Finite Group Databasebeta

← #29743441

Subgroups — #29743441

Order 15616.

740 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#21455666C211yesmaximal, characteristic, complemented
2#21454651C211yesmaximal, characteristic
3#21454416C211yesmaximal, characteristic, complemented
4#21447345C211yesmaximal, characteristic, complemented
5#21447324C211yesmaximal, characteristic, complemented
6#21447179C211yesmaximal, characteristic, complemented
7#21447161C211yesmaximal, characteristic, complemented
8#21446601C211yesmaximal, characteristic, complemented
9#21446420C211yesmaximal, characteristic, complemented
10#21446330C211yesmaximal, characteristic
11#21446192C211yesmaximal, characteristic
12#21446093C211yesmaximal, characteristic, complemented
13#21446042C211yesmaximal, characteristic, complemented
14#21446029C211yesmaximal, characteristic, complemented
15C61×#3081C211yesmaximal, characteristic, complemented
16C2×#14519803C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18#16840242C2211yescharacteristic
19#16840242C2211yescharacteristic
20#16840066—21no—
21#16840059—21no—
22#16840054—21no—
23#16840039—21no—
24#16839872—21no—
25#16839862—21no—
26C4×#11758200—21no—
27C4×#11758199—21no—
28C2×#14519712C2211yescharacteristic
29#16839805C2211yescharacteristic
30#16839805C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839805C2211yescharacteristic
33#16839801—21no—
34#16839801—21no—
35#16839799—21no—
36C2×#14519711C2211yescharacteristic
37C2×#14519710C2211yescharacteristic
38#107×D61—21no—
39#16839789C2211yescharacteristic
40C2×#14519703C2211yescharacteristic
41#16839761C2211yescharacteristic
42#16839756C2211yescharacteristic
43C2×#14519698C2211yescharacteristic
44C2×#14519694C2211yescharacteristic
45#16839718C2211yescharacteristic
46#16839665C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839530C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839527C2211yescharacteristic
52#16839521C2211yescharacteristic
53#16839521C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C61×#544C2211yescharacteristic
57C61×#518C2211yescharacteristic
58C61×#478—21no—
59C61×#478—21no—
60C61×#476—21no—
61C61×#474—21no—
62C61×#428C2211yescharacteristic
63C61×#427C2211yescharacteristic
64C61×#424C2211yescharacteristic
65C61×#418C2211yescharacteristic
66C2×#102×C61C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519803—22no—
69#14519803Q814yescomplemented
70C2×#11758200—21no—
71C2×#11758199—21no—
72#14519783—22no—
73#14519783—22no—
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C2×#11758189—21no—
78C22×#92418C2311yescharacteristic
79C2×C8×D61—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×D244—21no—
83C4×D244—21no—
84#14519737—41no—
85#14519737—41no—
86#14519735—21no—
87#14519735—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—41no—
91C4⋊C4×D61—21no—
92C4⋊C4×D61—21no—
93C2×C22⋊Dic61—41no—
94C2×C4×Dic61C2311yescharacteristic
95#14519712—22no—
96#14519711—21no—
97#14519711—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519708—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108C4⋊#92418—21no—
109C4⋊#92418—21no—
110#14519703—22no—
111#14519700—21no—
112#14519700D411yescharacteristic
113#14519700D411yescharacteristic
114#14519698—21no—
115#14519698—21no—
116#14519697—21no—
117#14519697D411yescharacteristic
118#14519697D411yescharacteristic
119#14519694—22no—
120#14519688—21no—
121C4×#92418—21no—
122C22×D4×C61C2311yescharacteristic
123C2×M16×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61—21no—
127#125×C61—21no—
128#125×C61—21no—
129C22⋊Q8×C61—41no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61—21no—
134#117×C61C2311yescharacteristic
135#117×C61C2311yescharacteristic
136#117×C61C2311yescharacteristic
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C22⋊C4×C61—41no—
139#107×C61—21no—
140#107×C61D411yescharacteristic
141#107×C61D411yescharacteristic
142#106×C61—21no—
143#106×C61D411yescharacteristic
144#106×C61D411yescharacteristic
145C4⋊C8×C61—21no—
146C4⋊C8×C61—21no—
147#102×C61—21no—
148#102×C61—21no—
149#102×C61—21no—
150#102×C61—21no—
151C24×C61—21no—
152C2×Q8×C61—41no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61C2×Q812yes—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C22×C4×C61C2411yescharacteristic
163C2×C22⋊D61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167#11758222—21no—
168#11758222—21no—
