Finite Group Databasebeta

← #30610995

Subgroups — #30610995

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462679C211yesmaximal, characteristic, complemented
2#21462258C211yesmaximal, characteristic, complemented
3#21461221C211yesmaximal, characteristic, complemented
4#21461218C212yesmaximal, complemented
5#21461211C211yesmaximal, characteristic, complemented
6#21461204C212yesmaximal, complemented
7#21461202C211yesmaximal, characteristic, complemented
8#21461195C212yesmaximal, complemented
9#21461190C211yesmaximal, characteristic, complemented
10C2×#16840615C212yesmaximal, complemented
11#21459859C211yesmaximal, characteristic
12C2×D4×#6923—21no—
13#16840796—21no—
14#16840791—22no—
15D4×#92420—21no—
16#16840784C2211yescharacteristic
17#16840781C2211yescharacteristic
18#16840733C2212yes—
19C2×#14519834C2212yes—
20C4⋊C4×#6923—21no—
21C4×C4⋊#6923C2211yescharacteristic
22C22⋊C4×#6923—22no—
23C4×C22⋊#6923C2212yes—
24#16840698C2211yescharacteristic
25C2×(C2×#6923)⋊C4C2212yes—
26#16840615C2211yescharacteristic
27#16840615C2212yes—
28#16840615C2212yes—
29#16840615C2212yes—
30#16840615C2211yescharacteristic
31#16840615C2212yes—
32#16840615C2212yes—
33#16840613C2211yescharacteristic
34#16840613C2211yescharacteristic
35#16840613C2212yes—
36#16840486C411yescharacteristic, complemented
37#16840470—21no—
38#16840447C412yescomplemented
39#16840435—22no—
40D4×Dic122C411yescharacteristic, complemented
41#16840418C412yescomplemented
42#16840416C411yescharacteristic, complemented
43#16840400—21no—
44#16840385—21no—
45#16840356—22no—
46#16840317C2211yescharacteristic
47#16840294C2211yescharacteristic
48#16840272C2212yes—
49C2×#14519757C2212yes—
50#16840245—21no—
51C4×D4×D61C2211yescharacteristic
52C61×#547C411yescharacteristic, complemented
53C23×#6923—22no—
54C2×#11758213—22no—
55C22×#92420—22no—
56D4×#6923—28no—
57C2×C4⋊#6923—21no—
58C2×C22⋊#6923—22no—
59C2×C4×#6923—21no—
60C2×C4×#6923—22no—
61C2×C4×#6923—21no—
62C4⋊#92420—21no—
63C4⋊#92420—21no—
64#14519834—22no—
65#14519834C2311yescharacteristic
66#14519834C2312yes—
67#14519834C2311yescharacteristic
68C22⋊#92420—22no—
69#14519832—21no—
70C4×#92420—21no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—21no—
73(C2×#6923)⋊C4C2311yescharacteristic
74(C2×#6923)⋊C4C2312yes—
75(C2×#6923)⋊C4C2311yescharacteristic
76C2×#11758224C2×C412yescomplemented
77C22×Dic122C2×C412yescomplemented
78C2×D4×D61D411yescharacteristic
79C22×C4×D61C2312yes—
80#14519777C2×C411yescharacteristic, complemented
81#14519776—21no—
82#14519772—22no—
83#14519772—22no—
84#14519770—21no—
85#14519768—21no—
86#14519768C2×C411yescharacteristic, complemented
87#14519767—22no—
88#14519766—21no—
89#14519764C2×C411yescharacteristic, complemented
90#14519763—22no—
91#14519761—22no—
92#14519760—22no—
93#14519759—22no—
94#14519759—21no—
95#14519757—22no—
96#14519757—22no—
97#14519757C2×C412yescomplemented
98#14519757C2×C412yescomplemented
99#14519757C2311yescharacteristic
100#14519757C2312yes—
101#14519756C2×C412yescomplemented
102#14519754—22no—
103#14519754—22no—
104#14519753—21no—
105#14519751—21no—
106#14519749—22no—
107#14519748C2311yescharacteristic
108#14519745—22no—
109Q8×Dic61—21no—
110Q8×Dic61—21no—
111C4×Dic122C2×C411yescharacteristic, complemented
112#14519742C2×C412yescomplemented
113C4×D244C2×C411yescharacteristic
114D4×Dic61—21no—
115D4×Dic61—22no—
116D4×Dic61—22no—
117D4×Dic61D411yescharacteristic
118#14519739D411yescharacteristic
119C4×C22⋊D61—21no—
120C4×C22⋊D61C2×C412yes—
121#14519737—22no—
122#14519737D412yes—
123#14519736—22no—
124#14519736—21no—
125#14519735—21no—
126#14519735C2×C412yescomplemented
127C2×C4⋊Dic61—22no—
