Finite Group Databasebeta

← #29836715

Subgroups — #29836715

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840193C211yesmaximal, characteristic, complemented
2#21457573C211yesmaximal, characteristic, complemented
3#21457535C211yesmaximal, characteristic, complemented
4#21448073C211yesmaximal, characteristic, complemented
5#21447915C211yesmaximal, characteristic, complemented
6#21447912C211yesmaximal, characteristic, complemented
7#21447867C211yesmaximal, characteristic, complemented
8#21447852C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447758C211yesmaximal, characteristic, complemented
11#21447742C211yesmaximal, characteristic, complemented
12#21447739C211yesmaximal, characteristic, complemented
13SD32×Dic61C211yesmaximal, characteristic, complemented
14C16×C22⋊D61C211yesmaximal, characteristic, complemented
15C2×C61×#508C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519796C2211yescharacteristic
18#16840431—21no—
19#16840411—21no—
20#16840193—24no—
21C2×#14519723C2211yescharacteristic
22C2×#14519722C2211yescharacteristic
23C2×#14519721C2211yescharacteristic
24C2×#14519720C2211yescharacteristic
25C2×#14519718C2211yescharacteristic
26C2×C16×D61C2211yescharacteristic
27#16840140C2211yescharacteristic
28#16840137C2211yescharacteristic
29#16840003C2211yescharacteristic
30#16840000C2211yescharacteristic
31#16839998C2211yescharacteristic
32#16839995C2211yescharacteristic
33#16839965—21no—
34#16839960—21no—
35#16839924—21no—
36#16839921—21no—
37#16839907—21no—
38#16839906—21no—
39Q16×Dic61C2211yescharacteristic
40D8×Dic61C2211yescharacteristic
41C8×C22⋊D61C2211yescharacteristic
42#16839590C2211yescharacteristic
43#16839580C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839563C2211yescharacteristic
50C22⋊#11758185C2211yescharacteristic
51#16839541C2211yescharacteristic
52C16×Dic61C2211yescharacteristic
53C2×#135×C61C2211yescharacteristic
54C2×#135×C61C2211yescharacteristic
55C61×#508—24no—
56C2×Q32×C61C2211yescharacteristic
57C2×SD32×C61C2211yescharacteristic
58C2×SD32×C61C2211yescharacteristic
59C2×D16×C61C2211yescharacteristic
60C22×C16×C61C2211yescharacteristic
61C2×#11758226—21no—
62C2×#11758224—21no—
63#14519800—22no—
64#14519800—41no—
65#14519796—22no—
66#14519796—41no—
67C2×Dic244C2311yescharacteristic
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758201—21no—
71C2×D488C2311yescharacteristic
72C2×#11758199—21no—
73C2×C8×D61C2311yescharacteristic
74#14519775—21no—
75#14519773—21no—
76#14519773—41no—
77#14519760—41no—
78#14519758—41no—
79#14519756—21no—
80#14519753—21no—
81#14519751—21no—
82#14519751—41no—
83#14519749—41no—
84#14519748—21no—
85#14519748—41no—
86Q8×Dic61—21no—
87D4×Dic61—21no—
88C4×C22⋊D61—41no—
89C4×C22⋊D61D411yescharacteristic
90C2×#11758196—41no—
91#14519723—22no—
92#14519722—22no—
93#14519721—41no—
94#14519720—41no—
95#14519718—22no—
96C2×#11758185C2311yescharacteristic
97C2×#11758185C2311yescharacteristic
98C16×D61—22no—
99#14519711C2311yescharacteristic
100#14519711C2311yescharacteristic
101#14519706D411yescharacteristic
102#14519702—21no—
103#14519701—21no—
104#14519701—21no—
105#14519699—21no—
106#14519698—21no—
107#14519698—21no—
108C22⋊#92418D411yescharacteristic
109#14519689D411yescharacteristic
110C8×Dic61C2311yescharacteristic
111C2×#41×C61—21no—
112C2×#41×C61—21no—
113#135×C61—22no—
114#135×C61—22no—
115#135×C61—41no—
116#135×C61—41no—
117C2×Q16×C61C2311yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124Q32×C61—22no—
125SD32×C61—22no—
126SD32×C61D414yescomplemented
127D16×C61—22no—
128C2×C16×C61—22no—
129C2×C16×C61C2311yescharacteristic
130C2×C16×C61C2311yescharacteristic
131#41×C61—42no—
132#41×C61—42no—
133#41×C61—81no—
134#41×C61—81no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61C2×D411yescharacteristic
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61—41no—
148C2×C22⋊D61—41no—
149C2×C22⋊D61D811yescharacteristic
150C22×Dic61—41no—
151C22×Dic61—41no—
152#11758226—42no—
153#11758226—81no—
154#11758224—42no—
155#11758224—81no—
156C2×D244—21no—
157C2×D244—41no—
158C2×C4×D61—41no—
159C2×C4×D61C2×D411yescharacteristic
160C2×Dic122—21no—
161Q16×C61—21no—
162Q16×C61—21no—
163Q16×C61—21no—
164Q16×C61C2×D412yes—
165SD16×C61—42no—
