Finite Group Databasebeta

← #29669444

Subgroups — #29669444

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758199C211yesmaximal, characteristic, complemented
2#21458450C211yesmaximal, characteristic
3Q16×D244C211yesmaximal, characteristic, complemented
4#21445488C211yesmaximal, characteristic, complemented
5#21445308C211yesmaximal, characteristic, complemented
6#21445261C211yesmaximal, characteristic, complemented
7#21445211C211yesmaximal, characteristic, complemented
8#21445157C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic, complemented
10#21445066C211yesmaximal, characteristic, complemented
11#21445012C211yesmaximal, characteristic, complemented
12#21444920C211yesmaximal, characteristic, complemented
13#21444646C211yesmaximal, characteristic, complemented
14#21444638C211yesmaximal, characteristic, complemented
15C61×#3278C211yesmaximal, characteristic, complemented
16C2×Q16×D61—21no—
17Q8×D244C2211yescharacteristic
18Q8×D244C2211yescharacteristic
19#16840165—21no—
20#16840165C2211yescharacteristic
21#16840165C2211yescharacteristic
22#16840142C2211yescharacteristic
23#16840108—21no—
24#16840108C2211yescharacteristic
25#16840108C2211yescharacteristic
26#16840042—21no—
27#16840042—21no—
28#16840042C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840039—21no—
31#16840039C2211yescharacteristic
32#16840004—21no—
33#16839972C2211yescharacteristic
34#16839972C2211yescharacteristic
35#16839954—21no—
36#16839954C2211yescharacteristic
37#16839919—21no—
38#16839919—21no—
39Q8×#92418C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41C4×#11758199—21no—
42C4×#11758199—21no—
43C4×#11758199C2211yescharacteristic
44C8×D244C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839509C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C2×#137×C61—21no—
57C61×#550C2211yescharacteristic
58D4×Q8×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#468—21no—
61C61×#462C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#452—21no—
64C61×#448—21no—
65C4×Q16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Q8×D61—21no—
69C2×Dic244—21no—
70C2×Dic244—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73Q16×D61—44no—
74C2×#11758199—21no—
75C2×#11758199D411yescharacteristic
76C2×C8×D61—21no—
77#14519777—21no—
78#14519777—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519757—21no—
82#14519757—21no—
83#14519757—41no—
84#14519757—41no—
85C4×D244—21no—
86C4×D244—21no—
87C4×D244C2311yescharacteristic
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—41no—
94#14519710D411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519700—21no—
101#14519700—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697D411yescharacteristic
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C22×Q8×C61—21no—
118#137×C61—44no—
119C2×Q16×C61—21no—
120C2×Q16×C61D411yescharacteristic
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×SD16×C61—21no—
124C2×M16×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61—21no—
128#127×C61C2311yescharacteristic
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131#121×C61—21no—
132#121×C61—41no—
133C4×Q8×C61C2311yescharacteristic
134C4×Q8×C61C2311yescharacteristic
135C4×D4×C61—21no—
136C4×D4×C61—21no—
137C4×D4×C61C2311yescharacteristic
138#107×C61D411yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#103×C61D411yescharacteristic
143#102×C61—21no—
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×Q8×C61C2×D411yescharacteristic
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61C2×D411yescharacteristic
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C22×C4×C61—41no—
163Q8×D61—42no—
164Q8×D61—42no—
165Q8×D61—81no—
166Q8×D61—81no—
167C2×D244C2×D411yescharacteristic
168C2×C4×D61—21no—
169C2×C4×D61—41no—
170C2×C4×D61—41no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—41no—
174C2×Dic122—41no—
175Q16×C61—42no—
176Q16×C61D814yescomplemented
177SD16×C61—42no—
178SD16×C61—42no—
179SD16×C61—81no—
