Finite Group Databasebeta

← #30547665

Subgroups — #30547665

Order 15616.

2090 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459632C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×Dic61C211yesmaximal, characteristic
3#21455674C213yesmaximal, complemented
4#21455668C213yesmaximal, complemented
5#21455652C213yesmaximal, complemented
6#21455646C213yesmaximal, complemented
7#21455630C213yesmaximal, complemented
8#21455613C213yesmaximal, complemented
9#21455603C213yesmaximal, complemented
10C2×#16840040C213yesmaximal, complemented
11#21454419C211yesmaximal, characteristic, complemented
12#21454417C211yesmaximal, characteristic, complemented
13#21454385C211yesmaximal, characteristic, complemented
14#21454345C211yesmaximal, characteristic
15C61×#2897C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17Q8×Dic122—21no—
18#16840488—21no—
19#16840471—23no—
20D4×Dic122—23no—
21#16840393C2213yescomplemented
22#16840390C2213yescomplemented
23#16840315C2213yescomplemented
24C2×#14519776C2213yescomplemented
25#16840245—21no—
26#16840245C2213yes—
27#16840245C2211yescharacteristic
28#16840245C2211yescharacteristic
29#16840237—23no—
30#16840237C2213yes—
31#41×Dic61—21no—
32#41×Dic61C2213yes—
33#41×Dic61C2211yescharacteristic
34C2×D4×Dic61—23no—
35C2×D4×Dic61C2213yes—
36#16840071C2213yescomplemented
37#16840067C2213yescomplemented
38#16840064C2216yescomplemented
39#16840064C2213yescomplemented
40#16840060C2213yescomplemented
41#16840060C2216yescomplemented
42#16840055C2216yescomplemented
43#16840055C2211yescharacteristic, complemented
44#16840050C2216yescomplemented
45#16840050C2213yescomplemented
46#16840045C2213yescomplemented
47#16840040C2216yescomplemented
48#16840040C2216yescomplemented
49#16840040C2216yescomplemented
50#16840040C2213yescomplemented
51#16839805C2213yescomplemented
52#16839804C2213yescomplemented
53#16839804C2213yescomplemented
54#16839802C2213yescomplemented
55#16839800C2213yescomplemented
56#16839798C2213yescomplemented
57C2×#14519711C2213yescomplemented
58C2×#14519710C2213yescomplemented
59#16839790C2213yescomplemented
60#16839788C2213yescomplemented
61#16839785C2213yescomplemented
62C2×#14519706C2213yescomplemented
63#16839767C2213yes—
64#16839763C2213yes—
65#16839758C2213yes—
66C2×#14519698C2213yes—
67C2×H(5,𝔽2)×C61C2211yescharacteristic
68C2×#131×C61C2211yescharacteristic
69C61×#520C2211yescharacteristic
70C61×#418C2213yes—
71C61×#417C2213yes—
72C61×#416C2213yes—
73C2×#102×C61C2213yes—
74C22×Dic122—23no—
75C23×Dic61—23no—
76#14519784—24no—
77C2×#11758194C2313yes—
78C22×#92418C2313yes—
79#14519777—23no—
80#14519776—23no—
81#14519776—23no—
82#14519776—23no—
83#14519776C2313yes—
84#14519776C2316yes—
85#14519776C2311yescharacteristic
86#14519769—23no—
87#14519768—23no—
88#14519768C2313yes—
89#14519768C2313yes—
90#14519756—26no—
91#14519754—23no—
92#14519754—23no—
93#14519754—26no—
94Q8×Dic61—21no—
95Q8×Dic61—21no—
96Q8×Dic61D411yescharacteristic
97Q8×Dic61D411yescharacteristic
98Q8×Dic61C2311yescharacteristic
99Q8×Dic61C2313yes—
100C4×Dic122—23no—
101C4×Dic122—23no—
102D4×Dic61—23no—
103D4×Dic61—23no—
104D4×Dic61—23no—
105D4×Dic61—23no—
106D4×Dic61—26no—
107D4×Dic61—23no—
108D4×Dic61—23no—
109D4×Dic61—23no—
110D4×Dic61—23no—
111D4×Dic61D413yes—
112D4×Dic61D413yes—
113D4×Dic61C2313yes—
114D4×Dic61C2316yes—
115D4×Dic61C2313yes—
