Finite Group Databasebeta

← #30564769

Subgroups — #30564769

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
2C2×#16840417C211yesmaximal, characteristic, complemented
3#21457748C214yesmaximal, complemented
4#21457737C214yesmaximal, complemented
5#21457551C214yesmaximal, complemented
6#21457541C214yesmaximal, complemented
7#21456769C211yesmaximal, characteristic, complemented
8#21456746C211yesmaximal, characteristic, complemented
9#21456602C214yesmaximal, complemented
10#21455828C211yesmaximal, characteristic, complemented
11#21455820C211yesmaximal, characteristic, complemented
12#21455041C214yesmaximal, complemented
13C61×#2972C211yesmaximal, characteristic, complemented
14C22×#11758222C2212yes—
15C2×#14519799C2218yescomplemented
16#16840495—22no—
17#16840482C2212yescomplemented
18#16840477C2212yescomplemented
19#16840462—22no—
20#16840433C2212yescomplemented
21#16840417—22no—
22#16840417—22no—
23#16840417C2214yes—
24#16840417C2214yes—
25#16840417C2212yescomplemented
26#16840350—22no—
27#16840329—21no—
28#16840319C2211yescharacteristic
29#16840312C2211yescharacteristic
30C2×#14519777C2211yescharacteristic
31#16840273C2214yes—
32C2×#14519759—22no—
33C2×#14519756C2212yes—
34C2×#14519748—21no—
35#16840243—21no—
36C2×C4×D244C2211yescharacteristic
37C2×C4⋊C4×D61—21no—
38#16840142C2212yescomplemented
39#16840135C2212yescomplemented
40#16840113C2212yescomplemented
41#16840113C2212yescomplemented
42#16840104C2218yescomplemented
43#16840093C2212yes—
44#16840076C2212yes—
45#16840053C2212yescomplemented
46#16840043C2212yescomplemented
47#16840003C2218yescomplemented
48#16839991C2218yescomplemented
49#16839973C2214yescomplemented
50#16839968C2214yescomplemented
51#16839968C2212yescomplemented
52#16839958C2212yescomplemented
53#16839926C2218yescomplemented
54#16839926C2218yescomplemented
55#16839920C2214yes—
56#16839915C2212yes—
57#16839909C2218yescomplemented
58#16839903C2214yes—
59#16839898C2212yes—
60C4×#11758190C2214yescomplemented
61C4×D488C2214yescomplemented
62#16839759C2218yescomplemented
63C2×C4×D4×C61C2211yescharacteristic
64C8×D4×C61C2214yescomplemented
65C8×D4×C61C2214yescomplemented
66C61×#431C2211yescharacteristic
67C61×#431C2211yescharacteristic
68C2×C22⋊C8×C61C2214yes—
69C2×#11758222—21no—
70C2×#11758222—24no—
71C2×#11758222—48no—
72C2×#11758222C2318yes—
73C2×#11758222C2314yes—
74C22×Dic122C2312yes—
75C22×D244C2311yescharacteristic
76C22×C22⋊D61—22no—
77C22×C4×D61—21no—
78C22×C4×D61—22no—
79#14519799—416no—
80C2×Dic244C2318yes—
81C2×#11758190C2314yes—
82C2×#11758190C2318yes—
83C2×D488C2314yes—
84#14519777—21no—
85#14519777—21no—
86#14519777C2312yes—
87#14519777C2311yescharacteristic
88#14519777C2312yes—
89#14519777C2311yescharacteristic
90#14519777C2311yescharacteristic
91#14519774C2311yescharacteristic
92#14519773—22no—
93#14519770—24no—
94#14519769—21no—
95#14519765—21no—
96#14519764—22no—
97#14519763—24no—
98#14519760—24no—
99#14519759—24no—
