Finite Group Databasebeta

← #29656591

Subgroups — #29656591

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457206C211yesmaximal, characteristic, complemented
2#21456193C211yesmaximal, characteristic, complemented
3#21455890C211yesmaximal, characteristic, complemented
4#21445019C212yesmaximal, complemented
5#21445012C212yesmaximal, complemented
6#21444998C212yesmaximal, complemented
7#21444998C212yesmaximal, complemented
8#21444487C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3498C211yesmaximal, characteristic
12#16840319C2211yescharacteristic, complemented
13#16840126—22no—
14#16840108C2212yescomplemented
15#16840108C2212yescomplemented
16#16840103—21no—
17#16840054—21no—
18#16840047C2211yescharacteristic, complemented
19#16840042C2211yescharacteristic, complemented
20#16840042C2212yescomplemented
21#16840006—22no—
22#16839972C2211yescharacteristic, complemented
23#16839967C2211yescharacteristic, complemented
24#16839967C2212yescomplemented
25#16839959—21no—
26#16839924—21no—
27C4×#11758202—21no—
28C4×#11758200—21no—
29#16839815C2211yescharacteristic, complemented
30#16839786C2211yescharacteristic, complemented
31#16839767—22no—
32C4×#11758194C2211yescharacteristic, complemented
33#16839506C2212yescomplemented
34#16839506C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38D4⋊#92418C2212yescomplemented
39C2×H(5,𝔽2)×C61—21no—
40C61×#550—21no—
41C61×#550C2212yes—
42D42×C61—22no—
43D42×C61—22no—
44D42×C61—21no—
45D42×C61C2212yes—
46C61×#535—22no—
47C61×#535—21no—
48C61×#532—21no—
49C61×#532C2211yescharacteristic
50C2×#127×C61—21no—
51C2×#127×C61C2211yescharacteristic
52C4×#41×C61C2211yescharacteristic
53C2×#11758202—21no—
54C2×#11758200—21no—
55C2×#11758200—22no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66#14519710—22no—
67#14519710—21no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2311yescharacteristic
71C4⋊#92418C2312yes—
72C4⋊#92418C2311yescharacteristic
73#14519701—21no—
74#14519700—22no—
75#14519700—21no—
76#14519700—21no—
77#14519698—22no—
78#14519698—24no—
79#14519698—24no—
80#14519698—21no—
81C22⋊#92418—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
85H(5,𝔽2)×C61—44no—
86H(5,𝔽2)×C61—44no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—21no—
90C2×#41×C61D411yescharacteristic
91C22×D4×C61—22no—
92C22×D4×C61—24no—
93C22×D4×C61—22no—
94C22×D4×C61—22no—
95C22×D4×C61—22no—
96C22×D4×C61—21no—
97C22×D4×C61—21no—
98C22×D4×C61D412yes—
99C4⋊Q8×C61D411yescharacteristic
100#127×C61—22no—
101#127×C61—22no—
102#127×C61—22no—
103#127×C61—21no—
104#127×C61—21no—
105#127×C61D411yescharacteristic
106#127×C61D412yes—
107#127×C61C2312yes—
108#124×C61—22no—
109#124×C61—21no—
110#121×C61—22no—
111#121×C61—24no—
112#121×C61—24no—
113#121×C61—21no—
114#121×C61—22no—
115#121×C61—22no—
116#121×C61—22no—
117#121×C61D412yes—
118#120×C61—24no—
119#120×C61—24no—
120#120×C61—24no—
121C4×Q8×C61D411yescharacteristic
122C4×Q8×C61C2311yescharacteristic
123C4×D4×C61—22no—
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—22no—
127C4×D4×C61—21no—
128C4×D4×C61D411yescharacteristic
129C4×D4×C61C2312yes—
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61D411yescharacteristic
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C24×C61—24no—
