Finite Group Databasebeta

← #30558416

Subgroups — #30558416

Order 15616.

2020 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459739C211yesmaximal, characteristic, complemented
2#21459599C211yesmaximal, characteristic, complemented
3#21457992C214yesmaximal, complemented
4#21457917C214yesmaximal, complemented
5#21457872C214yesmaximal, complemented
6#21457797C214yesmaximal, complemented
7C2×#16840131C211yesmaximal, characteristic, complemented
8#21455789C212yesmaximal, complemented
9#21455642C211yesmaximal, characteristic, complemented
10#21455510C212yesmaximal, complemented
11#21455307C211yesmaximal, characteristic, complemented
12#21455137C212yesmaximal, complemented
13#21454621C211yesmaximal, characteristic, complemented
14#21454353C212yesmaximal, complemented
15C61×#2932C211yesmaximal, characteristic, complemented
16C2×#14519824C2212yes—
17C2×#14519810C2212yes—
18C2×#14519799C2211yescharacteristic
19#16840458C2214yes—
20#16840455—24no—
21#16840452C2214yes—
22D4×Dic122C2214yes—
23#16840417C2214yes—
24#16840412—24no—
25#16840373—21no—
26#16840366—21no—
27#16840340C2212yes—
28#16840333C2212yes—
29#16840332—22no—
30#16840310—21no—
31C2×#14519777C2211yescharacteristic
32C4⋊Q8×D61—21no—
33#16840291—22no—
34C2×#14519769C2211yescharacteristic
35#16840245C2212yes—
36C4×#11758222C2211yescharacteristic
37C4×#11758222C2211yescharacteristic
38#16840131C2214yes—
39#16840131C2214yes—
40#16840104C2214yes—
41#16840104C2214yes—
42#16840098C2214yes—
43#16840093C2214yes—
44#16840070C2214yes—
45#16840063C2214yes—
46#16840020C2214yes—
47#16840008C2214yes—
48#16840003C2214yes—
49#16839991C2214yes—
50#16839973C2214yes—
51#16839968C2214yes—
52#16839926C2214yes—
53#16839920C2214yes—
54#16839909C2214yes—
55#16839903C2214yes—
56#16839875C2212yes—
57#16839870C2212yes—
58#16839864C2212yes—
59#16839864C2212yes—
60#16839804C2212yes—
61#16839800C2211yescharacteristic
62#16839772C2214yes—
63#16839772C2212yes—
64#16839764C2212yes—
65#16839764C2214yes—
66#16839759C2211yescharacteristic
67#16839759C2214yes—
68#16839759C2211yescharacteristic
69C2×#14519702C2212yes—
70C2×#14519702C2214yes—
71C2×#14519702C2212yes—
72C2×#131×C61C2212yes—
73C61×#532C2211yescharacteristic
74C61×#435C2214yes—
75C61×#435C2214yes—
76C61×#423C2211yescharacteristic
77C61×#408C2212yes—
78C2×#97×C61C2211yescharacteristic
79#14519824—28no—
80#14519824—44no—
81C2×#11758222C2311yescharacteristic
82C2×#11758222C2314yes—
83C2×#11758222C2311yescharacteristic
84C2×#11758224—22no—
85C2×#11758224—24no—
86C22×Dic122C2312yes—
87C22×Dic122C2314yes—
88C2×Q8×D61—22no—
89#14519810—44no—
