Finite Group Databasebeta

← #30547684

Subgroups — #30547684

Order 15616.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519823C211yesmaximal, characteristic
2#21455677C213yesmaximal, complemented
3#21455666C213yesmaximal, complemented
4#21455644C213yesmaximal, complemented
5#21455644C213yesmaximal, complemented
6#21455637C213yesmaximal, complemented
7#21455612C213yesmaximal, complemented
8#21455601C213yesmaximal, complemented
9C2×#16840059C213yesmaximal, complemented
10#21454417C211yesmaximal, characteristic, complemented
11#21454412C211yesmaximal, characteristic, complemented
12#21454387C211yesmaximal, characteristic, complemented
13#21454342C211yesmaximal, characteristic
14#21454342C211yesmaximal, characteristic
15C61×#3504C211yesmaximal, characteristic, complemented
16C2×#14519823C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840244—21no—
21#16840242C2213yes—
22#16840236C2213yes—
23#16840234—23no—
24C4×#11758226—21no—
25C4×#11758222C2213yes—
26C2×C4×D244C2213yes—
27C4×D4×D61—23no—
28#16840066C2213yescomplemented
29#16840066C2216yescomplemented
30#16840066C2211yescharacteristic, complemented
31#16840059C2216yescomplemented
32#16840059C2216yescomplemented
33#16840059C2216yescomplemented
34#16840059C2216yescomplemented
35#16840059C2213yescomplemented
36#16840059C2213yescomplemented
37#16840054C2213yescomplemented
38#16840049C2213yescomplemented
39#16840049C2216yescomplemented
40#16840044C2213yescomplemented
41#16840039C2213yescomplemented
42#16840039C2216yescomplemented
43#16839805C2213yescomplemented
44#16839805C2213yescomplemented
45#16839802C2213yescomplemented
46#16839802C2213yescomplemented
47#16839798C2213yescomplemented
48#16839798C2213yescomplemented
49C2×#14519710C2213yescomplemented
50C2×#14519708C2213yescomplemented
51#16839789C2213yescomplemented
52#16839789C2213yescomplemented
53#16839786C2213yescomplemented
54C2×C4⋊#92418C2213yescomplemented
55#16839766C2213yes—
56#16839766C2213yes—
57#16839761C2213yes—
58#16839761C2213yes—
59#16839756C2213yes—
60#16839756C2213yes—
61C2×#14519697C2213yes—
62C2×#14519697C2213yes—
63C61×#557—21no—
64C61×#557—21no—
65C61×#554—23no—
66C61×#554—23no—
67C61×#544C2213yescomplemented
68C61×#544C2213yescomplemented
69C61×#533C2213yescomplemented
70C2×#125×C61C2213yescomplemented
71C61×#520C2211yescharacteristic
72C61×#520C2211yescharacteristic
73C4×#41×C61C2211yescharacteristic
74#14519823—24no—
75#14519823Q818yescomplemented
76C2×#11758226—21no—
77C2×#11758222C2313yes—
78C22×D244C2313yes—
79C2×D4×D61—23no—
80C22×C4×D61—23no—
81#14519784—24no—
82#14519784—24no—
83C2×#11758194C2313yes—
84C2×#11758194C2313yes—
85C22×#92418C2313yes—
86C22×#92418C2313yes—
87Q8×Dic61—21no—
88C4×Dic122C2311yescharacteristic
89C4×Dic122C2313yes—
90C4×D244—23no—
91C4×D244—23no—
92C4×D244C2313yes—
93C4×D244C2316yes—
94C4×D244C2313yes—
95D4×Dic61—23no—
96#14519739—23no—
97#14519739—23no—
98C4×C22⋊D61—23no—
99C4×C22⋊D61—23no—
100#14519737—26no—
101#14519736C2313yes—
102#14519735—23no—
103#14519734—23no—
104C2×C4⋊Dic61C2313yes—
105C4⋊C4×D61—23no—
106C2×#11758198—23no—
107C22⋊C4×D61—26no—
108C42×D61—23no—
109#14519710C2311yescharacteristic
110#14519710C2313yes—
111#14519710C2316yes—
112#14519710C2313yes—
113#14519710C2313yes—
114#14519708C2313yes—
115#14519708C2313yes—
116#14519708C2316yes—
117#14519708C2311yescharacteristic
118#14519708C2313yes—