136C24×C61—22no—
137#41×C61—22no—
138#41×C61—22no—
139#41×C61—48no—
140#41×C61—42no—
141#41×C61—42no—
142#41×C61—44no—
143#41×C61—44no—
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D411yescharacteristic
146C2×D4×C61—21no—
147C2×D4×C61—24no—
148C2×D4×C61—24no—
149C2×D4×C61—22no—
150C2×D4×C61—24no—
151C2×D4×C61—24no—
152C2×D4×C61—22no—
153C2×D4×C61—21no—
154C2×D4×C61—22no—
155C2×D4×C61—21no—
156C2×D4×C61—22no—
157C2×D4×C61—22no—
158C2×D4×C61—22no—
159C2×D4×C61—48no—
160C2×D4×C61—42no—
161C2×D4×C61—42no—
162C2×D4×C61—44no—
163C2×D4×C61—416no—
164C2×D4×C61—44no—
165C2×D4×C61—44no—
166C2×D4×C61—81no—
167C2×D4×C61—81no—
168C2×D4×C61—82no—
169C2×D4×C61—82no—
170C2×D4×C61—84no—
171C2×D4×C61C2×D412yes—
172C2×D4×C61C2×D411yescharacteristic
173C2×D4×C61C2×D414yes—
174C22×C4×C61—21no—
175C22×C4×C61—24no—
176C22×C4×C61—21no—
177C22×C4×C61—21no—
178C22×C4×C61—22no—
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2×D411yescharacteristic
182C2×Dic122—21no—
183C2×Dic122—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D412yes—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—24no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—22no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61C2×D411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2×D411yescharacteristic
203C42×C61C2411yescharacteristic
204C22⋊Dic61—41no—
205#11758202—42no—
206#11758200—42no—
207#11758200—82no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212#11758196—41no—
213#11758196—41no—
214C4×Dic61—41no—
215#11758194—42no—
216#11758194—81no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—22no—
220C2×#92418—22no—
221C23×C61—21no—
222C23×C61—22no—
223C23×C61—24no—
224C23×C61—44no—
225C23×C61—48no—
226C23×C61—82no—
227C23×C61—82no—
228C23×C61—84no—
229C23×C61—84no—
230C23×C61#12011yescharacteristic
231Q8×C61—21no—
232Q8×C61—42no—
233Q8×C61AGL(1,ℤ/8)12yes—
234D4×C61—24no—
235D4×C61—21no—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—44no—
239D4×C61—44no—
240D4×C61—42no—
241D4×C61—48no—
242D4×C61—44no—
243D4×C61—48no—
244D4×C61—42no—
245D4×C61—42no—
246D4×C61—42no—
247D4×C61—42no—
248D4×C61—44no—
249D4×C61—42no—
250D4×C61—82no—
251D4×C61—84no—
252D4×C61—84no—
253D4×C61AGL(1,ℤ/8)12yes—
254C2×C4×C61—24no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—22no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—22no—
264C2×C4×C61—48no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—42no—
270C2×C4×C61#12011yescharacteristic
271C2×C4×C61#12011yescharacteristic
272C2×C4×C61#12011yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×Dic61—41no—
277C2×Dic61—41no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281Dic122—41no—
282#92418—42no—
283#92418—41no—
284#92418—41no—
285#92418—42no—
286#18574—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—22no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—44no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—44no—
297C22×C61—44no—
298C22×C61—42no—
299C22×C61—164no—
300C22×C61—168no—
301C22×C61C2×#12011yescharacteristic
302C4×C61—21no—
303C4×C61—22no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—42no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61#45512yes—
