Finite Group Databasebeta

← #30617766

Subgroups — #30617766

Order 15616.

1278 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462430C211yesmaximal, characteristic, complemented
2#21462419C211yesmaximal, characteristic, complemented
3#21461864C212yesmaximal, complemented
4#21461861C212yesmaximal, complemented
5#21461824C212yesmaximal, complemented
6#21461821C212yesmaximal, complemented
7#21461729C212yesmaximal, complemented
8#21461671C212yesmaximal, complemented
9#21459608C211yesmaximal, characteristic
10#16840842C2212yes—
11#16840829C2212yes—
12#16840761C2214yescomplemented
13#16840755C2214yescomplemented
14#16840748—22no—
15#16840748C2212yes—
16D61⋊#100—22no—
17D61⋊#100C2212yes—
18#16840743C2212yes—
19#16840741C2212yes—
20C2×#14519836—22no—
21D61⋊#99—22no—
22#16840633C2214yescomplemented
23#16840629C2214yescomplemented
24C2×#14519825C2211yescharacteristic
25C2×#14519823C2211yescharacteristic
26#16840422—22no—
27#16840404—22no—
28#16840404—22no—
29#16840403—22no—
30#16840337C411yescharacteristic, complemented
31#16840337C2211yescharacteristic
32#16840331C411yescharacteristic, complemented
33#16840330C411yescharacteristic, complemented
34#16840330C2211yescharacteristic
35#16840313C411yescharacteristic, complemented
36#16840313C2211yescharacteristic
37#16840308C411yescharacteristic, complemented
38C2×#14519766C411yescharacteristic, complemented
39#124×D61C411yescharacteristic, complemented
40#124×D61C2211yescharacteristic
41#16840230C2211yescharacteristic
42C61×#536C411yescharacteristic, complemented
43#14519865—44no—
44D61⋊M16—24no—
45D61⋊M16C2314yes—
46D4×#6923—44no—
47#14519856C2314yes—
48C2×C22⋊#6923—24no—
49#14519842—44no—
50#14519840—44no—
51#14519838—22no—
52#14519837—22no—
53#14519837—24no—
54#14519836—24no—
55#14519836—22no—
56#14519835—22no—
57#14519825—44no—
58#14519823—44no—
59C2×#11758226—21no—
60C2×#11758226C2×C411yescharacteristic
61C2×#11758226C2×C411yescharacteristic, complemented
62C2×#11758226C2311yescharacteristic
63C2×#11758226C2311yescharacteristic
64C2×#11758222—21no—
65C2×#11758222—21no—
66C2×#11758222C2×C411yescharacteristic, complemented
67C2×#11758222D412yes—
68C2×#11758222C2311yescharacteristic
69C2×Q8×D61—21no—
70C2×Q8×D61C2×C411yescharacteristic, complemented
71C2×Q8×D61C2311yescharacteristic
72C22×D244C2×C411yescharacteristic
73C22×D244D412yes—
74C2×D4×D61C2×C411yescharacteristic
75C2×D4×D61D412yes—
76C2×D4×D61D412yes—
77C2×D4×D61C2311yescharacteristic
78#14519775C2×C411yescharacteristic, complemented
79#14519774C2×C411yescharacteristic
80#14519774C2×C411yescharacteristic, complemented
81#14519773C2×C411yescharacteristic
82#14519766—22no—
83#14519766C2×C411yescharacteristic
84#14519766C2×C411yescharacteristic
85#14519766C2×C411yescharacteristic, complemented
86#14519766C2×C411yescharacteristic, complemented
87#14519766C2×C411yescharacteristic, complemented
88#14519765C2×C411yescharacteristic
89#14519765C2×C411yescharacteristic, complemented
90#14519764C2×C411yescharacteristic
91#14519763—22no—
92#14519763—22no—
93#14519762—22no—
94#14519762—22no—
95#14519755—22no—
96#14519755—22no—