119C4⋊#92418C2311yescharacteristic
120C4⋊#92418C2313yes—
121C4⋊#92418C2313yes—
122C4⋊#92418C2313yes—
123C4⋊#92418C2316yes—
124#14519700C2311yescharacteristic
125#14519700C2311yescharacteristic
126#14519700C2313yes—
127#14519700C2313yes—
128#14519697C2313yes—
129#14519697C2313yes—
130#14519697C2316yes—
131#14519697C2316yes—
132#14519697C2313yes—
133#14519697C2313yes—
134C2×#41×C61C2311yescharacteristic
135C4⋊Q8×C61—23no—
136C4⋊Q8×C61—23no—
137C4⋊Q8×C61C2313yes—
138C4⋊Q8×C61C2313yes—
139#125×C61—23no—
140#125×C61—23no—
141#125×C61—23no—
142#125×C61—23no—
143#125×C61C2311yescharacteristic
144#125×C61C2313yes—
145#125×C61C2316yes—
146C22⋊Q8×C61—23no—
147C22⋊Q8×C61—23no—
148C22⋊Q8×C61—26no—
149C22⋊Q8×C61—26no—
150C4×Q8×C61—21no—
151C4×Q8×C61—21no—
152C4×Q8×C61—23no—
153C4×Q8×C61—23no—
154C4×Q8×C61D411yescharacteristic
155C4×Q8×C61D411yescharacteristic
156C4×D4×C61—23no—
157C4×D4×C61—23no—
158C4×D4×C61D413yes—
159C4×D4×C61D413yes—
160#117×C61C2313yes—
161#117×C61C2313yes—
162#117×C61C2313yes—
163C2×C4⋊C4×C61—23no—
164C2×C4⋊C4×C61—23no—
165C2×C4⋊C4×C61C2313yes—
166C2×C4⋊C4×C61C2313yes—
167C2×C42×C61C2313yes—
168C23×D61—23no—
169#41×C61C2×Q814yes—
170#41×C61#4114yes—
171C2×Q8×C61—23no—
172C2×Q8×C61—23no—
173C2×Q8×C61C2×D411yescharacteristic
174C2×D4×C61C2×D413yes—
175C22×C4×C61—23no—
176C22×C4×C61—23no—
177C22×C4×C61C2×D413yes—
178C22×C4×C61C2×D413yes—
179C22×C4×C61C2413yes—
180C2×C22⋊D61—23no—
181C22×Dic61—23no—
182#11758226—28no—
183D4×D61—224no—
184#11758222—26no—
185#11758222C2×Q8112yes—
186C2×D244—23no—
187C2×D244—26no—
188C2×D244C2×Q8112yes—
189C2×D244C2413yes—
190C2×C4×D61—23no—
191C2×C4×D61—23no—
192C2×C4×D61—26no—
193C2×C4×D61—23no—
194C2×C4×D61—86no—
195C2×Dic122C2411yescharacteristic
196C4⋊C4×C61—23no—
197C4⋊C4×C61—23no—
198C4⋊C4×C61—26no—
199C4⋊C4×C61—26no—
200C4⋊C4×C61—23no—
201C4⋊C4×C61—23no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—23no—
205C4⋊C4×C61—23no—
206C4⋊C4×C61C2×D413yes—
207C4⋊C4×C61C2×D413yes—
208C4⋊C4×C61C2413yes—
209C4⋊C4×C61C2413yes—
210C4⋊C4×C61C2411yescharacteristic
211C4⋊C4×C61C2411yescharacteristic
212C22⋊C4×C61—23no—
213C22⋊C4×C61—23no—
214C22⋊C4×C61C2×D416yes—
215C42×C61—23no—
216C42×C61—23no—
217C42×C61C2×D413yes—
218C42×C61C2×D413yes—
219C42×C61C2411yescharacteristic
220C42×C61C2413yes—
221C22⋊Dic61—23no—
222#11758198—23no—
223#11758198—26no—
224#11758198—23no—
225C4⋊Dic61—23no—
226C4⋊Dic61C2413yes—
227C4⋊Dic61C2411yescharacteristic
228#11758196—23no—
229#11758196—21no—
230C4×Dic61—21no—
231C4×Dic61—23no—
232C4×Dic61—23no—
233#11758194—26no—
234#11758194—26no—
235C2×#92418—26no—
236C2×#92418—26no—
237C2×#92418C2411yescharacteristic
238C2×#92418C2413yes—
239C2×#92418C2413yes—
240C2×#92418C2411yescharacteristic
241C23×C61C22×D413yes—
242C22×D61—23no—
243C22×D61—212no—
244C22×D61—83no—
245C22×D61—83no—
246Q8×C61—83no—
247Q8×C61—83no—
248Q8×C61C22⋊Q814yes—
249D4×C61C22⋊Q8112yes—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—23no—
253C2×C4×C61—23no—
254C2×C4×C61—43no—
255C2×C4×C61—43no—
256C2×C4×C61—83no—
257C2×C4×C61—83no—
258C2×C4×C61C22×D413yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D413yes—
262C2×C4×C61C22×D413yes—
263C2×C4×C61C22×Q816yes—
264C2×C4×C61C2×#4116yes—
265C2×C4×C61C2511yescharacteristic
266C22⋊D61—212no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—23no—
270C2×Dic61—83no—
271D244—212no—
272D244—23no—
