Finite Group Databasebeta

← #30564765

Subgroups — #30564765

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#21459668C211yesmaximal, characteristic, complemented
2C2×#16840416C211yesmaximal, characteristic, complemented
3#21457749C214yesmaximal, complemented
4#21457733C214yesmaximal, complemented
5#21457552C214yesmaximal, complemented
6#21457540C214yesmaximal, complemented
7#21456769C211yesmaximal, characteristic, complemented
8#21456741C211yesmaximal, characteristic, complemented
9#21456599C214yesmaximal, complemented
10#21455827C211yesmaximal, characteristic, complemented
11#21455032C214yesmaximal, complemented
12#21454842C211yesmaximal, characteristic, complemented
13C61×#3100C211yesmaximal, characteristic, complemented
14C22×#11758224C2212yes—
15C2×#14519796C2218yescomplemented
16#16840494—22no—
17#16840486C2212yescomplemented
18#16840477C2212yescomplemented
19#16840458—22no—
20#16840432C2212yescomplemented
21#16840416—22no—
22#16840416—22no—
23#16840416C2214yes—
24#16840416C2214yes—
25#16840416C2212yescomplemented
26#16840349—22no—
27#16840326—21no—
28#16840319C2211yescharacteristic
29C2×#14519777C2211yescharacteristic
30#16840272C2214yes—
31C2×#14519761—22no—
32C2×#14519757C2212yes—
33C2×#14519751—21no—
34#16840245—21no—
35C2×C4×D244C2211yescharacteristic
36C22×C4⋊Dic61—21no—
37#16840142C2212yescomplemented
38#16840134C2212yescomplemented
39#16840114C2212yescomplemented
40#16840113C2212yescomplemented
41#16840105C2218yescomplemented
42#16840092C2212yes—
43#16840052C2212yescomplemented
44#16840004C2218yescomplemented
45#16839990C2218yescomplemented
46#16839973C2214yescomplemented
47#16839967C2214yescomplemented
48#16839967C2212yescomplemented
49#16839925C2218yescomplemented
50#16839925C2218yescomplemented
51#16839920C2214yes—
52#16839913C2212yes—
53#16839904C2218yescomplemented
54#16839900C2214yes—
55C4×#11758200C2214yescomplemented
56C8×D244C2214yescomplemented
57#16839827C2214yescomplemented
58#16839815C2211yescharacteristic
59#16839755C2218yescomplemented
60C22⋊C8×D61C2214yes—
61C61×#532C2211yescharacteristic
62C2×C4×D4×C61C2211yescharacteristic
63C61×#480C2212yes—
64C61×#474C2212yescomplemented
65C61×#459C2212yescomplemented
66C61×#447C2212yes—
67C4×D8×C61C2214yescomplemented
68C61×#431C2211yescharacteristic
69C2×#11758224—24no—
70C2×#11758224—48no—
71C2×#11758224C2318yes—
72C2×#11758224C2314yes—
73C22×Dic122C2312yes—
74C22×D244D411yescharacteristic
75C22×C22⋊D61—22no—
76C23×Dic61—22no—
77C22×C4×D61C2312yes—
78#14519796—416no—
79C2×Dic244C2318yes—
80C2×#11758200C2314yes—
81C2×#11758200C2318yes—
82C2×C8×D61C2318yes—
83#14519777—21no—
84#14519777—21no—
85#14519777C2312yes—
86#14519777C2311yescharacteristic
87#14519777C2312yes—
88#14519777C2311yescharacteristic
89#14519773—22no—
90#14519772—24no—
91#14519769—21no—
92#14519766—22no—
93#14519763—24no—
94#14519761—24no—
95#14519761—28no—
96#14519761—28no—
97#14519759—24no—
98#14519757—24no—
99#14519757C2314yes—
100#14519757C2318yes—
101#14519756—21no—
102#14519756C2312yes—
103#14519751—22no—
104#14519751—22no—
