211yescharacteristic
46#102×D61C2211yescharacteristic
47#102×D61C2211yescharacteristic
48C61×#416C411yescharacteristic, complemented
49C23×#6923—22no—
50C2×D61⋊C8C2312yes—
51D4×#6923—24no—
52D4×#6923—42no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923C2312yes—
55C2×C4⋊#6923C2312yes—
56C2×C22⋊#6923—22no—
57C2×C22⋊#6923—21no—
58C2×C22⋊#6923—21no—
59C2×C22⋊#6923D412yes—
60C2×C4×#6923—22no—
61C2×C4×#6923C2312yes—
62#14519845—42no—
63#14519840—42no—
64#14519839—24no—
65#14519834—21no—
66C22⋊#92420—21no—
67C8×#6923—42no—
68(C2×#6923)⋊C4—22no—
69(C2×#6923)⋊C4—22no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—21no—
73(C2×#6923)⋊C4—21no—
74(C2×#6923)⋊C4D411yescharacteristic
75(C2×#6923)⋊C4D411yescharacteristic
76C2×#11758226C2×C411yescharacteristic
77C2×#11758222—21no—
78C2×#11758222C2×C411yescharacteristic
79C2×#11758224—21no—
80C2×#11758224C2×C411yescharacteristic
81#41×D61—24no—
82#41×D61—42no—
83C2×Q8×D61C2311yescharacteristic
84C2×D4×D61—21no—
85C2×D4×D61C2311yescharacteristic
86C22×C4×D61—21no—
87C22×C4×D61C2311yescharacteristic
88C22×#92418C2×C411yescharacteristic, complemented
89C2×C8×D61—42no—
90C2×C8×D61C2×C411yescharacteristic, complemented
91C2×C8×D61C2×C411yescharacteristic, complemented
92C2×C8×D61C2311yescharacteristic
93C2×C8×D61C2311yescharacteristic
94#14519736C2×C411yescharacteristic, complemented
95#14519735C2×C411yescharacteristic, complemented
96#14519735C2×C411yescharacteristic, complemented
97#14519734—21no—
98#14519733—21no—
99C4⋊C4×D61C2311yescharacteristic
100C4⋊C4×D61C2311yescharacteristic
101C22⋊C4×D61—21no—
102C42×D61—21no—
103#14519702C2×C411yescharacteristic, complemented
104#14519702C2×C411yescharacteristic, complemented
105#14519701C2×C411yescharacteristic, complemented
106#14519701C2×C411yescharacteristic, complemented
107#14519700C2×C411yescharacteristic, complemented
108#14519700C2×C411yescharacteristic, complemented
109#14519699C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519698C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114#14519697C2×C411yescharacteristic
115C2×#41×C61C2×C411yescharacteristic
116C22×C8×C61C2×C411yescharacteristic, complemented
117#117×C61C2×C411yescharacteristic, complemented
118#103×C61C2×C411yescharacteristic, complemented
119#103×C61C2×C411yescharacteristic, complemented
120#102×C61C2×C411yescharacteristic
121#102×C61C2×C411yescharacteristic
122C23×D61—21no—
123C23×D61C2×D411yescharacteristic
124C22×#6923—22no—
125C22×#6923—22no—
126C22×#6923—44no—
127C22×#6923—44no—
128C22×#6923—82no—
129C22×#6923C2×D412yes—
130C22×#6923C2×D412yes—
131#41×C61—22no—
132#41×C61—41no—
133C2×Q8×C61C22×C411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61C22×C411yescharacteristic
136C22×C4×C61—21no—
137C22×C4×C61C22×C411yescharacteristic
138C2×C22⋊D61—21no—
139C2×C22⋊D61—21no—
140C2×C22⋊D61—21no—
141C22×Dic61—21no—
142C22×Dic61#4111yescharacteristic
143#11758226—22no—
144#11758226—41no—
145Q8×D61—22no—
146Q8×D61—22no—
147#11758224—22no—
148#11758224—41no—
149#11758224—44no—
150D4×D61—22no—
151D4×D61—44no—
152D4×D61#4114yes—
153#11758222—22no—
154#11758222—41no—
155#11758222—44no—
156C2×D244—21no—
157C2×D244C22×C411yescharacteristic
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—44no—