310C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
312C4×C61#57611yescharacteristic
313C4×C61#57711yescharacteristic
314Dic61—81no—
315Dic61—81no—
316#3498D6111yescharacteristic
317#3254—611no—
318#3138—611no—
319#3107—611no—
320#1642—612no—
321#1641—612no—
322#1639—612no—
323#1639—612no—
324#1545—611no—
325#1489—611no—
326#1484—611no—
327C2×C61—41no—
328C2×C61—42no—
329C2×C61—82no—
330C2×C61—84no—
331C2×C61#300311yescharacteristic
332C2×C61#301211yescharacteristic
333C2×C61C2×#45511yescharacteristic
334C2×H(5,𝔽2)—21no—
335#550—21no—
336#550C2×D6112yes—
337D42—22no—
338D42—22no—
339D42—21no—
340D42C2×D6112yes—
341#535—22no—
342#535—21no—
343#533—611no—
344#532—21no—
345#532C2×D6111yescharacteristic
346C2×#127—21no—
347C2×#127C2×D6111yescharacteristic
348C4×#41C2×D6111yescharacteristic
349#490—1222no—
350#486—612no—
351#486—612no—
352#485—1221no—
353#477—1221no—
354#475—611no—
355#474—611no—
356#474—612no—
357#468—1222no—
358#462—611no—
359#461—611no—
360#461—612no—
361#460—1221no—
362#452—1221no—
363C4×Q16—1221no—
364C4×SD16—1221no—
365#431—611no—
366#423—611no—
367#418—1222no—
368C4×M16—611no—
369#331—612no—
370#331—612no—
371Q8⋊C8—612no—
372D4⋊C8—612no—
373D4⋊C8—612no—
374D4⋊C8—612no—
375C61#1857411yescharacteristic, complemented, Sylow
376H(5,𝔽2)—44no—
377H(5,𝔽2)—44no—
378C2×#41—22no—
379C2×#41—22no—
380C2×#41—21no—
381C2×#41C22⋊D6111yescharacteristic
382C22×D4—22no—
383C22×D4—24no—
384C22×D4—22no—
385C22×D4—22no—
386C22×D4—22no—
387C22×D4—21no—
388C22×D4—21no—
389C22×D4C22⋊D6112yes—
390C2×Q16—1221no—
391C2×SD16—1222no—
392C2×SD16—1221no—
393C2×M16—1221no—
394C2×M16—1221no—
395C4⋊Q8—611no—
396C4⋊Q8—611no—
397C4⋊Q8C22⋊D6111yescharacteristic
398#127—22no—
399#127—22no—
400#127—22no—
401#127—21no—
402#127—21no—
403#127C22⋊D6111yescharacteristic
404#127C22⋊D6112yes—
405#127C22×D6112yes—
406#125—1221no—
407#124—22no—
408#124—21no—
409C22⋊Q8—1221no—
410C22⋊Q8—1221no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—21no—
415#121—22no—
416#121—22no—
417#121—22no—
418#121C22⋊D6112yes—
419#120—24no—
420#120—24no—
421#120—24no—
422C4×Q8—1221no—
423C4×Q8—1221no—
424C4×Q8C22⋊D6111yescharacteristic
425C4×Q8C22×D6111yescharacteristic
426C4×D4—22no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—22no—
430C4×D4—21no—
431C4×D4C22⋊D6111yescharacteristic
432C4×D4C22×D6111yescharacteristic
433C4×D4C22×D6112yes—
434#117—21no—
435#117—1221no—
436#117C22⋊D6111yescharacteristic
437C2×C42—21no—
438C2×C42C22×D6111yescharacteristic
439#107—1221no—
440#107—1222no—
441#106—1221no—
442C4⋊C8—611no—
443C4⋊C8—611no—
444C4⋊C8—612no—
445C4⋊C8—1221no—
446#103—1221no—
447#103—1221no—
448#103—1221no—
449#103—1222no—
450#102—1224no—
451#102—1224no—
452#102—1221no—
453#102—1222no—
454C22⋊C8—1221no—
455#97—1221no—
456C4×C8—612no—
457C4×C8—1221no—
458C24—24no—
459C24—22no—
460#41—22no—
461#41—22no—
462#41—48no—
463#41—42no—
464#41—42no—
465#41—44no—
466#41—44no—
467C2×Q8—21no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8C2×C22⋊D6111yescharacteristic
471C2×D4—21no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—48no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—44no—
488C2×D4—416no—
489C2×D4—44no—
490C2×D4—44no—
491C2×D4—81no—
492C2×D4—81no—
493C2×D4—82no—
494C2×D4—82no—
