Finite Group Databasebeta

← #30571121

Subgroups — #30571121

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
1C2×#16840434C211yesmaximal, characteristic, complemented
2#21458857C214yesmaximal, complemented
3#21458839C214yesmaximal, complemented
4#21458548C214yesmaximal, complemented
5#21458526C214yesmaximal, complemented
6#21456834C214yesmaximal, complemented
7#21456757C211yesmaximal, characteristic, complemented
8#21456734C211yesmaximal, characteristic, complemented
9#21455953C214yesmaximal, complemented
10#21455823C211yesmaximal, characteristic, complemented
11#21455815C211yesmaximal, characteristic, complemented
12#21454904C211yesmaximal, characteristic, complemented
13C61×#3507C211yesmaximal, characteristic, complemented
14C22×#11758226—21no—
15#16840434—22no—
16#16840434—22no—
17#16840434C2214yes—
18#16840434C2214yes—
19C2×#14519774C2211yescharacteristic
20C2×#14519774C2212yes—
21C2×#14519752—22no—
22C2×#14519752—21no—
23C2×C4×D244C2212yes—
24C2×C4×D244C2211yescharacteristic
25#16840152C2218yescomplemented
26#16840138C2212yescomplemented
27#16840136C2212yescomplemented
28#16840128C2218yescomplemented
29#16840110C2212yescomplemented
30#16840110C2212yescomplemented
31#16840090C2212yes—
32#16840073C2212yes—
33#16840059C2218yescomplemented
34#16840049C2212yescomplemented
35#16840039C2212yescomplemented
36#16839974C2218yescomplemented
37#16839964C2212yescomplemented
38#16839962C2214yescomplemented
39#16839962C2218yescomplemented
40#16839957C2214yescomplemented
41#16839957C2218yescomplemented
42#16839954C2212yescomplemented
43#16839912C2214yes—
44#16839912C2212yes—
45#16839895C2214yes—
46#16839895C2212yes—
47#16839893C2214yescomplemented
48#16839893C2214yescomplemented
49#16839872C2218yescomplemented
50#16839860C2218yescomplemented
51C4×#11758201C2214yescomplemented
52C4×#11758199C2214yescomplemented
53#16839819C2214yes—
54#16839815C2211yescharacteristic
55#16839815C2211yescharacteristic
56C61×#559C2212yescomplemented
57C61×#557C2212yescomplemented
58C61×#556—22no—
59C61×#556C2212yescomplemented
60C61×#554C2212yescomplemented
61C61×#552—22no—
62C61×#539—22no—
63C61×#536—21no—
64C61×#533C2211yescharacteristic
65C61×#532C2211yescharacteristic
66C61×#529C2214yes—
67C61×#520—21no—
68C2×C4×Q8×C61C2211yescharacteristic
69C2×#11758226—24no—
70C2×#11758226—24no—
71C2×#11758226—48no—
72C22×D244—22no—
73C22×D244—21no—
74C22×D244—22no—
75C22×D244—21no—
76C22×D244C2312yes—
77C22×D244C2311yescharacteristic
78C22×C4×D61—22no—
79C22×C4×D61—21no—
80C2×#11758201C2314yes—
81C2×#11758201C2318yes—
82C2×#11758199C2314yes—
83C2×#11758199C2318yes—
84#14519774—24no—
85#14519774—21no—
86#14519774—21no—
87#14519774C2312yes—
88#14519774C2318yes—
89#14519774C2312yes—
90#14519752—24no—
91#14519752—24no—
92#14519752—28no—
93#14519752—28no—
94C4×D244—24no—
95C4×D244—21no—
96C4×D244—21no—
97C4×D244C2312yes—
98C4×D244C2312yes—
99C4×D244C2318yes—
100C2×C4⋊Dic61C2311yescharacteristic
