Finite Group Databasebeta

← #30560133

Subgroups — #30560133

Order 15616.

1730 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic
2#21459629C211yesmaximal, characteristic, complemented
3#21458666C214yesmaximal, complemented
4#21458623C214yesmaximal, complemented
5#21458588C214yesmaximal, complemented
6#21458560C214yesmaximal, complemented
7#21456182C212yesmaximal, complemented
8#21456165C211yesmaximal, characteristic
9#21455686C212yesmaximal, complemented
10#21455624C211yesmaximal, characteristic, complemented
11#21455346C212yesmaximal, complemented
12#21455291C211yesmaximal, characteristic
13#21454759C211yesmaximal, characteristic, complemented
14#21454656C212yesmaximal, complemented
15C2×C61×#441C211yesmaximal, characteristic, complemented
16C2×#14519808C2212yes—
17#16840497C2212yes—
18Q8×Dic122—21no—
19Q8×Dic122C2212yes—
20#16840491—21no—
21#16840491C2212yes—
22#16840490C2212yes—
23#16840488—22no—
24#16840488C2214yes—
25#16840485—21no—
26#16840484C2214yes—
27#16840472—21no—
28#16840453—22no—
29#16840393—22no—
30#16840393C2212yes—
31#16840359—22no—
32#16840336C2212yes—
33#16840319—22no—
34#16840319C2211yescharacteristic
35#16840319C2211yescharacteristic
36#16840312C2211yescharacteristic
37#16840296—22no—
38#16840243C2212yes—
39#16840240C2212yes—
40#16840131C2212yes—
41#16840129C2214yes—
42#16840127C2212yes—
43#16840123C2214yes—
44#16840122C2212yes—
45#16840122C2212yes—
46#16840118C2214yes—
47#16840111C2214yes—
48#16840072C2214yes—
49#16840069C2214yes—
50#16840061C2214yes—
51#16840057C2212yes—
52#16840051C2214yes—
53#16840047C2212yes—
54#16840046C2214yes—
55#16839986C2214yes—
56#16839986C2214yes—
57#16839984C2214yes—
58#16839972C2212yes—
59#16839970C2214yes—
60#16839970C2214yes—
61#16839962C2212yes—
62#16839890C2214yes—
63#16839890C2214yes—
64#16839866C2214yes—
65#16839865C2214yes—
66#16839854C2214yes—
67#16839847C2214yes—
68#16839802C2212yes—
69#16839788C2212yes—
70#16839786C2211yescharacteristic
71#16839771C2212yes—
72#16839761C2212yes—
73#16839759C2212yes—
74#16839759C2212yes—
75#16839717C2212yes—
76C22×Q16×C61C2211yescharacteristic
77C2×C4×Q8×C61C2211yescharacteristic
78C2×C4×Q8×C61C2211yescharacteristic
79C61×#441C2214yes—
80C61×#441C2214yes—
81C2×#106×C61C2211yescharacteristic
82C2×#103×C61C2211yescharacteristic
83C2×#103×C61C2211yescharacteristic
84C2×#97×C61C2211yescharacteristic
85C2×#11758222C2312yes—
86#14519808—28no—
87C2×#11758202C2314yes—
88C2×#11758201C2314yes—
89C2×#11758194C2312yes—
90#14519777—22no—
91#14519777—22no—
92#14519777D411yescharacteristic
93#14519777D411yescharacteristic
94#14519777C2311yescharacteristic