180M16×C61—42no—
181C2×C8×C61—21no—
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—41no—
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C22⋊C4×C61—21no—
193C22⋊C4×C61—41no—
194C22⋊C4×C61—41no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61C2411yescharacteristic
198#11758202—42no—
199#11758202—42no—
200#11758199—22no—
201#11758199—42no—
202#11758198—21no—
203#11758198—41no—
204#11758198—41no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—41no—
210C4⋊Dic61—41no—
211C4⋊Dic61C2×D411yescharacteristic
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218Dic244—42no—
219Dic244—81no—
220C8×D61—42no—
221C23×C61—21no—
222C23×C61—41no—
223C22×D61—21no—
224Q8×C61—21no—
225Q8×C61—21no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—82no—
229Q8×C61C2×D812yes—
230Q8×C61C2×D812yes—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—82no—
238D4×C61—81no—
239D4×C61—161no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—82no—
249C2×C4×C61—82no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×Dic61—21no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256D244—21no—
257D244#13712yescomplemented
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—82no—
261C4×D61—82no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265Dic122—41no—
266Dic122—82no—
267Dic122—82no—
268C8×C61—21no—
269C8×C61—41no—
270C8×C61C2×D812yes—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#19761—611nomaximal, Sylow
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61D4211yescharacteristic
285C2×D61—22no—
286C2×D61—41no—
287C2×D61—41no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61C22×D811yescharacteristic
296C4×C61C2×#13711yescharacteristic
297C4×C61#57611yescharacteristic
298C4×C61#57811yescharacteristic
299Dic61—41no—
300Dic61—81no—
301Dic61—81no—
302Q8×D8—611no—
303#3358—611no—
304D4×Q16—611no—
305#3278D6111yescharacteristic, complemented
306#1722—611no—
307#1689—611no—
308#1682—611no—
309#1674—611no—
310#1665—611no—
311#1649—611no—
312#1649—611no—
313#1641—611no—
314#1627—611no—
315#1581—611no—
316#1579—611no—
317C2×C61—81no—
318C2×C61—161no—
319C2×C61#327011yescharacteristic
320C2×C61D4×D811yescharacteristic
321C2×C61#328911yescharacteristic
322D61—42no—
323C2×#137—21no—
324C22×Q16—1221no—
325#550C2×D6111yescharacteristic
326D4×Q8—611no—
327D4×Q8—611no—
328D4×Q8C2×D6111yescharacteristic
329#500—611no—
330#500—611no—
331#500—1221no—
332#494—611no—
333#492C2×D6111yescharacteristic
334#486—611no—
335#486—611no—
336#486—1221no—
337#474—611no—
338#474—611no—
339#474—611no—
340#474—1221no—
341#474—1221no—
342#474—1221no—
343#468—21no—
344#467—1221no—
345#462—611no—
346#462—611no—
347#462C2×D6111yescharacteristic
348#460C2×D6111yescharacteristic
349#459—611no—
350#459—1221no—
351#452—21no—
352#451—1221no—
353#451—1221no—
354#448—21no—
355C8×Q8—611no—
356C8×Q8—611no—
357C4×Q16C2×D6111yescharacteristic
358C4×D8—611no—
359C4×D8—1221no—
360C4×D8—1221no—
361#436C2×D6111yescharacteristic
362C8×D4—611no—
363#335—611no—
364#334—611no—
365#332—611no—
366#332—611no—
367#332—611no—
368#331—611no—
369Q8⋊C8—611no—
370Q8⋊C8—611no—
371D4⋊C8—611no—
372D4⋊C8—611no—
373#322—611no—
374C61#1976111yescharacteristic, complemented, Sylow
375C2×#41—21no—
376C22×Q8—21no—
377C22×Q8—1221no—
378C22×Q8—1221no—
379#137—44no—
380C2×Q16—21no—
381C2×Q16—1221no—
382C2×Q16—1221no—
383C2×Q16—1221no—
384C2×Q16—1221no—
385C2×Q16—2444no—
386C2×Q16C22⋊D6111yescharacteristic
387C2×SD16—21no—
388C2×SD16—21no—
389C2×SD16—21no—
390C2×D8—611no—
391C2×D8—1221no—
392C2×M16—21no—
393C22×C8—1221no—
394C4⋊Q8—21no—
395C4⋊Q8—611no—
396C4⋊Q8—611no—
397C4⋊Q8—1221no—
398C4⋊Q8—1221no—
399C4⋊Q8C22×D6111yescharacteristic
400#127—21no—
401#127C22×D6111yescharacteristic
402C22⋊Q8—21no—
403C22⋊Q8—41no—
404C22⋊Q8—1221no—
405C22⋊Q8—1221no—
406C22⋊Q8—2441no—
407C22⋊Q8—2441no—
408#121—21no—
409#121—41no—
410C4×Q8C22×D6111yescharacteristic