90#14519799—24no—
91C2×Dic244C2311yescharacteristic
92C2×Dic244C2314yes—
93C2×#11758190C2311yescharacteristic
94C2×#11758190C2311yescharacteristic
95C2×#11758190C2314yes—
96C2×D488C2311yescharacteristic
97#14519777—21no—
98#14519777—21no—
99#14519777C2314yes—
100#14519776—24no—
101#14519775—22no—
102#14519772—24no—
103#14519770—24no—
104#14519769—21no—
105#14519769—21no—
106#14519769C2314yes—
107#14519767—24no—
108#14519764—22no—
109#14519759—24no—
110#14519759—24no—
111#14519759—24no—
112#14519758—24no—
113#14519758—24no—
114#14519756C2314yes—
115#14519756C2314yes—
116#14519756C2314yes—
117#14519754—24no—
118#14519754—24no—
119#14519753—22no—
120#14519751C2314yes—
121#14519749—24no—
122#14519749—24no—
123#14519748—22no—
124#14519748—24no—
125Q8×Dic61—22no—
126C4×Dic122C2312yes—
127C4×Dic122C2312yes—
128C4×D244C2312yes—
129C4×D244C2312yes—
130D4×Dic61—24no—
131D4×Dic61—22no—
132D4×Dic61—24no—
133C4×C22⋊D61—22no—
134C4×C22⋊D61—22no—
135#14519737—24no—
136#14519736C2314yes—
137#14519736C2311yescharacteristic
138#14519736C2311yescharacteristic
139#14519734—21no—
140#14519734—21no—
141#14519733—24no—
142C2×C4⋊Dic61—22no—
143C2×C4⋊Dic61C2314yes—
144C2×C4⋊Dic61C2312yes—
145C2×#11758196—22no—
146C4⋊C4×D61—24no—
147C42×D61—21no—
148C42×D61—21no—
149#14519711C2314yes—
150#14519711C2312yes—
151#14519709C2314yes—
152#14519709C2312yes—
153#14519702C2314yes—
154#14519702C2314yes—
155#14519702C2314yes—
156#14519702C2312yes—
157#14519702C2314yes—
158#14519702C2312yes—
159#14519702C2314yes—
160#14519702C2314yes—
161#14519702C2314yes—
162#14519699C2312yes—
163#14519699C2312yes—
164#14519699C2314yes—
165C2×#41×C61C2312yes—
166#131×C61—44no—
167C2×M16×C61C2312yes—
168C2×M16×C61C2314yes—
169C22×C8×C61C2311yescharacteristic
170C22×C8×C61C2311yescharacteristic
171C22×C8×C61C2314yes—
172C4⋊Q8×C61D411yescharacteristic
173#127×C61D411yescharacteristic
174#124×C61D412yes—
175#121×C61D414yes—
176C4×D4×C61C2314yes—
177C2×C42×C61C2311yescharacteristic
178C4⋊C8×C61C2314yes—
179C22⋊C8×C61C2314yes—
180C22⋊C8×C61C2314yes—
181#97×C61C2312yes—
182#97×C61C2312yes—
183#41×C61—24no—
184#41×C61—24no—
185C2×Q8×C61C2×D412yes—
186C2×D4×C61C2×D414yes—
187C2×D4×C61C2×D412yes—
188C2×D4×C61C2×D414yes—
189C22×C4×C61C2×D412yes—
190C22×C4×C61C2414yes—
191C22×C4×C61C2411yescharacteristic
192C2×C22⋊D61—21no—
193C2×C22⋊D61—24no—
194C2×C22⋊D61—21no—
195C2×C22⋊D61—24no—
196C22×Dic61—22no—
197C22×Dic61—24no—
198C22×Dic61—22no—
199C22×Dic61—24no—
200Q8×D61—48no—
201#11758224—48no—
202#11758224—416no—
203#11758222—22no—
204#11758222—22no—
205#11758222—22no—
206#11758222—28no—