95#14519777C2311yescharacteristic
96#14519777C2311yescharacteristic
97#14519776—24no—
98#14519776—22no—
99#14519776—24no—
100#14519776C2314yes—
101#14519775C2314yes—
102#14519774C2311yescharacteristic
103#14519772—24no—
104#14519771—24no—
105#14519769—22no—
106#14519769—24no—
107#14519769D412yes—
108#14519769C2312yes—
109#14519768—22no—
110#14519768—24no—
111#14519768—24no—
112#14519768C2314yes—
113#14519767—24no—
114#14519766C2314yes—
115#14519765—22no—
116#14519765C2312yes—
117#14519762—24no—
118#14519760—24no—
119#14519756—21no—
120#14519756—21no—
121#14519756—24no—
122#14519756D412yes—
123#14519756D412yes—
124#14519755—24no—
125#14519753—22no—
126#14519753—22no—
127#14519751—21no—
128#14519751—21no—
129Q8×Dic61—22no—
130Q8×Dic61—22no—
131Q8×Dic61C2314yes—
132Q8×Dic61C2314yes—
133C4×Dic122—22no—
134C4×Dic122—22no—
135C4×Dic122—21no—
136C4×Dic122—21no—
137C4×Dic122—22no—
138C4×Dic122—24no—
139C4×Dic122C2312yes—
140C4×Dic122C2312yes—
141#14519742C2314yes—
142C4×D244—21no—
143C4×D244—21no—
144C4×D244—22no—
145#14519739C2314yes—
146C4×C22⋊D61—24no—
147#14519736—24no—
148#14519736C2312yes—
149#14519736C2312yes—
150#14519735—22no—
151#14519734—24no—
152C4⋊C4×D61—22no—
153#14519710C2314yes—
154#14519708C2314yes—
155C4⋊#92418C2312yes—
156C4⋊#92418C2312yes—
157#14519706C2314yes—
158#14519706C2314yes—
159#14519702C2312yes—
160#14519702C2312yes—
161#14519702C2312yes—
162#14519702C2312yes—
163#14519701C2314yes—
164#14519701C2314yes—
165#14519700C2314yes—
166#14519699C2312yes—
167#14519699C2312yes—
168#14519699C2312yes—
169#14519699C2312yes—
170#14519697C2314yes—
171#14519687C2314yes—
172C8×Dic61C2314yes—
173C22×Q8×C61C2311yescharacteristic
174C22×Q8×C61C2311yescharacteristic
175C2×Q16×C61—24no—
176C2×Q16×C61C2314yes—
177C22×C8×C61C2311yescharacteristic
178C22×C8×C61C2311yescharacteristic
179C4×Q8×C61—21no—
180C4×Q8×C61—21no—
181C4×Q8×C61—21no—
182C4×Q8×C61—21no—
183C4×Q8×C61C2312yes—
184C4×Q8×C61C2312yes—
185C4×Q8×C61C2312yes—
186C4×Q8×C61C2312yes—
187C2×C4⋊C4×C61—21no—
188C2×C4⋊C4×C61—21no—
189C2×C4⋊C4×C61C2311yescharacteristic
190C2×C4⋊C4×C61C2311yescharacteristic
191C2×C42×C61—21no—
192C2×C42×C61—21no—
193C2×C42×C61C2311yescharacteristic
194#106×C61C2314yes—
195#103×C61C2314yes—
196#103×C61C2314yes—
197#97×C61C2312yes—
198#97×C61C2312yes—
199C2×Q8×C61—21no—
200C2×Q8×C61—21no—
201C2×Q8×C61—22no—
202C2×Q8×C61—22no—
203C2×Q8×C61—22no—
204C2×Q8×C61#4114yes—
205C2×Q8×C61C2412yes—
206C2×Q8×C61C2412yes—
207C22×C4×C61—21no—
208C22×C4×C61—21no—
209C22×C4×C61—21no—
210C22×C4×C61—21no—
211C22×C4×C61C2×D411yescharacteristic
212C22×C4×C61C2×D411yescharacteristic