100#14519759—28no—
101#14519759—28no—
102#14519756—24no—
103#14519756—21no—
104#14519756C2314yes—
105#14519756C2318yes—
106#14519756C2312yes—
107#14519751—21no—
108#14519751C2314yes—
109#14519751C2312yes—
110#14519750—24no—
111#14519748—22no—
112#14519748—22no—
113#14519748—24no—
114#14519748—24no—
115#14519746—24no—
116C4×Dic122—21no—
117C4×Dic122C2312yes—
118#14519742—22no—
119C4×D244—21no—
120C4×D244—21no—
121C4×D244—21no—
122C4×D244C2312yes—
123C4×D244C2312yes—
124C4×D244C2312yes—
125#14519739—22no—
126C4×C22⋊D61—24no—
127C4×C22⋊D61—24no—
128#14519736—21no—
129#14519736C2314yes—
130#14519735—22no—
131C2×C4⋊Dic61—21no—
132C2×C4⋊Dic61—21no—
133C2×C4⋊Dic61C2311yescharacteristic
134C2×#11758196—22no—
135C4⋊C4×D61—22no—
136C4⋊C4×D61—24no—
137C4⋊C4×D61—24no—
138C2×C22⋊Dic61—22no—
139C2×#11758198—21no—
140#14519711C2314yes—
141#14519709C2314yes—
142#14519702C2314yes—
143#14519702C2318yes—
144#14519702C2318yes—
145#14519699C2314yes—
146#14519699C2314yes—
147#14519699C2314yes—
148C22×D4×C61D411yescharacteristic
149C23×C4×C61C2312yes—
150C22×C8×C61C2318yes—
151C4×D4×C61D412yes—
152C4×D4×C61D412yes—
153C4×D4×C61C2312yes—
154C4×D4×C61C2312yes—
155C2×C4⋊C4×C61D411yescharacteristic
156C2×C22⋊C4×C61D412yes—
157C2×C42×C61C2311yescharacteristic
158C4⋊C8×C61C2312yes—
159C4⋊C8×C61C2312yes—
160C22⋊C8×C61C2312yes—
161C22⋊C8×C61C2312yes—
162C22⋊C8×C61C2318yes—
163C4×C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C24×C61C2×D412yes—
166C23×D61—21no—
167C2×D4×C61—24no—
168C2×D4×C61C2×D412yes—
169C2×D4×C61C2×D412yes—
170C22×C4×C61—44no—
171C22×C4×C61C2×D412yes—
172C22×C4×C61C2×D414yes—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C22×C4×C61C2414yes—
177C2×C22⋊D61—28no—
178C2×C22⋊D61—22no—
179C2×C22⋊D61—24no—
180C2×C22⋊D61—24no—
181C2×C22⋊D61—48no—
182C22×Dic61—22no—
183C22×Dic61—21no—
184#11758222—216no—
185#11758222—48no—
186#11758222—48no—
187#11758222—416no—
188#11758222—44no—
189C2×D244—21no—
190C2×D244—24no—
191C2×D244—21no—
192C2×D244—21no—
193C2×D244C2412yes—
194C2×D244C2412yes—
195C2×C4×D61—28no—
196C2×C4×D61—22no—
197C2×C4×D61—22no—
198C2×C4×D61—22no—
199C2×C4×D61—24no—
200C2×C4×D61—48no—
201C2×C4×D61—44no—
202C2×Dic122—22no—
203C2×Dic122—21no—
204C2×Dic122—21no—
205C2×Dic122—44no—
206C2×Dic122C2414yes—
207C2×Dic122C2412yes—
208C2×C8×C61—48no—
209C2×C8×C61C2414yes—
210C2×C8×C61C2414yes—
211C4⋊C4×C61C2×D412yes—
212C4⋊C4×C61C2×D412yes—
213C22⋊C4×C61C2×D414yes—
214C22⋊C4×C61C2×D414yes—
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D411yescharacteristic
217C42×C61C2411yescharacteristic
218C42×C61C2411yescharacteristic
219C22⋊Dic61—28no—
220C22⋊Dic61—22no—
221C22⋊Dic61—24no—
222C22⋊Dic61—24no—
223#11758198—22no—
224#11758198—22no—
225#11758198—24no—
226#11758198—22no—
227#11758198—22no—
228C4⋊Dic61—22no—
229C4⋊Dic61—22no—
230C4⋊Dic61—21no—
231C4⋊Dic61—22no—
232C4⋊Dic61—22no—
233C4⋊Dic61—21no—
234C4⋊Dic61—21no—
235C4⋊Dic61—21no—
236C4⋊Dic61—21no—
237C4⋊Dic61C2412yes—
