Finite Group Databasebeta

← #30571126

Subgroups — #30571126

Order 15616.

2042 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic, complemented
2#21458861C214yesmaximal, complemented
3#21458835C214yesmaximal, complemented
4#21458544C214yesmaximal, complemented
5#21458522C214yesmaximal, complemented
6#21456837C214yesmaximal, complemented
7#21456760C211yesmaximal, characteristic, complemented
8#21456737C211yesmaximal, characteristic, complemented
9#21455957C214yesmaximal, complemented
10#21455827C211yesmaximal, characteristic, complemented
11#21455819C211yesmaximal, characteristic, complemented
12#21454904C211yesmaximal, characteristic, complemented
13C2×C61×#550C211yesmaximal, characteristic, complemented
14#16840497C2212yescomplemented
15#16840490C2212yescomplemented
16#16840485C2212yescomplemented
17#16840484—22no—
18#16840472C2212yescomplemented
19#16840453—22no—
20#16840359—22no—
21#16840336—21no—
22#16840319C2211yescharacteristic
23#16840312C2211yescharacteristic
24#16840296C2214yes—
25#16840243—21no—
26#16840149C2218yescomplemented
27#16840138C2212yescomplemented
28#16840134C2212yescomplemented
29#16840126C2218yescomplemented
30#16840108C2212yescomplemented
31#16840108C2212yescomplemented
32#16840092C2212yes—
33#16840075C2212yes—
34#16840062C2218yescomplemented
35#16840052C2212yescomplemented
36#16840042C2212yescomplemented
37#16839977C2218yescomplemented
38#16839967C2212yescomplemented
39#16839959C2214yescomplemented
40#16839959C2218yescomplemented
41#16839957C2212yescomplemented
42#16839954C2214yescomplemented
43#16839954C2218yescomplemented
44#16839913C2214yes—
45#16839913C2212yes—
46#16839896C2214yes—
47#16839896C2212yes—
48#16839893C2214yescomplemented
49#16839893C2214yescomplemented
50#16839872C2218yescomplemented
51#16839862C2218yescomplemented
52C4×#11758200C2214yescomplemented
53C4×#11758199C2214yescomplemented
54#16839819C2214yes—
55#16839815C2211yescharacteristic
56#16839815C2211yescharacteristic
57C22×#41×C61—21no—
58C61×#550—22no—
59C61×#550—22no—
60C61×#550C2214yes—
61C61×#550C2214yes—
62C2×#127×C61C2211yescharacteristic
63C2×#127×C61C2212yes—
64C2×#121×C61—21no—
65C2×#121×C61—22no—
66C2×C4×Q8×C61C2211yescharacteristic
67C2×C4×D4×C61C2212yes—
68C2×C4×D4×C61C2211yescharacteristic
69C2×#11758222—21no—
70C2×#11758200C2314yes—
71C2×#11758200C2318yes—
72C2×#11758199C2314yes—
73C2×#11758199C2318yes—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519775—22no—
78#14519774C2311yescharacteristic
79#14519772—24no—
80#14519771—24no—
81#14519769—21no—
82#14519769C2314yes—
83#14519767—24no—
84#14519766—22no—
85#14519765—21no—
86#14519765C2314yes—
87#14519762—24no—
88#14519760—24no—
89#14519756—21no—
90#14519756C2312yes—
91#14519755—24no—
92#14519753—22no—
93#14519753—22no—
94#14519751—21no—
95#14519751C2312yes—
96C4×Dic122—21no—
97C4×Dic122C2314yes—
98C4×Dic122C2312yes—
99#14519742—22no—
100C4×D244—21no—
101C4×D244C2312yes—
102C4×D244C2314yes—
103#14519739—22no—
104C4×C22⋊D61—24no—
105#14519736—21no—
106#14519735—22no—
107#14519734—24no—