213C22×C4×C61C2411yescharacteristic
214C2×C22⋊D61—22no—
215#11758222—24no—
216#11758222—24no—
217C2×D244—21no—
218C2×D244—21no—
219C2×D244C2412yes—
220C2×C4×D61—22no—
221C2×C4×D61—24no—
222C2×Dic122—21no—
223C2×Dic122—21no—
224C2×Dic122—24no—
225C2×Dic122C2×D412yes—
226C2×Dic122C2×D412yes—
227C2×Dic122C2412yes—
228Q16×C61—28no—
229C2×C8×C61—22no—
230C2×C8×C61—22no—
231C2×C8×C61C2412yes—
232C2×C8×C61C2412yes—
233C4⋊C4×C61—22no—
234C4⋊C4×C61—22no—
235C4⋊C4×C61—22no—
236C4⋊C4×C61—22no—
237C4⋊C4×C61—21no—
238C4⋊C4×C61—21no—
239C4⋊C4×C61C2412yes—
240C4⋊C4×C61C2412yes—
241C42×C61—22no—
242C42×C61—22no—
243C42×C61—22no—
244C42×C61—22no—
245C42×C61C2×D411yescharacteristic
246C42×C61C2×D411yescharacteristic
247C42×C61C2411yescharacteristic
248C42×C61C2411yescharacteristic
249C22⋊Dic61—22no—
250C22⋊Dic61C2×D414yes—
251#11758202—28no—
252#11758201—28no—
253#11758198—22no—
254#11758198—22no—
255#11758198—24no—
256#11758198—24no—
257#11758198—24no—
258C4⋊Dic61—21no—
259C4⋊Dic61—21no—
260C4⋊Dic61—21no—
261C4⋊Dic61—21no—
262C4⋊Dic61—24no—
263C4⋊Dic61—24no—
264C4⋊Dic61—24no—
265C4⋊Dic61C2×D412yes—
266C4⋊Dic61C2×D412yes—
267C4⋊Dic61C2×D412yes—
268C4⋊Dic61C2×D412yes—
269C4⋊Dic61C2412yes—
270C4⋊Dic61C2412yes—
271#11758196—22no—
272#11758196—22no—
273#11758196—24no—
274#11758196—24no—
275#11758196—24no—
276#11758196—24no—
277#11758196—24no—
278#11758196C2×D414yes—
279#11758196C2×D414yes—
280C4×Dic61—22no—
281C4×Dic61—24no—
282C4×Dic61—24no—
283C4×Dic61—24no—
284C4×Dic61C2414yes—
285#11758194—24no—
286C2×#92418C2414yes—
287C23×C61C22×D411yescharacteristic
288C22×D61—22no—
289Q8×C61—24no—
290Q8×C61—22no—
291Q8×C61—22no—
292Q8×C61—22no—
293Q8×C61—42no—
294Q8×C61C2×#4114yes—
295C2×C4×C61—24no—
296C2×C4×C61—22no—
297C2×C4×C61—22no—
298C2×C4×C61—22no—
299C2×C4×C61—22no—
300C2×C4×C61—41no—
301C2×C4×C61—41no—
302C2×C4×C61—42no—
303C2×C4×C61—81no—
304C2×C4×C61—81no—
305C2×C4×C61C22×D411yescharacteristic
306C2×C4×C61C22×D411yescharacteristic
307C2×C4×C61C22×D411yescharacteristic
308C2×C4×C61C22×D411yescharacteristic
309C2×C4×C61C22×D411yescharacteristic
310C2×C4×C61C2×#4112yes—
311C2×C4×C612−1+412yes—
312C2×C4×C61C2511yescharacteristic
313C22⋊D61—44no—
314C2×Dic61—22no—
315C2×Dic61—22no—
316C2×Dic61—22no—
317C2×Dic61C22×D414yes—
318C2×Dic61C22×D414yes—
319D244—22no—
320D244—22no—
321D244—42no—
322D244—81no—
323C4×D61—44no—
324C4×D61—84no—
325Dic122—22no—
326Dic122—22no—
327Dic122—42no—
328Dic122—42no—
329Dic122—42no—
330Dic122—84no—
331Dic122—81no—
332C8×C61—22no—
333C8×C61—22no—
334#92418—24no—
335#61335—611nomaximal, Sylow
336C22×C61—21no—
337C22×C61—21no—
338C22×C61#54812yes—
