Finite Group Databasebeta

← #30547693

Subgroups — #30547693

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
1#21459645C211yesmaximal, characteristic, complemented
2#21455681C213yesmaximal, complemented
3#21455670C213yesmaximal, complemented
4#21455650C213yesmaximal, complemented
5#21455650C213yesmaximal, complemented
6#21455641C213yesmaximal, complemented
7#21455616C213yesmaximal, complemented
8#21455607C213yesmaximal, complemented
9C2×#16840062C213yesmaximal, complemented
10#21454417C211yesmaximal, characteristic, complemented
11#21454412C211yesmaximal, characteristic, complemented
12#21454387C211yesmaximal, characteristic, complemented
13#21454345C211yesmaximal, characteristic
14#21454345C211yesmaximal, characteristic
15C4×H(5,𝔽2)×C61C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840488—21no—
19#16840488—21no—
20#16840471—23no—
21#16840471—23no—
22#16840390C2213yescomplemented
23#16840390C2213yescomplemented
24#16840319C2213yescomplemented
25C2×#14519769C2213yescomplemented
26#16840245C2211yescharacteristic
27#16840245C2211yescharacteristic
28#16840069C2213yescomplemented
29#16840069C2216yescomplemented
30#16840069C2211yescharacteristic, complemented
31#16840062C2213yescomplemented
32#16840062C2216yescomplemented
33#16840062C2216yescomplemented
34#16840062C2216yescomplemented
35#16840062C2216yescomplemented
36#16840062C2213yescomplemented
37#16840057C2213yescomplemented
38#16840052C2216yescomplemented
39#16840052C2213yescomplemented
40#16840047C2213yescomplemented
41#16840042C2216yescomplemented
42#16840042C2213yescomplemented
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#16839767C2213yes—
56#16839767C2213yes—
57#16839763C2213yes—
58#16839763C2213yes—
59#16839758C2213yes—
60#16839758C2213yes—
61C2×#14519698C2213yes—
62C2×#14519698C2213yes—
63C2×H(5,𝔽2)×C61C2211yescharacteristic
64C61×#520—21no—
65C61×#520C2213yes—
66C61×#520C2211yescharacteristic
67C61×#518—23no—
68C61×#518C2213yes—
69C4×#41×C61—21no—
70C4×#41×C61C2213yes—
71C4×#41×C61C2211yescharacteristic
72C2×C4×D4×C61—23no—
73C2×C4×D4×C61C2213yes—
74#14519784—24no—
75#14519784—24no—
76C2×#11758194C2313yes—
77C2×#11758194C2313yes—
78C22×#92418C2313yes—
79C22×#92418C2313yes—
80#14519777C2313yes—
81#14519777C2313yes—
82#14519776—23no—
83#14519776—23no—
84#14519769—23no—
85#14519769—23no—
86#14519769C2313yes—
87#14519769C2316yes—
88#14519769C2311yescharacteristic
89#14519768—23no—
90#14519768—23no—
91#14519756—23no—
92#14519756—23no—
93#14519754—26no—
94#14519754—26no—
95Q8×Dic61—21no—
96Q8×Dic61—21no—
97C4×Dic122—23no—
98C4×Dic122—23no—
99D4×Dic61—23no—
100D4×Dic61—23no—
101#14519736C2313yes—
102#14519736C2313yes—
103C2×C4⋊Dic61C2313yes—
104C2×C4⋊Dic61C2313yes—
105C2×#11758196—23no—
106C2×#11758196—23no—
107#14519710C2311yescharacteristic
108#14519710C2313yes—
109#14519710C2316yes—
110#14519710C2313yes—
111#14519710C2313yes—
112#14519708C2313yes—
113#14519708C2313yes—
114#14519708C2316yes—
115#14519708C2311yescharacteristic
116#14519708C2313yes—
117C4⋊#92418C2311yescharacteristic
118C4⋊#92418C2313yes—
119C4⋊#92418C2313yes—
120C4⋊#92418C2313yes—
121C4⋊#92418C2316yes—
122#14519701C2311yescharacteristic
123#14519701C2311yescharacteristic
124#14519701C2313yes—
125#14519701C2313yes—