116#14519736—23no—
117#14519736—23no—
118#14519736C2313yes—
119#14519736C2313yes—
120#14519736C2313yes—
121C2×C4⋊Dic61—23no—
122C2×C4⋊Dic61C2313yes—
123C2×C4⋊Dic61C2313yes—
124C2×#11758196—23no—
125C2×C22⋊Dic61—23no—
126C2×C22⋊Dic61—26no—
127C2×C4×Dic61—23no—
128C2×C4×Dic61C2313yes—
129#14519711C2313yes—
130#14519711C2316yes—
131#14519711C2311yescharacteristic
132#14519710C2313yes—
133#14519710C2316yes—
134#14519710C2311yescharacteristic
135#14519709C2313yes—
136#14519709C2313yes—
137#14519708C2313yes—
138#14519708C2313yes—
139#14519706C2311yescharacteristic
140#14519706C2313yes—
141#14519706C2313yes—
142#14519706C2313yes—
143#14519706C2316yes—
144#14519701C2311yescharacteristic
145#14519701C2313yes—
146#14519698C2313yes—
147#14519698C2316yes—
148#14519698C2313yes—
149H(5,𝔽2)×C61—24no—
150H(5,𝔽2)×C61Q818yescomplemented
151C2×#41×C61—21no—
152C2×#41×C61C2313yes—
153C2×#41×C61C2311yescharacteristic
154C22×D4×C61—23no—
155C22×D4×C61C2313yes—
156#131×C61—24no—
157C2×M16×C61C2313yes—
158C22×C8×C61C2313yes—
159C4×Q8×C61—21no—
160C4×D4×C61—23no—
161#117×C61C2313yes—
162C2×C4⋊C4×C61C2313yes—
163#103×C61C2313yes—
164#103×C61C2311yescharacteristic
165#102×C61C2313yes—
166#102×C61C2316yes—
167#102×C61C2313yes—
168C24×C61—23no—
169#41×C61—28no—
170#41×C61—26no—
171#41×C61C2×Q814yes—
172#41×C61C2×Q8112yescomplemented
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×Q8112yescomplemented
182C2×D4×C61C2413yes—
183C22×C4×C61—23no—
184C22×C4×C61—23no—
185C22×C4×C61C2413yes—
186C22×Dic61—23no—
187C22×Dic61—23no—
188C22×Dic61—23no—
189C22×Dic61—26no—
190C22×Dic61—23no—
191C22×Dic61—86no—
192C22×Dic61C2×D413yes—
193C22×Dic61C2×D413yes—
194C2×Dic122—23no—
195C2×Dic122—21no—
196C2×Dic122—26no—
197C2×Dic122—86no—
198M16×C61—26no—
199C2×C8×C61—26no—
200C2×C8×C61C2411yescharacteristic
201C2×C8×C61C2413yes—
202C4⋊C4×C61—23no—
203C4⋊C4×C61C2413yes—
204C4⋊C4×C61C2411yescharacteristic
205C22⋊C4×C61—23no—
206C42×C61—23no—
207C22⋊Dic61—23no—
208C22⋊Dic61—23no—
209C22⋊Dic61—23no—
210C22⋊Dic61—26no—
211C22⋊Dic61—23no—
212C22⋊Dic61C2×D416yes—
213C4⋊Dic61—23no—
214C4⋊Dic61—23no—
215C4⋊Dic61—23no—
216C4⋊Dic61—21no—
217C4⋊Dic61—23no—
218C4⋊Dic61—23no—
219C4⋊Dic61C2×D413yes—
220C4⋊Dic61C2×D413yes—
221C4⋊Dic61C2411yescharacteristic
222C4⋊Dic61C2413yes—
223C4⋊Dic61C2411yescharacteristic
224C4⋊Dic61C2413yes—
225#11758196—23no—
226#11758196—26no—
227#11758196—21no—
228#11758196—23no—
229#11758196—23no—
230C4×Dic61—23no—
231C4×Dic61—23no—
232C4×Dic61—21no—
233C4×Dic61—23no—
234C4×Dic61C2×D413yes—
235C4×Dic61C2×D413yes—
236C4×Dic61C2411yescharacteristic
237C4×Dic61C2413yes—
238#11758194—26no—
239C2×#92418—26no—
240C2×#92418C2413yes—
241C2×#92418C2411yescharacteristic
242C23×C61—23no—
243C23×C61—212no—
244C23×C61—83no—
245C23×C61—83no—
246C23×C61C22×D413yes—
247Q8×C61—21no—
248Q8×C61C22⋊Q814yes—
249Q8×C61C22×Q812yes—
250D4×C61—212no—
251D4×C61—212no—
252D4×C61—23no—
253D4×C61C22⋊Q8112yes—
254D4×C61C22×Q816yes—
255C2×C4×C61—21no—
256C2×C4×C61—212no—
257C2×C4×C61—21no—
258C2×C4×C61—23no—
259C2×C4×C61—83no—
260C2×C4×C61C22×D413yes—
261C2×C4×C61C22×Q816yes—
262C2×C4×C61C2×#4116yes—
263C2×C4×C61C2511yescharacteristic
264C2×Dic61—26no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—23no—
268C2×Dic61—23no—
269C2×Dic61—43no—
270C2×Dic61—46no—
271C2×Dic61—83no—
272C2×Dic61—86no—
273C2×Dic61—83no—
274C2×Dic61—83no—