105#14519751—24no—
106#14519751—24no—
107#14519749—24no—
108#14519748C2314yes—
109#14519745—24no—
110Q8×Dic61—22no—
111C4×Dic122—21no—
112C4×Dic122C2312yes—
113C4×D244D412yes—
114C4×D244D412yes—
115C4×D244C2312yes—
116C4×D244C2312yes—
117D4×Dic61—24no—
118D4×Dic61—22no—
119#14519737—24no—
120#14519736—21no—
121#14519736—22no—
122#14519735C2314yes—
123C2×C4⋊Dic61—21no—
124C2×C4⋊Dic61—21no—
125C2×C4⋊Dic61—22no—
126C2×C4⋊Dic61—22no—
127C2×C4⋊Dic61—24no—
128C2×C4⋊Dic61—24no—
129C2×C4⋊Dic61D411yescharacteristic
130C2×#11758198—22no—
131C2×#11758198D412yes—
132#14519708C2314yes—
133C4⋊#92418C2312yes—
134#14519702C2318yes—
135#14519700C2314yes—
136#14519700C2318yes—
137#14519698C2314yes—
138C22⋊#92418C2312yes—
139#14519689C2318yes—
140C4×#92418C2312yes—
141C2×#41×C61—21no—
142C22×D4×C61C2311yescharacteristic
143C23×C4×C61—21no—
144C2×D8×C61C2314yes—
145C4⋊Q8×C61C2311yescharacteristic
146#127×C61C2311yescharacteristic
147#124×C61—21no—
148#121×C61—21no—
149#121×C61C2312yes—
150C4×D4×C61—21no—
151C4×D4×C61—21no—
152C4×D4×C61—21no—
153C4×D4×C61C2312yes—
154C4×D4×C61C2312yes—
155C4×D4×C61C2312yes—
156C2×C4⋊C4×C61C2311yescharacteristic
157C2×C22⋊C4×C61—21no—
158C2×C42×C61C2311yescharacteristic
159#107×C61C2314yes—
160C4⋊C8×C61C2312yes—
161#102×C61C2314yes—
162#102×C61C2314yes—
163C22⋊C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C24×C61—21no—
166C23×D61C2×D412yes—
167#41×C61—44no—
168C2×Q8×C61—21no—
169C2×D4×C61—24no—
170C2×D4×C61—22no—
171C2×D4×C61—21no—
172C2×D4×C61—21no—
173C2×D4×C61—21no—
174C2×D4×C61C2412yes—
175C2×D4×C61C2412yes—
176C22×C4×C61—21no—
177C22×C4×C61—22no—
178C22×C4×C61—22no—
179C22×C4×C61—22no—
180C22×C4×C61—44no—
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2411yescharacteristic
184C2×C22⋊D61—28no—
185C2×C22⋊D61—24no—
186C2×C22⋊D61—24no—
187C2×C22⋊D61—48no—
188C22×Dic61—22no—
189C22×Dic61—28no—
190C22×Dic61—24no—
191C22×Dic61—48no—
192C22×Dic61C2×D412yes—
193#11758224—216no—
194#11758224—48no—
195#11758224—48no—
196#11758224—416no—
197C2×D244—24no—
198C2×D244C2×D412yes—
199C2×D244C2×D412yes—
200C2×C4×D61—44no—
201C2×C4×D61C2×D414yes—
202C2×C4×D61C2414yes—
203C2×Dic122—22no—
204C2×Dic122—21no—
205C2×Dic122—44no—
206C2×Dic122C2414yes—
207C2×Dic122C2412yes—
208D8×C61—28no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61C2412yes—
213C4⋊C4×C61C2412yes—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—22no—
217C22⋊C4×C61—22no—
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2411yescharacteristic
221C42×C61C2411yescharacteristic
222C22⋊Dic61—22no—
223C22⋊Dic61—24no—
224#11758200—28no—
225#11758200—48no—
226#11758198—28no—
227#11758198—24no—
228#11758198—24no—
229#11758198C2×D414yes—
230#11758198C2×D414yes—
231C4⋊Dic61—22no—
232C4⋊Dic61—22no—
233C4⋊Dic61—21no—
234C4⋊Dic61—22no—
235C4⋊Dic61—22no—
236C4⋊Dic61—21no—
237C4⋊Dic61—21no—
238C4⋊Dic61—28no—
239C4⋊Dic61—24no—
240C4⋊Dic61C2×D412yes—
241C4⋊Dic61C2×D412yes—
242C4⋊Dic61C2412yes—
243#11758196—22no—
244#11758196—22no—
245C4×Dic61—22no—
246C4×Dic61—24no—
247C2×#92418C2414yes—
248Dic244—48no—