97#14519752—22no—
98#14519752—22no—
99#14519752—22no—
100#14519750—22no—
101#14519750—22no—
102#14519749—22no—
103#14519747—22no—
104#14519747—22no—
105#14519747—22no—
106#14519742—21no—
107#14519739—21no—
108#14519739—22no—
109#14519739—22no—
110C4×C22⋊D61—22no—
111#14519737—21no—
112#14519737—21no—
113#14519734C2×C411yescharacteristic, complemented
114#14519734C2311yescharacteristic
115C2×#11758198—22no—
116C22⋊C4×D61—22no—
117C22⋊C4×D61—22no—
118C2×C4×Dic61C2×C411yescharacteristic, complemented
119C42×D61C2×C411yescharacteristic, complemented
120C42×D61C2311yescharacteristic
121C2×#41×C61C2×C411yescharacteristic
122C22×Q8×C61C2×C411yescharacteristic, complemented
123C4⋊Q8×C61C2×C411yescharacteristic, complemented
124C4⋊Q8×C61C2×C411yescharacteristic, complemented
125#124×C61C2×C411yescharacteristic
126#124×C61C2×C411yescharacteristic
127#123×C61—22no—
128C22⋊Q8×C61—22no—
129#117×C61C2×C411yescharacteristic, complemented
130C23×D61C22⋊C412yes—
131C23×D61C22⋊C412yes—
132C23×D61C2×D412yes—
133C22×#6923—44no—
134#41×C61—42no—
135C2×Q8×C61—21no—
136C2×Q8×C61—42no—
137C2×Q8×C61C22×C411yescharacteristic
138C2×Q8×C61C22×C411yescharacteristic
139C2×Q8×C61C22×C411yescharacteristic
140C2×D4×C61C22⋊C412yes—
141C2×D4×C61C22×C411yescharacteristic
142C22×C4×C61—21no—
143C22×C4×C61C22⋊C412yes—
144C22×C4×C61C22×C411yescharacteristic
145C2×C22⋊D61—21no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61C22⋊C412yes—
148C2×C22⋊D61C2×D411yescharacteristic
149C2×C22⋊D61C2×D411yescharacteristic
150C2×C22⋊D61C2×D412yes—
151C22×Dic61—21no—
152#11758226—24no—
153#11758226—24no—
154#11758226—42no—
155#11758226—42no—
156#11758226—44no—
157Q8×D61—24no—
158Q8×D61—42no—
159Q8×D61—44no—
160D4×D61—24no—
161D4×D61—42no—
162D4×D61—44no—
163D4×D61—44no—
164#11758222—24no—
165#11758222—44no—
166#11758222—42no—
167#11758222—44no—
168#11758222—44no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—22no—
172C2×D244—22no—
173C2×D244—42no—
174C2×D244—44no—
175C2×D244C22⋊C412yes—
176C2×D244C22⋊C412yes—
177C2×D244C22×C411yescharacteristic
178C2×D244C22×C411yescharacteristic
179C2×D244C22×C411yescharacteristic
180C2×D244C2×D412yes—
181C2×D244C2×D411yescharacteristic
182C2×C4×D61—21no—
183C2×C4×D61—21no—
184C2×C4×D61—21no—
185C2×C4×D61—21no—
186C2×C4×D61—21no—
187C2×C4×D61—21no—
188C2×C4×D61—21no—
189C2×C4×D61C22⋊C412yes—
190C2×C4×D61C22×C411yescharacteristic
191C2×C4×D61C2×D412yes—
192C2×C4×D61C2411yescharacteristic
193C2×Dic122—21no—
194C2×Dic122—21no—
195C2×Dic122C22×C411yescharacteristic
196C2×Dic122C2×D411yescharacteristic
197C22⋊#6923—24no—
198C22⋊#6923—44no—
199#11758217—24no—
200C4⋊#6923—24no—
201C4×#6923—24no—
202#11758213—24no—
203#11758213—44no—
204D61⋊C8—24no—
205D61⋊C8—44no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61—22no—
208C4⋊C4×C61—22no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C42×C61C22×C411yescharacteristic
213C42×C61C22×C411yescharacteristic
214C22⋊Dic61—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198—22no—
218#11758198—22no—
219#11758198—22no—
220#11758198—22no—
221#11758198—21no—
222#11758198—21no—
223#11758198—22no—