275C2×Dic61C22×D413yes—
276C2×Dic61C22×D411yescharacteristic
277C2×Dic61C22×D413yes—
278C2×Dic61C22×D411yescharacteristic
279Dic122—42no—
280Dic122—46no—
281Dic122—86no—
282C8×C61—21no—
283C8×C61—23no—
284#92418—21no—
285#92418—23no—
286#60494—611nomaximal, Sylow
287C22×C61—23no—
288C22×C61—23no—
289C22×C61—26no—
290C22×C61—43no—
291C22×C61—43no—
292C22×C61—812no—
293C22×C61C2×C22⋊Q816yes—
294C22×C61C23×D411yescharacteristic
295C4×C61—22no—
296C4×C61—41no—
297C4×C61—43no—
298C4×C61C2×C22⋊Q816yes—
299C4×C61C23×Q811yescharacteristic
300C4×C61C22×#4111yescharacteristic
301Dic61—21no—
302Dic61—23no—
303Dic61—23no—
304Dic61—41no—
305Dic61—41no—
306Dic61—43no—
307Dic61—43no—
308Dic61#57612yes—
309#3503—611no—
310C4×H(5,𝔽2)—611no—
311#3083—613no—
312#3083—613no—
313#3081—613no—
314#3081—613no—
315#3079—613no—
316#3077—613no—
317#3076—613no—
318C2×#474—613no—
319#2913—611no—
320#2913—611no—
321#2906—611no—
322#2897—611no—
323#2897D6111yescharacteristic
324C2×C61—26no—
325C2×C61—46no—
326C2×C61C22×C22⋊Q811yescharacteristic
327C2×C61C2×#57611yescharacteristic
328C2×C61#358611yescharacteristic
329C2×H(5,𝔽2)C2×D6111yescharacteristic
330C2×#131—611no—
331C2×#131C2×D6111yescharacteristic
332Q82—1221no—
333#557—1221no—
334#554—1223no—
335D4×Q8—1223no—
336#544—613no—
337#544—613no—
338#533—613no—
339C2×C4⋊Q8—613no—
340#520—611no—
341#520—611no—
342#520—613no—
343#520—1221no—
344#520C2×D6111yescharacteristic
345#518—613no—
346#518—1223no—
347C4×#41—611no—
348C4×#41—613no—
349C4×#41—1221no—
350C2×C4×D4—613no—
351C2×C4×D4—1223no—
352#479—613no—
353#479—613no—
354#478—613no—
355#478—613no—
356#478—616no—
357#478—616no—
358#477—611no—
359#477—616no—
360#476—613no—
361#476—616no—
362#475—613no—
363#474—613no—
364#474—616no—
365#474—616no—
366#474—616no—
367#428—613no—
368#428—613no—
369#428—613no—
370#428—613no—
371#427—613no—
372#427—613no—
373C2×#107—613no—
374C2×#107—613no—
375#424—613no—
376#424—613no—
377#423—613no—
378C2×C4⋊C8—613no—
379#418—613no—
380#418C2×D6113yes—
381#417—613no—
382#417C2×D6113yes—
383#416—613no—
384#416C2×D6113yes—
385C2×#102—613no—
386C2×#102C2×D6113yes—
387C61#6049411yescharacteristic, complemented, Sylow
388H(5,𝔽2)—24no—
389H(5,𝔽2)Dic12218yescomplemented
390C2×#41—21no—
391C2×#41C22×D6113yes—
392C2×#41C22×D6111yescharacteristic
393C22×Q8—1223no—
394C22×D4—23no—
395C22×D4C22×D6113yes—
396C23×C4—1223no—
397#131—24no—
398#131—1224no—
399C2×M16—613no—
400C2×M16C22×D6113yes—
401C22×C8—613no—
402C22×C8C22×D6113yes—
403C4⋊Q8—613no—
404C4⋊Q8—616no—
405C4⋊Q8—611no—
406C4⋊Q8—1223no—
407C4⋊Q8—1223no—
408C4⋊Q8—1223no—
409C4⋊Q8—1223no—
410#125—613no—
411#125—613no—
412#125—1223no—
413#125—1223no—
414C22⋊Q8—1226no—
415C22⋊Q8—1223no—
416C22⋊Q8—1223no—
417C22⋊Q8—1226no—
418C4×Q8—21no—
419C4×Q8—611no—
420C4×Q8—611no—
421C4×Q8—611no—
422C4×Q8—613no—
423C4×Q8—1223no—
424C4×Q8—1223no—
425C4×Q8—1221no—
426C4×Q8—1221no—
427C4×D4—23no—
428C4×D4—613no—
429C4×D4—613no—
430C4×D4—613no—
431C4×D4—616no—
432C4×D4—613no—
433C4×D4—1223no—
434C4×D4—1223no—
435C4×D4—1226no—
436C4×D4—1223no—
437C4×D4—1223no—
438C4×D4—1223no—
439C4×D4—1223no—
440C4×D4—1223no—
441C4×D4—1223no—
442#117—613no—
443#117—613no—
444#117—613no—
445#117—1223no—
446#117—1223no—
447#117C22×D6113yes—
448C2×C4⋊C4—613no—
449C2×C4⋊C4—613no—
450C2×C4⋊C4—1223no—
451C2×C4⋊C4—1223no—
452C2×C4⋊C4C22×D6113yes—