126#14519698C2313yes—
127#14519698C2313yes—
128#14519698C2316yes—
129#14519698C2316yes—
130#14519698C2313yes—
131#14519698C2313yes—
132H(5,𝔽2)×C61—24no—
133H(5,𝔽2)×C61Q818yescomplemented
134C2×#41×C61—21no—
135C2×#41×C61C2313yes—
136C2×#41×C61C2311yescharacteristic
137C22×D4×C61—23no—
138C22×D4×C61C2313yes—
139C23×C4×C61—23no—
140C4×Q8×C61—21no—
141C4×Q8×C61D411yescharacteristic
142C4×Q8×C61D411yescharacteristic
143C4×Q8×C61C2313yes—
144C4×Q8×C61C2311yescharacteristic
145C4×D4×C61—23no—
146C4×D4×C61—23no—
147C4×D4×C61—23no—
148C4×D4×C61—23no—
149C4×D4×C61—23no—
150C4×D4×C61—26no—
151C4×D4×C61—23no—
152C4×D4×C61—23no—
153C4×D4×C61D413yes—
154C4×D4×C61D413yes—
155C4×D4×C61C2316yes—
156C4×D4×C61C2313yes—
157C4×D4×C61C2313yes—
158#117×C61—23no—
159#117×C61—23no—
160#117×C61C2313yes—
161#117×C61C2313yes—
162C2×C4⋊C4×C61—23no—
163C2×C4⋊C4×C61C2313yes—
164C2×C22⋊C4×C61—23no—
165C2×C22⋊C4×C61—26no—
166C2×C42×C61—23no—
167C2×C42×C61C2313yes—
168C24×C61—23no—
169#41×C61—28no—
170#41×C61—26no—
171#41×C61C2×Q8112yes—
172#41×C61C2×Q814yes—
173#41×C61#4114yes—
174C2×Q8×C61C2×D411yescharacteristic
175C2×Q8×C61C2411yescharacteristic
176C2×D4×C61—23no—
177C2×D4×C61—224no—
178C2×D4×C61—23no—
179C2×D4×C61—26no—
180C2×D4×C61C2×D413yes—
181C2×D4×C61C2×Q8112yes—
182C2×D4×C61C2413yes—
183C22×C4×C61—23no—
184C22×C4×C61—23no—
185C22×C4×C61—23no—
186C22×C4×C61—23no—
187C22×C4×C61—26no—
188C22×C4×C61—86no—
189C22×C4×C61C2×D413yes—
190C22×C4×C61C2×D413yes—
191C22×C4×C61C2413yes—
192C22×Dic61—23no—
193C22×Dic61—23no—
194C2×Dic122—23no—
195C2×Dic122—23no—
196C4⋊C4×C61—23no—
197C4⋊C4×C61—23no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61C2×D413yes—
200C4⋊C4×C61C2×D413yes—
201C4⋊C4×C61C2413yes—
202C4⋊C4×C61C2411yescharacteristic
203C22⋊C4×C61—23no—
204C22⋊C4×C61—23no—
205C22⋊C4×C61—23no—
206C22⋊C4×C61—26no—
207C22⋊C4×C61C2×D416yes—
208C42×C61—21no—
209C42×C61—23no—
210C42×C61—23no—
211C42×C61C2×D413yes—
212C42×C61C2×D413yes—
213C42×C61C2411yescharacteristic
214C42×C61C2413yes—
215C22⋊Dic61—23no—
216C22⋊Dic61—23no—
217C4⋊Dic61—23no—
218C4⋊Dic61—23no—
219C4⋊Dic61—23no—
220C4⋊Dic61—23no—
221C4⋊Dic61C2413yes—
222C4⋊Dic61C2411yescharacteristic
223C4⋊Dic61C2413yes—
224C4⋊Dic61C2411yescharacteristic
225#11758196—23no—
226#11758196—23no—
227#11758196—26no—
228#11758196—26no—
229#11758196—21no—
230#11758196—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×C61—23no—
242C23×C61—212no—
243C23×C61—83no—
244C23×C61—83no—
245C23×C61C22×D413yes—
246Q8×C61—21no—
247Q8×C61C22⋊Q814yes—
248Q8×C61C22×Q812yes—
249D4×C61—212no—
250D4×C61—212no—
251D4×C61—23no—
252D4×C61C22⋊Q8112yes—
253D4×C61C22×Q816yes—
254C2×C4×C61—21no—
255C2×C4×C61—212no—
256C2×C4×C61—21no—
257C2×C4×C61—23no—
258C2×C4×C61—43no—
259C2×C4×C61—43no—
260C2×C4×C61—46no—
261C2×C4×C61—83no—
262C2×C4×C61—86no—
263C2×C4×C61—83no—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D413yes—
267C2×C4×C61C22×D413yes—
268C2×C4×C61C22×D413yes—
269C2×C4×C61C22×Q816yes—
270C2×C4×C61C2×#4116yes—
271C2×C4×C61C2511yescharacteristic
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—23no—
275C2×Dic61—23no—
276C2×Dic61—83no—
277C2×Dic61—83no—
278Dic122—83no—
279Dic122—83no—
280#92418—21no—
281#92418—21no—
282#92418—23no—
283#92418—23no—
