Finite Group Databasebeta

← #30558418

Subgroups — #30558418

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#21459741C211yesmaximal, characteristic, complemented
2#21459602C211yesmaximal, characteristic, complemented
3#21457997C214yesmaximal, complemented
4#21457922C214yesmaximal, complemented
5#21457877C214yesmaximal, complemented
6#21457802C214yesmaximal, complemented
7C2×#16840130C211yesmaximal, characteristic, complemented
8#21455786C212yesmaximal, complemented
9#21455642C211yesmaximal, characteristic, complemented
10#21455504C212yesmaximal, complemented
11#21455133C212yesmaximal, complemented
12#21454617C211yesmaximal, characteristic, complemented
13#21454536C211yesmaximal, characteristic, complemented
14#21454351C212yesmaximal, complemented
15C61×#3039C211yesmaximal, characteristic, complemented
16C2×#14519825C2212yes—
17C2×#14519797C2211yescharacteristic
18C2×#14519785C2212yes—
19#16840462C2214yes—
20#16840461—24no—
21#16840453C2214yes—
22#16840427C2214yes—
23#16840415C2214yes—
24#16840413—24no—
25#16840374—21no—
26#16840367—21no—
27#16840340C2212yes—
28#16840337—22no—
29#16840336C2212yes—
30#16840313—21no—
31#16840312C2211yescharacteristic
32C4⋊Q8×D61—21no—
33#16840296—22no—
34C2×#14519769C2211yescharacteristic
35#16840242C2212yes—
36C4×#11758222C2211yescharacteristic
37#16840130C2214yes—
38#16840130C2214yes—
39#16840102C2214yes—
40#16840101C2214yes—
41#16840096C2214yes—
42#16840091C2214yes—
43#16840070C2214yes—
44#16840063C2214yes—
45#16840017C2214yes—
46#16840005C2214yes—
47#16840000C2214yes—
48#16839988C2214yes—
49#16839923C2214yes—
50#16839916C2214yes—
51#16839906C2214yes—
52#16839899C2214yes—
53#16839872C2212yes—
54#16839867C2212yes—
55#16839862C2212yes—
56#16839860C2212yes—
57#16839832C2214yes—
58#16839832C2214yes—
59#16839804C2212yes—
60#16839798C2211yescharacteristic
61#16839769C2214yes—
62#16839764C2212yes—
63#16839762C2214yes—
64#16839758C2211yescharacteristic
65#16839757C2214yes—
66#16839756C2211yescharacteristic
67C2×#14519701C2212yes—
68C2×#14519701C2212yes—
69#103×D61C2214yes—
70#16839730C2212yes—
71C2×#14519688C2211yescharacteristic
72C2×#137×C61C2212yes—
73C2×C4⋊Q8×C61C2211yescharacteristic
74C4×#41×C61C2211yescharacteristic
75C61×#462C2214yes—
76C61×#461C2214yes—
77C61×#423C2211yescharacteristic
78C61×#419C2212yes—
79#14519825—28no—
80#14519825—44no—
81C2×#11758226—22no—
82C2×#11758226C2314yes—
83C2×#11758222—24no—
84C2×#11758222C2312yes—
85C2×#11758222C2311yescharacteristic
86C2×Q8×D61—22no—
87C2×Q8×D61C2314yes—
88#14519797—24no—
89C2×#11758202C2311yescharacteristic
90C2×#11758200C2311yescharacteristic
91C2×#11758201C2311yescharacteristic
92C2×#11758199C2311yescharacteristic
93#14519785—44no—
94C2×#11758189C2314yes—
95C22×#92418C2311yescharacteristic
96C22×#92418C2311yescharacteristic
97C2×C8×D61C2314yes—
98#14519777D411yescharacteristic
99#14519776—24no—
100#14519775—22no—
101#14519774D411yescharacteristic
102#14519771—24no—