128C2×C4⋊Dic61—21no—
129C4⋊C4×D61D411yescharacteristic
130C2×#11758198—22no—
131C22⋊C4×D61D412yes—
132C42×D61C2311yescharacteristic
133C2×#41×C61—21no—
134C4⋊Q8×C61C2×C411yescharacteristic, complemented
135#123×C61—22no—
136C22⋊Q8×C61—21no—
137#121×C61C2×C412yescomplemented
138C4×D4×C61C2×C411yescharacteristic
139C4×D4×C61C2×C411yescharacteristic, complemented
140C2×C4⋊C4×C61C2×C412yescomplemented
141C23×D61C2×D412yes—
142C22×#6923—21no—
143C22×#6923—21no—
144C22×#6923—22no—
145C22×#6923—28no—
146C22×#6923—82no—
147C22×#6923—82no—
148#41×C61—44no—
149C2×Q8×C61—21no—
150C2×D4×C61—22no—
151C2×D4×C61C22⋊C411yescharacteristic
152C2×D4×C61C22×C411yescharacteristic
153C22×C4×C61—22no—
154C22×C4×C61—21no—
155C22×C4×C61C22×C412yes—
156C2×C22⋊D61—21no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61C22⋊C412yes—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61C2×D412yes—
162#11758224—82no—
163D4×D61—24no—
164C2×D244C22⋊C411yescharacteristic
165C2×C4×D61—82no—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D412yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2411yescharacteristic
172C2×Dic122—22no—
173C2×Dic122—82no—
174C2×Dic122C22×C411yescharacteristic
175C2×Dic122C22×C412yes—
176C22⋊#6923—28no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420—22no—
180C2×#92420—82no—
181C4⋊#6923—24no—
182C4×#6923—24no—
183C4×#6923—81no—
184C4×#6923—82no—
185#11758213—82no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22⋊C411yescharacteristic
189C4⋊C4×C61C22×C411yescharacteristic
190C4⋊C4×C61C22×C412yes—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61C22⋊C412yes—
194C42×C61C22×C411yescharacteristic
195C22⋊Dic61—22no—
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—21no—
199C22⋊Dic61C22⋊C412yes—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—21no—
204#11758198—22no—
205#11758198C22⋊C412yes—
206#11758198C22⋊C412yes—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—22no—
210C4⋊Dic61—22no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—22no—
214C4⋊Dic61C22⋊C411yescharacteristic
215C4⋊Dic61C22×C411yescharacteristic
216#11758196—22no—
217#11758196—21no—
218#11758196—21no—
219#11758196—22no—
220#11758196C22⋊C412yes—
221C4×Dic61—21no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C23×C61—21no—
228C23×C61C2×C22⋊C412yes—
229C22×D61—24no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—42no—
233C22×D61C2×C22⋊C412yes—
234C22×D61C22×D411yescharacteristic
235C2×#6923—22no—
236C2×#6923—22no—
237C2×#6923—24no—
238C2×#6923—41no—
239C2×#6923—42no—
240C2×#6923—41no—
241C2×#6923—42no—
242C2×#6923—41no—
243C2×#6923—41no—
244C2×#6923—88no—
245Q8×C61—42no—
246D4×C61—41no—
247D4×C61—44no—
248D4×C61#10414yes—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—42no—
252C2×C4×C61—44no—
253C2×C4×C61—42no—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C22⋊D61—24no—
259C22⋊D61—41no—
260C22⋊D61—41no—
261C22⋊D61—82no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—42no—
266C2×Dic61—82no—
267C2×Dic61—82no—
268C2×Dic61C2×C22⋊C411yescharacteristic
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C22×D411yescharacteristic
272D244—22no—
273C4×D61—21no—
274C4×D61—21no—
275C4×D61—22no—
276C4×D61—42no—
277C4×D61—42no—
278C4×D61—41no—
279C4×D61—42no—
280Dic122—41no—
281Dic122—42no—
282Dic122—82no—
283#92420—41no—
284#92420—42no—
285#92420—41no—
286#17548—611nomaximal, Sylow
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61C2×#10414yes—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—44no—
300C2×D61—44no—
301C2×D61#57711yescharacteristic
302C2×D61#57811yescharacteristic
303#6923—42no—
304#6923—44no—
305#6923—42no—