238C4⋊Dic61C2412yes—
239C4⋊Dic61C2412yes—
240#11758196—28no—
241#11758196—24no—
242#11758196—22no—
243#11758196—22no—
244C4×Dic61—24no—
245C4×Dic61—22no—
246Dic244—48no—
247D488—28no—
248#11758190—28no—
249#11758190—48no—
250C23×C61—24no—
251C23×C61—42no—
252C23×C61—42no—
253C23×C61C22×D414yes—
254C23×C61C22×D411yescharacteristic
255C22×D61—24no—
256C22×D61—22no—
257C22×D61—22no—
258C22×D61—42no—
259C22×D61—42no—
260D4×C61—24no—
261D4×C61—24no—
262C2×C4×C61—28no—
263C2×C4×C61—22no—
264C2×C4×C61—42no—
265C2×C4×C61—44no—
266C2×C4×C61—41no—
267C2×C4×C61#13514yescomplemented
268C2×C4×C61C22×D414yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2511yescharacteristic
275C22⋊D61—416no—
276C22⋊D61—44no—
277C22⋊D61—84no—
278C22⋊D61—84no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—24no—
284C2×Dic61—44no—
285C2×Dic61—42no—
286D244—24no—
287D244—22no—
288D244—24no—
289D244—42no—
290D244—81no—
291D244H(5,𝔽2)14yes—
292C4×D61—416no—
293C4×D61—44no—
294C4×D61—44no—
295C4×D61—44no—
296C4×D61—84no—
297Dic122—24no—
298Dic122—28no—
299Dic122—44no—
300Dic122—42no—
301Dic122—81no—
302C8×C61—24no—
303C8×C61—24no—
304#61565—611nomaximal, Sylow
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—24no—
308C22×C61—24no—
309C22×C61—24no—
310C22×C61—84no—
311C22×C61C2×#13512yes—
312C22×C61C23×D411yescharacteristic
313C2×D61—24no—
314C2×D61—48no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—24no—
324C4×C61C2×#13512yes—
325C4×C61C2×#13512yes—
326C4×C61C2×H(5,𝔽2)12yes—
327Dic61—42no—
328Dic61—44no—
329Dic61—42no—
330Dic61—42no—
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#3100—611no—
343#3002—614no—
344#2972D6111yescharacteristic, complemented
345C2×C61—22no—
346C2×C61—44no—
347C2×C61#348411yescharacteristic
348C2×C61C22×#13511yescharacteristic
349C2×C61#358611yescharacteristic
350D61—44no—
351D61—82no—
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#532—611no—
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#480—612no—
384#476—612no—
385#474—612no—
386#467—618no—
387#465—618no—
388#462—614no—
389#461—612no—
390#461—614no—
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×D4C2×D6114yescomplemented
402C8×D4C2×D6114yescomplemented
403#431C2×D6111yescharacteristic
404#431C2×D6111yescharacteristic
405#416—618no—
406C2×C22⋊C8C2×D6114yes—
407C61#6156511yescharacteristic, complemented, Sylow
408C2×#41—618no—
409C2×#41—614no—
410C2×#41—1221no—
411C2×#41—1224no—
412C2×#41—2448no—
413C22×Q8—612no—
414C22×D4—611no—
415C22×D4—1222no—
416C22×D4D24411yescharacteristic
417C23×C4—1221no—
418C23×C4—1222no—
419C23×C4C22×D6112yes—
420#135—24416no—
421C2×Q16—618no—
422C2×SD16—614no—
423C2×SD16—618no—
424C2×D8—614no—
425C22×C8C22×D6118yes—
426C4⋊Q8—611no—
427C4⋊Q8—611no—
428C4⋊Q8—612no—
429C4⋊Q8—612no—
430C4⋊Q8—611no—
431C4⋊Q8—1221no—
432C4⋊Q8—1221no—
433#127—611no—
434#127—1222no—
435#126—1224no—
436#125—1221no—
437#124—1222no—
438#124—1224no—
439#124—1221no—
440#123—1224no—