103#14519770—24no—
104#14519769—21no—
105#14519769—21no—
106#14519769C2314yes—
107#14519767—24no—
108#14519766—22no—
109#14519765D412yes—
110#14519762—24no—
111#14519762—24no—
112#14519762—24no—
113#14519760—24no—
114#14519760—24no—
115#14519756—22no—
116#14519755—24no—
117#14519755—24no—
118#14519753C2314yes—
119#14519753C2314yes—
120#14519753C2314yes—
121#14519752—24no—
122#14519751—22no—
123#14519751D414yes—
124#14519750—24no—
125#14519749—24no—
126#14519748C2314yes—
127C4×Dic122—22no—
128C4×Dic122C2312yes—
129C4×D244—22no—
130C4×D244C2314yes—
131C4×D244C2312yes—
132#14519739—24no—
133C4×C22⋊D61—22no—
134C4×C22⋊D61—24no—
135#14519737—24no—
136#14519736C2311yescharacteristic
137#14519735C2314yes—
138#14519734—21no—
139#14519734—24no—
140C2×C4⋊Dic61C2312yes—
141C2×#11758196—22no—
142C4⋊C4×D61—24no—
143C4⋊C4×D61C2314yes—
144C42×D61—21no—
145#14519711C2314yes—
146#14519710C2312yes—
147#14519709C2314yes—
148#14519708C2312yes—
149#14519702C2314yes—
150#14519701C2314yes—
151#14519701C2312yes—
152#14519701C2314yes—
153#14519701C2314yes—
154#14519701C2314yes—
155#14519700C2312yes—
156#14519699C2314yes—
157#14519698C2312yes—
158#14519697C2312yes—
159#14519689C2314yes—
160#14519689C2314yes—
161#14519688C2312yes—
162#14519688C2312yes—
163C2×#41×C61C2311yescharacteristic
164C22×Q8×C61C2312yes—
165#137×C61—44no—
166C2×Q16×C61C2314yes—
167C2×SD16×C61C2314yes—
168C2×M16×C61C2312yes—
169C4⋊Q8×C61—21no—
170C4⋊Q8×C61—21no—
171C4⋊Q8×C61C2314yes—
172C4×Q8×C61C2312yes—
173C4×D4×C61—22no—
174C4×D4×C61C2312yes—
175#117×C61—21no—
176#117×C61C2311yescharacteristic
177C2×C4⋊C4×C61—22no—
178C2×C42×C61—21no—
179C2×C42×C61C2311yescharacteristic
180C4⋊C8×C61C2314yes—
181#103×C61C2314yes—
182#103×C61C2314yes—
183#41×C61—22no—
184#41×C61—22no—
185C2×Q8×C61—22no—
186C2×Q8×C61—24no—
187C2×Q8×C61—24no—
188C2×Q8×C61C2414yes—
189C2×Q8×C61C2411yescharacteristic
190C2×D4×C61—21no—
191C2×D4×C61C2411yescharacteristic
192C22×C4×C61—22no—
193C22×C4×C61—21no—
194C22×C4×C61—22no—
195C22×C4×C61C2×D412yes—
196C22×C4×C61C2411yescharacteristic
197C2×C22⋊D61—21no—
198C2×C22⋊D61—24no—
199C2×C22⋊D61C2×D414yes—
200C22×Dic61—22no—
201#11758226—28no—
202#11758226—44no—
203#11758226—48no—
204Q8×D61—28no—
205Q8×D61—44no—
206Q8×D61—48no—
207#11758222—24no—
208#11758222—22no—
209#11758222—24no—
210#11758222—416no—
211#11758222#4114yes—
212C2×D244—24no—
213C2×D244C2×D412yes—
214C2×D244C2×D414yes—
215C2×D244C2411yescharacteristic
216C2×C4×D61—21no—
217C2×C4×D61—24no—
218C2×C4×D61—22no—
219C2×C4×D61—24no—
220C2×C4×D61—24no—
221C2×C4×D61C2414yes—
222C2×Dic122—24no—
223C2×Dic122C2×D412yes—
224C2×Dic122C2411yescharacteristic
225Q16×C61—44no—
226SD16×C61—44no—
227M16×C61—24no—
228C2×C8×C61C2414yes—
229C4⋊C4×C61—24no—
230C4⋊C4×C61—24no—
231C4⋊C4×C61—22no—
232C4⋊C4×C61C2412yes—
233C22⋊C4×C61—21no—
234C22⋊C4×C61—22no—
235C42×C61—21no—
236C42×C61—22no—
237C42×C61C2×D411yescharacteristic
238C42×C61C2×D411yescharacteristic
239C42×C61C2412yes—