207#11758222—44no—
208#11758222#4114yes—
209C2×D244C2411yescharacteristic
210C2×D244C2411yescharacteristic
211C2×C4×D61—21no—
212C2×C4×D61—24no—
213C2×C4×D61—21no—
214C2×C4×D61—22no—
215C2×C4×D61—22no—
216C2×Dic122—24no—
217C2×Dic122—24no—
218C2×Dic122—28no—
219C2×Dic122—24no—
220C2×Dic122—22no—
221C2×Dic122—44no—
222C2×Dic122C2414yes—
223C2×Dic122C2411yescharacteristic
224C2×Dic122C2411yescharacteristic
225M16×C61—24no—
226M16×C61—44no—
227C2×C8×C61—22no—
228C2×C8×C61—22no—
229C2×C8×C61—44no—
230C2×C8×C61C2411yescharacteristic
231C2×C8×C61C2411yescharacteristic
232C2×C8×C61C2412yes—
233C2×C8×C61C2414yes—
234C4⋊C4×C61C2×D414yes—
235C22⋊C4×C61C2×D414yes—
236C22⋊C4×C61C2×D414yes—
237C42×C61C2×D411yescharacteristic
238C42×C61C2×D411yescharacteristic
239C42×C61C2412yes—
240C22⋊Dic61—24no—
241C22⋊Dic61—21no—
242C22⋊Dic61—21no—
243C22⋊Dic61—24no—
244C22⋊Dic61—24no—
245C22⋊Dic61—24no—
246#11758198—22no—
247#11758198—22no—
248#11758198—24no—
249C4⋊Dic61—24no—
250C4⋊Dic61—24no—
251C4⋊Dic61—22no—
252C4⋊Dic61—24no—
253C4⋊Dic61C2412yes—
254C4⋊Dic61C2414yes—
255C4⋊Dic61C2412yes—
256#11758196—24no—
257#11758196—24no—
258#11758196—24no—
259#11758196—24no—
260#11758196—24no—
261#11758196—22no—
262#11758196—22no—
263C4×Dic61—22no—
264C4×Dic61—21no—
265C4×Dic61—21no—
266C4×Dic61—22no—
267C4×Dic61—24no—
268Dic244—22no—
269Dic244—44no—
270D488—22no—
271#11758190—22no—
272#11758190—22no—
273#11758190—44no—
274C23×C61C22×D414yes—
275C23×C61C22×D411yescharacteristic
276C22×D61—21no—
277C22×D61—21no—
278Q8×C61—24no—
279D4×C61—28no—
280D4×C61—24no—
281D4×C61—44no—
282C2×C4×C61—24no—
283C2×C4×C61—24no—
284C2×C4×C61—42no—
285C2×C4×C61C22×D412yes—
286C2×C4×C61C22×D412yes—
287C2×C4×C61C22×D414yes—
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C2×#4112yes—
290C2×C4×C61H(5,𝔽2)12yes—
291C2×C4×C61C2511yescharacteristic
292C22⋊D61—24no—
293C22⋊D61—48no—
294C22⋊D61—42no—
295C22⋊D61—84no—
296C2×Dic61—21no—
297C2×Dic61—22no—
298C2×Dic61—21no—
299C2×Dic61—22no—
300C2×Dic61—24no—
301C2×Dic61—24no—
302C2×Dic61—48no—
303C2×Dic61—44no—
304C2×Dic61—84no—
305C2×Dic61—82no—
306D244—21no—
307D244—21no—
308D244—21no—
309D244C2×#4112yes—
310C4×D61—24no—
311C4×D61—48no—
312C4×D61—44no—
313C4×D61—42no—
314C4×D61—82no—
315Dic122—21no—
316Dic122—21no—
317Dic122—24no—
318Dic122—24no—
319Dic122—21no—
320Dic122—48no—
321Dic122—44no—
322Dic122C2×#4112yes—
323C8×C61—21no—
324C8×C61—21no—
325C8×C61—24no—
326C8×C61—22no—
327#61224—611nomaximal, Sylow
328C22×C61—22no—
329C22×C61—24no—
330C22×C61—24no—
331C22×C61—24no—
332C22×C61#54612yes—
333C22×C61C23×D411yescharacteristic
