Finite Group Databasebeta

← #30564767

Subgroups — #30564767

Order 15616.

2142 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459671C211yesmaximal, characteristic, complemented
2#21457747C214yesmaximal, complemented
3#21457736C214yesmaximal, complemented
4#21457545C214yesmaximal, complemented
5#21457537C214yesmaximal, complemented
6#21456767C211yesmaximal, characteristic, complemented
7#21456744C211yesmaximal, characteristic, complemented
8#21456600C214yesmaximal, complemented
9#21455827C211yesmaximal, characteristic, complemented
10#21455819C211yesmaximal, characteristic, complemented
11#21455037C214yesmaximal, complemented
12#21454843C211yesmaximal, characteristic, complemented
13C2×C61×#547C211yesmaximal, characteristic, complemented
14C2×#14519797C2218yescomplemented
15#16840495—22no—
16#16840482C2212yescomplemented
17#16840477C2212yescomplemented
18#16840462—22no—
19#16840433C2212yescomplemented
20#16840417C2212yescomplemented
21#16840350—22no—
22#16840329—21no—
23#16840319C2211yescharacteristic
24#16840312C2211yescharacteristic
25#16840273C2214yes—
26#16840243—21no—
27#16840141C2212yescomplemented
28#16840136C2212yescomplemented
29#16840112C2212yescomplemented
30#16840112C2212yescomplemented
31#16840100C2218yescomplemented
32#16840092C2212yes—
33#16840075C2212yes—
34#16840052C2212yescomplemented
35#16840042C2212yescomplemented
36#16840001C2218yescomplemented
37#16839993C2218yescomplemented
38#16839969C2214yescomplemented
39#16839967C2212yescomplemented
40#16839964C2214yescomplemented
41#16839957C2212yescomplemented
42#16839924C2218yescomplemented
43#16839922C2218yescomplemented
44#16839918C2214yes—
45#16839913C2212yes—
46#16839907C2218yescomplemented
47#16839901C2214yes—
48#16839896C2212yes—
49C4×#11758200C2214yescomplemented
50C4×#11758199C2214yescomplemented
51D4×#92418C2214yescomplemented
52D4×#92418C2214yescomplemented
53#16839815C2211yescharacteristic
54#16839815C2211yescharacteristic
55#16839758C2218yescomplemented
56C2×C22⋊#92418C2214yes—
57C22×#41×C61C2212yes—
58C61×#547—22no—
59C61×#547—22no—
60C61×#547C2214yes—
61C61×#547C2214yes—
62C2×C4⋊Q8×C61C2211yescharacteristic
63C2×#123×C61—22no—
64C2×C22⋊Q8×C61C2212yes—
65C2×#121×C61—21no—
66C2×C4×D4×C61C2211yescharacteristic
67C2×C4×D4×C61C2211yescharacteristic
68C22×C4⋊C4×C61—21no—
69C2×#11758222—21no—
70C2×#11758222C2314yes—
71#14519797—416no—
72C2×#11758202C2318yes—
73C2×#11758200C2314yes—
74C2×#11758201C2318yes—
75C2×#11758199C2314yes—
76C22×#92418C2318yes—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519773—22no—
82#14519770—24no—
83#14519769—21no—
84#14519765—21no—
85#14519764—22no—
86#14519763—24no—
87#14519760—24no—
88#14519759—24no—
89#14519756—21no—
90#14519756C2314yes—
91#14519756C2312yes—
92#14519751—21no—
93#14519751C2314yes—
94#14519751C2312yes—
95#14519750—24no—
96#14519748—22no—
97#14519748—22no—
98#14519746—24no—
99C4×Dic122—21no—
100C4×Dic122C2312yes—
101#14519742—22no—
102C4×D244—21no—
103C4×D244C2312yes—
104#14519739—22no—
105C4×C22⋊D61—24no—
106C4×C22⋊D61—24no—
107#14519736—21no—
108#14519736C2314yes—
109#14519735—22no—
110C4⋊C4×D61—22no—