284#60496—611nomaximal, Sylow
285C22×C61—23no—
286C22×C61—23no—
287C22×C61—26no—
288C22×C61—43no—
289C22×C61—43no—
290C22×C61—812no—
291C22×C61C2×C22⋊Q816yes—
292C22×C61C23×D411yescharacteristic
293C4×C61—23no—
294C4×C61—21no—
295C4×C61—23no—
296C4×C61—21no—
297C4×C61—22no—
298C4×C61—41no—
299C4×C61—43no—
300C4×C61C2×C22⋊Q816yes—
301C4×C61C23×Q811yescharacteristic
302C4×C61C22×#4111yescharacteristic
303Dic61—41no—
304Dic61—43no—
305Dic61—41no—
306Dic61—43no—
307#3504—611no—
308C4×H(5,𝔽2)D6111yescharacteristic
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×C61—46no—
324C2×C61C22×C22⋊Q811yescharacteristic
325C2×C61#358411yescharacteristic
326C2×C61#358511yescharacteristic
327C2×H(5,𝔽2)C2×D6111yescharacteristic
328C2×#131—611no—
329C2×#131—611no—
330#557—1221no—
331#557—1221no—
332#554—1223no—
333#554—1223no—
334#544—613no—
335#544—613no—
336#533—613no—
337C2×#125—613no—
338#520—21no—
339#520—611no—
340#520—611no—
341#520C2×D6113yes—
342#520C2×D6111yescharacteristic
343#518—23no—
344#518C2×D6113yes—
345C4×#41—21no—
346C4×#41C2×D6113yes—
347C4×#41C2×D6111yescharacteristic
348C2×C4×D4—23no—
349C2×C4×D4C2×D6113yes—
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)—24no—
387H(5,𝔽2)Dic12218yescomplemented
388C2×#41—21no—
389C2×#41C22×D6113yes—
390C2×#41C22×D6111yescharacteristic
391C22×D4—23no—
392C22×D4C22×D6113yes—
393C23×C4—23no—
394#131—1224no—
395#131—1224no—
396C2×M16—613no—
397C2×M16—613no—
398C22×C8—613no—
399C22×C8—613no—
400C4⋊Q8—613no—
401C4⋊Q8—613no—
402C4⋊Q8—1223no—
403C4⋊Q8—1223no—
404#125—613no—
405#125—616no—
406#125—611no—
407#125—1223no—
408#125—1223no—
409#125—1223no—
410#125—1223no—
411C22⋊Q8—1223no—
412C22⋊Q8—1223no—
413C22⋊Q8—1226no—
414C22⋊Q8—1226no—
415C4×Q8—21no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418C4×Q8—1223no—
419C4×Q8—1223no—
420C4×Q8C22⋊D6111yescharacteristic
421C4×Q8C22⋊D6111yescharacteristic
422C4×Q8C22×D6111yescharacteristic
423C4×Q8C22×D6113yes—
424C4×D4—23no—
425C4×D4—23no—
426C4×D4—23no—
427C4×D4—23no—
428C4×D4—23no—
429C4×D4—26no—
430C4×D4—23no—
431C4×D4—23no—
432C4×D4—1223no—
433C4×D4—1223no—
434C4×D4C22⋊D6113yes—
435C4×D4C22⋊D6113yes—
436C4×D4C22×D6113yes—
437C4×D4C22×D6116yes—
438C4×D4C22×D6113yes—
439#117—23no—
440#117—23no—
441#117—613no—
442#117—613no—
443#117C22×D6113yes—
444#117C22×D6113yes—
445C2×C4⋊C4—23no—
446C2×C4⋊C4—613no—
447C2×C4⋊C4—613no—
448C2×C4⋊C4—1223no—
449C2×C4⋊C4—1223no—
450C2×C4⋊C4C22×D6113yes—
451C2×C22⋊C4—23no—
452C2×C22⋊C4—26no—
453C2×C42—23no—
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—23no—
481#41—28no—
482#41—26no—
483#41C2×Dic122112yes—
484#41C2×Dic12214yes—
485#41#1175822214yes—
486C2×Q8—1223no—
487C2×Q8—1223no—
488C2×Q8C2×C22⋊D6111yescharacteristic
489C2×Q8C23×D6111yescharacteristic
490C2×D4—23no—
491C2×D4—224no—
492C2×D4—23no—
493C2×D4—26no—
494C2×D4C2×Dic122112yes—
495C2×D4C2×C22⋊D6113yes—
496C2×D4C23×D6113yes—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—23no—
501C22×C4—26no—
502C22×C4—86no—
503C22×C4—1223no—
504C22×C4—1223no—
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—21no—
519C4⋊C4—613no—
520C4⋊C4—611no—
521C4⋊C4—613no—
522C4⋊C4—611no—
523C4⋊C4—1223no—
524C4⋊C4—1223no—
525C4⋊C4—1223no—
526C4⋊C4—1223no—
527C4⋊C4—1226no—