249C8×D61—48no—
250C23×C61—24no—
251C23×C61—22no—
252C23×C61—22no—
253C23×C61—42no—
254C23×C61—42no—
255C23×C61C22×D411yescharacteristic
256C22×D61—24no—
257C22×D61—42no—
258C22×D61—42no—
259C22×D61C22×D414yes—
260Q8×C61—42no—
261D4×C61—24no—
262D4×C61—22no—
263D4×C61—24no—
264D4×C61—44no—
265D4×C61—42no—
266D4×C61—81no—
267D4×C61H(5,𝔽2)14yes—
268C2×C4×C61—22no—
269C2×C4×C61—22no—
270C2×C4×C61—22no—
271C2×C4×C61—41no—
272C2×C4×C61—42no—
273C2×C4×C61—44no—
274C2×C4×C61—44no—
275C2×C4×C61—44no—
276C2×C4×C61#13514yescomplemented
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C2511yescharacteristic
283C22⋊D61—416no—
284C22⋊D61—84no—
285C22⋊D61—84no—
286C2×Dic61—22no—
287C2×Dic61—22no—
288C2×Dic61—24no—
289C2×Dic61—42no—
290C2×Dic61—416no—
291C2×Dic61—44no—
292C2×Dic61—84no—
293C2×Dic61C22×D414yes—
294D244—24no—
295D244—24no—
296C4×D61—28no—
297C4×D61—44no—
298Dic122—24no—
299Dic122—28no—
300Dic122—44no—
301Dic122—81no—
302C8×C61—24no—
303#92418—24no—
304#61565—611nomaximal, Sylow
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—24no—
308C22×C61—48no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61C2×#13512yes—
315C22×C61C23×D411yescharacteristic
316C2×D61—24no—
317C2×D61—24no—
318C2×D61—24no—
319C2×D61—84no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—42no—
324C4×C61—42no—
325C4×C61C2×#13512yes—
326C4×C61C2×#13512yes—
327C4×C61C2×H(5,𝔽2)12yes—
328Dic61—24no—
329Dic61—42no—
330Dic61—44no—
331Dic61—42no—
332#3506—611no—
333C2×#547—611no—
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#3100D6111yescharacteristic, complemented
343#3002—614no—
344#2972—611no—
345C2×C61—22no—
346C2×C61—44no—
347C2×C61—82no—
348C2×C61#348411yescharacteristic
349C2×C61C22×#13511yescharacteristic
350C2×C61#358611yescharacteristic
351D61—44no—
352C22×#41—612no—
353C2×#135—618no—
354#559—1222no—
355#556—612no—
356#555—612no—
357#553—1222no—
358#550—612no—
359#547—614no—
360#547—614no—
361#547—612no—
362#547—1222no—
363#547—1222no—
364#538—1222no—
365#535—1221no—
366#533—611no—
367#532C2×D6111yescharacteristic
368C2×C4⋊Q8—611no—
369#525—614no—
370C2×#123—1222no—
371C2×C22⋊Q8—612no—
372C2×#121—1221no—
373#520—1221no—
374C2×C4×D4—611no—
375C2×C4×D4C2×D6111yescharacteristic
376C22×C4⋊C4—1221no—
377#494—612no—
378#492—612no—
379#487—612no—
380#487—612no—
381#485—618no—
382#483—612no—
383#480C2×D6112yes—
384#476—612no—
385#474C2×D6112yescomplemented
386#467—618no—
387#465—618no—
388#462—614no—
389#461—612no—
390#461—614no—
391#459C2×D6112yescomplemented
392#452—618no—
393#452—618no—
394#451—614no—
395#450—612no—
396#449—618no—
397#448—614no—
398#447C2×D6112yes—
399C4×SD16—614no—
400C4×D8C2×D6114yescomplemented
401C8×D4—614no—
402C8×D4—614no—
403#431—611no—
404#431C2×D6111yescharacteristic
405#416—618no—
406C2×C22⋊C8—614no—
407C61#6156511yescharacteristic, complemented, Sylow
408C2×#41—21no—
409C2×#41—618no—
410C2×#41—614no—
411C2×#41—1224no—
412C2×#41—2448no—
413C22×Q8—612no—
414C22×D4—611no—
415C22×D4—1222no—
416C22×D4C22×D6111yescharacteristic
417C23×C4—21no—