108C4⋊C4×D61—22no—
109#14519710C2314yes—
110#14519708C2314yes—
111C4⋊#92418C2312yes—
112C4⋊#92418C2318yes—
113C4⋊#92418C2312yes—
114#14519700C2314yes—
115#14519698C2314yes—
116#14519698C2318yes—
117#14519698C2318yes—
118#14519698C2314yes—
119#14519697C2314yes—
120C22⋊#92418C2312yes—
121C22⋊#92418C2312yes—
122#14519688C2318yes—
123C4×#92418C2312yes—
124C4×#92418C2312yes—
125C2×#41×C61—24no—
126C2×#41×C61—24no—
127C2×#41×C61—48no—
128C22×Q8×C61D411yescharacteristic
129C22×D4×C61—22no—
130C22×D4×C61—21no—
131C22×D4×C61—22no—
132C22×D4×C61—21no—
133C22×D4×C61C2312yes—
134C22×D4×C61C2311yescharacteristic
135C23×C4×C61—22no—
136C23×C4×C61—21no—
137#127×C61—24no—
138#127×C61—21no—
139#127×C61—21no—
140#127×C61C2312yes—
141#127×C61C2318yes—
142#127×C61C2312yes—
143#121×C61—24no—
144#121×C61—24no—
145#121×C61—28no—
146#121×C61—28no—
147C4×Q8×C61D412yes—
148C4×Q8×C61D412yes—
149C4×Q8×C61C2312yes—
150C4×Q8×C61C2312yes—
151C4×D4×C61—24no—
152C4×D4×C61—21no—
153C4×D4×C61—21no—
154C4×D4×C61C2312yes—
155C4×D4×C61C2312yes—
156C4×D4×C61C2318yes—
157C2×C4⋊C4×C61D411yescharacteristic
158C2×C4⋊C4×C61D412yes—
159C2×C4⋊C4×C61C2311yescharacteristic
160C2×C22⋊C4×C61—22no—
161C2×C22⋊C4×C61—21no—
162C2×C42×C61C2312yes—
163C2×C42×C61C2311yescharacteristic
164C24×C61—22no—
165C24×C61—21no—
166#41×C61—416no—
167#41×C61—416no—
168C2×Q8×C61—24no—
169C2×Q8×C61C2×D412yes—
170C2×Q8×C61C2×D412yes—
171C2×D4×C61—22no—
172C2×D4×C61—21no—
173C2×D4×C61—28no—
174C2×D4×C61—22no—
175C2×D4×C61—22no—
176C2×D4×C61—28no—
177C2×D4×C61—22no—
178C2×D4×C61—22no—
179C2×D4×C61—42no—
180C2×D4×C61—44no—
181C2×D4×C61—48no—
182C2×D4×C61—44no—
183C2×D4×C61—82no—
184C2×D4×C61—84no—
185C2×D4×C61C2412yes—
186C2×D4×C61C2414yes—
187C22×C4×C61—21no—
188C22×C4×C61—28no—
189C22×C4×C61—22no—
190C22×C4×C61—22no—
191C22×C4×C61—48no—
192C22×C4×C61—44no—
193C22×C4×C61C2×D412yes—
194C22×C4×C61C2×D412yes—
195C22×C4×C61C2×D411yescharacteristic
196C22×C4×C61C2×D411yescharacteristic
197C22×C4×C61C2411yescharacteristic
198C2×C22⋊D61—22no—
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—21no—
209C4⋊C4×C61C2×D412yes—
210C4⋊C4×C61C2×D412yes—
211C4⋊C4×C61C2×D414yes—
212C4⋊C4×C61C2×D414yes—
213C4⋊C4×C61C2412yes—
214C22⋊C4×C61—28no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—22no—
217C42×C61C2×D414yes—
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2411yescharacteristic
221C42×C61C2411yescharacteristic
222C42×C61C2414yes—
223C22⋊Dic61—22no—
224#11758200—44no—
225#11758200—48no—
226#11758199—48no—
227#11758199—44no—
228#11758198—22no—
229#11758198—24no—
230#11758198—24no—
231#11758198—22no—
232#11758198—24no—
233C4⋊Dic61—21no—
234C4⋊Dic61—21no—
235C4⋊Dic61—21no—
236C4⋊Dic61—21no—
237C4⋊Dic61C2412yes—
238C4⋊Dic61C2412yes—
239#11758196—24no—
240#11758196—22no—
241#11758196—24no—
242#11758196—24no—
243#11758196—22no—
244C4×Dic61—24no—
245C4×Dic61—22no—
246C2×#92418C2414yes—
247C2×#92418C2414yes—
248C23×C61—22no—
249C23×C61—24no—
250C23×C61—42no—
251C23×C61—44no—
252C23×C61—44no—
253C23×C61—42no—
254C23×C61—44no—