528C4⋊C4—1226no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1223no—
532C4⋊C4—1223no—
533C4⋊C4C2×C22⋊D6113yes—
534C4⋊C4C2×C22⋊D6113yes—
535C4⋊C4C23×D6113yes—
536C4⋊C4C23×D6111yescharacteristic
537C22⋊C4—23no—
538C22⋊C4—23no—
539C22⋊C4—26no—
540C22⋊C4—23no—
541C22⋊C4—1223no—
542C22⋊C4—1223no—
543C22⋊C4C2×C22⋊D6116yes—
544C42—21no—
545C42—23no—
546C42—23no—
547C42—1223no—
548C42—1223no—
549C42C2×C22⋊D6113yes—
550C42C2×C22⋊D6113yes—
551C42C23×D6111yescharacteristic
552C42C23×D6113yes—
553C23—23no—
554C23—212no—
555C23—83no—
556C23—83no—
557C23C22×C22⋊D6113yes—
558Q8—21no—
559Q8—4883no—
560Q8—4883no—
561Q8#1451975614yes—
562Q8C22×Dic12212yes—
563D4—212no—
564D4—212no—
565D4—23no—
566D4#14519756112yes—
567D4C22×Dic12216yes—
568C2×C4—21no—
569C2×C4—212no—
570C2×C4—21no—
571C2×C4—23no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—46no—
575C2×C4—83no—
576C2×C4—86no—
577C2×C4—83no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1223no—
581C2×C4—1223no—
582C2×C4—4883no—
583C2×C4—4883no—
584C2×C4C22×C22⋊D6111yescharacteristic
585C2×C4C22×C22⋊D6113yes—
586C2×C4C22×C22⋊D6111yescharacteristic
587C2×C4C22×C22⋊D6113yes—
588C2×C4C22×C22⋊D6113yes—
589C2×C4C22×Dic12216yes—
590C2×C4C2×#1175822216yes—
591C2×C4C24×D6111yescharacteristic
592C8—1221no—
593C8—1221no—
594C8—1223no—
595C8—1223no—
596C22—23no—
597C22—23no—
598C22—26no—
599C22—43no—
600C22—43no—
601C22—812no—
602C22C2×#1451975616yes—
603C22C23×C22⋊D6111yescharacteristic
604C4—22no—
605C4—21no—
606C4—23no—
607C4—21no—
608C4—23no—
609C4—41no—
610C4—43no—
611C4—2443no—
612C4—2441no—
613C4—2443no—
614C4—2441no—
615C4C2×#1451975616yes—
616C4C23×Dic12211yescharacteristic
617C4C22×#1175822211yescharacteristic
618C2—26no—
619C2—46no—
620C2C22×#1451975611yescharacteristic
621C2#2146059011yescharacteristic
622C2#2146059911yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC4×H(5,𝔽2)×C61C2
socleC22×C61C23×D4
prime radicalC2×H(5,𝔽2)×C61C22
hypercenterC4×H(5,𝔽2)D61
hypocenterC61#60496
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #21459645C2semidirect
2. #21455681C2semidirect
3. #21455670C2semidirect
4. #21455650C2semidirect
5. #21455650C2semidirect
6. #21455641C2semidirect
7. #21455616C2semidirect
8. #21455607C2semidirect
9. C2×#16840062C2semidirect
10. #21454417C2semidirect
11. #21454412C2semidirect
12. #21454387C2semidirect
22. #16840390C22semidirect
23. #16840390C22semidirect
24. #16840319C22semidirect
25. C2×#14519769C22semidirect
28. #16840069C22semidirect
29. #16840069C22semidirect
30. #16840069C22semidirect
31. #16840062C22semidirect
32. #16840062C22semidirect
33. #16840062C22semidirect
34. #16840062C22semidirect
35. #16840062C22semidirect
36. #16840062C22semidirect
37. #16840057C22semidirect
38. #16840052C22semidirect
39. #16840052C22semidirect
40. #16840047C22semidirect
41. #16840042C22semidirect
42. #16840042C22semidirect
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
133. H(5,𝔽2)×C61Q8semidirect
385. C61#60496semidirect
387. H(5,𝔽2)Dic122semidirect

Maximal subgroups

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

subgroupindex
#214596452
#214556812
#214556702
#214556502
#214556412
#214556162
#214556072
C2×#168400622
#214544172
#214544122
#214543872
#214543452
C4×H(5,𝔽2)×C612
#6049661