418C23×C4—612no—
419C23×C4—1222no—
420#135—24416no—
421C2×Q16—618no—
422C2×SD16—614no—
423C2×SD16—618no—
424C2×D8C22×D6114yes—
425C22×C8—618no—
426C4⋊Q8—611no—
427C4⋊Q8—612no—
428C4⋊Q8—612no—
429C4⋊Q8—611no—
430C4⋊Q8—1221no—
431C4⋊Q8—1221no—
432C4⋊Q8C22×D6111yescharacteristic
433#127—1222no—
434#127C22×D6111yescharacteristic
435#126—1224no—
436#125—1221no—
437#124—21no—
438#124—1224no—
439#124—1222no—
440#123—1228no—
441#123—1228no—
442#123—1224no—
443#123—1224no—
444C22⋊Q8—612no—
445C22⋊Q8—618no—
446C22⋊Q8—614no—
447C22⋊Q8—1224no—
448C22⋊Q8—1221no—
449#121—21no—
450#121—614no—
451#121—1224no—
452#121—1224no—
453#121—1224no—
454#121—1222no—
455#121—1222no—
456#121C22×D6112yes—
457#120—1224no—
458C4×Q8—612no—
459C4×Q8—1222no—
460C4×Q8—1221no—
461C4×D4—21no—
462C4×D4—21no—
463C4×D4—21no—
464C4×D4—612no—
465C4×D4—612no—
466C4×D4—612no—
467C4×D4—612no—
468C4×D4—1224no—
469C4×D4—1222no—
470C4×D4—1224no—
471C4×D4C22×D6112yes—
472C4×D4C22×D6112yes—
473C4×D4C22×D6112yes—
474#117—614no—
475#117—1222no—
476#117—1221no—
477C2×C4⋊C4—611no—
478C2×C4⋊C4—1224no—
479C2×C4⋊C4—1224no—
480C2×C4⋊C4—1222no—
481C2×C4⋊C4—1222no—
482C2×C4⋊C4—1221no—
483C2×C4⋊C4—1221no—
484C2×C4⋊C4C22×D6111yescharacteristic
485C2×C22⋊C4—21no—
486C2×C22⋊C4—612no—
487C2×C22⋊C4—1222no—
488C2×C42C22×D6111yescharacteristic
489#107C22×D6114yes—
490#106—614no—
491C4⋊C8—612no—
492C4⋊C8C22×D6112yes—
493#103—618no—
494#103—618no—
495#103—614no—
496#102—614no—
497#102C22×D6114yes—
498#102C22×D6114yes—
499C22⋊C8—612no—
500C22⋊C8—618no—
501C22⋊C8C22×D6112yes—
502C4×C8—612no—
503C4×C8C22×D6112yes—
504C24—21no—
505C24—612no—
506#41—44no—
507#41—12216no—
508#41—2448no—
509#41—2448no—
510#41—24416no—
511C2×Q8—21no—
512C2×Q8—614no—
513C2×Q8—612no—
514C2×Q8—1221no—
515C2×Q8—1222no—
516C2×Q8—2444no—
517C2×D4—24no—
518C2×D4—22no—
519C2×D4—21no—
520C2×D4—21no—
521C2×D4—21no—
522C2×D4—612no—
523C2×D4—612no—
524C2×D4—1224no—
525C2×D4—1224no—
526C2×D4—1228no—
527C2×D4—1224no—
528C2×D4—2448no—
529C2×D4C23×D6112yes—
530C2×D4C23×D6112yes—
531C22×C4—21no—
532C22×C4—22no—
533C22×C4—22no—
534C22×C4—22no—
535C22×C4—44no—
536C22×C4—612no—
537C22×C4—614no—
538C22×C4—614no—
539C22×C4—1224no—
540C22×C4—1228no—
541C22×C4—1222no—
542C22×C4—2444no—
543C22×C4—2448no—
544C22×C4D4×D6111yescharacteristic
545C22×C4D4×D6111yescharacteristic
546C22×C4C23×D6111yescharacteristic
547Q16—2448no—
548SD16—1228no—
549SD16—2448no—
550D8—28no—
551C2×C8—614no—
552C2×C8—2448no—
553C2×C8C23×D6114yes—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—612no—
557C4⋊C4—612no—
558C4⋊C4—612no—
559C4⋊C4—1228no—
560C4⋊C4—1224no—
561C4⋊C4—1222no—
562C4⋊C4—1222no—
563C4⋊C4—1221no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C4⋊C4—1222no—
567C4⋊C4—1222no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4C23×D6112yes—
571C4⋊C4C23×D6112yes—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—614no—
577C22⋊C4—614no—
578C22⋊C4—1228no—
579C22⋊C4—1224no—
580C22⋊C4—1224no—
581C22⋊C4—1224no—
582C22⋊C4—1222no—
583C42—1224no—
584C42—1222no—
585C42D4×D6111yescharacteristic