453C2×C22⋊C4—1226no—
454C2×C22⋊C4—1223no—
455C2×C42—613no—
456C2×C42—1223no—
457#107—613no—
458#107—616no—
459#107—611no—
460#107—613no—
461#107—616no—
462#107—611no—
463#106—613no—
464#106—613no—
465#106—613no—
466#106—613no—
467C4⋊C8—611no—
468C4⋊C8—613no—
469C4⋊C8—613no—
470C4⋊C8—613no—
471C4⋊C8—616no—
472#103—611no—
473#103—613no—
474#103C22×D6111yescharacteristic
475#103C22×D6113yes—
476#102—613no—
477#102—616no—
478#102—613no—
479#102C22×D6113yes—
480#102C22×D6116yes—
481#102C22×D6113yes—
482C24—23no—
483#41—28no—
484#41—26no—
485#41C2×Dic12214yes—
486#41C2×Dic122112yescomplemented
487#41#1175822414yes—
488C2×Q8—1226no—
489C2×Q8—1221no—
490C2×Q8—1223no—
491C2×Q8—4886no—
492C2×Q8D4×D6111yescharacteristic
493C2×Q8C23×D6111yescharacteristic
494C2×D4—23no—
495C2×D4—224no—
496C2×D4—23no—
497C2×D4—26no—
498C2×D4C2×Dic122112yescomplemented
499C2×D4D4×D6113yes—
500C2×D4C23×D6113yes—
501C22×C4—23no—
502C22×C4—23no—
503C22×C4—613no—
504C22×C4—613no—
505C22×C4—1223no—
506C22×C4—1226no—
507C22×C4—1223no—
508C22×C4—1223no—
509C22×C4—1223no—
510C22×C4—4886no—
511C22×C4C23×D6113yes—
512M16—26no—
513M16—1226no—
514C2×C8—26no—
515C2×C8—613no—
516C2×C8—611no—
517C2×C8—1226no—
518C2×C8C23×D6113yes—
519C2×C8C23×D6111yescharacteristic
520C4⋊C4—23no—
521C4⋊C4—613no—
522C4⋊C4—613no—
523C4⋊C4—611no—
524C4⋊C4—613no—
525C4⋊C4—611no—
526C4⋊C4—613no—
527C4⋊C4—1223no—
528C4⋊C4—1223no—
529C4⋊C4—1226no—
530C4⋊C4—1221no—
531C4⋊C4—1223no—
532C4⋊C4—1223no—
533C4⋊C4—1221no—
534C4⋊C4—1223no—
535C4⋊C4—1223no—
536C4⋊C4—1223no—
537C4⋊C4—1223no—
538C4⋊C4C23×D6111yescharacteristic
539C4⋊C4C23×D6113yes—
540C22⋊C4—23no—
541C22⋊C4—616no—
542C22⋊C4—1223no—
543C22⋊C4—1226no—
544C22⋊C4—1223no—
545C22⋊C4—1223no—
546C22⋊C4—1223no—
547C42—23no—
548C42—613no—
549C42—613no—
550C42—611no—
551C42—613no—
552C42—1221no—
553C42—1223no—
554C42—1223no—
555C42—1223no—
556C23—23no—
557C23—212no—
558C23—83no—
559C23—83no—
560C23C2×D4×D6113yes—
561Q8—21no—
562Q8—4886no—
563Q8—2446no—
564Q8—2442no—
565Q8#1451975414yes—
566Q8C22×Dic12212yes—
567D4—212no—
568D4—212no—
569D4—23no—
570D4#14519754112yes—
571D4C22×Dic12216yes—
572C2×C4—21no—
573C2×C4—212no—
574C2×C4—21no—
575C2×C4—23no—
576C2×C4—83no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—613no—
580C2×C4—613no—
581C2×C4—1226no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1223no—
585C2×C4—1223no—
586C2×C4—4883no—
587C2×C4—4886no—
588C2×C4—4883no—
589C2×C4—4883no—
590C2×C4—2446no—
591C2×C4—2443no—
592C2×C4C2×D4×D6113yes—
593C2×C4C22×Dic12216yes—
594C2×C4C2×#1175822416yes—
595C2×C4C24×D6111yescharacteristic
596C8—21no—
597C8—23no—
598C8—1221no—
599C8—1223no—
600C22—23no—
601C22—23no—
602C22—26no—
603C22—43no—
604C22—43no—
605C22—812no—
606C22C2×#1451975416yes—
607C22C22×D4×D6111yescharacteristic
608C4—22no—
609C4—43no—
610C4—41no—
611C4—612no—
612C4—1223no—
613C4—1223no—
614C4—1221no—
615C4—2443no—
616C4—2443no—
617C4—2441no—
618C4—2441no—
619C4C2×#1451975416yes—
620C4C23×Dic12211yescharacteristic
621C4C22×#1175822411yescharacteristic
622C2—26no—
623C2—46no—
624C2C22×#1451975411yescharacteristic
625C2#576×D6111yescharacteristic
626C2#2146060111yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2897C2
socleC22×C61C23×D4
prime radicalC2×H(5,𝔽2)×C61C22
hypercenter#2897D61
hypocenterC61#60494
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2897