101C2×#11758198—22no—
102C2×#11758198—21no—
103#14519710C2314yes—
104#14519708C2314yes—
105C4⋊#92418C2312yes—
106C4⋊#92418C2312yes—
107C4⋊#92418C2318yes—
108#14519701C2314yes—
109#14519698C2314yes—
110#14519697C2314yes—
111#14519697C2318yes—
112#14519697C2318yes—
113#14519697C2314yes—
114C22⋊#92418C2312yes—
115C22⋊#92418C2312yes—
116#14519688C2318yes—
117C4×#92418C2312yes—
118C4×#92418C2312yes—
119C2×#41×C61—21no—
120C22×Q8×C61D411yescharacteristic
121C4⋊Q8×C61—22no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C4⋊Q8×C61C2311yescharacteristic
125#127×C61C2311yescharacteristic
126#126×C61—24no—
127#126×C61—24no—
128#125×C61—24no—
129#125×C61—21no—
130#125×C61C2314yes—
131#124×C61—22no—
132#124×C61—21no—
133#124×C61C2314yes—
134#123×C61—24no—
135#123×C61—24no—
136C22⋊Q8×C61—22no—
137C22⋊Q8×C61—22no—
138C22⋊Q8×C61—21no—
139C22⋊Q8×C61—24no—
140C22⋊Q8×C61C2312yes—
141#121×C61—21no—
142#121×C61C2312yes—
143C4×Q8×C61—22no—
144C4×Q8×C61—21no—
145C4×Q8×C61D412yes—
146C4×Q8×C61D412yes—
147C4×Q8×C61C2312yes—
148C4×Q8×C61C2314yes—
149C4×Q8×C61C2312yes—
150C4×Q8×C61C2312yes—
151C4×D4×C61—24no—
152C4×D4×C61—22no—
153C4×D4×C61—21no—
154C4×D4×C61C2312yes—
155C4×D4×C61C2314yes—
156#117×C61—21no—
157#117×C61—22no—
158#117×C61—24no—
159C2×C4⋊C4×C61—22no—
160C2×C4⋊C4×C61D412yes—
161C2×C4⋊C4×C61D411yescharacteristic
162C2×C42×C61C2311yescharacteristic
163C2×C42×C61C2312yes—
164C23×D61—22no—
165C23×D61—21no—
166#41×C61—44no—
167C2×Q8×C61—24no—
168C2×Q8×C61—21no—
169C2×Q8×C61—21no—
170C2×Q8×C61C2×D412yes—
171C2×Q8×C61C2×D412yes—
172C2×Q8×C61C2412yes—
173C2×D4×C61—22no—
174C2×D4×C61—21no—
175C2×D4×C61—21no—
176C2×D4×C61C2412yes—
177C22×C4×C61—22no—
178C22×C4×C61—24no—
179C22×C4×C61C2×D412yes—
180C22×C4×C61C2×D412yes—
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2411yescharacteristic
184C22×Dic61—21no—
185#11758226—416no—
186#11758226—416no—
187C2×D244—28no—
188C2×D244—22no—
189C2×D244—22no—
190C2×D244—22no—
191C2×D244—21no—
192C2×D244—28no—
193C2×D244—22no—
194C2×D244—22no—
195C2×D244—42no—
196C2×D244—44no—
197C2×D244—48no—
198C2×D244—44no—
199C2×D244—84no—
200C2×D244—82no—
201C2×D244C2412yes—
202C2×D244C2414yes—
203C2×C4×D61—28no—
204C2×C4×D61—22no—
205C2×C4×D61—22no—
206C2×C4×D61—48no—
207C2×C4×D61—44no—
208C4⋊C4×C61—21no—
209C4⋊C4×C61—24no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—22no—
212C4⋊C4×C61—24no—
213C4⋊C4×C61—21no—
214C4⋊C4×C61—21no—
215C4⋊C4×C61—22no—
216C4⋊C4×C61—24no—
217C4⋊C4×C61C2×D412yes—
218C4⋊C4×C61C2×D412yes—
219C4⋊C4×C61C2×D414yes—
220C4⋊C4×C61C2×D414yes—
221C4⋊C4×C61C2412yes—
222C4⋊C4×C61C2412yes—
223C22⋊C4×C61—22no—
224C22⋊C4×C61—22no—
225C22⋊C4×C61—24no—
226C22⋊C4×C61—24no—
227C22⋊C4×C61—22no—
228C22⋊C4×C61—24no—
229C42×C61—24no—
230C42×C61—22no—
231C42×C61C2×D414yes—
232C42×C61C2×D411yescharacteristic
233C42×C61C2×D411yescharacteristic
234C42×C61C2411yescharacteristic
235C42×C61C2414yes—
236C42×C61C2411yescharacteristic