240C22⋊Dic61—21no—
241C22⋊Dic61—24no—
242#11758202—22no—
243#11758201—22no—
244#11758200—22no—
245#11758199—22no—
246#11758198—24no—
247#11758198—24no—
248#11758198—24no—
249#11758198—24no—
250#11758198—22no—
251#11758198C2×D414yes—
252#11758198C2×D414yes—
253C4⋊Dic61—22no—
254C4⋊Dic61C2×D414yes—
255C4⋊Dic61C2412yes—
256C4⋊Dic61C2414yes—
257#11758196—24no—
258#11758196—24no—
259#11758196—22no—
260#11758196—24no—
261#11758196—24no—
262#11758196—24no—
263#11758196—24no—
264C4×Dic61—21no—
265C4×Dic61—22no—
266C4×Dic61—24no—
267C2×#92418—22no—
268C2×#92418—22no—
269C2×#92418C2411yescharacteristic
270C2×#92418C2411yescharacteristic
271C2×#92418C2412yes—
272#11758189—44no—
273C8×D61—44no—
274C23×C61—21no—
275C23×C61C22×D411yescharacteristic
276C22×D61—21no—
277C22×D61C22×D414yes—
278Q8×C61—21no—
279Q8×C61—24no—
280Q8×C61—24no—
281Q8×C61—21no—
282Q8×C61—44no—
283D4×C61—21no—
284D4×C61—21no—
285D4×C61—42no—
286C2×C4×C61—22no—
287C2×C4×C61—21no—
288C2×C4×C61—24no—
289C2×C4×C61—42no—
290C2×C4×C61—44no—
291C2×C4×C61—42no—
292C2×C4×C61—82no—
293C2×C4×C61C22×D412yes—
294C2×C4×C61C22×D412yes—
295C2×C4×C61C22×D411yescharacteristic
296C2×C4×C61C2×#4112yes—
297C2×C4×C61H(5,𝔽2)12yes—
298C2×C4×C61C2511yescharacteristic
299C22⋊D61—28no—
300C22⋊D61—24no—
301C22⋊D61—84no—
302C2×Dic61—21no—
303C2×Dic61—22no—
304C2×Dic61—24no—
305C2×Dic61—82no—
306C2×Dic61C22×D414yes—
307D244—24no—
308D244—21no—
309D244—44no—
310D244—48no—
311D244C2×#4112yes—
312C4×D61—24no—
313C4×D61—24no—
314C4×D61—24no—
315C4×D61—48no—
316C4×D61—48no—
317C4×D61—44no—
318C4×D61—84no—
319Dic122—24no—
320Dic122—21no—
321Dic122—48no—
322Dic122C2×#4112yes—
323C8×C61—24no—
324#92418—21no—
325#92418—21no—
326#92418—22no—
327#61224—611nomaximal, Sylow
328C22×C61—22no—
329C22×C61—42no—
330C22×C61—41no—
331C22×C61#54612yes—
332C22×C61C23×D411yescharacteristic
333C2×D61—24no—
334C2×D61—24no—
335C2×D61—22no—
336C2×D61—24no—
337C2×D61—42no—
338C4×C61—22no—
339C4×C61—22no—
340C4×C61—42no—
341C4×C61—44no—
342C4×C61—41no—
343C4×C61#54612yes—
344C4×C61C22×#4111yescharacteristic
345C4×C61C2×H(5,𝔽2)11yescharacteristic
346Dic61—24no—
347Dic61—22no—
348Dic61—42no—
349Dic61—44no—
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#3039D6111yescharacteristic, complemented
361#3013—612no—
362#2943—611no—
363#2932—611no—
364#2898—612no—
365C2×C61—22no—
366C2×C61—81no—
367C2×C61C2×#54611yescharacteristic
368C2×C61#358612yes—
369D61—44no—
370D61—42no—
371C2×2−1+4—612no—
372C2×#137C2×D6112yes—
373C2×#135—611no—
374C2×#131—612no—
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—611no—
387#532—1221no—
388C2×C4⋊Q8—1221no—
389C2×C4⋊Q8C2×D6111yescharacteristic
390#529—1222no—
391C2×#125—611no—
392#520—612no—
393C4×#41—611no—
394C4×#41C2×D6111yescharacteristic
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#462C2×D6114yes—
408#461C2×D6114yes—
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#435—614no—
418#435—614no—
419#428—612no—