255C23×C61—42no—
256C23×C61C22×D411yescharacteristic
257C22×D61—22no—
258Q8×C61—24no—
259Q8×C61—24no—
260D4×C61—22no—
261D4×C61—28no—
262D4×C61—22no—
263D4×C61—42no—
264D4×C61—44no—
265D4×C61—416no—
266D4×C61—44no—
267D4×C61—44no—
268D4×C61—82no—
269D4×C61—88no—
270D4×C61—82no—
271C2×C4×C61—24no—
272C2×C4×C61—22no—
273C2×C4×C61—22no—
274C2×C4×C61—41no—
275C2×C4×C61—42no—
276C2×C4×C61—416no—
277C2×C4×C61—42no—
278C2×C4×C61—44no—
279C2×C4×C61—44no—
280C2×C4×C61AGL(1,ℤ/8)14yes—
281C2×C4×C61C22×D414yes—
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C22×D414yes—
285C2×C4×C61C22×D411yescharacteristic
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C2511yescharacteristic
289C22⋊D61—44no—
290C2×Dic61—22no—
291C2×Dic61—22no—
292C2×Dic61—22no—
293D244—24no—
294D244—42no—
295D244—81no—
296C4×D61—44no—
297C4×D61—84no—
298Dic122—24no—
299Dic122—42no—
300Dic122—81no—
301#92418—24no—
302#92418—24no—
303#61890—611nomaximal, Sylow
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—44no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—44no—
310C22×C61—44no—
311C22×C61—42no—
312C22×C61—88no—
313C22×C61—84no—
314C22×C61C2×AGL(1,ℤ/8)12yes—
315C22×C61C23×D411yescharacteristic
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C4×C61—21no—
320C4×C61—24no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—24no—
324C4×C61—42no—
325C4×C61C2×AGL(1,ℤ/8)12yes—
326C4×C61C2×AGL(1,ℤ/8)12yes—
327C4×C61#58512yes—
328Dic61—42no—
329Dic61—42no—
330Dic61—42no—
331#3507—611no—
332C2×#550D6111yescharacteristic, complemented
333#3399—614no—
334#3397—614no—
335#3368—614no—
336#3366—614no—
337#3217—614no—
338#3208—611no—
339#3205—611no—
340#3114—614no—
341#3101—611no—
342#3100—611no—
343#2983—611no—
344C2×C61—22no—
345C2×C61—82no—
346C2×C61—84no—
347C2×C61#348711yescharacteristic
348C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
349C2×C61#358511yescharacteristic
350D61—82no—
351C22×#41—21no—
352#559—612no—
353#557—612no—
354#556—612no—
355#556—1222no—
356#554—612no—
357#552—1222no—
358#550—22no—
359#550—22no—
360#550C2×D6114yes—
361#550C2×D6114yes—
362#539—1222no—
363#536—1221no—
364#533—611no—
365#532—611no—
366C2×#127C2×D6111yescharacteristic
367C2×#127C2×D6112yes—
368#529—614no—
369C2×#121—21no—
370C2×#121—22no—
371#520—1221no—
372C2×C4×Q8C2×D6111yescharacteristic
373C2×C4×D4C2×D6111yescharacteristic
374C2×C4×D4C2×D6112yes—
375#496—618no—
376#493—612no—
377#492—612no—
378#490—618no—
379#486—612no—
380#486—612no—
381#483—612no—
382#480—612no—
383#478—618no—
384#476—612no—
385#474—612no—
386#463—618no—
387#461—612no—
388#460—618no—
389#460—614no—
390#459—614no—
391#459—618no—
392#459—612no—
393#450—612no—
394#450—614no—
395#447—612no—
396#447—614no—
397#446—614no—
398#446—614no—
399#442—618no—
400#440—618no—
401C4×SD16—614no—
402C4×D8—614no—
403#432—614no—
404#431—611no—
405#431—611no—
406C61#6189011yescharacteristic, complemented, Sylow
407C2×#41—24no—
408C2×#41—24no—
409C2×#41—48no—
410C2×#41—1221no—
411C22×Q8C22⋊D6111yescharacteristic
412C22×D4—22no—