169#11758222C2×Q812yes—
170#11758222#4112yes—
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×C4×D61—41no—
174C2×C4×D61—41no—
175C2×Dic122C2×D411yescharacteristic
176M16×C61—21no—
177M16×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C4⋊C4×C61C2×D411yescharacteristic
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—41no—
198C22⋊C4×C61—41no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C22⋊Dic61—82no—
203#11758200—24no—
204#11758199—24no—
205#11758198—41no—
206#11758198—41no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211#11758196—41no—
212#11758196—41no—
213#11758196—82no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—41no—
217C4×Dic61—41no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194C2×Q812yes—
223#11758194#4112yes—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418C2×D411yescharacteristic
230C2×#92418C2×D411yescharacteristic
231#11758189—24no—
232C8×D61—24no—
233C23×C61—22no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—81no—
237C23×C61—81no—
238C23×C61H(5,𝔽2)11yescharacteristic
239C22×D61—21no—
240Q8×C61—161no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—41no—
244D4×C61—41no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—81no—
253C2×C4×C61—161no—
254C2×C4×C61—161no—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×Q811yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C22⋊D61—22no—
262C22⋊D61—22no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—81no—
270C2×Dic61—81no—
271D244—21no—
272D244C22⋊Q812yes—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—161no—
276C4×D61—161no—
277Dic122—21no—
278Dic122C22⋊Q812yes—
279C8×C61—21no—
280C8×C61—21no—
281C8×C61—41no—
282#92418—22no—
283#92418—22no—
284#92418—21no—
285#92418—21no—
286#92418—41no—
287#26400—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—42no—
291C22×C61—82no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—161no—
295C22×C61#53911yescharacteristic
296C22×C61D4211yescharacteristic
297C22×C61#55411yescharacteristic
298C2×D61—22no—
299C2×D61—41no—
300C2×D61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—81no—
308C4×C61C2×C22⋊Q811yescharacteristic
309C4×C61#52511yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61D4×Q811yescharacteristic
312Dic61—22no—
313Dic61—42no—
314Dic61—81no—
315Dic61—81no—
316#3083—611no—
317#3081D6111yescharacteristic, complemented
318#2946—611no—
319#2913—611no—
320#2093—611no—
321#2089—611no—
322#2063—611no—
323#2060—611no—
324#1952—611no—
325#1922—611no—
326#1907—611no—
327#1883—611no—
328#1866—611no—
329#1859—611no—
330#1856—611no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—82no—
334C2×C61#244911yescharacteristic
335C2×C61#328911yescharacteristic
336C2×C61#329211yescharacteristic
337D61—42no—
338C2×AGL(1,ℤ/8)—611no—
339C2×#131—611no—
340#544C2×D6111yescharacteristic
341#520—611no—
342#520—611no—
343#518C2×D6111yescharacteristic
344#479—1221no—
345#478—21no—
346#478—21no—
347#478—1221no—
348#477—1221no—
349#476—21no—
350#474—21no—
351#474—1221no—
352#442—1221no—
353#440—1221no—
354C4×SD16—1221no—
355C4×D8—1221no—
356C2×#108—611no—
357#428—611no—
358#428—611no—
359#428—611no—
360#428—611no—
361#428—1221no—
362#428—1221no—
363#428C2×D6111yescharacteristic
364#427—1221no—
365#427C2×D6111yescharacteristic
366C2×#107—611no—
367C2×#107—611no—
368C2×#107—1221no—
369#424—611no—
370#424C2×D6111yescharacteristic
371C2×#104—611no—
372#418C2×D6111yescharacteristic
373#417—611no—
374#416—611no—
375C2×#102—611no—
376C2×#102C2×D6111yescharacteristic
377C2×#100—611no—
378#405—611no—
379#390—611no—
380#384—611no—
381#343—611no—
382#343—611no—
383#343—611no—
384#341—611no—
385#339—611no—
386#339—611no—
387#339—611no—
388#336—611no—
389C61#2640011yescharacteristic, complemented, Sylow
390C2×#41—611no—
391C22×D4C22×D6111yescharacteristic
392AGL(1,ℤ/8)—614no—
393AGL(1,ℤ/8)—1222no—
394C2×SD16—1221no—
395C2×D8—1221no—
396#131—1222no—
397#131—1222no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—611no—
401C2×M16—1221no—
402C2×M16C22×D6111yescharacteristic
403C22×C8—611no—
404C22×C8—1221no—
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—21no—
407C4⋊Q8—21no—
408#125—21no—
409#125—21no—
410C22⋊Q8—41no—
411C4×Q8—1221no—
412C4×Q8—1221no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—2441no—
420C4×D4—2441no—
421#117—1221no—
422#117—1221no—
423#117C22×D6111yescharacteristic