495C2×D4—84no—
496C2×D4C2×C22⋊D6112yes—
497C2×D4C2×C22⋊D6111yescharacteristic
498C2×D4C2×C22⋊D6114yes—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507Q16—2442no—
508SD16—4882no—
509SD16—2442no—
510M16—4881no—
511M16—2442no—
512C2×C8—1222no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1222no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4C2×C22⋊D6111yescharacteristic
526C4⋊C4C2×C22⋊D6111yescharacteristic
527C4⋊C4C2×C22⋊D6112yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—22no—
535C22⋊C4—2441no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—2441no—
540C42C2×C22⋊D6111yescharacteristic
541C42C2×C22⋊D6111yescharacteristic
542C42C2×C22⋊D6111yescharacteristic
543C42C23×D6111yescharacteristic
544C23—21no—
545C23—22no—
546C23—24no—
547C23—44no—
548C23—48no—
549C23—82no—
550C23—82no—
551C23—84no—
552C23—84no—
553C23#1451974611yescharacteristic
554Q8—21no—
555Q8—42no—
556Q8—2441no—
557Q8—2441no—
558Q8—2441no—
559Q8—2441no—
560Q8#1451980312yes—
561D4—24no—
562D4—21no—
563D4—22no—
564D4—22no—
565D4—44no—
566D4—44no—
567D4—42no—
568D4—48no—
569D4—44no—
570D4—48no—
571D4—44no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—82no—
578D4—84no—
579D4—84no—
580D4#1451980312yes—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4#1451974611yescharacteristic
600C2×C4#1451974611yescharacteristic
601C2×C4#1451974611yescharacteristic
602C2×C4C22×C22⋊D6111yescharacteristic
603C2×C4C22×C22⋊D6111yescharacteristic
604C2×C4C22×C22⋊D6111yescharacteristic
605C8—2442no—
606C8—2441no—
607C8—2441no—
608C8—2442no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—164no—
622C22—168no—
623C22C2×#1451974611yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—4881no—
632C4—4881no—
633C4#1683994012yes—
634C4C2×#1451980311yescharacteristic
635C4C2×#1451980311yescharacteristic
636C4#1684056511yescharacteristic
637C4#1684057311yescharacteristic
638C2—41no—
639C2—42no—
640C2—82no—
641C2—84no—
642C2#2145504611yescharacteristic
643C2#2145512311yescharacteristic
644C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3498C2
socleC22×C61C2×#120
hypercenter#3498D61
hypocenterC61#18574
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3498

Complements

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

NH ≅ G/Ntype
1. #21457206C2semidirect
2. #21456193C2semidirect
3. #21455890C2semidirect
4. #21445019C2semidirect
5. #21445012C2semidirect
6. #21444998C2semidirect
7. #21444998C2semidirect
8. #21444487C2semidirect
9. #21444258C2semidirect
10. #21444236C2semidirect
12. #16840319C22semidirect
14. #16840108C22semidirect
15. #16840108C22semidirect
18. #16840047C22semidirect
19. #16840042C22semidirect
20. #16840042C22semidirect
22. #16839972C22semidirect
23. #16839967C22semidirect
24. #16839967C22semidirect
29. #16839815C22semidirect
30. #16839786C22semidirect
32. C4×#11758194C22semidirect
33. #16839506C22semidirect
34. #16839506C22semidirect
35. Q8⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
38. D4⋊#92418C22semidirect
375. C61#18574semidirect

Maximal subgroups

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

subgroupindex
#214572062
#214561932
#214558902
#214450192
#214450122
#214449982
#214444872
#214442582
#214442362
C61×#34982
#1857461