111#14519710C2314yes—
112#14519708C2314yes—
113C4⋊#92418C2312yes—
114C4⋊#92418C2312yes—
115#14519701C2318yes—
116#14519701C2318yes—
117#14519700C2314yes—
118#14519698C2314yes—
119#14519698C2314yes—
120#14519697C2314yes—
121C22⋊#92418C2312yes—
122C22⋊#92418C2312yes—
123C22⋊#92418C2318yes—
124C4×#92418C2312yes—
125C4×#92418C2312yes—
126C2×#41×C61—24no—
127C2×#41×C61—48no—
128C2×#41×C61C2318yes—
129C22×Q8×C61C2312yes—
130C22×D4×C61—22no—
131C22×D4×C61D411yescharacteristic
132C22×D4×C61C2311yescharacteristic
133C23×C4×C61—22no—
134C23×C4×C61—21no—
135C23×C4×C61C2312yes—
136C4⋊Q8×C61—21no—
137C4⋊Q8×C61—21no—
138C4⋊Q8×C61C2312yes—
139C4⋊Q8×C61C2312yes—
140#123×C61—28no—
141#123×C61—28no—
142C22⋊Q8×C61—24no—
143C22⋊Q8×C61C2318yes—
144#121×C61—24no—
145#121×C61—24no—
146C4×D4×C61—21no—
147C4×D4×C61—21no—
148C4×D4×C61D412yes—
149C4×D4×C61D412yes—
150C4×D4×C61C2312yes—
151C4×D4×C61C2312yes—
152C4×D4×C61C2312yes—
153C4×D4×C61C2312yes—
154C2×C4⋊C4×C61—21no—
155C2×C4⋊C4×C61—21no—
156C2×C4⋊C4×C61—22no—
157C2×C4⋊C4×C61—24no—
158C2×C4⋊C4×C61—24no—
159C2×C4⋊C4×C61D411yescharacteristic
160C2×C4⋊C4×C61C2311yescharacteristic
161C2×C22⋊C4×C61—22no—
162C2×C22⋊C4×C61—21no—
163C2×C22⋊C4×C61D412yes—
164C2×C42×C61C2311yescharacteristic
165C24×C61—21no—
166C24×C61C2×D412yes—
167#41×C61—216no—
168#41×C61—48no—
169#41×C61—416no—
170C2×Q8×C61—22no—
171C2×Q8×C61—44no—
172C2×Q8×C61C2414yes—
173C2×D4×C61—24no—
174C2×D4×C61—28no—
175C2×D4×C61—24no—
176C2×D4×C61—21no—
177C2×D4×C61—48no—
178C2×D4×C61C2×D412yes—
179C2×D4×C61C2×D412yes—
180C2×D4×C61C2412yes—
181C22×C4×C61—21no—
182C22×C4×C61—22no—
183C22×C4×C61—22no—
184C22×C4×C61—22no—
185C22×C4×C61—28no—
186C22×C4×C61—44no—
187C22×C4×C61—48no—
188C22×C4×C61—44no—
189C22×C4×C61C2×D411yescharacteristic
190C22×C4×C61C2×D412yes—
191C22×C4×C61C2×D414yes—
192C22×C4×C61C2×D411yescharacteristic
193C22×C4×C61C2414yes—
194C22×C4×C61C2411yescharacteristic
195C2×C22⋊D61—22no—
196C2×C22⋊D61—24no—
197C2×C22⋊D61—24no—
198#11758222—48no—
199#11758222—44no—
200C2×D244—21no—
201C2×D244—21no—
202C2×D244C2412yes—
203C2×C4×D61—22no—
204C2×C4×D61—24no—
205C2×Dic122—21no—
206C2×Dic122—21no—
207C2×Dic122C2412yes—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—22no—
210C4⋊C4×C61—28no—
211C4⋊C4×C61—22no—
212C4⋊C4×C61—22no—
213C4⋊C4×C61—21no—
214C4⋊C4×C61C2×D412yes—
215C4⋊C4×C61C2×D412yes—
216C4⋊C4×C61C2412yes—
217C22⋊C4×C61—28no—
218C22⋊C4×C61—22no—
219C22⋊C4×C61—22no—
220C22⋊C4×C61C2×D414yes—
221C22⋊C4×C61C2×D414yes—
222C42×C61C2×D411yescharacteristic
223C42×C61C2×D411yescharacteristic
224C42×C61C2411yescharacteristic
225C42×C61C2411yescharacteristic
226C22⋊Dic61—22no—
227C22⋊Dic61—24no—
228C22⋊Dic61—24no—
229#11758202—48no—
230#11758201—48no—
231#11758200—28no—
232#11758199—28no—
233#11758198—24no—
234#11758198—22no—
235#11758198—22no—
236C4⋊Dic61—21no—
237C4⋊Dic61—21no—
238C4⋊Dic61—21no—
239C4⋊Dic61—21no—
240C4⋊Dic61C2412yes—
241C4⋊Dic61C2412yes—
242#11758196—24no—
243#11758196—22no—
244#11758196—22no—
245C4×Dic61—24no—
246C4×Dic61—22no—
247C2×#92418—48no—
248C2×#92418C2414yes—
249C2×#92418C2414yes—
250C23×C61—24no—
251C23×C61—22no—