339C22×C61C23×D411yescharacteristic
340C2×D61—42no—
341C2×D61—42no—
342C2×D61—42no—
343C4×C61—21no—
344C4×C61—21no—
345C4×C61—21no—
346C4×C61—21no—
347C4×C61—24no—
348C4×C61—42no—
349C4×C61—42no—
350C4×C61—42no—
351C4×C61#54812yes—
352C4×C61C22×#4111yescharacteristic
353C4×C61C2×2−1+411yescharacteristic
354Dic61—24no—
355Dic61—24no—
356Dic61—42no—
357Dic61—42no—
358Dic61—42no—
359#3507—611no—
360#3502—611no—
361#3377—614no—
362#3374—614no—
363#3371—614no—
364#3369—614no—
365#3137—612no—
366#3135—611no—
367#3084—612no—
368#3078—611no—
369#3044—612no—
370#3037—611no—
371#2961—611no—
372#2947—612no—
373C2×#441D6111yescharacteristic, complemented
374C2×C61—22no—
375C2×C61C2×#54811yescharacteristic
376C2×C61#358611yescharacteristic
377C2×C61#358611yescharacteristic
378D61—82no—
379C2×#137—612no—
380C22×Q16C2×D6111yescharacteristic
381#559—612no—
382Q82—612no—
383Q82—1221no—
384#557—614no—
385#557—612no—
386#557—612no—
387#557—1222no—
388#557—1221no—
389#556—614no—
390#556—1221no—
391#554—1221no—
392#552—1222no—
393#544—612no—
394#544—1222no—
395#539—1222no—
396#536—612no—
397#533—611no—
398#533—611no—
399#533—1222no—
400#532—611no—
401#529—1222no—
402#520—612no—
403#519—612no—
404C2×C4×Q8C2×D6111yescharacteristic
405C2×C4×Q8C2×D6111yescharacteristic
406#491—614no—
407#491—612no—
408#490—612no—
409#489—614no—
410#489—612no—
411#489—612no—
412#488—614no—
413#487—614no—
414#479—614no—
415#479—614no—
416#478—614no—
417#477—612no—
418#476—614no—
419#475—612no—
420#475—614no—
421#464—614no—
422#464—614no—
423#464—614no—
424#462—614no—
425#462—614no—
426#462—612no—
427#460—612no—
428#446—614no—
429#446—614no—
430#441—614no—
431#441C2×D6114yes—
432#441C2×D6114yes—
433#440—614no—
434C4×Q16—614no—
435C4×SD16—614no—
436#428—612no—
437C2×#106C2×D6111yescharacteristic
438#424—612no—
439#423—611no—
440#419—612no—
441#417—612no—
442#416—612no—
443#416—612no—
444C2×#103C2×D6111yescharacteristic
445C2×#103C2×D6111yescharacteristic
446#405—612no—
447C2×#97C2×D6111yescharacteristic
448C61#6133511yescharacteristic, complemented, Sylow
449C2×#41—612no—
450C22×Q8C22×D6111yescharacteristic
451C22×Q8C22×D6111yescharacteristic
452#137—1228no—
453C2×Q16—24no—
454C2×Q16—614no—
455C2×Q16C22×D6114yes—
456C2×SD16—614no—
457C2×M16—612no—
458C22×C8C22×D6111yescharacteristic
459C22×C8C22×D6111yescharacteristic
460C4⋊Q8—614no—
461C4⋊Q8—614no—
462C4⋊Q8—611no—
463C4⋊Q8—611no—
464C4⋊Q8—611no—
465C4⋊Q8—611no—
466C4⋊Q8—611no—
467C4⋊Q8—1222no—
468C4⋊Q8—1222no—
469C4⋊Q8—1222no—
470C4⋊Q8—1224no—
471C4⋊Q8—1224no—
472#127—611no—
473#126—1224no—
474#126—1224no—
475#125—614no—
476#125—612no—