411C4×Q8C22×D6111yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—611no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4C22×D6111yescharacteristic
418#107—611no—
419#107—1221no—
420#107—1221no—
421#107—1221no—
422#107—1221no—
423#107—1221no—
424#107—2441no—
425#107C22⋊D6111yescharacteristic
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432#103—21no—
433#103—1221no—
434#103—1221no—
435#103C22⋊D6111yescharacteristic
436#103C22⋊D6111yescharacteristic
437#102—21no—
438#102—611no—
439#102—611no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—2441no—
447C22⋊C8—21no—
448C22⋊C8—1221no—
449C4×C8—611no—
450C4×C8—611no—
451C4×C8—1221no—
452C4×C8—1221no—
453C4×C8C22×D6111yescharacteristic
454#41—42no—
455#41—81no—
456C2×Q8—21no—
457C2×Q8—41no—
458C2×Q8—42no—
459C2×Q8—81no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—4881no—
463C2×Q8—4881no—
464C2×Q8—2441no—
465C2×Q8—2441no—
466C2×Q8—2442no—
467C2×Q8—2442no—
468C2×Q8C2×C22⋊D6111yescharacteristic
469C2×Q8C2×C22⋊D6111yescharacteristic
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—611no—
476C2×D4D4×D6111yescharacteristic
477C22×C4—21no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—1221no—
481C22×C4—2441no—
482C22×C4—2441no—
483Q16—42no—
484Q16—4881no—
485Q16—2442no—
486Q16—2442no—
487Q16—2442no—
488Q16#1175819914yescomplemented
489SD16—42no—
490SD16—42no—
491SD16—81no—
492D8—1222no—
493D8—2442no—
494M16—42no—
495C2×C8—21no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—2442no—
503C2×C8C2×C22⋊D6111yescharacteristic
504C2×C8C2×C22⋊D6111yescharacteristic
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4D4×D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—1221no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—21no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—41no—
531C23—1221no—
532Q8—21no—
533Q8—21no—
534Q8—41no—
535Q8—41no—
536Q8—82no—
537Q8—4882no—
538Q8—4882no—
539Q8—2441no—
540Q8—2441no—
541Q8—2441no—
542Q8—2441no—
543Q8C2×#1175819912yes—
544Q8C2×#1175819912yes—
545D4—21no—
546D4—21no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—82no—
552D4—81no—
553D4—161no—
554D4—612no—
555D4—1221no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—1221no—
567C2×C4—4882no—
568C2×C4—4882no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#1451982211yescharacteristic
576C8—21no—
577C8—41no—
578C8—1221no—
579C8—1221no—
580C8—2441no—
581C8—2441no—
582C8—2441no—
583C8—2441no—
584C8C2×#1175819912yes—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—1222no—
592C22—2441no—
593C22—2441no—
594C22D4×C22⋊D6111yescharacteristic
595C4—22no—
596C4—22no—
597C4—21no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—4881no—
603C4—4881no—
604C4—2441no—
605C4C22×#1175819911yescharacteristic
606C4#137×D6111yescharacteristic
607C4#1684056511yescharacteristic
608C4#1684057811yescharacteristic
609C2—81no—
610C2—161no—
611C2—2442no—
612C2#2145738311yescharacteristic
613C2D4×#1175819911yescharacteristic
614C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3278C2
socleC22×C61D42
prime radical#21458450C2
hypercenter#3278D61
hypocenterC61#19761
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3278

Complements

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

NH ≅ G/Ntype
1. Q8×#11758199C2semidirect
3. Q16×D244C2semidirect
4. #21445488C2semidirect
5. #21445308C2semidirect
6. #21445261C2semidirect
7. #21445211C2semidirect
8. #21445157C2semidirect
9. #21445068C2semidirect
10. #21445066C2semidirect
11. #21445012C2semidirect
12. #21444920C2semidirect
13. #21444646C2semidirect
14. #21444638C2semidirect
15. C61×#3278C2semidirect
176. Q16×C61D8semidirect
257. D244#137semidirect
305. #3278D61semidirect
374. C61#19761semidirect
488. Q16#11758199semidirect

Maximal subgroups

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

subgroupindex
Q8×#117581992
#214584502
Q16×D2442
#214454882
#214453082
#214452612
#214452112
#214451572
#214450682
#214450662
#214450122
#214449202
#214446462
#214446382
C61×#32782
#1976161