252C23×C61—24no—
253C23×C61—42no—
254C23×C61—42no—
255C23×C61—42no—
256C23×C61—42no—
257C23×C61C22×D414yes—
258C23×C61C22×D411yescharacteristic
259C22×D61—22no—
260Q8×C61—28no—
261Q8×C61—44no—
262D4×C61—24no—
263D4×C61—24no—
264D4×C61—24no—
265D4×C61—22no—
266D4×C61—416no—
267D4×C61H(5,𝔽2)14yes—
268C2×C4×C61—28no—
269C2×C4×C61—22no—
270C2×C4×C61—22no—
271C2×C4×C61—24no—
272C2×C4×C61—42no—
273C2×C4×C61—44no—
274C2×C4×C61—41no—
275C2×C4×C61—416no—
276C2×C4×C61—44no—
277C2×C4×C61—42no—
278C2×C4×C61—44no—
279C2×C4×C61—44no—
280C2×C4×C61#13514yescomplemented
281C2×C4×C61C22×D414yes—
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C22×D411yescharacteristic
285C2×C4×C61C22×D411yescharacteristic
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C2511yescharacteristic
288C22⋊D61—44no—
289C22⋊D61—84no—
290C22⋊D61—84no—
291C2×Dic61—22no—
292C2×Dic61—22no—
293C2×Dic61—22no—
294D244—24no—
295D244—42no—
296D244—81no—
297C4×D61—44no—
298C4×D61—84no—
299Dic122—24no—
300Dic122—42no—
301Dic122—81no—
302#92418—24no—
303#92418—24no—
304#61565—611nomaximal, Sylow
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—24no—
308C22×C61—24no—
309C22×C61—24no—
310C22×C61—24no—
311C22×C61—48no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—84no—
315C22×C61C2×#13512yes—
316C22×C61C23×D411yescharacteristic
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—24no—
323C4×C61—21no—
324C4×C61—44no—
325C4×C61—42no—
326C4×C61C2×#13512yes—
327C4×C61C2×#13512yes—
328C4×C61C2×H(5,𝔽2)12yes—
329Dic61—42no—
330Dic61—42no—
331Dic61—42no—
332#3506—611no—
333C2×#547D6111yescharacteristic, complemented
334#3303—614no—
335#3302—614no—
336#3284—614no—
337#3283—614no—
338#3209—611no—
339#3206—611no—
340#3190—614no—
341#3101—611no—
342#3100—611no—
343#3002—614no—
344#2972—611no—
345C2×C61—22no—
346C2×C61—44no—
347C2×C61—44no—
348C2×C61#348411yescharacteristic
349C2×C61C22×#13511yescharacteristic
350C2×C61#358611yescharacteristic
351D61—82no—
352C22×#41C2×D6112yes—
353C2×#135—618no—
354#559—1222no—
355#556—612no—
356#555—612no—
357#553—1222no—
358#550—612no—
359#547—22no—
360#547—22no—
361#547—612no—
362#547C2×D6114yes—
363#547C2×D6114yes—
364#538—1222no—
365#535—1221no—
366#533—611no—
367#532—611no—
368C2×C4⋊Q8C2×D6111yescharacteristic
369#525—614no—
370C2×#123—22no—
371C2×C22⋊Q8C2×D6112yes—
372C2×#121—21no—
373#520—1221no—
374C2×C4×D4C2×D6111yescharacteristic
375C2×C4×D4C2×D6111yescharacteristic
376C22×C4⋊C4—21no—
377#494—612no—
378#492—612no—
379#487—612no—
380#487—612no—
381#485—618no—
382#483—612no—
383#480—612no—
384#476—612no—
385#474—612no—
386#467—618no—
387#465—618no—
388#462—614no—
389#461—614no—
390#461—612no—
391#459—612no—
392#452—618no—
393#452—618no—
394#451—614no—
395#450—612no—
396#449—618no—
397#448—614no—
398#447—612no—
399C4×SD16—614no—
400C4×D8—614no—
401C8×D4—614no—
402C8×D4—614no—
403#431—611no—
404#431—611no—
405#416—618no—
406C2×C22⋊C8—614no—
407C61#6156511yescharacteristic, complemented, Sylow
408C2×#41—24no—
409C2×#41—48no—
410C2×#41—614no—
411C2×#41—1221no—
412C2×#41C22×D6118yes—
413C22×Q8C22×D6112yes—
414C22×D4—22no—
