Finite Group Databasebeta

← #30622273

Subgroups — #30622273

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462578C211yesmaximal, characteristic, complemented
2#21462576C212yesmaximal, complemented
3#21462474C211yesmaximal, characteristic, complemented
4#21461966C211yesmaximal, characteristic, complemented
5#21461937C211yesmaximal, characteristic, complemented
6#21461905C212yesmaximal, complemented
7#21461819C212yesmaximal, complemented
8#21461774C212yesmaximal, complemented
9#21461624C211yesmaximal, characteristic, complemented
10#21461593C211yesmaximal, characteristic, complemented
11#472×D61C211yesmaximal, characteristic
12#16840831C2212yes—
13#16840829C2212yes—
14#16840803—22no—
15#16840803—22no—
16#16840801—22no—
17SD16×#6923—22no—
18D8×#6923—21no—
19D8×#6923—21no—
20D8×#6923—22no—
21#16840773—22no—
22#16840773C2212yes—
23#16840773C2212yes—
24#16840772—21no—
25#16840772C2211yescharacteristic
26C2×#14519846—21no—
27C2×#14519846C2211yescharacteristic
28D61⋊#104C2211yescharacteristic
29#16840754C2212yes—
30#16840752C2211yescharacteristic
31C2×#14519839C2212yes—
32#16840728C2212yes—
33#16840725C2211yescharacteristic
34C2×C8×#6923C2211yescharacteristic
35#16840714C2212yes—
36#16840641C2212yes—
37#16840637C2211yescharacteristic
38#16840634C2211yescharacteristic
39#16840634C2212yes—
40AGL(1,ℤ/8)×D61C2212yes—
41C2×D8×D61C2211yescharacteristic
42#131×D61C2211yescharacteristic
43#16840033C411yescharacteristic, complemented
44#16840032C411yescharacteristic, complemented
45#16840031C412yescomplemented
46#16840030C412yescomplemented
47#16840029C411yescharacteristic, complemented
48#108×D61C2211yescharacteristic
49#100×D61C2212yes—
50C61×#472C411yescharacteristic, complemented
51D61⋊M16C2311yescharacteristic
52C2×D61⋊C8C2311yescharacteristic
53Q8×#6923—22no—
54D4×#6923—22no—
55D4×#6923—22no—
56D4×#6923—22no—
57D4×#6923—22no—
58D4×#6923—22no—
59D4×#6923—42no—
60#14519856—22no—
61#14519856C2311yescharacteristic
62#14519856C2312yes—
63C2×C4⋊#6923—22no—
64C2×C4⋊#6923C2312yes—
65C2×C22⋊#6923—42no—
66C2×C22⋊#6923—42no—
67C2×C4×#6923C2311yescharacteristic
68#14519846—22no—
69#14519846—22no—
70#14519846—21no—
71#14519846—21no—
72#14519846D411yescharacteristic
73#14519846D411yescharacteristic
74#14519845—22no—
75#14519845—22no—
76#14519845D412yes—
77#14519842—21no—
78#14519841—21no—
79#14519840D411yescharacteristic
80#14519840D411yescharacteristic
81#14519839—22no—
82#14519839—22no—
83#14519839—22no—
84#14519839—22no—
85#14519839D411yescharacteristic
86#14519839D411yescharacteristic
87#14519831—22no—
88#14519831—21no—
89C8×#6923—22no—
90C8×#6923—21no—
91C8×#6923—21no—
92C8×#6923—21no—
93#41×D61C2311yescharacteristic
94C2×D4×D61C2312yes—
95#14519805C2×C412yescomplemented
96#14519804—22no—
97#14519803C2×C412yescomplemented
98#14519802C2×C412yescomplemented
99#14519801—22no—
100SD16×D61—22no—
101C2×D488C2×C411yescharacteristic, complemented
102C2×#11758199C2×C412yescomplemented
103D8×D61—22no—
104D8×D61—21no—
105D8×D61—21no—
106D8×D61—21no—
107D8×D61—21no—
108#14519785—21no—
109#14519785C2×C411yescharacteristic, complemented
110#14519784C2×C411yescharacteristic, complemented
111#14519783—21no—
112#14519783C2×C411yescharacteristic, complemented
113M16×D61—21no—
114M16×D61C2312yes—
115M16×D61C2311yescharacteristic
116C2×C8×D61—21no—
117C2×C8×D61C2311yescharacteristic
118#14519713C2×C411yescharacteristic, complemented
119#14519712C2×C412yescomplemented
120#14519694C2×C412yescomplemented
121#14519693C2×C412yescomplemented
122#14519693C2×C412yescomplemented
123AGL(1,ℤ/8)×C61C2×C412yescomplemented