586C42D4×D6111yescharacteristic
587C42C23×D6111yescharacteristic
588C42C23×D6111yescharacteristic
589C23—24no—
590C23—22no—
591C23—22no—
592C23—42no—
593C23—42no—
594C23—614no—
595C23—1224no—
596C23—2442no—
597C23—2442no—
598C23C2×D4×D6111yescharacteristic
599Q8—42no—
600Q8—1224no—
601Q8—1228no—
602Q8—4881no—
603Q8—2444no—
604D4—24no—
605D4—22no—
606D4—24no—
607D4—44no—
608D4—42no—
609D4—81no—
610D4—1224no—
611D4—1224no—
612D4—4884no—
613D4—4884no—
614D4—24416no—
615D4#1451982314yes—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—614no—
625C2×C4—1228no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1224no—
629C2×C4—4884no—
630C2×C4—2444no—
631C2×C4—2442no—
632C2×C4—24416no—
633C2×C4—2444no—
634C2×C4#1451979614yescomplemented
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C24×D6111yescharacteristic
641C8—24no—
642C8—1224no—
643C22—21no—
644C22—21no—
645C22—24no—
646C22—48no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—1224no—
653C22—1224no—
654C22—1224no—
655C22—4884no—
656C22C2×#1451979612yes—
657C22C22×D4×D6111yescharacteristic
658C4—21no—
659C4—21no—
660C4—21no—
661C4—42no—
662C4—42no—
663C4—1224no—
664C4—2442no—
665C4—2442no—
666C4—2444no—
667C4C2×#1451979612yes—
668C4C2×#1451979612yes—
669C4C2×#1451982312yes—
670C2—22no—
671C2—44no—
672C2—82no—
673C2—2444no—
674C2#2145945211yescharacteristic
675C2C22×#1451979611yescharacteristic
676C2#2146060311yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3100C2
socleC22×C61C23×D4
hypercenter#3100D61
hypocenterC61#61565
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #21459668C2semidirect
2. C2×#16840416C2semidirect
3. #21457749C2semidirect
4. #21457733C2semidirect
5. #21457552C2semidirect
6. #21457540C2semidirect
7. #21456769C2semidirect
8. #21456741C2semidirect
9. #21456599C2semidirect
10. #21455827C2semidirect
11. #21455032C2semidirect
12. #21454842C2semidirect
13. C61×#3100C2semidirect
15. C2×#14519796C22semidirect
17. #16840486C22semidirect
18. #16840477C22semidirect
20. #16840432C22semidirect
25. #16840416C22semidirect
37. #16840142C22semidirect
38. #16840134C22semidirect
39. #16840114C22semidirect
40. #16840113C22semidirect
41. #16840105C22semidirect
43. #16840052C22semidirect
44. #16840004C22semidirect
45. #16839990C22semidirect
46. #16839973C22semidirect
47. #16839967C22semidirect
48. #16839967C22semidirect
49. #16839925C22semidirect
50. #16839925C22semidirect
53. #16839904C22semidirect
55. C4×#11758200C22semidirect
56. C8×D244C22semidirect
57. #16839827C22semidirect
59. #16839755C22semidirect
64. C61×#474C22semidirect
65. C61×#459C22semidirect
67. C4×D8×C61C22semidirect
276. C2×C4×C61#135semidirect
342. #3100D61semidirect
385. #474C2×D61semidirect
391. #459C2×D61semidirect
400. C4×D8C2×D61semidirect
407. C61#61565semidirect
634. C2×C4#14519796semidirect

Maximal subgroups

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

subgroupindex
#214596682
C2×#168404162
#214577492
#214577332
#214575522
#214575402
#214567692
#214567412
#214565992
#214558272
#214550322
#214548422
C61×#31002
#6156561