441#123—1228no—
442#123—1228no—
443#123—1224no—
444C22⋊Q8—618no—
445C22⋊Q8—612no—
446C22⋊Q8—614no—
447C22⋊Q8—1224no—
448C22⋊Q8—1221no—
449#121—612no—
450#121—614no—
451#121—1224no—
452#121—1224no—
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—612no—
462C4×D4—612no—
463C4×D4—612no—
464C4×D4—1224no—
465C4×D4—1224no—
466C4×D4—1222no—
467C4×D4—1221no—
468C4×D4—1221no—
469C4×D4—1221no—
470C4×D4D24412yes—
471C4×D4D24412yes—
472C4×D4C22×D6112yes—
473C4×D4C22×D6112yes—
474#117—614no—
475#117—1221no—
476#117—1222no—
477C2×C4⋊C4—611no—
478C2×C4⋊C4—1221no—
479C2×C4⋊C4—1222no—
480C2×C4⋊C4—1221no—
481C2×C4⋊C4—1224no—
482C2×C4⋊C4—1224no—
483C2×C4⋊C4—1222no—
484C2×C4⋊C4D24411yescharacteristic
485C2×C22⋊C4—1222no—
486C2×C22⋊C4—1221no—
487C2×C22⋊C4D24412yes—
488C2×C42C22×D6111yescharacteristic
489#107—614no—
490#106—614no—
491C4⋊C8C22×D6112yes—
492C4⋊C8C22×D6112yes—
493#103—618no—
494#103—618no—
495#103—614no—
496#102—614no—
497#102—614no—
498#102—614no—
499C22⋊C8C22×D6112yes—
500C22⋊C8C22×D6112yes—
501C22⋊C8C22×D6118yes—
502C4×C8C22×D6112yes—
503C4×C8C22×D6112yes—
504C24—1221no—
505C24C2×D24412yes—
506#41—12216no—
507#41—2448no—
508#41—2448no—
509#41—24416no—
510#41—2444no—
511C2×Q8—614no—
512C2×Q8—612no—
513C2×Q8—1221no—
514C2×Q8—1221no—
515C2×Q8—1222no—
516C2×Q8—2444no—
517C2×D4—24no—
518C2×D4—612no—
519C2×D4—612no—
520C2×D4—1224no—
521C2×D4—1224no—
522C2×D4—1224no—
523C2×D4—1222no—
524C2×D4—1228no—
525C2×D4—1221no—
526C2×D4—1221no—
527C2×D4—1221no—
528C2×D4—2448no—
529C2×D4C2×D24412yes—
530C2×D4C2×D24412yes—
531C22×C4—44no—
532C22×C4—1221no—
533C22×C4—1222no—
534C22×C4—1224no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1228no—
538C22×C4—1222no—
539C22×C4—2448no—
540C22×C4—2444no—
541C22×C4C2×D24411yescharacteristic
542C22×C4C2×D24412yes—
543C22×C4C2×D24414yes—
544C22×C4C2×D24411yescharacteristic
545C22×C4C23×D6111yescharacteristic
546C22×C4C23×D6114yes—
547Q16—2448no—
548SD16—1228no—
549SD16—2448no—
550D8—1228no—
551C2×C8—48no—
552C2×C8C23×D6114yes—
553C2×C8C23×D6114yes—
554C4⋊C4—612no—
555C4⋊C4—612no—
556C4⋊C4—612no—
557C4⋊C4—1221no—
558C4⋊C4—1228no—
559C4⋊C4—1222no—
560C4⋊C4—1222no—
561C4⋊C4—1221no—
562C4⋊C4—1224no—
563C4⋊C4—1222no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C4⋊C4—1222no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4C2×D24412yes—
571C4⋊C4C2×D24412yes—
572C22⋊C4—1228no—
573C22⋊C4—1224no—
574C22⋊C4—1222no—
575C22⋊C4—1224no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C22⋊C4—1224no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C22⋊C4C2×D24414yes—
582C22⋊C4C2×D24414yes—
583C42—1224no—
584C42—1222no—
585C42C2×D24411yescharacteristic
586C42C2×D24411yescharacteristic
587C42C23×D6111yescharacteristic
588C42C23×D6111yescharacteristic
589C23—24no—
590C23—42no—
591C23—42no—
592C23—1224no—
593C23—1222no—
594C23—1222no—
595C23—2442no—
596C23—2442no—
597C23C22×D24414yes—
598C23C22×D24411yescharacteristic
599Q8—1224no—
600Q8—1228no—