124C2×D8×C61C2×C411yescharacteristic, complemented
125#131×C61C2×C411yescharacteristic, complemented
126#108×C61C2×C411yescharacteristic, complemented
127#100×C61C2×C412yescomplemented
128C23×D61—22no—
129C22×#6923—21no—
130C22×#6923—22no—
131C22×#6923—42no—
132C22×#6923—42no—
133C22×#6923—42no—
134#41×C61C22×C411yescharacteristic
135C2×D4×C61C22×C412yes—
136C2×C22⋊D61—22no—
137#11758226#4111yescharacteristic
138Q8×D61C2×D411yescharacteristic
139#11758224—21no—
140#11758224#4111yescharacteristic
141D4×D61—21no—
142D4×D61—22no—
143D4×D61—22no—
144D4×D61—22no—
145D4×D61—22no—
146D4×D61C2×D411yescharacteristic
147#11758222—21no—
148#11758222C22×C411yescharacteristic
149C2×D244C22×C412yes—
150C2×C4×D61—21no—
151C2×C4×D61C2411yescharacteristic
152C22⋊#6923—21no—
153C22⋊#6923—22no—
154C22⋊#6923—42no—
155C22⋊#6923—42no—
156C22⋊#6923—42no—
157C22⋊#6923—84no—
158#11758217—21no—
159C2×#92420—21no—
160C4⋊#6923—22no—
161C4⋊#6923—22no—
162C4⋊#6923—42no—
163C4⋊#6923C2×D412yes—
164C4⋊#6923C2×D412yes—
165C4⋊#6923C2×D411yescharacteristic
166C4⋊#6923C2×D411yescharacteristic
167C4×#6923—21no—
168C4×#6923—22no—
169C4×#6923—21no—
170C4×#6923—21no—
171C4×#6923—42no—
172#11758213—21no—
173D61⋊C8—21no—
174D61⋊C8—21no—
175D61⋊C8C2×D411yescharacteristic
176D61⋊C8C2×D411yescharacteristic
177SD16×C61—22no—
178D8×C61—22no—
179D8×C61—21no—
180D8×C61—21no—
181M16×C61—21no—
182M16×C61C22×C411yescharacteristic
183M16×C61C22×C412yes—
184C2×C8×C61—21no—
185C2×C8×C61C22×C411yescharacteristic
186#11758201—22no—
187#11758200—22no—
188#11758199—22no—
189#11758199—22no—
190#11758199—22no—
191#11758199—22no—
192#11758194—21no—
193#11758194C22×C411yescharacteristic
194#11758194C22×C412yes—
195C2×#92418—21no—
196C2×#92418C22×C411yescharacteristic
197D488—22no—
198D488—21no—
199D488—21no—
200#11758190—22no—
201#11758189—21no—
202#11758189—22no—
203#11758189—21no—
204#11758189—21no—
205C8×D61—22no—
206C8×D61—21no—
207C8×D61—21no—
208C8×D61—21no—
209C8×D61C2×D411yescharacteristic
210C8×D61C2×D411yescharacteristic
211C8×D61#4111yescharacteristic
212C8×D61#4111yescharacteristic
213C23×C61—22no—
214C22×D61—21no—
215C22×D61—22no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61H(5,𝔽2)11yescharacteristic
221C2×#6923—22no—
222C2×#6923—22no—
223C2×#6923—22no—
224C2×#6923—21no—
225C2×#6923—21no—
226C2×#6923—41no—
227C2×#6923—41no—
228C2×#6923—42no—
229C2×#6923—82no—
230C2×#6923—162no—
231C2×#6923—162no—
232C2×#6923—162no—
233Q8×C61C4×D411yescharacteristic
234D4×C61—21no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61C4×D411yescharacteristic
238C2×C4×C61—21no—
239C2×C4×C61C23×C411yescharacteristic
240C22⋊D61—21no—
241C22⋊D61—21no—
242C22⋊D61—21no—
243C22⋊D61—42no—
244C22⋊D61—42no—
245C2×Dic61—21no—
246C2×Dic612−1+411yescharacteristic
247D244—21no—
248D244—22no—
249D244—22no—
250D244C4×D411yescharacteristic
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61C22×D411yescharacteristic
255C4×D61C22×D411yescharacteristic
256C4×D61C2×#4111yescharacteristic
257C4×D61C2×#4111yescharacteristic
258Dic122—21no—
259Dic122C4×D411yescharacteristic
260#92420—21no—
261#92420—21no—
262#92420—41no—
263C8×C61—21no—
264C8×C61—22no—
265C8×C61C4×D411yescharacteristic
266C8×C61C4×D411yescharacteristic
267#92418—21no—
268#92418—22no—
269#92418C4×D411yescharacteristic
270#92418C4×D411yescharacteristic
271#23046—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—42no—
274C22×C61—82no—
275C22×C61#52011yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—42no—
282C2×D61—82no—
283C2×D61—82no—
284C2×D61—82no—