420#427—611no—
421#423C2×D6111yescharacteristic
422#419—614no—
423#419C2×D6112yes—
424#417—612no—
425#417—614no—
426#416—611no—
427#416—611no—
428#416—614no—
429C2×#103—612no—
430C2×#103—612no—
431C2×#103—614no—
432#408—612no—
433C2×#97—611no—
434C61#6122411yescharacteristic, complemented, Sylow
4352−1+4—1228no—
4362−1+4—2444no—
437C2×#41—612no—
438C2×#41—614no—
439C2×#41—611no—
440C2×#41—1224no—
441C2×#41—1222no—
442C2×#41C22×D6111yescharacteristic
443C22×Q8—614no—
444C22×Q8—1222no—
445C22×Q8C22×D6112yes—
446#137—44no—
447#135—1224no—
448C2×Q16—611no—
449C2×Q16C22×D6114yes—
450C2×SD16—611no—
451C2×SD16—611no—
452C2×SD16C22×D6114yes—
453C2×D8—611no—
454#131—2444no—
455C2×M16—614no—
456C2×M16C22×D6112yes—
457C22×C8—611no—
458C22×C8—611no—
459C22×C8—614no—
460C4⋊Q8—21no—
461C4⋊Q8—21no—
462C4⋊Q8—611no—
463C4⋊Q8—1224no—
464C4⋊Q8—1222no—
465C4⋊Q8C22×D6114yes—
466#127—611no—
467#126—1224no—
468#126—1224no—
469#125—614no—
470#125—1224no—
471#125—1221no—
472#125—1221no—
473#124—612no—
474#124—1222no—
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—1224no—
484C22⋊Q8—1224no—
485C22⋊Q8—1222no—
486#121—614no—
487#121—614no—
488#121—1224no—
489#121—1224no—
490#121—1222no—
491#121—1224no—
492C4×Q8—612no—
493C4×Q8—1222no—
494C4×Q8C22×D6112yes—
495C4×D4—22no—
496C4×D4—614no—
497C4×D4—612no—
498C4×D4—1222no—
499C4×D4—1224no—
500C4×D4—1224no—
501C4×D4—1224no—
502C4×D4—1222no—
503C4×D4C22×D6112yes—
504#117—21no—
505#117—611no—
506#117—614no—
507#117—1224no—
508#117—1221no—
509#117C22×D6111yescharacteristic
510C2×C4⋊C4—22no—
511C2×C4⋊C4—612no—
512C2×C4⋊C4—614no—
513C2×C4⋊C4—1224no—
514C2×C4⋊C4—1222no—
515C2×C42—21no—
516C2×C42—1221no—
517C2×C42C22×D6111yescharacteristic
518#107—614no—
519#107—612no—
520#106—614no—
521#106—612no—
522C4⋊C8C22×D6114yes—
523#103—614no—
524#103—614no—
525#103—614no—
526#103—614no—
527#103—612no—
528#103—614no—
529#103—612no—
530#103C22×D6114yes—
531#103C22×D6114yes—
532#102—612no—
533#102—614no—
534#102—612no—
535C22⋊C8—614no—
536C22⋊C8—614no—
537#97—612no—
538#97—612no—
539#41—22no—
540#41—22no—
541#41—614no—
542#41—1224no—
543#41—1224no—
544#41—1222no—
545#41—1228no—
546#41—24416no—
547#41—2444no—
548#41—2448no—
549C2×Q8—22no—
550C2×Q8—24no—
551C2×Q8—24no—
552C2×Q8—612no—
553C2×Q8—611no—
554C2×Q8—1224no—
555C2×Q8—1228no—
556C2×Q8—2444no—
557C2×Q8—2448no—
558C2×Q8C23×D6114yes—
559C2×Q8C23×D6111yescharacteristic
560C2×D4—21no—
561C2×D4—614no—
562C2×D4—612no—
563C2×D4—614no—
564C2×D4—611no—
565C2×D4—1224no—
566C2×D4—1221no—
567C2×D4—1224no—
568C2×D4C23×D6111yescharacteristic
569C22×C4—22no—
570C22×C4—21no—
571C22×C4—22no—
572C22×C4—614no—
573C22×C4—1224no—
574C22×C4—1224no—
575C22×C4—1224no—
576C22×C4—1221no—
577C22×C4—1222no—
578C22×C4—1222no—
579C22×C4D4×D6112yes—
580C22×C4C23×D6111yescharacteristic
581Q16—44no—
582Q16—1222no—
583SD16—44no—
584SD16—1222no—
585SD16—1222no—
586D8—1222no—
587M16—24no—
588M16—2444no—
589C2×C8—611no—
590C2×C8—611no—