424#117C22×D6111yescharacteristic
425#117C22×D6111yescharacteristic
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—2441no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—41no—
433C2×C22⋊C4—2441no—
434C2×C42—611no—
435#108—1222no—
436#107—21no—
437#107—1221no—
438#107—1221no—
439#107—1221no—
440#107—1221no—
441#107—1221no—
442#107—1221no—
443#107—1221no—
444#107—1221no—
445#107C22⋊D6111yescharacteristic
446#107C22⋊D6111yescharacteristic
447#106—21no—
448#106—1221no—
449#106—1221no—
450#106—1221no—
451#106—1221no—
452#106C22⋊D6111yescharacteristic
453#106C22⋊D6111yescharacteristic
454C4⋊C8—21no—
455C4⋊C8—21no—
456C4⋊C8—1221no—
457C4⋊C8—1221no—
458#104—1222no—
459#103—611no—
460#103—611no—
461#103—1221no—
462#102—21no—
463#102—21no—
464#102—21no—
465#102—21no—
466#102—611no—
467#102—611no—
468#102—1221no—
469#102—1221no—
470#102—1221no—
471#100—1222no—
472#97—1221no—
473C4×C8—1221no—
474C24—21no—
475#41—612no—
476#41—612no—
477#41—1221no—
478#41—1221no—
479C2×Q8—41no—
480C2×Q8—611no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—611no—
487C2×D4—1221no—
488C2×D4Q8×D6112yes—
489C22×C4—21no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—2441no—
497C22×C4—2441no—
498C22×C4C23×D6111yescharacteristic
499SD16—1224no—
500D8—1224no—
501M16—21no—
502M16—21no—
503M16—612no—
504M16—612no—
505M16—1224no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509M16—1221no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—1224no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8C2×C22⋊D6111yescharacteristic
522C2×C8C2×C22⋊D6111yescharacteristic
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—4882no—
533C4⋊C4—2441no—
534C4⋊C4—2441no—
535C4⋊C4D4×D6111yescharacteristic
536C4⋊C4D4×D6111yescharacteristic
537C4⋊C4C2×C22⋊D6111yescharacteristic
538C4⋊C4C2×C22⋊D6111yescharacteristic
539C4⋊C4C2×C22⋊D6111yescharacteristic
540C4⋊C4C2×C22⋊D6111yescharacteristic
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—4882no—
547C22⋊C4—2441no—
548C22⋊C4—2441no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—1221no—
553C42—1221no—
554C42—2441no—
555C42—2441no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—81no—
560C23—81no—
561C23—1221no—
562C23#1451982211yescharacteristic
563Q8—161no—
564Q8—612no—
565Q8—1221no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—612no—
571D4—1222no—
572D4—1222no—
573D4—1221no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—81no—
582C2×C4—161no—
583C2×C4—161no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—9761no—
589C2×C4—9761no—
590C2×C4—4881no—
591C2×C4—4881no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C2×Q8×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4#1451982511yescharacteristic
602C8—21no—
603C8—21no—
604C8—41no—
605C8—1222no—
606C8—1222no—
607C8—1221no—
608C8—1221no—
609C8—2441no—
610C22—21no—
611C22—21no—
612C22—42no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—161no—
617C22—1222no—
618C22—2441no—
619C22—2441no—
620C22#1684035811yescharacteristic
621C22D4×C22⋊D6111yescharacteristic
622C22#1684046611yescharacteristic
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—1222no—
631C4—4881no—
632C4—4881no—
633C4—2442no—
634C4C22⋊Q8×D6111yescharacteristic
635C4#1684026911yescharacteristic
636C4#1684041411yescharacteristic
637C4Q8×C22⋊D6111yescharacteristic
638C2—21no—
639C2—21no—
640C2—82no—
641C2—2442no—
642C2#2144962511yescharacteristic
643C2#2145759211yescharacteristic
644C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3081C2
socleC22×C61D42
prime radicalC2×#14519803C22
hypercenter#3081D61
hypocenterC61#26400
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3081

Complements

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

NH ≅ G/Ntype
1. #21455666C2semidirect
3. #21454416C2semidirect
4. #21447345C2semidirect
5. #21447324C2semidirect
6. #21447179C2semidirect
7. #21447161C2semidirect
8. #21446601C2semidirect
9. #21446420C2semidirect
12. #21446093C2semidirect
13. #21446042C2semidirect
14. #21446029C2semidirect
15. C61×#3081C2semidirect
69. #14519803Q8semidirect
317. #3081D61semidirect
389. C61#26400semidirect

Maximal subgroups

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

subgroupindex
#214556662
#214546512
#214544162
#214473452
#214473242
#214471792
#214471612
#214466012
#214464202
#214463302
#214461922
#214460932
#214460422
#214460292
C61×#30812
#2640061