334C2×D61—22no—
335C2×D61—42no—
336C2×D61—42no—
337C2×D61—41no—
338C4×C61—22no—
339C4×C61—22no—
340C4×C61—24no—
341C4×C61#54612yes—
342C4×C61C22×#4111yescharacteristic
343C4×C61C2×H(5,𝔽2)11yescharacteristic
344Dic61—22no—
345Dic61—42no—
346Dic61—42no—
347Dic61—44no—
348Dic61—44no—
349Dic61—41no—
350#3512—611no—
351#3499—611no—
352#3320—614no—
353#3315—614no—
354#3312—614no—
355#3307—614no—
356C2×#491—611no—
357#3097—612no—
358#3080—611no—
359#3063—612no—
360#3039—611no—
361#3013—612no—
362#2943—611no—
363#2932D6111yescharacteristic, complemented
364#2898—612no—
365C2×C61—22no—
366C2×C61—44no—
367C2×C61C2×#54611yescharacteristic
368C2×C61#358612yes—
369D61—42no—
370D61—81no—
371C2×2−1+4—612no—
372C2×#137—612no—
373C2×#135—611no—
374C2×#131C2×D6112yes—
375#553—614no—
376#553—1224no—
377#552—614no—
378D4×Q8—614no—
379#547—614no—
380#547—1224no—
381#541—1221no—
382#540—1221no—
383#537—612no—
384#536—612no—
385#536—1222no—
386#532—1221no—
387#532C2×D6111yescharacteristic
388C2×C4⋊Q8—611no—
389C2×C4⋊Q8—1221no—
390#529—1222no—
391C2×#125—611no—
392#520—612no—
393C4×#41—611no—
394C4×#41—611no—
395#491—614no—
396#491—614no—
397#485—614no—
398#485—614no—
399#484—614no—
400#483—614no—
401#479—614no—
402#478—614no—
403#470—614no—
404#468—614no—
405#467—614no—
406#465—614no—
407#462—614no—
408#461—614no—
409#452—614no—
410#451—614no—
411#449—614no—
412#448—614no—
413#442—612no—
414#441—612no—
415#440—612no—
416#440—612no—
417#435C2×D6114yes—
418#435C2×D6114yes—
419#428—612no—
420#427—611no—
421#423C2×D6111yescharacteristic
422#419—612no—
423#419—614no—
424#417—612no—
425#417—614no—
426#416—611no—
427#416—614no—
428#416—611no—
429C2×#103—612no—
430C2×#103—612no—
431C2×#103—614no—
432#408C2×D6112yes—
433C2×#97C2×D6111yescharacteristic
434C61#6122411yescharacteristic, complemented, Sylow
4352−1+4—1228no—
4362−1+4—2444no—
437C2×#41—611no—
438C2×#41—611no—
439C2×#41—614no—
440C2×#41—1222no—
441C2×#41—1224no—
442C2×#41C22×D6112yes—
443C22×Q8—612no—
444C22×Q8—614no—
445C22×Q8—1222no—
446#137—2444no—
447#135—1224no—
448C2×Q16—611no—
449C2×Q16—614no—
450C2×SD16—611no—
451C2×SD16—611no—
452C2×SD16—614no—
453C2×D8—611no—
454#131—44no—
455C2×M16C22×D6112yes—
456C2×M16C22×D6114yes—
457C22×C8C22×D6111yescharacteristic
458C22×C8C22×D6111yescharacteristic
459C22×C8C22×D6114yes—
460C4⋊Q8—614no—
461C4⋊Q8—1222no—
462C4⋊Q8—1224no—
463C4⋊Q8—1221no—
464C4⋊Q8—1221no—
465C4⋊Q8D24411yescharacteristic
466#127D24411yescharacteristic
467#126—1224no—
468#126—1224no—
469#125—614no—
470#125—1224no—
471#125—1221no—
472#125—1221no—
473#124—1222no—
474#124D24412yes—