413C22×D4—21no—
414C22×D4—22no—
415C22×D4—21no—
416C22×D4C22×D6111yescharacteristic
417C22×D4C22×D6112yes—
418C23×C4—22no—
419C23×C4—21no—
420C2×SD16—614no—
421C2×SD16—618no—
422C2×D8—614no—
423C2×D8—618no—
424C4⋊Q8—611no—
425C4⋊Q8—611no—
426C4⋊Q8—611no—
427C4⋊Q8—1222no—
428#127—24no—
429#127—21no—
430#127—21no—
431#127—611no—
432#127C22×D6112yes—
433#127C22×D6118yes—
434#127C22×D6112yes—
435#126—1224no—
436#126—1224no—
437#125—614no—
438#125—1224no—
439#125—1221no—
440#124—614no—
441#124—1222no—
442#124—1221no—
443#123—1224no—
444#123—1224no—
445C22⋊Q8—612no—
446C22⋊Q8—1222no—
447C22⋊Q8—1222no—
448C22⋊Q8—1221no—
449C22⋊Q8—1224no—
450#121—24no—
451#121—24no—
452#121—28no—
453#121—28no—
454#121—612no—
455#121—1221no—
456C4×Q8—614no—
457C4×Q8—612no—
458C4×Q8—1222no—
459C4×Q8—1221no—
460C4×Q8C22⋊D6112yes—
461C4×Q8C22⋊D6112yes—
462C4×Q8C22×D6112yes—
463C4×Q8C22×D6112yes—
464C4×D4—24no—
465C4×D4—21no—
466C4×D4—21no—
467C4×D4—614no—
468C4×D4—612no—
469C4×D4—1224no—
470C4×D4—1222no—
471C4×D4—1221no—
472C4×D4C22×D6112yes—
473C4×D4C22×D6112yes—
474C4×D4C22×D6118yes—
475#117—1221no—
476#117—1222no—
477#117—1224no—
478C2×C4⋊C4—1222no—
479C2×C4⋊C4C22⋊D6111yescharacteristic
480C2×C4⋊C4C22⋊D6112yes—
481C2×C4⋊C4C22×D6111yescharacteristic
482C2×C22⋊C4—22no—
483C2×C22⋊C4—21no—
484C2×C42C22×D6112yes—
485C2×C42C22×D6111yescharacteristic
486#107—614no—
487#106—614no—
488C4⋊C8—612no—
489C4⋊C8—612no—
490C4⋊C8—618no—
491#103—614no—
492#102—614no—
493#102—614no—
494#102—618no—
495#102—618no—
496#102—614no—
497C22⋊C8—612no—
498C22⋊C8—612no—
499#97—618no—
500C4×C8—612no—
501C4×C8—612no—
502C24—22no—
503C24—21no—
504#41—416no—
505#41—416no—
506#41—2444no—
507C2×Q8—24no—
508C2×Q8—612no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8C2×C22⋊D6112yes—
512C2×Q8C2×C22⋊D6112yes—
513C2×D4—22no—
514C2×D4—21no—
515C2×D4—28no—
516C2×D4—22no—
517C2×D4—22no—
518C2×D4—28no—
519C2×D4—22no—
520C2×D4—22no—
521C2×D4—42no—
522C2×D4—44no—
523C2×D4—48no—
524C2×D4—44no—
525C2×D4—84no—
526C2×D4—82no—
527C2×D4—612no—
528C2×D4—1222no—
529C2×D4—1221no—
530C2×D4—1221no—
531C2×D4C23×D6112yes—
532C2×D4C23×D6114yes—
533C22×C4—21no—
534C22×C4—28no—
535C22×C4—22no—
536C22×C4—22no—
537C22×C4—48no—
538C22×C4—44no—
539C22×C4—1224no—
540C22×C4—1222no—
541C22×C4C2×C22⋊D6112yes—
542C22×C4C2×C22⋊D6112yes—
543C22×C4C2×C22⋊D6111yescharacteristic
544C22×C4C2×C22⋊D6111yescharacteristic
545C22×C4C23×D6111yescharacteristic
546SD16—2444no—
547SD16—2448no—
548D8—2448no—
549D8—2444no—
550C2×C8—614no—
551C2×C8—614no—
552C4⋊C4—21no—
553C4⋊C4—612no—
554C4⋊C4—612no—
555C4⋊C4—1221no—
556C4⋊C4—1224no—
557C4⋊C4—1222no—
558C4⋊C4—1224no—
559C4⋊C4—1222no—
560C4⋊C4—1224no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4C2×C22⋊D6112yes—
565C4⋊C4C2×C22⋊D6112yes—
566C4⋊C4C2×C22⋊D6114yes—
567C4⋊C4C2×C22⋊D6114yes—
568C4⋊C4C23×D6112yes—
569C22⋊C4—28no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4—1224no—
575C22⋊C4—1224no—
576C22⋊C4—1222no—
577C22⋊C4—1224no—
578C42—1224no—
579C42—1222no—
580C42C2×C22⋊D6114yes—