237#11758201—44no—
238#11758201—48no—
239#11758199—48no—
240#11758199—44no—
241#11758198—28no—
242#11758198—22no—
243#11758198—22no—
244C4⋊Dic61—21no—
245C4⋊Dic61C2412yes—
246C2×#92418C2414yes—
247C2×#92418C2414yes—
248C23×C61—22no—
249C23×C61C22×D411yescharacteristic
250C22×D61—22no—
251C22×D61—24no—
252C22×D61—42no—
253C22×D61—44no—
254C22×D61—42no—
255C22×D61—44no—
256C22×D61—44no—
257C22×D61—42no—
258Q8×C61—24no—
259Q8×C61—24no—
260Q8×C61—24no—
261Q8×C61—42no—
262Q8×C61—81no—
263D4×C61—24no—
264D4×C61—44no—
265D4×C61—42no—
266D4×C61—81no—
267C2×C4×C61—24no—
268C2×C4×C61—22no—
269C2×C4×C61—22no—
270C2×C4×C61—22no—
271C2×C4×C61—22no—
272C2×C4×C61—42no—
273C2×C4×C61—41no—
274C2×C4×C61—44no—
275C2×C4×C61—84no—
276C2×C4×C61AGL(1,ℤ/8)14yes—
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D414yes—
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C22×D414yes—
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C2511yescharacteristic
285C2×Dic61—22no—
286C2×Dic61—42no—
287D244—22no—
288D244—28no—
289D244—22no—
290D244—42no—
291D244—44no—
292D244—416no—
293D244—44no—
294D244—44no—
295D244—88no—
296D244—82no—
297D244—82no—
298C4×D61—416no—
299C4×D61—44no—
300C4×D61—44no—
301#92418—24no—
302#92418—24no—
303#61890—611nomaximal, Sylow
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61C2×AGL(1,ℤ/8)12yes—
310C22×C61C23×D411yescharacteristic
311C2×D61—42no—
312C2×D61—44no—
313C2×D61—42no—
314C2×D61—44no—
315C2×D61—44no—
316C2×D61—42no—
317C2×D61—88no—
318C2×D61—84no—
319C4×C61—24no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—24no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61C2×AGL(1,ℤ/8)12yes—
328C4×C61C2×AGL(1,ℤ/8)12yes—
329C4×C61#58512yes—
330Dic61—42no—
331#3507D6111yescharacteristic, complemented
332C2×#550—611no—
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#348711yescharacteristic
347C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
348C2×C61#358511yescharacteristic
349D61—82no—
350D61—84no—
351C22×#41—1221no—
352#559C2×D6112yescomplemented
353#557C2×D6112yescomplemented
354#556—22no—
355#556C2×D6112yescomplemented
356#554C2×D6112yescomplemented
357#552—22no—
358#550—614no—
359#550—614no—
360#550—1222no—
361#550—1222no—
362#539—22no—
363#536—21no—
364#533C2×D6111yescharacteristic
365#532C2×D6111yescharacteristic
366C2×#127—611no—
367C2×#127—612no—
368#529C2×D6114yes—
369C2×#121—1221no—
370C2×#121—1222no—
371#520—21no—
372C2×C4×Q8C2×D6111yescharacteristic
373C2×C4×D4—611no—
374C2×C4×D4—612no—
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—612no—
391#459—614no—
392#459—618no—
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—21no—
408C2×#41—1224no—
409C2×#41—1224no—
410C2×#41—2448no—
411C22×Q8C22⋊D6111yescharacteristic
412C22×D4—611no—
413C22×D4—612no—
414C22×D4—1222no—
415C22×D4—1221no—
416C22×D4—1222no—
417C22×D4—1221no—
418C23×C4—1222no—
419C23×C4—1221no—
420C2×SD16—614no—
421C2×SD16—618no—