475#123—1224no—
476#123—1224no—
477#123—1224no—
478#123—1224no—
479#123—1224no—
480C22⋊Q8—614no—
481C22⋊Q8—614no—
482C22⋊Q8—614no—
483C22⋊Q8—1222no—
484C22⋊Q8—1224no—
485C22⋊Q8—1224no—
486#121—614no—
487#121—1222no—
488#121—1224no—
489#121—1224no—
490#121—1224no—
491#121D24414yes—
492C4×Q8—612no—
493C4×Q8—612no—
494C4×Q8—1222no—
495C4×D4—612no—
496C4×D4—612no—
497C4×D4—1222no—
498C4×D4—1224no—
499C4×D4—1224no—
500C4×D4—1222no—
501C4×D4—1222no—
502C4×D4—1224no—
503C4×D4C22×D6114yes—
504#117—611no—
505#117—611no—
506#117—614no—
507#117—1224no—
508#117—1221no—
509#117—1221no—
510C2×C4⋊C4—612no—
511C2×C4⋊C4—614no—
512C2×C4⋊C4—1222no—
513C2×C4⋊C4—1222no—
514C2×C4⋊C4—1224no—
515C2×C42—1221no—
516C2×C42—1221no—
517C2×C42C22×D6111yescharacteristic
518#107—612no—
519#107—614no—
520#106—612no—
521#106—614no—
522C4⋊C8C22×D6114yes—
523#103—612no—
524#103—614no—
525#103—614no—
526#103—614no—
527#103—614no—
528#103—614no—
529#103—614no—
530#103—614no—
531#103—612no—
532#102—612no—
533#102—614no—
534#102—612no—
535C22⋊C8C22×D6114yes—
536C22⋊C8C22×D6114yes—
537#97C22×D6112yes—
538#97C22×D6112yes—
539#41—24no—
540#41—24no—
541#41—614no—
542#41—1228no—
543#41—1222no—
544#41—1222no—
545#41—1222no—
546#41—2444no—
547#41—2448no—
548#41—24416no—
549C2×Q8—611no—
550C2×Q8—614no—
551C2×Q8—611no—
552C2×Q8—1228no—
553C2×Q8—1224no—
554C2×Q8—1222no—
555C2×Q8—1224no—
556C2×Q8—1224no—
557C2×Q8—2444no—
558C2×Q8—2448no—
559C2×Q8C2×D24412yes—
560C2×D4—611no—
561C2×D4—611no—
562C2×D4—1221no—
563C2×D4—1224no—
564C2×D4—1221no—
565C2×D4—1224no—
566C2×D4C2×D24414yes—
567C2×D4C2×D24412yes—
568C2×D4C2×D24414yes—
569C22×C4—1221no—
570C22×C4—1222no—
571C22×C4—1224no—
572C22×C4—1222no—
573C22×C4—1224no—
574C22×C4—1222no—
575C22×C4—1222no—
576C22×C4—1221no—
577C22×C4—1224no—
578C22×C4C2×D24412yes—
579C22×C4C23×D6114yes—
580C22×C4C23×D6111yescharacteristic
581Q16—1222no—
582Q16—2444no—
583SD16—1222no—
584SD16—1222no—
585SD16—2444no—
586D8—1222no—
587M16—24no—
588M16—44no—
589C2×C8—22no—
590C2×C8—22no—
591C2×C8—44no—
592C2×C8C23×D6111yescharacteristic
593C2×C8C23×D6111yescharacteristic
594C2×C8C23×D6112yes—
595C2×C8C23×D6114yes—
596C4⋊C4—612no—
597C4⋊C4—614no—
598C4⋊C4—612no—
599C4⋊C4—1222no—
600C4⋊C4—1224no—
601C4⋊C4—1224no—
602C4⋊C4—1224no—
603C4⋊C4—1224no—
604C4⋊C4—1224no—
605C4⋊C4—1224no—
606C4⋊C4—1224no—
607C4⋊C4—1222no—
608C4⋊C4—1222no—
609C4⋊C4—1224no—
610C4⋊C4C2×D24414yes—
611C22⋊C4—1221no—
612C22⋊C4—1224no—
613C22⋊C4—1221no—
614C22⋊C4—1224no—
615C22⋊C4—1224no—
616C22⋊C4—1224no—
617C22⋊C4—1222no—
618C22⋊C4—1222no—
619C22⋊C4—1224no—
620C22⋊C4C2×D24414yes—