Complements

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

NH ≅ G/Ntype
1. #21459632C2semidirect
3. #21455674C2semidirect
4. #21455668C2semidirect
5. #21455652C2semidirect
6. #21455646C2semidirect
7. #21455630C2semidirect
8. #21455613C2semidirect
9. #21455603C2semidirect
10. C2×#16840040C2semidirect
11. #21454419C2semidirect
12. #21454417C2semidirect
13. #21454385C2semidirect
21. #16840393C22semidirect
22. #16840390C22semidirect
23. #16840315C22semidirect
24. C2×#14519776C22semidirect
36. #16840071C22semidirect
37. #16840067C22semidirect
38. #16840064C22semidirect
39. #16840064C22semidirect
40. #16840060C22semidirect
41. #16840060C22semidirect
42. #16840055C22semidirect
43. #16840055C22semidirect
44. #16840050C22semidirect
45. #16840050C22semidirect
46. #16840045C22semidirect
47. #16840040C22semidirect
48. #16840040C22semidirect
49. #16840040C22semidirect
50. #16840040C22semidirect
51. #16839805C22semidirect
52. #16839804C22semidirect
53. #16839804C22semidirect
54. #16839802C22semidirect
55. #16839800C22semidirect
56. #16839798C22semidirect
57. C2×#14519711C22semidirect
58. C2×#14519710C22semidirect
59. #16839790C22semidirect
60. #16839788C22semidirect
61. #16839785C22semidirect
62. C2×#14519706C22semidirect
150. H(5,𝔽2)×C61Q8semidirect
172. #41×C61C2×Q8semidirect
181. C2×D4×C61C2×Q8semidirect
387. C61#60494semidirect
389. H(5,𝔽2)Dic122semidirect
486. #41C2×Dic122semidirect
498. C2×D4C2×Dic122semidirect

Maximal subgroups

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

subgroupindex
#214596322
H(5,𝔽2)×Dic612
#214556742
#214556682
#214556522
#214556462
#214556302
#214556132
#214556032
C2×#168400402
#214544192
#214544172
#214543852
#214543452
C61×#28972
#6049461