422C2×D8—614no—
423C2×D8—618no—
424C4⋊Q8—22no—
425C4⋊Q8C22×D6111yescharacteristic
426C4⋊Q8C22×D6111yescharacteristic
427C4⋊Q8C22×D6111yescharacteristic
428#127—612no—
429#127—618no—
430#127—612no—
431#127—1224no—
432#127—1221no—
433#127—1221no—
434#127C22×D6111yescharacteristic
435#126—24no—
436#126—24no—
437#125—24no—
438#125—21no—
439#125C22×D6114yes—
440#124—22no—
441#124—21no—
442#124C22×D6114yes—
443#123—24no—
444#123—24no—
445C22⋊Q8—22no—
446C22⋊Q8—22no—
447C22⋊Q8—21no—
448C22⋊Q8—24no—
449C22⋊Q8C22×D6112yes—
450#121—21no—
451#121—1224no—
452#121—1224no—
453#121—1228no—
454#121—1228no—
455#121C22×D6112yes—
456C4×Q8—22no—
457C4×Q8—21no—
458C4×Q8C22⋊D6112yes—
459C4×Q8C22⋊D6112yes—
460C4×Q8C22×D6114yes—
461C4×Q8C22×D6112yes—
462C4×Q8C22×D6112yes—
463C4×Q8C22×D6112yes—
464C4×D4—24no—
465C4×D4—22no—
466C4×D4—21no—
467C4×D4—612no—
468C4×D4—612no—
469C4×D4—618no—
470C4×D4—1224no—
471C4×D4—1221no—
472C4×D4—1221no—
473C4×D4C22×D6114yes—
474C4×D4C22×D6112yes—
475#117—21no—
476#117—22no—
477#117—24no—
478C2×C4⋊C4—22no—
479C2×C4⋊C4—611no—
480C2×C4⋊C4C22⋊D6111yescharacteristic
481C2×C4⋊C4C22⋊D6112yes—
482C2×C22⋊C4—1222no—
483C2×C22⋊C4—1221no—
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—614no—
495#102—618no—
496#102—618no—
497C22⋊C8—612no—
498C22⋊C8—612no—
499#97—618no—
500C4×C8—612no—
501C4×C8—612no—
502C24—1222no—
503C24—1221no—
504#41—44no—
505#41—24416no—
506#41—24416no—
507C2×Q8—21no—
508C2×Q8—21no—
509C2×Q8—24no—
510C2×Q8C2×C22⋊D6112yes—
511C2×Q8C2×C22⋊D6112yes—
512C2×Q8C23×D6112yes—
513C2×D4—22no—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4—612no—
517C2×D4—614no—
518C2×D4—1228no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—1222no—
522C2×D4—1221no—
523C2×D4—1228no—
524C2×D4—1222no—
525C2×D4—1222no—
526C2×D4—4884no—
527C2×D4—4882no—
528C2×D4—2442no—
529C2×D4—2444no—
530C2×D4—2448no—
531C2×D4—2444no—
532C2×D4C23×D6112yes—
533C22×C4—22no—
534C22×C4—24no—
535C22×C4—1221no—
536C22×C4—1228no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—2448no—
540C22×C4—2444no—
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—24no—
554C4⋊C4—21no—
555C4⋊C4—22no—
556C4⋊C4—24no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—24no—
561C4⋊C4—612no—
562C4⋊C4—1221no—
563C4⋊C4C2×C22⋊D6112yes—
564C4⋊C4C2×C22⋊D6112yes—
565C4⋊C4C2×C22⋊D6114yes—
566C4⋊C4C2×C22⋊D6114yes—
567C4⋊C4C23×D6112yes—
568C4⋊C4C23×D6112yes—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—22no—
574C22⋊C4—24no—
575C22⋊C4—1228no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C42—24no—
579C42—22no—
580C42C2×C22⋊D6114yes—
581C42C2×C22⋊D6111yescharacteristic
582C42C2×C22⋊D6111yescharacteristic
583C42C23×D6111yescharacteristic
584C42C23×D6111yescharacteristic
585C42C23×D6114yes—
586C23—22no—
587C23—1222no—
588C23—1224no—
589C23—2444no—
590C23—2442no—
591C23—2442no—
592C23—2444no—
593C23—2444no—
594C23—2442no—
595C23C22×C22⋊D6111yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—24no—