621C22⋊C4C2×D24414yes—
622C42—1221no—
623C42—1221no—
624C42—1222no—
625C42—1222no—
626C42—1224no—
627C42C2×D24411yescharacteristic
628C42C2×D24411yescharacteristic
629C42C23×D6112yes—
630C23—1221no—
631C23—1221no—
632C23C22×D24414yes—
633C23C22×D24411yescharacteristic
634Q8—24no—
635Q8—612no—
636Q8—1221no—
637Q8—1221no—
638Q8—1224no—
639Q8—1224no—
640Q8—1221no—
641Q8—2448no—
642Q8—2444no—
643D4—28no—
644D4—24no—
645D4—44no—
646D4—612no—
647D4—1224no—
648D4—1221no—
649D4—1221no—
650D4—1221no—
651D4—4884no—
652D4—2448no—
653D4—2442no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—42no—
657C2×C4—1221no—
658C2×C4—1224no—
659C2×C4—1222no—
660C2×C4—1222no—
661C2×C4—1221no—
662C2×C4—1224no—
663C2×C4—1224no—
664C2×C4—4882no—
665C2×C4—4884no—
666C2×C4—4882no—
667C2×C4—2448no—
668C2×C4—2444no—
669C2×C4—2448no—
670C2×C4—2444no—
671C2×C4—2442no—
672C2×C4C22×D24412yes—
673C2×C4C22×D24412yes—
674C2×C4C22×D24414yes—
675C2×C4C22×D24411yescharacteristic
676C2×C4#41×D6112yes—
677C2×C4#1451982212yes—
678C2×C4C24×D6111yescharacteristic
679C8—21no—
680C8—21no—
681C8—22no—
682C8—24no—
683C22—22no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—1222no—
688C22—2441no—
689C22—2442no—
690C22—2442no—
691C22#1684040712yes—
692C22C23×D24411yescharacteristic
693C4—24no—
694C4—22no—
695C4—22no—
696C4—1222no—
697C4—2444no—
698C4—2441no—
699C4—2442no—
700C4—2444no—
701C4—2442no—
702C4#1684040712yes—
703C4C2×#41×D6111yescharacteristic
704C4C2×#1451982211yescharacteristic
705C2—22no—
706C2—44no—
707C2—4881no—
708C2—2442no—
709C2C2×#1684040711yescharacteristic
710C2#2146060712yes—
subgroupG/H
centerC22C23×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2932C2
socleC22×C61C23×D4
hypercenter#2932D61
hypocenterC61#61224
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #532 ◁ #2932

Complements

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

NH ≅ G/Ntype
1. #21459739C2semidirect
2. #21459599C2semidirect
3. #21457992C2semidirect
4. #21457917C2semidirect
5. #21457872C2semidirect
6. #21457797C2semidirect
7. C2×#16840131C2semidirect
8. #21455789C2semidirect
9. #21455642C2semidirect
10. #21455510C2semidirect
11. #21455307C2semidirect
12. #21455137C2semidirect
13. #21454621C2semidirect
14. #21454353C2semidirect
15. C61×#2932C2semidirect
363. #2932D61semidirect
434. C61#61224semidirect

Maximal subgroups

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

subgroupindex
#214597392
#214595992
#214579922
#214579172
#214578722
#214577972
C2×#168401312
#214557892
#214556422
#214555102
#214553072
#214551372
#214546212
#214543532
C61×#29322
#6122461