Finite Group Databasebeta

← #30611406

Subgroups — #30611406

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462682C211yesmaximal, characteristic, complemented
2#21462332C211yesmaximal, characteristic, complemented
3#21461252C211yesmaximal, characteristic, complemented
4#21461248C211yesmaximal, characteristic, complemented
5#21461235C212yesmaximal, complemented
6#21461234C212yesmaximal, complemented
7#21461205C212yesmaximal, complemented
8#21461204C212yesmaximal, complemented
9#21461201C211yesmaximal, characteristic, complemented
10#21461199C211yesmaximal, characteristic, complemented
11#21459663C211yesmaximal, characteristic
12C2×D4×#6923—21no—
13#16840797—21no—
14#16840796—21no—
15#16840795—22no—
16#16840781C2211yescharacteristic
17#16840780C2211yescharacteristic
18#16840734C2212yes—
19#16840733C2212yes—
20#16840713—21no—
21#16840709—22no—
22#16840706C2211yescharacteristic
23#16840704C2212yes—
24C4×C4⋊#6923C2211yescharacteristic
25C2×(C2×#6923)⋊C4C2212yes—
26#16840616C2212yes—
27#16840616C2211yescharacteristic
28#16840616C2211yescharacteristic
29#16840615C2212yes—
30#16840615C2211yescharacteristic
31#16840615C2211yescharacteristic
32#16840613C2212yes—
33#16840613C2212yes—
34#16840613C2212yes—
35#16840613C2212yes—
36#16840498—21no—
37#16840495—21no—
38#16840485C411yescharacteristic, complemented
39#16840482C411yescharacteristic, complemented
40#16840476C412yescomplemented
41#16840455—22no—
42#16840432C411yescharacteristic, complemented
43#16840411C412yescomplemented
44#16840344—22no—
45#16840325—21no—
46#16840316C2211yescharacteristic
47#16840310C2211yescharacteristic
48#16840272C2212yes—
49#16840269C2212yes—
50#16840241—21no—
51C4×D4×D61C2211yescharacteristic
52C61×#550C411yescharacteristic, complemented
53C23×#6923—22no—
54C2×#11758213—22no—
55C2×#11758217—22no—
56D4×#6923—44no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—22no—
60C2×C22⋊#6923—22no—
61C2×C4×#6923—21no—
62#14519844—21no—
63C4⋊#92420—21no—
64#14519834—22no—
65#14519834C2311yescharacteristic
66#14519834C2312yes—
67#14519834C2311yescharacteristic
68C22⋊#92420—22no—
69C4×#92420—21no—
70C4×#92420—21no—
71(C2×#6923)⋊C4—21no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4C2311yescharacteristic
74(C2×#6923)⋊C4C2312yes—
75(C2×#6923)⋊C4C2311yescharacteristic
76C2×#11758224—21no—
77C2×#11758224C2×C412yescomplemented
78C2×D4×D61D411yescharacteristic
79C22×C4×D61C2312yes—
80#14519777C2×C411yescharacteristic, complemented
81#14519776C2×C411yescharacteristic, complemented
82#14519775C2×C411yescharacteristic, complemented
83#14519773—21no—
84#14519772—22no—
85#14519770—22no—
86#14519767—21no—
87#14519765—21no—
88#14519764—21no—
89#14519764—22no—
90#14519764—21no—
91#14519763—22no—
92#14519760—22no—
93#14519759—22no—
94#14519758—22no—
95#14519758—22no—
96#14519757C2311yescharacteristic
97#14519756C2×C412yescomplemented
98#14519755—21no—
99#14519753C2×C412yescomplemented
100#14519753C2311yescharacteristic
101#14519749—22no—
102#14519749—22no—
103#14519749—22no—
104#14519748—21no—
105#14519748C2×C412yescomplemented
106#14519748C2312yes—
107#14519745—22no—
108#14519745—22no—
109C4×Dic122—21no—
110C4×Dic122—21no—
111C4×Dic122C2×C411yescharacteristic, complemented
112#14519742—21no—
113C4×D244C2×C411yescharacteristic
114D4×Dic61—22no—
115D4×Dic61—21no—
116D4×Dic61—21no—
117D4×Dic61C2×C412yescomplemented
118D4×Dic61D411yescharacteristic
119#14519739D411yescharacteristic
120C4×C22⋊D61—22no—
121C4×C22⋊D61—22no—
122C4×C22⋊D61C2×C412yes—
123#14519737—22no—
124#14519737D412yes—
125#14519736C2×C412yescomplemented
126#14519735C2×C412yescomplemented
127#14519734—21no—
128#14519733—22no—
129C2×#11758196—22no—
130C4⋊C4×D61D411yescharacteristic
131C22⋊C4×D61D412yes—