166C2×C4×D61C22×C411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61#4111yescharacteristic
169C2×C4×D61C2411yescharacteristic
170C2×Dic122—21no—
171C2×Dic122C22×C411yescharacteristic
172C22⋊#6923—24no—
173C22⋊#6923—44no—
174C22⋊#6923—81no—
175C22⋊#6923—81no—
176C2×#92420—22no—
177C4⋊#6923—22no—
178C4⋊#6923—22no—
179C4⋊#6923—22no—
180C4⋊#6923—22no—
181C4⋊#6923—82no—
182C4×#6923—22no—
183C4×#6923—22no—
184C4×#6923—44no—
185D61⋊C8—22no—
186D61⋊C8—22no—
187C2×C8×C61—41no—
188C2×C8×C61C22×C411yescharacteristic
189C2×C8×C61C22×C411yescharacteristic
190C4⋊C4×C61C22×C411yescharacteristic
191C4⋊C4×C61C22×C411yescharacteristic
192C22⋊C4×C61—21no—
193C42×C61—21no—
194C22⋊Dic61—21no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61C22×C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199#11758196—21no—
200#11758196—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C2×#92418—41no—
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C8×D61—22no—
208C8×D61—22no—
209C8×D61—41no—
210C8×D61—41no—
211C23×C61—21no—
212C23×C61C4×D411yescharacteristic
213C22×D61—21no—
214C22×D61—24no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—42no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61C4×D411yescharacteristic
223C22×D61C22×D411yescharacteristic
224C2×#6923—22no—
225C2×#6923—22no—
226C2×#6923—22no—
227C2×#6923—22no—
228C2×#6923—22no—
229C2×#6923—42no—
230C2×#6923—42no—
231C2×#6923—42no—
232C2×#6923—42no—
233C2×#6923—42no—
234C2×#6923—42no—
235C2×#6923—88no—
236C2×#6923#13514yes—
237Q8×C61—21no—
238Q8×C61—21no—
239D4×C61—21no—
240D4×C61—42no—
241D4×C61#11712yes—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—42no—
248C2×C4×C61C4×D411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C22⋊D61—24no—
251C22⋊D61—42no—
252C22⋊D61—42no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—42no—
258C2×Dic61C4×D411yescharacteristic
259C2×Dic61C2×#4111yescharacteristic
260D244—21no—
261D244—42no—
262D244#11712yes—
263C4×D61—22no—
264C4×D61—21no—
265C4×D61—21no—
266C4×D61—22no—
267C4×D61—42no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61—42no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61C2×#4112yes—
274C4×D61H(5,𝔽2)12yes—
275Dic122—21no—
276Dic122—21no—
277Dic122—42no—
278#92420—42no—
279C8×C61—21no—
280C8×C61—21no—
281#92418—21no—
282#92418—21no—
283#22572—611nomaximal, Sylow
284C22×C61—21no—
285C22×C61—22no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61C2×C4×D411yescharacteristic
291C2×D61—21no—
292C2×D61—22no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—44no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—44no—
301C2×D61#54612yes—
302C2×D61C2×#13512yes—
303C2×D61#57711yescharacteristic
304C2×D61#57711yescharacteristic
305#6923—24no—
306#6923—42no—
307#6923—82no—
308#6923—82no—
309C4×C61—21no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61C2×#11711yescharacteristic
314C4×C61#51811yescharacteristic
315Dic61—21no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319Dic61#55112yes—
320C2×#416—611no—
321#2346—611no—
322#2339—611no—
323#1933—612no—
324#1927—612no—
325#1864—612no—
326#1825—611no—
327#1807—611no—
328#1788—611no—
329C4×#102—612no—
330C8×C22⋊C4—611no—
331C2×C61—41no—
332C2×C61—42no—
333C2×C61#234211yescharacteristic