224#11758196—22no—
225#11758196—21no—
226#11758196—21no—
227C4×Dic61—21no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C22×C411yescharacteristic
232C23×C61C2×C22⋊C412yes—
233C23×C61C2×C22⋊C411yescharacteristic
234C22×D61—24no—
235C22×D61—21no—
236C22×D61—42no—
237C22×D61—42no—
238C22×D61—82no—
239C22×D61—82no—
240C22×D61—82no—
241C22×D61C2×C22⋊C412yes—
242C22×D61C2×C22⋊C412yes—
243C22×D61C2×C22⋊C411yescharacteristic
244C22×D61C22×D411yescharacteristic
245C2×#6923—44no—
246C2×#6923—164no—
247Q8×C61—22no—
248Q8×C61—22no—
249Q8×C61—22no—
250Q8×C61—42no—
251D4×C61—22no—
252D4×C61—42no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C412yes—
262C2×C4×C61C23×C411yescharacteristic
263C22⋊D61—22no—
264C22⋊D61—22no—
265C22⋊D61—24no—
266C22⋊D61—42no—
267C22⋊D61—42no—
268C22⋊D61—82no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—81no—
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C22×D411yescharacteristic
275D244—22no—
276D244—24no—
277D244—22no—
278D244—22no—
279D244—22no—
280D244—42no—
281D244—42no—
282D244—42no—
283D244—42no—
284D244—82no—
285D244—82no—
286C4×D61—22no—
287C4×D61—24no—
288C4×D61—42no—
289C4×D61—42no—
290C4×D61—42no—
291C4×D61—42no—
292C4×D61—42no—
293C4×D61—42no—
294C4×D61—81no—
295C4×D61—81no—
296C4×D61H(5,𝔽2)14yes—
297Dic122—22no—
298Dic122—22no—
299Dic122—42no—
300Dic122—42no—
301#92420—44no—
302#21784—611nomaximal, Sylow
303C22×C61—21no—
304C22×C61—22no—
305C22×C61—41no—
306C22×C61—42no—
307C22×C61C22×C22⋊C411yescharacteristic
308C2×D61—21no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—41no—
317C2×D61—84no—
318C2×D61—162no—
319C2×D61—162no—
320C2×D61#53412yes—
321#6923—84no—
322C4×C61—21no—
323C4×C61—22no—
324C4×C61—41no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61#51812yes—
329Dic61—21no—
330Dic61—41no—
331Dic61—41no—
332Dic61—41no—
333Dic61#53412yes—
334#3500—611no—
335#2888—611no—
336#2883—611no—
337#1911—612no—
338#1909—612no—
339#1885—612no—
340#1884—612no—
341#1827—612no—
342#1796—612no—
343C2×C61—21no—
344C2×C61—42no—
345C2×C61—41no—
346C2×C61#230811yescharacteristic
347D61—22no—
348D61—42no—
349D61—41no—
350D61—42no—
351D61—81no—
352C2×2−1+4—611no—
353C2×H(5,𝔽2)—611no—
354#568—612no—
355#548—1222no—
356#546—1222no—
357#546—1222no—
358#546—1222no—
359#536—611no—
360#536—611no—
361#536#692311yescharacteristic, complemented
362#535—611no—
363#535—611no—
364#535—611no—
365#532—611no—
366#532—611no—
367#532—611no—
368C2×#124—611no—
369C2×#124—611no—
370C2×#124—611no—
371#518—612no—
372C4×#41—611no—
373#421—614no—
374#419—614no—
375#413—612no—
376#413—1222no—
377C2×#100—612no—
378C2×#100—1222no—
379#410—612no—
380#410—612no—
381C2×#99—1222no—
382C2×#99—1222no—
383#339—614no—
384H(3,ℤ/4)—614no—
385C61#2178411yescharacteristic, complemented, Sylow
3862−1+4—2444no—
387H(5,𝔽2)—2444no—
388C2×#41—611no—
389C2×#41—611no—
390C2×#41—612no—
391C2×#41—611no—
392C2×#41—611no—
393C2×#41—611no—
394C2×#41—611no—
395C2×#41—1221no—
396C2×#41—1221no—
397C2×#41—1221no—
398C2×#41C2×#692311yescharacteristic
399C22×Q8—611no—
400C22×Q8—611no—
401C22×Q8—1221no—