599Q8—42no—
600Q8—81no—
601D4—24no—
602D4—44no—
603D4—42no—
604D4—81no—
605D4—1222no—
606D4—1228no—
607D4—1222no—
608D4—4888no—
609D4—4882no—
610D4—4882no—
611D4—2442no—
612D4—2444no—
613D4—24416no—
614D4—2444no—
615D4—2444no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—84no—
625C2×C4—1222no—
626C2×C4—24416no—
627C2×C4—2442no—
628C2×C4—2444no—
629C2×C4—2444no—
630C2×C4#1451980214yes—
631C2×C4C22×C22⋊D6114yes—
632C2×C4C22×C22⋊D6111yescharacteristic
633C2×C4C22×C22⋊D6111yescharacteristic
634C2×C4C22×C22⋊D6114yes—
635C2×C4C22×C22⋊D6111yescharacteristic
636C2×C4C22×C22⋊D6111yescharacteristic
637C2×C4C22×C22⋊D6111yescharacteristic
638C2×C4C24×D6111yescharacteristic
639C8—1224no—
640C8—1224no—
641C22—21no—
642C22—21no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—4888no—
647C22—4884no—
648C22—2442no—
649C22—2444no—
650C22—2444no—
651C22—2442no—
652C22—2442no—
653C22—2444no—
654C22C2×#1451980212yes—
655C22C23×C22⋊D6111yescharacteristic
656C4—21no—
657C4—24no—
658C4—24no—
659C4—21no—
660C4—21no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—2442no—
665C4C2×#1451980212yes—
666C4C2×#1451980212yes—
667C4#1684059912yes—
668C2—22no—
669C2—82no—
670C2—4882no—
671C2—4884no—
672C2#2145947411yescharacteristic
673C2C22×#1451980211yescharacteristic
674C2#2146059811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3507C2
socleC22×C61C23×D4
hypercenter#3507D61
hypocenterC61#61890
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. C2×#16840434C2semidirect
2. #21458857C2semidirect
3. #21458839C2semidirect
4. #21458548C2semidirect
5. #21458526C2semidirect
6. #21456834C2semidirect
7. #21456757C2semidirect
8. #21456734C2semidirect
9. #21455953C2semidirect
10. #21455823C2semidirect
11. #21455815C2semidirect
12. #21454904C2semidirect
13. C61×#3507C2semidirect
25. #16840152C22semidirect
26. #16840138C22semidirect
27. #16840136C22semidirect
28. #16840128C22semidirect
29. #16840110C22semidirect
30. #16840110C22semidirect
33. #16840059C22semidirect
34. #16840049C22semidirect
35. #16840039C22semidirect
36. #16839974C22semidirect
37. #16839964C22semidirect
38. #16839962C22semidirect
39. #16839962C22semidirect
40. #16839957C22semidirect
41. #16839957C22semidirect
42. #16839954C22semidirect
47. #16839893C22semidirect
48. #16839893C22semidirect
49. #16839872C22semidirect
50. #16839860C22semidirect
51. C4×#11758201C22semidirect
52. C4×#11758199C22semidirect
56. C61×#559C22semidirect
57. C61×#557C22semidirect
59. C61×#556C22semidirect
60. C61×#554C22semidirect
331. #3507D61semidirect
352. #559C2×D61semidirect
353. #557C2×D61semidirect
355. #556C2×D61semidirect
356. #554C2×D61semidirect
406. C61#61890semidirect

Maximal subgroups

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

subgroupindex
C2×#168404342
#214588572
#214588392
#214585482
#214585262
#214568342
#214567572
#214567342
#214559532
#214558232
#214558152
#214549042
C61×#35072
#6189061