285C2×D61—162no—
286C2×D61D4211yescharacteristic
287C2×D61#55311yescharacteristic
288#6923—42no—
289#6923—42no—
290#6923—41no—
291#6923—42no—
292#6923—41no—
293#6923—82no—
294C4×C61—21no—
295C4×C61C2×C4×D411yescharacteristic
296C4×C61C2×C4×D411yescharacteristic
297Dic61—21no—
298Dic61#54711yescharacteristic
299Dic61#54811yescharacteristic
300C2×#472—611no—
301#2948—611no—
302#2946—612no—
303#2913—611no—
304#1970—611no—
305#1952—611no—
306#1931—612no—
307#1883—612no—
308#1856—612no—
309#1772—611no—
310#1764—611no—
311C2×C61—21no—
312C2×C61—41no—
313C2×C61—82no—
314C2×C61#236111yescharacteristic
315D61—21no—
316D61—41no—
317D61—82no—
318D61#329011yescharacteristic
319D61#329211yescharacteristic
320C2×AGL(1,ℤ/8)—612no—
321C22×D8—611no—
322C2×#131—611no—
323#520—612no—
324#518—612no—
325#472—612no—
326#472—611no—
327#472—611no—
328#472—611no—
329#472—612no—
330#472#692311yescharacteristic, complemented
331#442—1222no—
332#442—1222no—
333#440—1222no—
334C4×SD16—1222no—
335C4×D8—1222no—
336C4×D8—1221no—
337C4×D8—1221no—
338C2×#108—611no—
339#428—612no—
340#428—612no—
341#428—1222no—
342#427—611no—
343#427—1221no—
344C2×#107—611no—
345C2×#107—1221no—
346C2×#104—611no—
347#418—612no—
348#416—611no—
349C2×#102—612no—
350C2×#100—612no—
351#405—611no—
352#405—612no—
353C2×C4×C8—611no—
354#390—612no—
355#343—612no—
356#341—611no—
357#339—611no—
358#339—612no—
359C61#2304611yescharacteristic, complemented, Sylow
360C2×#41—611no—
361C22×D4—612no—
362AGL(1,ℤ/8)—612no—
363AGL(1,ℤ/8)—612no—
364AGL(1,ℤ/8)—612no—
365AGL(1,ℤ/8)—1222no—
366AGL(1,ℤ/8)—1222no—
367AGL(1,ℤ/8)C2×#692312yescomplemented
368C2×SD16—1222no—
369C2×D8—612no—
370C2×D8—611no—
371C2×D8—1221no—
372C2×D8—1221no—
373C2×D8—1221no—
374C2×D8—1221no—
375C2×D8—1222no—
376C2×D8C2×#692311yescharacteristic, complemented
377#131—611no—
378#131—611no—
379#131—611no—
380#131—1221no—
381#131—1221no—
382#131C2×#692311yescharacteristic, complemented
383C2×M16—611no—
384C2×M16—612no—
385C2×M16—611no—
386C2×M16—1221no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8—1221no—
390C4×Q8—1222no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1222no—
396C4×D4—2442no—
397#117—611no—
398#117—612no—
399#117—1222no—
400C2×C4⋊C4—612no—
401C2×C4⋊C4—1222no—
402C2×C22⋊C4—2442no—
403C2×C22⋊C4—2442no—
404C2×C42—611no—
405#108—612no—
406#108—611no—
407#108C2×#692311yescharacteristic, complemented
408#107—611no—
409#107—611no—
410#107—1222no—
411#107—1222no—
412#107—1221no—
413#107—1221no—
414#106—612no—
415#106—1222no—
416#106—1222no—
417#104—1221no—
418#104—1221no—
419#103—611no—
420#103—611no—
421#102—611no—
422#102—611no—
423#102—1222no—
424#102—1222no—
425#102—1222no—
426#102—1222no—
427#100—612no—
428#100—612no—
429#100—612no—
430#100C2×#692312yescomplemented
431#97—1222no—
432#97—1221no—
433C4×C8—1221no—
434C4×C8—1221no—
435C4×C8—1221no—
436C4×C8—1222no—
437C24—1222no—
438#41—611no—
439#41—611no—
440#41—611no—
441#41—1221no—
442#41—1221no—
443#41C22×#692311yescharacteristic
444C2×Q8—611no—
445C2×D4—611no—
446C2×D4—612no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—1221no—
453C2×D4C22×#692312yes—
454C22×C4—611no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1222no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4—2442no—
461SD16—22no—
462SD16—1222no—
463SD16—1222no—
464SD16—1222no—
465D8—22no—
466D8—21no—
467D8—21no—
468D8—1222no—
469D8—1222no—
470D8—1222no—
471D8—1222no—
472D8—1222no—
473D8—1221no—
474D8—1221no—
475M16—21no—
476M16—611no—
477M16—612no—
478M16—1221no—
479M16—1221no—
480M16—1222no—
481M16—1221no—
482M16—1221no—