402C22×Q8C2×#692311yescharacteristic, complemented
403C22×D4—611no—
404C22×D4—612no—
405C22×D4—612no—
406C22×D4—611no—
407C22×D4—612no—
408C22×D4—611no—
409#131—2444no—
410C2×M16—614no—
411C2×M16—1224no—
412C4⋊Q8—611no—
413C4⋊Q8C2×#692311yescharacteristic, complemented
414C4⋊Q8C2×#692311yescharacteristic, complemented
415#127—611no—
416#127—611no—
417#127—611no—
418#124—611no—
419#124—611no—
420#124—611no—
421#124—611no—
422#124—611no—
423#124—611no—
424#124—611no—
425#124—611no—
426#124—1222no—
427#124—1222no—
428#124—1222no—
429#124C2×#692311yescharacteristic
430#124C2×#692311yescharacteristic
431#123—22no—
432#123—1222no—
433#123—1222no—
434C22⋊Q8—22no—
435C22⋊Q8—1222no—
436C22⋊Q8—1222no—
437#121—1222no—
438#121—1222no—
439#121—1222no—
440#121—1222no—
441#121—1222no—
442#121—1222no—
443#120—1222no—
444#120—1222no—
445#120—1222no—
446C4×Q8—1221no—
447C4×D4—1221no—
448C4×D4—1221no—
449C4×D4—1222no—
450C4×D4—1222no—
451C4×D4—1221no—
452C4×D4—1222no—
453C4×D4—2444no—
454#117—611no—
455#117—611no—
456#117—614no—
457#117C2×#692311yescharacteristic, complemented
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1222no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1224no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465#104—2444no—
466#103—2444no—
467#101—1222no—
468#100—1224no—
469#100—1222no—
470#99—1222no—
471#99—1222no—
472#99—1224no—
473C24—612no—
474C24—612no—
475C24—612no—
476#41—42no—
477#41—1224no—
478#41—1224no—
479#41—1224no—
480#41—2442no—
481#41—2444no—
482#41—2444no—
483#41—2444no—
484#41—2442no—
485#41—2442no—
486#41—2444no—
487C2×Q8—21no—
488C2×Q8—42no—
489C2×Q8—611no—
490C2×Q8—611no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1224no—
494C2×Q8—2442no—
495C2×Q8—2444no—
496C2×Q8C22×#692311yescharacteristic
497C2×Q8C22×#692311yescharacteristic
498C2×Q8C22×#692311yescharacteristic
499C2×D4—611no—
500C2×D4—611no—
501C2×D4—612no—
502C2×D4—612no—
503C2×D4—611no—
504C2×D4—612no—
505C2×D4—612no—
506C2×D4—612no—
507C2×D4—611no—
508C2×D4—611no—
509C2×D4—611no—
510C2×D4—1222no—
511C2×D4—1222no—
512C2×D4—1221no—
513C2×D4—1221no—
514C2×D4—1224no—
515C2×D4—1222no—
516C2×D4—1221no—
517C2×D4—2442no—
518C2×D4—2444no—
519C2×D4—2442no—
520C2×D4—2444no—
521C2×D4—2444no—
522C2×D4C22⋊#692312yes—
523C2×D4C22×#692311yescharacteristic
524C22×C4—21no—
525C22×C4—611no—
526C22×C4—612no—
527C22×C4—611no—
528C22×C4—612no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—1221no—
537C22×C4—2444no—
538C22×C4C22⋊#692312yes—
539C22×C4C22×#692311yescharacteristic
540M16—1224no—
541M16—1224no—
542M16—2444no—
543C2×C8—1224no—
544C2×C8—2444no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1224no—
551C4⋊C4—1222no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—21no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—1221no—
562C22⋊C4—1221no—
563C22⋊C4—1222no—
564C22⋊C4—1222no—
565C22⋊C4—1224no—
566C22⋊C4—2444no—
567C42—611no—
568C42—611no—
569C42—1221no—
570C42—1221no—
571C42—1224no—
572C42—1221no—
573C42C22×#692311yescharacteristic
574C42C22×#692311yescharacteristic
575C23—611no—
576C23—612no—
577C23—612no—
578C23—611no—
579C23—1224no—