591C2×C8—612no—
592C2×C8—1222no—
593C2×C8—1222no—
594C2×C8—2444no—
595C2×C8C23×D6114yes—
596C4⋊C4—24no—
597C4⋊C4—24no—
598C4⋊C4—22no—
599C4⋊C4—614no—
600C4⋊C4—614no—
601C4⋊C4—612no—
602C4⋊C4—1224no—
603C4⋊C4—1224no—
604C4⋊C4—1224no—
605C4⋊C4—1224no—
606C4⋊C4—1222no—
607C4⋊C4—1224no—
608C4⋊C4—1224no—
609C4⋊C4—1222no—
610C4⋊C4C23×D6112yes—
611C22⋊C4—21no—
612C22⋊C4—22no—
613C22⋊C4—614no—
614C22⋊C4—614no—
615C22⋊C4—1224no—
616C22⋊C4—1224no—
617C22⋊C4—1224no—
618C22⋊C4—1224no—
619C22⋊C4—1224no—
620C22⋊C4—1221no—
621C22⋊C4—1222no—
622C42—21no—
623C42—22no—
624C42—1224no—
625C42—1221no—
626C42—1222no—
627C42D4×D6111yescharacteristic
628C42D4×D6111yescharacteristic
629C42C23×D6112yes—
630C23—21no—
631C23—614no—
632C23—1221no—
633C23C2×D4×D6111yescharacteristic
634Q8—21no—
635Q8—24no—
636Q8—24no—
637Q8—21no—
638Q8—44no—
639Q8—612no—
640Q8—1221no—
641Q8—1224no—
642Q8—2448no—
643D4—21no—
644D4—21no—
645D4—42no—
646D4—612no—
647D4—1228no—
648D4—1224no—
649D4—1221no—
650D4—1224no—
651D4—4884no—
652D4—2444no—
653D4—2448no—
654C2×C4—22no—
655C2×C4—21no—
656C2×C4—24no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—82no—
661C2×C4—614no—
662C2×C4—1221no—
663C2×C4—1224no—
664C2×C4—1224no—
665C2×C4—1224no—
666C2×C4—1222no—
667C2×C4—1224no—
668C2×C4—4884no—
669C2×C4—4882no—
670C2×C4—2444no—
671C2×C4—2448no—
672C2×C4—2448no—
673C2×C4C2×D4×D6112yes—
674C2×C4C2×D4×D6112yes—
675C2×C4C2×D4×D6111yescharacteristic
676C2×C4#41×D6112yes—
677C2×C4#1451982212yes—
678C2×C4C24×D6111yescharacteristic
679C8—24no—
680C8—1221no—
681C8—1221no—
682C8—1222no—
683C22—22no—
684C22—41no—
685C22—42no—
686C22—1222no—
687C22—1224no—
688C22—1224no—
689C22—1224no—
690C22—2442no—
691C22#1684040912yes—
692C22C22×D4×D6111yescharacteristic
693C4—22no—
694C4—22no—
695C4—44no—
696C4—41no—
697C4—42no—
698C4—1224no—
699C4—1222no—
700C4—2444no—
701C4—2442no—
702C4#1684040912yes—
703C4C2×#41×D6111yescharacteristic
704C4C2×#1451982211yescharacteristic
705C2—22no—
706C2—81no—
707C2—2444no—
708C2—2442no—
709C2C2×#1684040911yescharacteristic
710C2#2146060212yes—
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3039C2
socleC22×C61C23×D4
hypercenter#3039D61
hypocenterC61#61224
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
1. #21459741C2semidirect
2. #21459602C2semidirect
3. #21457997C2semidirect
4. #21457922C2semidirect
5. #21457877C2semidirect
6. #21457802C2semidirect
7. C2×#16840130C2semidirect
8. #21455786C2semidirect
9. #21455642C2semidirect
10. #21455504C2semidirect
11. #21455133C2semidirect
12. #21454617C2semidirect
13. #21454536C2semidirect
14. #21454351C2semidirect
15. C61×#3039C2semidirect
360. #3039D61semidirect
434. C61#61224semidirect

Maximal subgroups

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

subgroupindex
#214597412
#214596022
#214579972
#214579222
#214578772
#214578022
C2×#168401302
#214557862
#214556422
#214555042
#214551332
#214546172
#214545362
#214543512
C61×#30392
#6122461