132C42×D61C2311yescharacteristic
133C2×#41×C61C2×C412yescomplemented
134#127×C61—21no—
135#127×C61C2×C411yescharacteristic, complemented
136#121×C61—22no—
137#121×C61C2×C412yescomplemented
138C4×Q8×C61C2×C411yescharacteristic, complemented
139C4×D4×C61—21no—
140C4×D4×C61C2×C411yescharacteristic
141C23×D61C2×D412yes—
142C22×#6923—21no—
143C22×#6923—22no—
144C22×#6923—21no—
145C22×#6923—44no—
146C22×#6923—44no—
147C22×#6923—82no—
148#41×C61—82no—
149C2×Q8×C61C22×C411yescharacteristic
150C2×D4×C61—22no—
151C2×D4×C61—22no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61C22⋊C411yescharacteristic
155C2×D4×C61C22×C412yes—
156C22×C4×C61—22no—
157C22×C4×C61C22×C412yes—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61C22⋊C412yes—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61C2×D412yes—
165#11758224—42no—
166#11758224—42no—
167#11758224—82no—
168D4×D61—22no—
169D4×D61—22no—
170C2×D244C22⋊C411yescharacteristic
171C2×C4×D61—82no—
172C2×C4×D61C22×C412yes—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D411yescharacteristic
177C2×C4×D61C2411yescharacteristic
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122C22×C411yescharacteristic
181C22⋊#6923—44no—
182#11758217—82no—
183C2×#92420—21no—
184C2×#92420—22no—
185C2×#92420—21no—
186C4⋊#6923—42no—
187C4⋊#6923—42no—
188C4⋊#6923—82no—
189C4×#6923—42no—
190#11758213—82no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61C22⋊C411yescharacteristic
193C4⋊C4×C61C22×C411yescharacteristic
194C22⋊C4×C61—22no—
195C22⋊C4×C61C22⋊C412yes—
196C42×C61—21no—
197C42×C61C22×C411yescharacteristic
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61—22no—
203C22⋊Dic61C22⋊C412yes—
204#11758198—22no—
205#11758198—22no—
206#11758198—21no—
207#11758198—21no—
208#11758198C22⋊C412yes—
209#11758198C22⋊C412yes—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61C22⋊C411yescharacteristic
213C4⋊Dic61C22×C412yes—
214C4⋊Dic61C22×C411yescharacteristic
215#11758196—22no—
216#11758196—21no—
217#11758196—21no—
218#11758196—22no—
219#11758196—21no—
220#11758196—21no—
221#11758196—21no—
222#11758196C22⋊C412yes—
223C4×Dic61—22no—
224C4×Dic61—21no—
225C4×Dic61—22no—
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C23×C61—22no—
230C23×C61C2×C22⋊C412yes—
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61C2×C22⋊C412yes—
236C22×D61C22×D411yescharacteristic
237C2×#6923—22no—
238C2×#6923—41no—
239C2×#6923—41no—
240C2×#6923—41no—
241C2×#6923—41no—
242C2×#6923—42no—
243C2×#6923—42no—
244C2×#6923—41no—
245C2×#6923—42no—
246C2×#6923—88no—
247Q8×C61—41no—
248D4×C61—22no—
249D4×C61—42no—
250D4×C61—42no—
251D4×C61—81no—
252D4×C61—82no—
253D4×C61—82no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—42no—
257C2×C4×C61—82no—
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C23×C411yescharacteristic
262C22⋊D61—24no—
263C22⋊D61—42no—
264C22⋊D61—42no—
265C22⋊D61—82no—
266C22⋊D61—82no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—42no—
271C2×Dic61—44no—
272C2×Dic61—82no—
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C22×D411yescharacteristic
277D244—22no—
278C4×D61—42no—
279C4×D61—41no—
280C4×D61—42no—
281C4×D61—41no—
282C4×D61—42no—
283C4×D61AGL(1,ℤ/8)12yes—
284C4×D61#13712yes—
285Dic122—42no—
286Dic122—41no—
287Dic122—81no—
288#92420—41no—
289#92420—42no—
290#92420—41no—
291#17630—611nomaximal, Sylow
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61C22×C22⋊C411yescharacteristic
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—44no—
305C2×D61—44no—
306C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
307C2×D61C2×#13711yescharacteristic
308#6923—42no—
309#6923—81no—
310#6923—82no—
311C4×C61—21no—
312C4×C61—22no—
313C4×C61—21no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61#41712yes—