581C42C2×C22⋊D6111yescharacteristic
582C42C2×C22⋊D6111yescharacteristic
583C42C23×D6111yescharacteristic
584C42C23×D6111yescharacteristic
585C42C23×D6114yes—
586C23—22no—
587C23—24no—
588C23—42no—
589C23—44no—
590C23—42no—
591C23—44no—
592C23—44no—
593C23—42no—
594C23—1222no—
595C23C22×C22⋊D6111yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—1224no—
599Q8—4881no—
600Q8—2442no—
601D4—22no—
602D4—28no—
603D4—22no—
604D4—42no—
605D4—44no—
606D4—416no—
607D4—44no—
608D4—44no—
609D4—88no—
610D4—82no—
611D4—82no—
612D4—1224no—
613D4—4881no—
614D4—2444no—
615D4—2442no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—416no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—4884no—
629C2×C4—2444no—
630C2×C4#1451980314yes—
631C2×C4C22×C22⋊D6114yes—
632C2×C4C22×C22⋊D6111yescharacteristic
633C2×C4C22×C22⋊D6111yescharacteristic
634C2×C4C22×C22⋊D6111yescharacteristic
635C2×C4C22×C22⋊D6114yes—
636C2×C4C22×C22⋊D6111yescharacteristic
637C2×C4C22×C22⋊D6111yescharacteristic
638C2×C4C24×D6111yescharacteristic
639C8—1224no—
640C8—1224no—
641C22—21no—
642C22—21no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—88no—
650C22—84no—
651C22—2442no—
652C22—2442no—
653C22—2442no—
654C22C2×#1451980312yes—
655C22C23×C22⋊D6111yescharacteristic
656C4—21no—
657C4—24no—
658C4—24no—
659C4—21no—
660C4—21no—
661C4—42no—
662C4—2442no—
663C4—2442no—
664C4—2442no—
665C4C2×#1451980312yes—
666C4C2×#1451980312yes—
667C4#1684059812yes—
668C2—22no—
669C2—82no—
670C2—84no—
671C2—4882no—
672C2#2145947711yescharacteristic
673C2C22×#1451980311yescharacteristic
674C2#2146059911yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#550C2
socleC22×C61C23×D4
hypercenterC2×#550D61
hypocenterC61#61890
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #21459684C2semidirect
2. #21458861C2semidirect
3. #21458835C2semidirect
4. #21458544C2semidirect
5. #21458522C2semidirect
6. #21456837C2semidirect
7. #21456760C2semidirect
8. #21456737C2semidirect
9. #21455957C2semidirect
10. #21455827C2semidirect
11. #21455819C2semidirect
12. #21454904C2semidirect
13. C2×C61×#550C2semidirect
14. #16840497C22semidirect
15. #16840490C22semidirect
16. #16840485C22semidirect
18. #16840472C22semidirect
26. #16840149C22semidirect
27. #16840138C22semidirect
28. #16840134C22semidirect
29. #16840126C22semidirect
30. #16840108C22semidirect
31. #16840108C22semidirect
34. #16840062C22semidirect
35. #16840052C22semidirect
36. #16840042C22semidirect
37. #16839977C22semidirect
38. #16839967C22semidirect
39. #16839959C22semidirect
40. #16839959C22semidirect
41. #16839957C22semidirect
42. #16839954C22semidirect
43. #16839954C22semidirect
48. #16839893C22semidirect
49. #16839893C22semidirect
50. #16839872C22semidirect
51. #16839862C22semidirect
52. C4×#11758200C22semidirect
53. C4×#11758199C22semidirect
332. C2×#550D61semidirect
406. C61#61890semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214588612
#214588352
#214585442
#214585222
#214568372
#214567602
#214567372
#214559572
#214558272
#214558192
#214549042
C2×C61×#5502
#6189061