415C22×D4C22⋊D6111yescharacteristic
416C22×D4C22×D6111yescharacteristic
417C23×C4—21no—
418C23×C4—22no—
419C23×C4C22×D6112yes—
420#135—24416no—
421C2×Q16—618no—
422C2×SD16—614no—
423C2×SD16—618no—
424C2×D8—614no—
425C22×C8—618no—
426C4⋊Q8—21no—
427C4⋊Q8—21no—
428C4⋊Q8—611no—
429C4⋊Q8—611no—
430C4⋊Q8—611no—
431C4⋊Q8C22×D6112yes—
432C4⋊Q8C22×D6112yes—
433#127—611no—
434#127—1222no—
435#126—1224no—
436#125—1221no—
437#124—1222no—
438#124—1224no—
439#124—1221no—
440#123—28no—
441#123—28no—
442#123—1224no—
443#123—1224no—
444C22⋊Q8—24no—
445C22⋊Q8—612no—
446C22⋊Q8—614no—
447C22⋊Q8—1221no—
448C22⋊Q8C22×D6118yes—
449#121—24no—
450#121—24no—
451#121—612no—
452#121—614no—
453#121—1222no—
454#121—1222no—
455#121—1221no—
456#121—1224no—
457#120—1224no—
458C4×Q8—612no—
459C4×Q8—1222no—
460C4×Q8—1221no—
461C4×D4—21no—
462C4×D4—21no—
463C4×D4—612no—
464C4×D4—1224no—
465C4×D4—1224no—
466C4×D4—1222no—
467C4×D4—1221no—
468C4×D4C22⋊D6112yes—
469C4×D4C22⋊D6112yes—
470C4×D4C22×D6112yes—
471C4×D4C22×D6112yes—
472C4×D4C22×D6112yes—
473C4×D4C22×D6112yes—
474#117—614no—
475#117—1221no—
476#117—1222no—
477C2×C4⋊C4—21no—
478C2×C4⋊C4—22no—
479C2×C4⋊C4—21no—
480C2×C4⋊C4—24no—
481C2×C4⋊C4—24no—
482C2×C4⋊C4—1222no—
483C2×C4⋊C4C22⋊D6111yescharacteristic
484C2×C4⋊C4C22×D6111yescharacteristic
485C2×C22⋊C4—22no—
486C2×C22⋊C4—21no—
487C2×C22⋊C4C22⋊D6112yes—
488C2×C42C22×D6111yescharacteristic
489#107—614no—
490#106—614no—
491C4⋊C8—612no—
492C4⋊C8—612no—
493#103—618no—
494#103—618no—
495#103—614no—
496#102—614no—
497#102—614no—
498#102—614no—
499C22⋊C8—612no—
500C22⋊C8—612no—
501C22⋊C8—618no—
502C4×C8—612no—
503C4×C8—612no—
504C24—21no—
505C24C2×C22⋊D6112yes—
506#41—216no—
507#41—48no—
508#41—416no—
509#41—2448no—
510#41—2444no—
511C2×Q8—22no—
512C2×Q8—44no—
513C2×Q8—612no—
514C2×Q8—1221no—
515C2×Q8—1221no—
516C2×Q8C23×D6114yes—
517C2×D4—24no—
518C2×D4—28no—
519C2×D4—21no—
520C2×D4—24no—
521C2×D4—48no—
522C2×D4—612no—
523C2×D4—1224no—
524C2×D4—1224no—
525C2×D4—1222no—
526C2×D4—1221no—
527C2×D4—1221no—
528C2×D4C2×C22⋊D6112yes—
529C2×D4C2×C22⋊D6112yes—
530C2×D4C23×D6112yes—
531C22×C4—21no—
532C22×C4—22no—
533C22×C4—22no—
534C22×C4—22no—
535C22×C4—28no—
536C22×C4—44no—
537C22×C4—48no—
538C22×C4—44no—
539C22×C4—1224no—
540C22×C4—1222no—
541C22×C4C2×C22⋊D6111yescharacteristic
542C22×C4C2×C22⋊D6112yes—
543C22×C4C2×C22⋊D6114yes—
544C22×C4C2×C22⋊D6111yescharacteristic
545C22×C4C23×D6111yescharacteristic
546C22×C4C23×D6114yes—
547Q16—2448no—
548SD16—1228no—
549SD16—2448no—
550D8—1228no—
551C2×C8—614no—
552C2×C8—614no—
553C2×C8—2448no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—28no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—612no—
561C4⋊C4—612no—
562C4⋊C4—1221no—
563C4⋊C4—1224no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4C2×C22⋊D6112yes—
570C4⋊C4C2×C22⋊D6112yes—
571C4⋊C4C23×D6112yes—
572C22⋊C4—28no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—1224no—
576C22⋊C4—1222no—
577C22⋊C4—1224no—
578C22⋊C4—1224no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C22⋊C4C2×C22⋊D6114yes—
582C22⋊C4C2×C22⋊D6114yes—
583C42—1224no—
584C42—1222no—
585C42C2×C22⋊D6111yescharacteristic