273D244C22×Q816yes—
274C4×D61—212no—
275C4×D61—46no—
276C4×D61—83no—
277C4×D61—86no—
278Dic122—21no—
279Dic122C22×Q812yes—
280#92418—21no—
281#92418—21no—
282#92418—23no—
283#92418—23no—
284#60496—611nomaximal, Sylow
285C22×C61—23no—
286C22×C61—23no—
287C22×C61C2×C22⋊Q816yes—
288C22×C61C23×D411yescharacteristic
289C2×D61—26no—
290C2×D61—43no—
291C2×D61—43no—
292C2×D61—812no—
293C4×C61—23no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—23no—
297C4×C61—43no—
298C4×C61—41no—
299C4×C61—43no—
300C4×C61—41no—
301C4×C61C2×C22⋊Q816yes—
302C4×C61C23×Q811yescharacteristic
303C4×C61C22×#4111yescharacteristic
304Dic61—22no—
305Dic61—41no—
306Dic61—43no—
307#3504D6111yescharacteristic, complemented
308C4×H(5,𝔽2)—611no—
309#3084—613no—
310#3083—613no—
311#3081—613no—
312#3081—613no—
313#3080—613no—
314#3077—613no—
315#3076—613no—
316C2×#478—613no—
317#2913—611no—
318#2912—611no—
319#2906—611no—
320#2897—611no—
321#2897—611no—
322C2×C61—26no—
323C2×C61C22×C22⋊Q811yescharacteristic
324C2×C61#358411yescharacteristic
325C2×C61#358511yescharacteristic
326D61—46no—
327C2×H(5,𝔽2)—611no—
328C2×#131—611no—
329C2×#131—611no—
330#557—21no—
331#557—21no—
332#554—23no—
333#554—23no—
334#544C2×D6113yescomplemented
335#544C2×D6113yescomplemented
336#533C2×D6113yescomplemented
337C2×#125C2×D6113yescomplemented
338#520—611no—
339#520—613no—
340#520—1221no—
341#520C2×D6111yescharacteristic
342#520C2×D6111yescharacteristic
343#518—613no—
344#518—1223no—
345C4×#41—613no—
346C4×#41—1221no—
347C4×#41C2×D6111yescharacteristic
348C2×C4×D4—613no—
349C2×C4×D4—1223no—
350#479—611no—
351#479—613no—
352#479—616no—
353#478—613no—
354#478—616no—
355#478—616no—
356#478—616no—
357#478—613no—
358#478—616no—
359#477—613no—
360#476—613no—
361#476—616no—
362#475—613no—
363#474—613no—
364#474—616no—
365#428—613no—
366#428—613no—
367#428—613no—
368#428—613no—
369#427—613no—
370#427—613no—
371C2×#107—613no—
372C2×#106—613no—
373#424—613no—
374#424—613no—
375#423—613no—
376C2×C4⋊C8—613no—
377#418—613no—
378#418—613no—
379#417—613no—
380#417—613no—
381#416—613no—
382#416—613no—
383C2×#102—613no—
384C2×#102—613no—
385C61#6049611yescharacteristic, complemented, Sylow
386H(5,𝔽2)—618no—
387H(5,𝔽2)—1224no—
388C2×#41—613no—
389C2×#41—1221no—
390C2×#41C22×D6111yescharacteristic
391C22×D4—613no—
392C22×D4—1223no—
393C23×C4—1223no—
394#131—1224no—
395#131—1224no—
396C2×M16—613no—
397C2×M16—613no—
398C22×C8—613no—
399C22×C8—613no—
400C4⋊Q8—23no—
401C4⋊Q8—23no—
402C4⋊Q8C22×D6113yes—
403C4⋊Q8C22×D6113yes—
404#125—23no—
405#125—23no—
406#125—23no—
407#125—23no—
408#125C22×D6113yes—
409#125C22×D6116yes—
410#125C22×D6111yescharacteristic
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—26no—
414C22⋊Q8—26no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—23no—
418C4×Q8—23no—
419C4×Q8—611no—
420C4×Q8—613no—
421C4×Q8—1221no—
422C4×Q8C22⋊D6111yescharacteristic
423C4×Q8C22⋊D6111yescharacteristic
424C4×D4—23no—
425C4×D4—23no—
426C4×D4—613no—
427C4×D4—616no—
428C4×D4—613no—
429C4×D4—1223no—
430C4×D4—1223no—
431C4×D4—1223no—
432C4×D4—1223no—
433C4×D4—1223no—
434C4×D4—1226no—
435C4×D4—1223no—
436C4×D4—1223no—
437C4×D4C22⋊D6113yes—
438C4×D4C22⋊D6113yes—
439#117—613no—
440#117—1223no—
441#117—1223no—
442#117C22×D6113yes—
443#117C22×D6113yes—
444#117C22×D6113yes—
445C2×C4⋊C4—23no—
446C2×C4⋊C4—23no—
447C2×C4⋊C4—613no—
448C2×C4⋊C4—1223no—