580C23—1221no—
581C23—4882no—
582C23—4882no—
583C23—4882no—
584C23—2442no—
585C23—2442no—
586C23C2×C22⋊#692312yes—
587C23C2×C22⋊#692311yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—22no—
591Q8—42no—
592Q8—1222no—
593Q8—1222no—
594Q8—2442no—
595Q8—2442no—
596D4—22no—
597D4—42no—
598D4—1224no—
599D4—1222no—
600D4—1224no—
601D4—1222no—
602D4—1222no—
603D4—1222no—
604D4—1222no—
605D4—1222no—
606D4—4882no—
607D4—4882no—
608D4—4882no—
609D4—2442no—
610D4—2442no—
611D4—2442no—
612D4—2442no—
613D4—2442no—
614D4—2442no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—611no—
623C2×C4—614no—
624C2×C4—611no—
625C2×C4—1224no—
626C2×C4—1222no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—9764no—
631C2×C4—4881no—
632C2×C4—4881no—
633C2×C4—4881no—
634C2×C4—2442no—
635C2×C4—2442no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2444no—
641C2×C4C2×C22⋊#692311yescharacteristic
642C2×C4C2×C22⋊#692312yes—
643C2×C4C23×#692311yescharacteristic
644C8—2444no—
645C22—21no—
646C22—22no—
647C22—41no—
648C22—42no—
649C22—612no—
650C22—1221no—
651C22—1222no—
652C22—1222no—
653C22—9762no—
654C22—9762no—
655C22—4884no—
656C22—2441no—
657C22—2442no—
658C22—2441no—
659C22—2442no—
660C22—2442no—
661C22—2441no—
662C22C22×C22⋊#692311yescharacteristic
663C4—21no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—612no—
670C4—1221no—
671C4—4884no—
672C4—2441no—
673C4—2441no—
674C4—2441no—
675C4#1684082912yes—
676C2—21no—
677C2—41no—
678C2—42no—
679C2—1222no—
680C2—4881no—
681C2—2441no—
682C2—2442no—
683C2—2442no—
684C2#2146229111yescharacteristic
subgroupG/H
centerC2#21462291
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC2×C61#2308
prime radical#21459608C2
cosocleC2×C4×D61C24
hypercenter#536#6923
hypocenterC61#21784
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #21462430C2semidirect
2. #21462419C2semidirect
3. #21461864C2semidirect
4. #21461861C2semidirect
5. #21461824C2semidirect
6. #21461821C2semidirect
7. #21461729C2semidirect
8. #21461671C2semidirect
12. #16840761C22semidirect
13. #16840755C22semidirect
22. #16840633C22semidirect
23. #16840629C22semidirect
30. #16840337C4semidirect
32. #16840331C4semidirect
33. #16840330C4semidirect
35. #16840313C4semidirect
37. #16840308C4semidirect
38. C2×#14519766C4semidirect
39. #124×D61C4semidirect
42. C61×#536C4semidirect
61. C2×#11758226C2×C4semidirect
66. C2×#11758222C2×C4semidirect
70. C2×Q8×D61C2×C4semidirect
78. #14519775C2×C4semidirect
80. #14519774C2×C4semidirect
85. #14519766C2×C4semidirect
86. #14519766C2×C4semidirect
87. #14519766C2×C4semidirect
89. #14519765C2×C4semidirect
113. #14519734C2×C4semidirect
118. C2×C4×Dic61C2×C4semidirect
119. C42×D61C2×C4semidirect
122. C22×Q8×C61C2×C4semidirect
123. C4⋊Q8×C61C2×C4semidirect
124. C4⋊Q8×C61C2×C4semidirect
129. #117×C61C2×C4semidirect
361. #536#6923semidirect
385. C61#21784semidirect
402. C22×Q8C2×#6923semidirect
413. C4⋊Q8C2×#6923semidirect
414. C4⋊Q8C2×#6923semidirect
457. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624302
#214624192
#214618642
#214618612
#214618242
#214618212
#214617292
#214616712
#214596082
#2178461