306C4×C61—21no—
307C4×C61—22no—
308C4×C61—21no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61C2×#10412yes—
312Dic61—21no—
313Dic61—22no—
314Dic61—21no—
315Dic61—21no—
316Dic61—41no—
317Dic61—42no—
318Dic61—41no—
319#3520—611no—
320#2974—611no—
321#2275—611no—
322#1509—611no—
323#1508—612no—
324#1506—611no—
325#1504—612no—
326#1503—611no—
327#1501—612no—
328#1500—611no—
329C2×#328—612no—
330C2×C61—24no—
331C2×C61—81no—
332C2×C61#288311yescharacteristic
333C2×C61#289811yescharacteristic
334C2×C61C22×#10411yescharacteristic
335D61—21no—
336D61—21no—
337D61—42no—
338#556—611no—
339#554—1221no—
340#552—612no—
341#551—1222no—
342D4×Q8—611no—
343#548—612no—
344#547—611no—
345#547#692311yescharacteristic, complemented
346#546—1221no—
347#542—1221no—
348#539—1222no—
349#533—611no—
350#529—611no—
351#525—612no—
352C2×C22⋊Q8—612no—
353#520—1221no—
354C2×C4×D4—611no—
355C2×C4×D4—1221no—
356#436—1221no—
357#435—1222no—
358C8×D4—1221no—
359#432—611no—
360#431—611no—
361#407—612no—
362C2×C22⋊C8—612no—
363C4×C4⋊C4—611no—
364C4×C4⋊C4—1221no—
365C4×C22⋊C4—612no—
366C4×C22⋊C4—1222no—
367#376—611no—
368C2×#95—612no—
369#328—611no—
370#328—612no—
371#328—612no—
372#328—612no—
373#328—611no—
374#328—612no—
375#328—612no—
376#327—611no—
377#327—611no—
378#327—612no—
379C61#1754811yescharacteristic, complemented, Sylow
380C2×#41—21no—
381C2×#41—612no—
382C22×Q8—612no—
383C22×D4—611no—
384C23×C4—612no—
385C23×C4—1222no—
386C2×M16—1222no—
387C22×C8—1222no—
388C4⋊Q8—611no—
389C4⋊Q8—1221no—
390C4⋊Q8C2×#692311yescharacteristic, complemented
391#126—1221no—
392#126—1222no—
393#126—1222no—
394#125—611no—
395#125—1221no—
396#125—1222no—
397#124—611no—
398#124—1222no—
399#124—1221no—
400#123—22no—
401#123—1222no—
402#123—1222no—
403#123—1221no—
404#123—1222no—
405C22⋊Q8—21no—
406C22⋊Q8—612no—
407C22⋊Q8—611no—
408C22⋊Q8—612no—
409C22⋊Q8—612no—
410C22⋊Q8—612no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414C22⋊Q8—1221no—
415C22⋊Q8—1222no—
416#121—611no—
417#121—1221no—
418#121—1222no—
419#121C2×#692312yescomplemented
420#120—1222no—
421C4×Q8—612no—
422C4×Q8—611no—
423C4×Q8—1221no—
424C4×Q8—1221no—
425C4×D4—611no—
426C4×D4—612no—
427C4×D4—611no—
428C4×D4—612no—
429C4×D4—611no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1221no—
433C4×D4—1222no—
434C4×D4—1228no—
435C4×D4—1221no—
436C4×D4C2×#692311yescharacteristic
437C4×D4C2×#692311yescharacteristic, complemented
438#117—612no—
439#117—1221no—
440#117—1222no—
441#117—1221no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C2×#692312yescomplemented
447C2×C22⋊C4—612no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1222no—
450C2×C42—611no—
451C2×C42—1221no—
452C2×C42—1222no—
453C2×C42—1221no—
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456C22⋊C8—611no—
457C22⋊C8—612no—
458C22⋊C8—611no—
459C22⋊C8—1222no—
460C22⋊C8—1222no—
461#97—1221no—
462C4×C8—1221no—
463#95—611no—
464#95—612no—
465#95—611no—
466#95—1222no—
467#95—1221no—
468C24—612no—
469#41—44no—
470#41—4882no—
471C2×Q8—21no—
472C2×Q8—611no—
473C2×Q8—612no—
474C2×Q8—1222no—
475C2×Q8—4882no—
476C2×D4—22no—
477C2×D4—612no—
478C2×D4—611no—
479C2×D4—1224no—
480C2×D4—1222no—
481C2×D4—1221no—
482C2×D4C22⋊#692311yescharacteristic
483C2×D4C22×#692311yescharacteristic
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—612no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—611no—
492C22×C4—612no—
493C22×C4—1222no—
494C22×C4—1228no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—4882no—
500C22×C4—4882no—
501C22×C4—4882no—
502C22×C4C22×#692312yes—
503M16—4882no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1222no—