334C2×C61C4×#13511yescharacteristic
335C2×C61#295211yescharacteristic
336D61—41no—
337D61—42no—
338D61#329612yes—
339D61#330312yes—
340C22×#41—611no—
341C23×C8—611no—
342C2×C4×D4—612no—
343C2×#117—611no—
344C2×#106—612no—
345#416—611no—
346#416—611no—
347#416—611no—
348#416—611no—
349#416—611no—
350#416—611no—
351#416—611no—
352#416#692311yescharacteristic, complemented
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—612no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—612no—
359C2×C22⋊C8—611no—
360C2×C4×C8—612no—
361#391—612no—
362#389—612no—
363#388—1222no—
364#386—1222no—
365#382—1221no—
366#381—1221no—
367#380—611no—
368#380—611no—
369C4×C4⋊C4—612no—
370C4×C22⋊C4—611no—
371#376—1221no—
372C2×#95—1221no—
373#340—612no—
374#340—612no—
375#340—612no—
376#336—612no—
377C61#2257211yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C2×#41—611no—
380C2×#41—611no—
381C2×#41—1221no—
382C2×#41—1221no—
383C2×#41—1224no—
384C2×#41—2442no—
385C2×#41C2×#692311yescharacteristic
386C22×Q8—611no—
387C22×D4—611no—
388C22×D4—1221no—
389C23×C4—611no—
390C23×C4—1221no—
391C23×C4—1222no—
392C22×C8—611no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—612no—
398C22×C8—2442no—
399C22×C8C2×#692311yescharacteristic, complemented
400C4×D4—1224no—
401C4×D4—2442no—
402#117—611no—
403#117—611no—
404#117—611no—
405#117—1221no—
406#117—1221no—
407#117C2×#692311yescharacteristic, complemented
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—1222no—
413C2×C22⋊C4—612no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1222no—
418C2×C42—612no—
419C2×C42—1221no—
420C2×C42—1222no—
421#106—2442no—
422#103—611no—
423#103—611no—
424#103—611no—
425#103—611no—
426#103—611no—
427#103—611no—
428#103—2442no—
429#103C2×#692311yescharacteristic, complemented
430#103C2×#692311yescharacteristic, complemented
431#102—611no—
432#102—611no—
433#102—611no—
434#102—611no—
435#102—611no—
436#102—611no—
437#102—1224no—
438#102C2×#692311yescharacteristic
439#102C2×#692311yescharacteristic
440C22⋊C8—1221no—
441C22⋊C8—1221no—
442C4×C8—2442no—
443#95—611no—
444#95—611no—
445#95—1221no—
446#95—1222no—
447#95—1221no—
448#95—1222no—
449#95—1222no—
450#95—1222no—
451C24—611no—
452C24—1221no—
453#41—22no—
454#41—41no—
455#41—1222no—
456#41—1222no—
457#41—1222no—
458#41—2441no—
459#41—2444no—
460#41—2441no—
461#41—2444no—
462#41—2441no—
463C2×Q8—611no—
464C2×Q8—1221no—
465C2×Q8—1222no—
466C2×Q8—1222no—
467C2×Q8C22×#692311yescharacteristic
468C2×D4—21no—
469C2×D4—614no—
470C2×D4—611no—
471C2×D4—1221no—
472C2×D4—1222no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—2444no—
477C2×D4C22×#692311yescharacteristic
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—612no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—612no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—4882no—
497C22×C4—2444no—
498C22×C4—2444no—
499C22×C4—2444no—
500C22×C4C22×#692311yescharacteristic
501C2×C8—41no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—2441no—
510C2×C8—2441no—
511C2×C8—2441no—
512C2×C8C22×#692311yescharacteristic
513C2×C8C22×#692311yescharacteristic
514C4⋊C4—611no—
515C4⋊C4—611no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—4882no—
523C4⋊C4C22×#692311yescharacteristic
524C4⋊C4C22×#692311yescharacteristic