477#125—612no—
478#125—1222no—
479#125—1224no—
480#125—1224no—
481#125—1222no—
482#125—1224no—
483#125—1224no—
484#124—614no—
485#124—612no—
486#124—1222no—
487#123—1224no—
488#123—1224no—
489C22⋊Q8—612no—
490C22⋊Q8—612no—
491C22⋊Q8—1224no—
492C22⋊Q8—1224no—
493C22⋊Q8—1222no—
494C22⋊Q8—1222no—
495C22⋊Q8—1221no—
496C22⋊Q8—1221no—
497#121—1221no—
498#121—1221no—
499C4×Q8—21no—
500C4×Q8—21no—
501C4×Q8—21no—
502C4×Q8—21no—
503C4×Q8—612no—
504C4×Q8—612no—
505C4×Q8—614no—
506C4×Q8—614no—
507C4×Q8—614no—
508C4×Q8—1224no—
509C4×Q8—1222no—
510C4×Q8—1222no—
511C4×Q8—1222no—
512C4×Q8—1222no—
513C4×Q8—1222no—
514C4×Q8—1221no—
515C4×Q8—1221no—
516C4×Q8C22×D6112yes—
517C4×Q8C22×D6112yes—
518C4×Q8C22×D6112yes—
519C4×Q8C22×D6112yes—
520C4×D4—614no—
521C4×D4—1222no—
522C4×D4—1224no—
523C4×D4—1221no—
524C4×D4—1221no—
525#117—612no—
526#117—612no—
527#117—1224no—
528#117—1224no—
529#117—1222no—
530C2×C4⋊C4—21no—
531C2×C4⋊C4—21no—
532C2×C4⋊C4—1222no—
533C2×C4⋊C4C22×D6111yescharacteristic
534C2×C4⋊C4C22×D6111yescharacteristic
535C2×C42—21no—
536C2×C42—21no—
537C2×C42C22×D6111yescharacteristic
538#107—614no—
539#106—614no—
540#106C22×D6114yes—
541C4⋊C8—614no—
542C4⋊C8—614no—
543C4⋊C8—612no—
544C4⋊C8—612no—
545#103—614no—
546#103—614no—
547#103—612no—
548#103—612no—
549#103—612no—
550#103—612no—
551#103—614no—
552#103C22×D6114yes—
553#103C22×D6114yes—
554#102—612no—
555#102—612no—
556#102—612no—
557#102—612no—
558#102—614no—
559#97—614no—
560#97C22×D6112yes—
561#97C22×D6112yes—
562C4×C8—614no—
563#41—1224no—
564#41—1224no—
565C2×Q8—21no—
566C2×Q8—21no—
567C2×Q8—22no—
568C2×Q8—22no—
569C2×Q8—22no—
570C2×Q8—612no—
571C2×Q8—612no—
572C2×Q8—612no—
573C2×Q8—1224no—
574C2×Q8—1221no—
575C2×Q8—1221no—
576C2×Q8#1175822214yes—
577C2×Q8C23×D6112yes—
578C2×Q8C23×D6112yes—
579C2×D4—612no—
580C2×D4—1222no—
581C2×D4—1221no—
582C2×D4—1221no—
583C22×C4—21no—
584C22×C4—21no—
585C22×C4—21no—
586C22×C4—21no—
587C22×C4—1224no—
588C22×C4—1222no—
589C22×C4D4×D6111yescharacteristic
590C22×C4D4×D6111yescharacteristic
591C22×C4C23×D6111yescharacteristic
592Q16—28no—
593Q16—1228no—
594SD16—1228no—
595M16—1224no—
596C2×C8—22no—
597C2×C8—22no—
598C2×C8—614no—
599C2×C8C23×D6112yes—
600C2×C8C23×D6112yes—
601C4⋊C4—22no—
602C4⋊C4—22no—
603C4⋊C4—22no—
604C4⋊C4—22no—
605C4⋊C4—21no—
606C4⋊C4—21no—
607C4⋊C4—612no—
608C4⋊C4—614no—
609C4⋊C4—614no—
610C4⋊C4—612no—
611C4⋊C4—612no—
612C4⋊C4—612no—
613C4⋊C4—612no—
614C4⋊C4—612no—
615C4⋊C4—1224no—
616C4⋊C4—1224no—
617C4⋊C4—1224no—
618C4⋊C4—1224no—
619C4⋊C4—1224no—
620C4⋊C4—1224no—
621C4⋊C4—1224no—
622C4⋊C4—1224no—
623C4⋊C4—1222no—
624C4⋊C4—1222no—
625C4⋊C4—1221no—