507C2×C8—4882no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—611no—
511C4⋊C4—612no—
512C4⋊C4—611no—
513C4⋊C4—1221no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1222no—
519C4⋊C4—1221no—
520C4⋊C4—1224no—
521C4⋊C4—1221no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4C22⋊#692311yescharacteristic
526C4⋊C4C22×#692311yescharacteristic
527C4⋊C4C22×#692312yes—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—612no—
531C22⋊C4—612no—
532C22⋊C4—612no—
533C22⋊C4—1221no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—1228no—
542C22⋊C4—1222no—
543C22⋊C4C22⋊#692312yes—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—1222no—
548C42—1222no—
549C42—1224no—
550C42—1221no—
551C42—4881no—
552C42—4882no—
553C42C22×#692311yescharacteristic
554C23—21no—
555C23—611no—
556C23—612no—
557C23—1222no—
558C23—1222no—
559C23—1224no—
560C23—2442no—
561C23C2×C22⋊#692312yes—
562Q8—42no—
563Q8—4882no—
564Q8—2441no—
565Q8—2442no—
566D4—41no—
567D4—44no—
568D4—1224no—
569D4—1222no—
570D4—4882no—
571D4—2441no—
572D4—2441no—
573D4#1451984114yes—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—612no—
582C2×C4—611no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1224no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1222no—
592C2×C4—4882no—
593C2×C4—4882no—
594C2×C4—4888no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2442no—
605C2×C4—2441no—
606C2×C4C2×C22⋊#692311yescharacteristic
607C2×C4C2×C22⋊#692312yes—
608C2×C4C2×C22⋊#692311yescharacteristic
609C2×C4C23×#692311yescharacteristic
610C8—2441no—
611C8—2442no—
612C8—2441no—
613C22—22no—
614C22—22no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—611no—
619C22—611no—
620C22—1221no—
621C22—1221no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—2444no—
626C22—2444no—
627C22C2×#1451984114yes—
628C22C22×C22⋊#692311yescharacteristic
629C4—21no—
630C4—22no—
631C4—21no—
632C4—41no—
633C4—42no—
634C4—1221no—
635C4—1222no—
636C4—1221no—
637C4—1221no—
638C4—2442no—
639C4—2444no—
640C4—2442no—
641C4—2442no—
642C4—2441no—
643C4—2441no—
644C4C2×#1451984112yes—
645C2—24no—
646C2—81no—
647C2—1221no—
648C2—1221no—
649C2—2442no—
650C2#2146241911yescharacteristic
651C2#2146244111yescharacteristic
652C2C22×#1451984111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#547C4
socleC22×C61C22×C22⋊C4
prime radical#21459859C2
cosocleC2×C4×D61C24
hypercenter#547#6923
hypocenterC61#17548
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #21462679C2semidirect
2. #21462258C2semidirect
3. #21461221C2semidirect
4. #21461218C2semidirect
5. #21461211C2semidirect
6. #21461204C2semidirect
7. #21461202C2semidirect
8. #21461195C2semidirect
9. #21461190C2semidirect
10. C2×#16840615C2semidirect
36. #16840486C4semidirect
38. #16840447C4semidirect
40. D4×Dic122C4semidirect
41. #16840418C4semidirect
42. #16840416C4semidirect
52. C61×#547C4semidirect
76. C2×#11758224C2×C4semidirect
77. C22×Dic122C2×C4semidirect
80. #14519777C2×C4semidirect
86. #14519768C2×C4semidirect
89. #14519764C2×C4semidirect
97. #14519757C2×C4semidirect
98. #14519757C2×C4semidirect
101. #14519756C2×C4semidirect
111. C4×Dic122C2×C4semidirect
112. #14519742C2×C4semidirect
126. #14519735C2×C4semidirect
134. C4⋊Q8×C61C2×C4semidirect
137. #121×C61C2×C4semidirect
139. C4×D4×C61C2×C4semidirect
140. C2×C4⋊C4×C61C2×C4semidirect
345. #547#6923semidirect
379. C61#17548semidirect
390. C4⋊Q8C2×#6923semidirect
419. #121C2×#6923semidirect
437. C4×D4C2×#6923semidirect
446. C2×C4⋊C4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214626792
#214622582
#214612212
#214612182
#214612112
#214612042
#214612022
#214611952
#214611902
C2×#168406152
#214598592
#1754861