586C42C2×C22⋊D6111yescharacteristic
587C42C23×D6111yescharacteristic
588C42C23×D6111yescharacteristic
589C23—24no—
590C23—22no—
591C23—24no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—1222no—
597C23C22×C22⋊D6114yes—
598C23C22×C22⋊D6111yescharacteristic
599Q8—28no—
600Q8—44no—
601Q8—1224no—
602Q8—4881no—
603Q8—2442no—
604D4—24no—
605D4—24no—
606D4—24no—
607D4—22no—
608D4—416no—
609D4—1224no—
610D4—4881no—
611D4—4884no—
612D4—4884no—
613D4—2444no—
614D4—2442no—
615D4#1451982214yes—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—416no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—4884no—
632C2×C4—2444no—
633C2×C4#1451979714yescomplemented
634C2×C4C22×C22⋊D6114yes—
635C2×C4C22×C22⋊D6111yescharacteristic
636C2×C4C22×C22⋊D6111yescharacteristic
637C2×C4C22×C22⋊D6111yescharacteristic
638C2×C4C22×C22⋊D6111yescharacteristic
639C2×C4C22×C22⋊D6111yescharacteristic
640C2×C4C24×D6111yescharacteristic
641C8—1224no—
642C8—1224no—
643C22—21no—
644C22—21no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—48no—
650C22—42no—
651C22—42no—
652C22—84no—
653C22—2442no—
654C22—2442no—
655C22—2442no—
656C22C2×#1451979712yes—
657C22C23×C22⋊D6111yescharacteristic
658C4—21no—
659C4—21no—
660C4—24no—
661C4—21no—
662C4—42no—
663C4—44no—
664C4—2442no—
665C4—2442no—
666C4—2442no—
667C4C2×#1451979712yes—
668C4C2×#1451979712yes—
669C4C2×#1451982212yes—
670C2—22no—
671C2—44no—
672C2—44no—
673C2—4882no—
674C2#2145945411yescharacteristic
675C2C22×#1451979711yescharacteristic
676C2#2146060611yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#547C2
socleC22×C61C23×D4
hypercenterC2×#547D61
hypocenterC61#61565
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #21459671C2semidirect
2. #21457747C2semidirect
3. #21457736C2semidirect
4. #21457545C2semidirect
5. #21457537C2semidirect
6. #21456767C2semidirect
7. #21456744C2semidirect
8. #21456600C2semidirect
9. #21455827C2semidirect
10. #21455819C2semidirect
11. #21455037C2semidirect
12. #21454843C2semidirect
13. C2×C61×#547C2semidirect
14. C2×#14519797C22semidirect
16. #16840482C22semidirect
17. #16840477C22semidirect
19. #16840433C22semidirect
20. #16840417C22semidirect
27. #16840141C22semidirect
28. #16840136C22semidirect
29. #16840112C22semidirect
30. #16840112C22semidirect
31. #16840100C22semidirect
34. #16840052C22semidirect
35. #16840042C22semidirect
36. #16840001C22semidirect
37. #16839993C22semidirect
38. #16839969C22semidirect
39. #16839967C22semidirect
40. #16839964C22semidirect
41. #16839957C22semidirect
42. #16839924C22semidirect
43. #16839922C22semidirect
46. #16839907C22semidirect
49. C4×#11758200C22semidirect
50. C4×#11758199C22semidirect
51. D4×#92418C22semidirect
52. D4×#92418C22semidirect
55. #16839758C22semidirect
280. C2×C4×C61#135semidirect
333. C2×#547D61semidirect
407. C61#61565semidirect
633. C2×C4#14519797semidirect

Maximal subgroups

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

subgroupindex
#214596712
#214577472
#214577362
#214575452
#214575372
#214567672
#214567442
#214566002
#214558272
#214558192
#214550372
#214548432
C2×C61×#5472
#6156561