317Dic61—21no—
318Dic61—22no—
319Dic61—21no—
320Dic61—41no—
321Dic61—42no—
322Dic61—41no—
323Dic61#41712yes—
324#3506—611no—
325#2975—611no—
326#2349—611no—
327#1518—611no—
328#1517—611no—
329#1513—612no—
330#1513—612no—
331#1504—612no—
332#1504—612no—
333#1503—611no—
334#1502—611no—
335C2×C61—42no—
336C2×C61—82no—
337C2×C61#288311yescharacteristic
338C2×C61C2×#41711yescharacteristic
339C2×C61#290111yescharacteristic
340D61—21no—
341D61—21no—
342D61—42no—
343#559—1221no—
344#559—1221no—
345#556—611no—
346#556—611no—
347#555—612no—
348#553—1222no—
349#550—611no—
350#550#692311yescharacteristic, complemented
351#547—612no—
352#538—1222no—
353#535—1221no—
354#533—611no—
355#532—611no—
356#525—612no—
357#525—612no—
358#520—1221no—
359C2×C4×D4—611no—
360C2×C4×D4—1221no—
361#436—1222no—
362#436—1221no—
363#436—1221no—
364#431—611no—
365#431—611no—
366#407—612no—
367#407—612no—
368#389—1221no—
369#385—1222no—
370#382—611no—
371#380—612no—
372C4×C4⋊C4—611no—
373C2×#95—612no—
374#328—611no—
375#328—611no—
376#328—612no—
377#328—611no—
378#328—611no—
379#328—612no—
380#327—612no—
381#327—612no—
382#327—612no—
383#327—612no—
384C61#1763011yescharacteristic, complemented, Sylow
385C2×#41—612no—
386C2×#41—1221no—
387C2×#41C2×#692312yescomplemented
388C22×D4—611no—
389C23×C4—612no—
390C23×C4—1222no—
391C2×M16—1222no—
392C2×M16—1222no—
393C4⋊Q8—611no—
394C4⋊Q8—611no—
395C4⋊Q8—611no—
396#127—21no—
397#127—1221no—
398#127C2×#692311yescharacteristic, complemented
399#126—1222no—
400#126—1222no—
401#125—1221no—
402#124—1221no—
403#124—1221no—
404#124—1222no—
405#124—1222no—
406#124—1221no—
407#123—1222no—
408#123—1222no—
409#123—1222no—
410#123—1222no—
411C22⋊Q8—611no—
412C22⋊Q8—612no—
413C22⋊Q8—612no—
414C22⋊Q8—611no—
415C22⋊Q8—1221no—
416#121—22no—
417#121—612no—
418#121—612no—
419#121—1222no—
420#121—1221no—
421#121—1222no—
422#121—1222no—
423#121C2×#692312yescomplemented
424#120—1222no—
425#120—1222no—
426C4×Q8—611no—
427C4×Q8—1221no—
428C4×Q8—1221no—
429C4×Q8—1221no—
430C4×Q8C2×#692311yescharacteristic, complemented
431C4×D4—21no—
432C4×D4—611no—
433C4×D4—612no—
434C4×D4—611no—
435C4×D4—612no—
436C4×D4—612no—
437C4×D4—611no—
438C4×D4—1221no—
439C4×D4—1222no—
440C4×D4—1221no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1222no—
444C4×D4—2444no—
445C4×D4C2×#692311yescharacteristic
446#117—612no—
447#117—612no—
448#117—1221no—
449#117—1222no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—1222no—
452C2×C4⋊C4—1222no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C22⋊C4—612no—
456C2×C22⋊C4—1222no—
457C2×C42—611no—
458C2×C42—1221no—
459C4⋊C8—1221no—
460C4⋊C8—1221no—
461C22⋊C8—611no—
462C22⋊C8—612no—
463C22⋊C8—611no—
464C22⋊C8—1222no—
465C22⋊C8—1222no—
466C4×C8—1221no—
467C4×C8—1221no—
468#95—611no—
469#95—612no—
470#95—611no—
471#95—1222no—
472#95—1221no—
473C24—612no—
474#41—82no—
475#41—4882no—
476#41—2442no—
477#41—2442no—
478C2×Q8—611no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8C22×#692311yescharacteristic
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—612no—
487C2×D4—611no—
488C2×D4—1222no—
489C2×D4—1222no—
490C2×D4—1222no—
491C2×D4—1222no—
492C2×D4—1222no—
493C2×D4C22⋊#692311yescharacteristic
494C2×D4C22×#692312yes—
495C22×C4—22no—
496C22×C4—611no—
497C22×C4—612no—
498C22×C4—611no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—611no—
502C22×C4—612no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1221no—
506C22×C4—1222no—
507C22×C4—1221no—
508C22×C4—4882no—
509C22×C4—4882no—
510C22×C4—2444no—
511C22×C4—2444no—
512C22×C4C22×#692312yes—
513M16—4882no—
514M16—4882no—
515C2×C8—1221no—
516C2×C8—1222no—
517C2×C8—1221no—
518C4⋊C4—21no—
519C4⋊C4—612no—
520C4⋊C4—611no—
521C4⋊C4—612no—