449C2×C4⋊C4C22×D6113yes—
450C2×C4⋊C4C22×D6113yes—
451C2×C22⋊C4—1223no—
452C2×C22⋊C4—1226no—
453C2×C42—1223no—
454C2×C42C22×D6113yes—
455#107—613no—
456#107—616no—
457#107—611no—
458#107—613no—
459#107—613no—
460#106—613no—
461#106—616no—
462#106—611no—
463#106—613no—
464#106—613no—
465C4⋊C8—611no—
466C4⋊C8—613no—
467C4⋊C8—613no—
468C4⋊C8—613no—
469C4⋊C8—616no—
470#103—611no—
471#103—611no—
472#103—613no—
473#103—613no—
474#102—613no—
475#102—613no—
476#102—616no—
477#102—616no—
478#102—613no—
479#102—613no—
480C24—1223no—
481#41—6112no—
482#41—1228no—
483#41—1226no—
484#41Q8×D6114yes—
485#41#1175822614yes—
486C2×Q8—23no—
487C2×Q8—23no—
488C2×Q8—611no—
489C2×Q8C2×C22⋊D6111yescharacteristic
490C2×D4—6112no—
491C2×D4—613no—
492C2×D4—1223no—
493C2×D4—12224no—
494C2×D4—1223no—
495C2×D4—1226no—
496C2×D4C2×C22⋊D6113yes—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—1223no—
500C22×C4—1223no—
501C22×C4—1223no—
502C22×C4—1226no—
503C22×C4—1223no—
504C22×C4—4886no—
505C22×C4C2×C22⋊D6113yes—
506C22×C4C2×C22⋊D6113yes—
507C22×C4C23×D6113yes—
508M16—1226no—
509M16—1226no—
510C2×C8—611no—
511C2×C8—613no—
512C2×C8—613no—
513C2×C8—611no—
514C2×C8—1226no—
515C2×C8—1226no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—26no—
519C4⋊C4—26no—
520C4⋊C4—23no—
521C4⋊C4—23no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—23no—
525C4⋊C4—23no—
526C4⋊C4—613no—
527C4⋊C4—611no—
528C4⋊C4—1223no—
529C4⋊C4—1223no—
530C4⋊C4—1221no—
531C4⋊C4C2×C22⋊D6113yes—
532C4⋊C4C2×C22⋊D6113yes—
533C4⋊C4C23×D6113yes—
534C4⋊C4C23×D6111yescharacteristic
535C4⋊C4C23×D6113yes—
536C4⋊C4C23×D6111yescharacteristic
537C22⋊C4—23no—
538C22⋊C4—23no—
539C22⋊C4—1223no—
540C22⋊C4—1223no—
541C22⋊C4—1226no—
542C22⋊C4—1223no—
543C22⋊C4C2×C22⋊D6116yes—
544C42—23no—
545C42—23no—
546C42—1221no—
547C42—1223no—
548C42—1223no—
549C42C2×C22⋊D6113yes—
550C42C2×C22⋊D6113yes—
551C42C23×D6111yescharacteristic
552C42C23×D6113yes—
553C23—12212no—
554C23—1223no—
555C23—4883no—
556C23—4883no—
557C23C22×C22⋊D6113yes—
558Q8—83no—
559Q8—83no—
560Q8—612no—
561Q8—1221no—
562Q8#1451975314yes—
563D4—616no—
564D4—12212no—
565D4—12212no—
566D4—1223no—
567D4#14519753112yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—83no—
575C2×C4—83no—
576C2×C4—1221no—
577C2×C4—12212no—
578C2×C4—1221no—
579C2×C4—1223no—
580C2×C4—4883no—
581C2×C4—4886no—
582C2×C4—4883no—
583C2×C4—2446no—
584C2×C4C22×C22⋊D6111yescharacteristic
585C2×C4C22×C22⋊D6113yes—
586C2×C4C22×C22⋊D6111yescharacteristic
587C2×C4C22×C22⋊D6113yes—
588C2×C4C22×C22⋊D6113yes—
589C2×C4C2×Q8×D6116yes—
590C2×C4C2×#1175822616yes—
591C2×C4C24×D6111yescharacteristic
592C8—1221no—
593C8—1221no—
594C8—1223no—
595C8—1223no—
596C22—23no—
597C22—23no—
598C22—1226no—
599C22—48812no—
600C22—2443no—
601C22—2443no—
602C22C2×#1451975316yes—
603C22C23×C22⋊D6111yescharacteristic
604C4—21no—
605C4—23no—
606C4—21no—
607C4—23no—
608C4—43no—
609C4—41no—
610C4—43no—
611C4—41no—
612C4—1222no—
613C4—2441no—
614C4—2443no—
615C4C2×#1451975316yes—
616C4C22×Q8×D6111yescharacteristic
617C4C22×#1175822611yescharacteristic
618C2—26no—
619C2—2446no—
620C2C22×#1451975311yescharacteristic
621C2#2146059011yescharacteristic
622C2#2146059811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3504C2
socleC22×C61C23×D4
prime radicalC2×#14519823C22
hypercenter#3504D61
hypocenterC61#60496
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3504