483M16—1221no—
484M16—1221no—
485M16C22×#692311yescharacteristic
486M16C22×#692312yes—
487C2×C8—21no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—611no—
494C2×C8—611no—
495C2×C8—1221no—
496C2×C8—1222no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8C22×#692311yescharacteristic
504C4⋊C4—611no—
505C4⋊C4—611no—
506C4⋊C4—612no—
507C4⋊C4—612no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—2442no—
511C22⋊C4—1221no—
512C22⋊C4—1222no—
513C22⋊C4—4884no—
514C22⋊C4—2442no—
515C22⋊C4—2442no—
516C22⋊C4—2442no—
517C42—1221no—
518C42—1222no—
519C42—1221no—
520C42—1221no—
521C42—2442no—
522C23—22no—
523C23—611no—
524C23—1221no—
525C23—1222no—
526C23—4882no—
527C23—4882no—
528C23—2442no—
529C23—2442no—
530Q8—611no—
531Q8—1221no—
532Q8D4×#692311yescharacteristic
533D4—21no—
534D4—22no—
535D4—22no—
536D4—611no—
537D4—1221no—
538D4—1221no—
539D4—1221no—
540D4—1221no—
541D4—1222no—
542D4—1222no—
543D4—2442no—
544D4—2442no—
545D4D4×#692311yescharacteristic
546C2×C4—21no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—9762no—
562C2×C4—9762no—
563C2×C4—9762no—
564C2×C4—4882no—
565C2×C4—2442no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4C23×#692311yescharacteristic
569C8—21no—
570C8—22no—
571C8—611no—
572C8—611no—
573C8—1221no—
574C8—1221no—
575C8—1221no—
576C8—1222no—
577C8—2441no—
578C8D4×#692311yescharacteristic
579C8D4×#692311yescharacteristic
580C22—21no—
581C22—42no—
582C22—82no—
583C22—611no—
584C22—611no—
585C22—1221no—
586C22—1221no—
587C22—1221no—
588C22—9762no—
589C22—4882no—
590C22—4882no—
591C22—4882no—
592C22—2441no—
593C22—2441no—
594C22—2442no—
595C22#1684083111yescharacteristic
596C4—21no—
597C4—611no—
598C4—611no—
599C4—1221no—
600C4—4882no—
601C4—2441no—
602C4—2442no—
603C4—2441no—
604C4—2442no—
605C4—2442no—
606C4C2×D4×#692311yescharacteristic
607C4C2×D4×#692311yescharacteristic
608C2—21no—
609C2—41no—
610C2—82no—
611C2—611no—
612C2—611no—
613C2—1221no—
614C2—4882no—
615C2—2441no—
616C2#2146234411yescharacteristic
subgroupG/H
centerC2#21462344
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#472C4
socleC2×C61#2361
prime radical#472×D61C2
cosocleC2×C4×D61C24
hypercenter#472#6923
hypocenterC61#23046
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #21462578C2semidirect
2. #21462576C2semidirect
3. #21462474C2semidirect
4. #21461966C2semidirect
5. #21461937C2semidirect
6. #21461905C2semidirect
7. #21461819C2semidirect
8. #21461774C2semidirect
9. #21461624C2semidirect
10. #21461593C2semidirect
43. #16840033C4semidirect
44. #16840032C4semidirect
45. #16840031C4semidirect
46. #16840030C4semidirect
47. #16840029C4semidirect
50. C61×#472C4semidirect
95. #14519805C2×C4semidirect
97. #14519803C2×C4semidirect
98. #14519802C2×C4semidirect
101. C2×D488C2×C4semidirect
102. C2×#11758199C2×C4semidirect
109. #14519785C2×C4semidirect
110. #14519784C2×C4semidirect
112. #14519783C2×C4semidirect
118. #14519713C2×C4semidirect
119. #14519712C2×C4semidirect
120. #14519694C2×C4semidirect
121. #14519693C2×C4semidirect
122. #14519693C2×C4semidirect
123. AGL(1,ℤ/8)×C61C2×C4semidirect
124. C2×D8×C61C2×C4semidirect
125. #131×C61C2×C4semidirect
126. #108×C61C2×C4semidirect
127. #100×C61C2×C4semidirect
330. #472#6923semidirect
359. C61#23046semidirect
367. AGL(1,ℤ/8)C2×#6923semidirect
376. C2×D8C2×#6923semidirect
382. #131C2×#6923semidirect
407. #108C2×#6923semidirect
430. #100C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625782
#214625762
#214624742
#214619662
#214619372
#214619052
#214618192
#214617742
#214616242
#214615932
#472×D612
#2304661