626C4⋊C4—1221no—
627C4⋊C4—1221no—
628C4⋊C4—1221no—
629C4⋊C4C23×D6112yes—
630C4⋊C4C23×D6112yes—
631C22⋊C4—614no—
632C22⋊C4—1224no—
633C22⋊C4—1224no—
634C22⋊C4—1224no—
635C22⋊C4—1222no—
636C22⋊C4—1222no—
637C22⋊C4—1222no—
638C42—22no—
639C42—22no—
640C42—22no—
641C42—22no—
642C42—614no—
643C42—1224no—
644C42—1224no—
645C42—1224no—
646C42—1222no—
647C42D4×D6111yescharacteristic
648C42D4×D6111yescharacteristic
649C42C23×D6111yescharacteristic
650C42C23×D6111yescharacteristic
651C23—1222no—
652C23C2×D4×D6111yescharacteristic
653Q8—22no—
654Q8—24no—
655Q8—22no—
656Q8—22no—
657Q8—42no—
658Q8—1222no—
659Q8—1222no—
660Q8—4881no—
661Q8—4884no—
662Q8—2442no—
663Q8—2442no—
664Q8—2442no—
665Q8C2×#1175822214yes—
666D4—1222no—
667D4—1222no—
668D4—4881no—
669D4—2442no—
670D4—2444no—
671C2×C4—22no—
672C2×C4—24no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—81no—
680C2×C4—81no—
681C2×C4—614no—
682C2×C4—614no—
683C2×C4—1222no—
684C2×C4—1222no—
685C2×C4—1222no—
686C2×C4—4884no—
687C2×C4—2444no—
688C2×C4C2×D4×D6111yescharacteristic
689C2×C4C2×D4×D6111yescharacteristic
690C2×C4C2×D4×D6111yescharacteristic
691C2×C4C2×D4×D6111yescharacteristic
692C2×C4C2×D4×D6111yescharacteristic
693C2×C4C2×#1175822212yes—
694C2×C4#1451982412yes—
695C2×C4C24×D6111yescharacteristic
696C8—22no—
697C8—22no—
698C8—1224no—
699C22—21no—
700C22—21no—
701C22—2442no—
702C22—2442no—
703C22—2442no—
704C22#1684042112yes—
705C22C22×D4×D6111yescharacteristic
706C4—24no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—21no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—1224no—
715C4—1224no—
716C4—2442no—
717C4—2442no—
718C4—2442no—
719C4#1684042112yes—
720C4C22×#1175822211yescharacteristic
721C4C2×#1451982411yescharacteristic
722C2—22no—
723C2—4882no—
724C2C2×#1684042111yescharacteristic
725C2#2146060211yescharacteristic
726C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#441C2
socleC22×C61C23×D4
prime radical#16840312C22
hypercenterC2×#441D61
hypocenterC61#61335
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
2. #21459629C2semidirect
3. #21458666C2semidirect
4. #21458623C2semidirect
5. #21458588C2semidirect
6. #21458560C2semidirect
7. #21456182C2semidirect
9. #21455686C2semidirect
10. #21455624C2semidirect
11. #21455346C2semidirect
13. #21454759C2semidirect
14. #21454656C2semidirect
15. C2×C61×#441C2semidirect
373. C2×#441D61semidirect
448. C61#61335semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214596292
#214586662
#214586232
#214585882
#214585602
#214561822
#214561652
#214556862
#214556242
#214553462
#214552912
#214547592
#214546562
C2×C61×#4412
#6133561