525C22⋊C4—21no—
526C22⋊C4—1221no—
527C22⋊C4—1221no—
528C22⋊C4—1224no—
529C22⋊C4—1221no—
530C22⋊C4—4881no—
531C22⋊C4—4881no—
532C22⋊C4—2444no—
533C42—21no—
534C42—1221no—
535C42—1221no—
536C42—1222no—
537C42—1222no—
538C42—1221no—
539C42—2444no—
540C23—21no—
541C23—611no—
542C23—611no—
543C23—1222no—
544C23—1221no—
545C23—1221no—
546C23—1224no—
547C23—1222no—
548C23—1221no—
549C23—2441no—
550C23—2442no—
551C23—2441no—
552C23D4×#692311yescharacteristic
553Q8—21no—
554Q8—21no—
555Q8—1221no—
556Q8—1221no—
557Q8—2442no—
558D4—21no—
559D4—42no—
560D4—612no—
561D4—1224no—
562D4—1221no—
563D4—2442no—
564D4—2442no—
565D4—2442no—
566D4#1451985612yes—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—42no—
573C2×C4—611no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—614no—
577C2×C4—611no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—4888no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4D4×#692311yescharacteristic
606C2×C4C23×#692311yescharacteristic
607C8—21no—
608C8—21no—
609C8—1221no—
610C8—1221no—
611C8—2442no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—21no—
616C22—41no—
617C22—41no—
618C22—612no—
619C22—612no—
620C22—611no—
621C22—611no—
622C22—1221no—
623C22—1221no—
624C22—1222no—
625C22—1221no—
626C22—2444no—
627C22—2442no—
628C22—2442no—
629C22—2444no—
630C22—2441no—
631C22—2441no—
632C22C2×D4×#692311yescharacteristic
633C4—21no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—612no—
638C4—1224no—
639C4—1221no—
640C4—4882no—
641C4—4882no—
642C4—2442no—
643C4—2441no—
644C4—2441no—
645C4—2441no—
646C4C2×#1451985611yescharacteristic
647C4#1684082911yescharacteristic
648C2—41no—
649C2—42no—
650C2—612no—
651C2—612no—
652C2—2441no—
653C2—2442no—
654C2#2146232511yescharacteristic
655C2#135×#692311yescharacteristic
656C2#2146258211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#416C4
socleC22×C61C2×C4×D4
prime radicalC2×#41×D61C22
cosocleC2×C4×D61C24
hypercenter#416#6923
hypocenterC61#22572
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #21462329C2semidirect
3. #21461908C2semidirect
4. #21461898C2semidirect
5. #21461784C2semidirect
6. #21461724C2semidirect
7. #21461695C2semidirect
9. #102×#6923C2semidirect
10. C8×C22⋊#6923C2semidirect
37. #16839759C4semidirect
38. #16839758C4semidirect
39. #16839757C4semidirect
40. #16839757C4semidirect
41. #16839756C4semidirect
42. #16839755C4semidirect
43. #16839755C4semidirect
48. C61×#416C4semidirect
88. C22×#92418C2×C4semidirect
90. C2×C8×D61C2×C4semidirect
91. C2×C8×D61C2×C4semidirect
94. #14519736C2×C4semidirect
95. #14519735C2×C4semidirect
96. #14519735C2×C4semidirect
103. #14519702C2×C4semidirect
104. #14519702C2×C4semidirect
105. #14519701C2×C4semidirect
106. #14519701C2×C4semidirect
107. #14519700C2×C4semidirect
108. #14519700C2×C4semidirect
116. C22×C8×C61C2×C4semidirect
117. #117×C61C2×C4semidirect
118. #103×C61C2×C4semidirect
119. #103×C61C2×C4semidirect
352. #416#6923semidirect
377. C61#22572semidirect
399. C22×C8C2×#6923semidirect
407. #117C2×#6923semidirect
429. #103C2×#6923semidirect
430. #103C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623292
#214623222
#214619082
#214618982
#214617842
#214617242
#214616952
#214616592
#102×#69232
C8×C22⋊#69232
#416×D612
#2257261