522C4⋊C4—611no—
523C4⋊C4—1221no—
524C4⋊C4—1222no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1222no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—4882no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C4⋊C4C22⋊#692311yescharacteristic
536C4⋊C4C22×#692311yescharacteristic
537C22⋊C4—22no—
538C22⋊C4—612no—
539C22⋊C4—612no—
540C22⋊C4—612no—
541C22⋊C4—1222no—
542C22⋊C4—1221no—
543C22⋊C4—1221no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—2444no—
551C22⋊C4C22⋊#692312yes—
552C42—21no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—1222no—
557C42—1222no—
558C42—1221no—
559C42—2442no—
560C42C22×#692311yescharacteristic
561C23—22no—
562C23—611no—
563C23—612no—
564C23—1222no—
565C23—1222no—
566C23—2442no—
567C23—2442no—
568C23C2×C22⋊#692312yes—
569Q8—41no—
570Q8—4881no—
571Q8—2441no—
572Q8—2442no—
573D4—22no—
574D4—42no—
575D4—42no—
576D4—81no—
577D4—82no—
578D4—82no—
579D4—1224no—
580D4—1222no—
581D4—4882no—
582D4—4882no—
583D4—2442no—
584D4—2442no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—42no—
588C2×C4—82no—
589C2×C4—611no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—611no—
593C2×C4—612no—
594C2×C4—611no—
595C2×C4—1222no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—4882no—
600C2×C4—4888no—
601C2×C4—2442no—
602C2×C4—2441no—
603C2×C4—2442no—
604C2×C4—2441no—
605C2×C4—2442no—
606C2×C4—2444no—
607C2×C4—2442no—
608C2×C4—2441no—
609C2×C4—2442no—
610C2×C4—2441no—
611C2×C4—2442no—
612C2×C4—2441no—
613C2×C4—2442no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4C2×C22⋊#692311yescharacteristic
617C2×C4C2×C22⋊#692312yes—
618C2×C4C2×C22⋊#692311yescharacteristic
619C2×C4C23×#692311yescharacteristic
620C8—2441no—
621C8—2442no—
622C8—2441no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—611no—
630C22—611no—
631C22—1221no—
632C22—1221no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—2444no—
637C22—2444no—
638C22C22×C22⋊#692311yescharacteristic
639C4—21no—
640C4—22no—
641C4—21no—
642C4—41no—
643C4—41no—
644C4—612no—
645C4—1221no—
646C4—1222no—
647C4—1221no—
648C4—4882no—
649C4—4881no—
650C4—2442no—
651C4—2441no—
652C4—2442no—
653C4—2441no—
654C4#1684075312yes—
655C2—42no—
656C2—82no—
657C2—1221no—
658C2—1221no—
659C2—2442no—
660C2#2146241811yescharacteristic
661C2C2×#1684075311yescharacteristic
662C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#550C4
socleC22×C61C22×C22⋊C4
prime radical#21459663C2
cosocleC2×C4×D61C24
hypercenter#550#6923
hypocenterC61#17630
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #21462682C2semidirect
2. #21462332C2semidirect
3. #21461252C2semidirect
4. #21461248C2semidirect
5. #21461235C2semidirect
6. #21461234C2semidirect
7. #21461205C2semidirect
8. #21461204C2semidirect
9. #21461201C2semidirect
10. #21461199C2semidirect
38. #16840485C4semidirect
39. #16840482C4semidirect
40. #16840476C4semidirect
42. #16840432C4semidirect
43. #16840411C4semidirect
52. C61×#550C4semidirect
77. C2×#11758224C2×C4semidirect
80. #14519777C2×C4semidirect
81. #14519776C2×C4semidirect
82. #14519775C2×C4semidirect
97. #14519756C2×C4semidirect
99. #14519753C2×C4semidirect
105. #14519748C2×C4semidirect
111. C4×Dic122C2×C4semidirect
117. D4×Dic61C2×C4semidirect
125. #14519736C2×C4semidirect
126. #14519735C2×C4semidirect
133. C2×#41×C61C2×C4semidirect
135. #127×C61C2×C4semidirect
137. #121×C61C2×C4semidirect
138. C4×Q8×C61C2×C4semidirect
350. #550#6923semidirect
384. C61#17630semidirect
387. C2×#41C2×#6923semidirect
398. #127C2×#6923semidirect
423. #121C2×#6923semidirect
430. C4×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214626822
#214623322
#214612522
#214612482
#214612352
#214612342
#214612052
#214612042
#214612012
#214611992
#214596632
#1763061