601Q8—4881no—
602Q8—2444no—
603Q8—2442no—
604D4—24no—
605D4—24no—
606D4—614no—
607D4—1224no—
608D4—1222no—
609D4—1224no—
610D4—4881no—
611D4—4884no—
612D4—4884no—
613D4—24416no—
614D4—2444no—
615D4—2442no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1224no—
626C2×C4—4884no—
627C2×C4—24416no—
628C2×C4—2444no—
629C2×C4—2442no—
630C2×C4—2444no—
631C2×C4—2444no—
632C2×C4—2444no—
633C2×C4#1451979914yescomplemented
634C2×C4C22×D24414yes—
635C2×C4C22×D24411yescharacteristic
636C2×C4C22×D24411yescharacteristic
637C2×C4C22×D24411yescharacteristic
638C2×C4C22×D24411yescharacteristic
639C2×C4C22×D24411yescharacteristic
640C2×C4C24×D6111yescharacteristic
641C8—24no—
642C8—24no—
643C22—21no—
644C22—21no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—84no—
649C22—1224no—
650C22—2448no—
651C22—2442no—
652C22—2442no—
653C22—2442no—
654C22—2442no—
655C22—2442no—
656C22C2×#1451979912yes—
657C22C23×D24411yescharacteristic
658C4—21no—
659C4—21no—
660C4—21no—
661C4—24no—
662C4—2442no—
663C4—2442no—
664C4—2442no—
665C4—2442no—
666C4—2444no—
667C4C2×#1451979912yes—
668C4C2×#1451979912yes—
669C4H(5,𝔽2)×D6112yes—
670C2—22no—
671C2—44no—
672C2—4882no—
673C2—2444no—
674C2#2145945511yescharacteristic
675C2C22×#1451979911yescharacteristic
676C2#2146060711yescharacteristic
subgroupG/H
centerC22C23×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2972C2
socleC22×C61C23×D4
hypercenter#2972D61
hypocenterC61#61565
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #21459671C2semidirect
2. C2×#16840417C2semidirect
3. #21457748C2semidirect
4. #21457737C2semidirect
5. #21457551C2semidirect
6. #21457541C2semidirect
7. #21456769C2semidirect
8. #21456746C2semidirect
9. #21456602C2semidirect
10. #21455828C2semidirect
11. #21455820C2semidirect
12. #21455041C2semidirect
13. C61×#2972C2semidirect
15. C2×#14519799C22semidirect
17. #16840482C22semidirect
18. #16840477C22semidirect
20. #16840433C22semidirect
25. #16840417C22semidirect
38. #16840142C22semidirect
39. #16840135C22semidirect
40. #16840113C22semidirect
41. #16840113C22semidirect
42. #16840104C22semidirect
45. #16840053C22semidirect
46. #16840043C22semidirect
47. #16840003C22semidirect
48. #16839991C22semidirect
49. #16839973C22semidirect
50. #16839968C22semidirect
51. #16839968C22semidirect
52. #16839958C22semidirect
53. #16839926C22semidirect
54. #16839926C22semidirect
57. #16839909C22semidirect
60. C4×#11758190C22semidirect
61. C4×D488C22semidirect
62. #16839759C22semidirect
64. C8×D4×C61C22semidirect
65. C8×D4×C61C22semidirect
267. C2×C4×C61#135semidirect
344. #2972D61semidirect
401. C8×D4C2×D61semidirect
402. C8×D4C2×D61semidirect
407. C61#61565semidirect
633. C2×C4#14519799semidirect

Maximal subgroups

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

subgroupindex
#214596712
C2×#168404172
#214577482
#214577372
#214575512
#214575412
#214567692
#214567462
#214566022
#214558282
#214558202
#214550412
C61×#29722
#6156561