Complements

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

NH ≅ G/Ntype
2. #21455677C2semidirect
3. #21455666C2semidirect
4. #21455644C2semidirect
5. #21455644C2semidirect
6. #21455637C2semidirect
7. #21455612C2semidirect
8. #21455601C2semidirect
9. C2×#16840059C2semidirect
10. #21454417C2semidirect
11. #21454412C2semidirect
12. #21454387C2semidirect
15. C61×#3504C2semidirect
28. #16840066C22semidirect
29. #16840066C22semidirect
30. #16840066C22semidirect
31. #16840059C22semidirect
32. #16840059C22semidirect
33. #16840059C22semidirect
34. #16840059C22semidirect
35. #16840059C22semidirect
36. #16840059C22semidirect
37. #16840054C22semidirect
38. #16840049C22semidirect
39. #16840049C22semidirect
40. #16840044C22semidirect
41. #16840039C22semidirect
42. #16840039C22semidirect
43. #16839805C22semidirect
44. #16839805C22semidirect
45. #16839802C22semidirect
46. #16839802C22semidirect
47. #16839798C22semidirect
48. #16839798C22semidirect
49. C2×#14519710C22semidirect
50. C2×#14519708C22semidirect
51. #16839789C22semidirect
52. #16839789C22semidirect
53. #16839786C22semidirect
54. C2×C4⋊#92418C22semidirect
67. C61×#544C22semidirect
68. C61×#544C22semidirect
69. C61×#533C22semidirect
70. C2×#125×C61C22semidirect
75. #14519823Q8semidirect
307. #3504D61semidirect
334. #544C2×D61semidirect
335. #544C2×D61semidirect
336. #533C2×D61semidirect
337. C2×#125C2×D61semidirect
385. C61#60496semidirect

Maximal subgroups

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

subgroupindex
C4×#145198232
#214556772
#214556662
#214556442
#214556372
#214556122
#214556012
C2×#168400592
#214544172
#214544122
#214543872
#214543422
C61×#35042
#6049661