166SD16×C61—42no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61—21no—
170D8×C61C2×D412yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2411yescharacteristic
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61D811yescharacteristic
179#11758202—81no—
180#11758201—42no—
181#11758200—42no—
182#11758199—81no—
183#11758198—41no—
184#11758198—41no—
185#11758198D811yescharacteristic
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—41no—
189#11758196—41no—
190#11758196—41no—
191#11758196—41no—
192#11758196—41no—
193#11758196D811yescharacteristic
194C4×Dic61—41no—
195C4×Dic61C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197C2×#92418C2×D411yescharacteristic
198Dic244—21no—
199Dic244—21no—
200D488—21no—
201D488—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C2×D811yescharacteristic
211C22×D61—41no—
212C22×D61C2×D811yescharacteristic
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217D4×C61—22no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—82no—
222D4×C61—82no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—82no—
228C2×C4×C61—82no—
229C2×C4×C61C2×D811yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C22⋊D61—22no—
232C22⋊D61—82no—
233C22⋊D61—82no—
234C22⋊D61—161no—
235C22⋊D61—161no—
236C22⋊D61—161no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241C2×Dic61—82no—
242C2×Dic61C2×D811yescharacteristic
243C2×Dic61C2×D811yescharacteristic
244C2×Dic61C2×D811yescharacteristic
245D244—41no—
246D244—41no—
247D244—82no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—82no—
251Dic122—41no—
252Dic122—41no—
253C8×C61—21no—
254C8×C61C22×D411yescharacteristic
255C8×C61H(5,𝔽2)11yescharacteristic
256#92418—21no—
257#92418—21no—
258#33850—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—42no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61C22×D811yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C4×C61—21no—
276C4×C61—42no—
277C4×C61—81no—
278C4×C61D4211yescharacteristic
279C4×C61#57611yescharacteristic
280Dic61—21no—
281Dic61—21no—
282Dic61—81no—
283Dic61—81no—
284Dic61—81no—
285Dic61#50812yescomplemented
286C2×#508—611no—
287C2×#508D6111yescharacteristic, complemented
288#3287—611no—
289#3283—611no—
290#2248—611no—
291#2217—611no—
292#2217—611no—
293#2209—611no—
294#2206—611no—
295#2189—611no—
296#2189—611no—
297#2186—611no—
298#2186—611no—
299C4×SD32—611no—
300D4×C16—611no—
301C2×C61—41no—
302C2×C61—82no—
303C2×C61—161no—
304C2×C61D4×D811yescharacteristic
305C2×C61C2×#50811yescharacteristic
306C2×C61#341611yescharacteristic
307D61—41no—
308D61—161no—
309C2×#135—611no—
310C2×#135—611no—
311C2×#135C2×D6111yescharacteristic
312C2×#135C2×D6111yescharacteristic
313#550—1221no—
314#547—1221no—
315#508—24no—
316#508—1224no—
317C2×Q32—611no—
318C2×Q32C2×D6111yescharacteristic
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32C2×D6111yescharacteristic
323C2×SD32C2×D6111yescharacteristic
324C2×D16—611no—
325C2×D16C2×D6111yescharacteristic
326C22×C16—611no—
327C22×C16C2×D6111yescharacteristic
328#494—611no—
329#493—611no—
330#467—611no—
331#467—611no—
332#466—611no—
333#466—611no—
334#461—1221no—
335#460—1221no—
336#452—1221no—
337#452—1221no—
338#449—1221no—
339#449—1221no—
340C4×Q16—611no—
341C4×D8—611no—
342C8×D4—611no—
343#367—611no—
344C4⋊C16—611no—
345#358—611no—
346#358—611no—
347#358—611no—
348#357—611no—
349#357—611no—
350#357—611no—
351C22⋊C16—611no—
352C22⋊C16—611no—
353C4×C16—611no—
354C61#3385011yescharacteristic, complemented, Sylow
355C2×#41—21no—
356C2×#41—21no—
357C2×#41—1221no—
358C2×#41—1221no—
359#135—22no—
360#135—22no—
361#135—41no—
362#135—41no—
363#135—1222no—
364#135—1222no—
365#135—2441no—
366#135—2441no—
367C2×Q16—611no—
368C2×Q16—1221no—
369C2×Q16C22×D6111yescharacteristic
370C2×Q16C22×D6111yescharacteristic
371C2×SD16—21no—
372C2×SD16—21no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×D8—611no—
376C2×D8—1221no—
377C2×D8C22×D6111yescharacteristic
378C2×D8C22×D6111yescharacteristic
379C22×C8—611no—
380C22×C8C22×D6111yescharacteristic
381C4⋊Q8—1221no—
382#127—1221no—
383#127—2441no—
384#123—2441no—
385#123—2441no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388#121—1221no—
389#121—1221no—
390#121—2441no—
391#121—2441no—
392#121—2441no—
393C4×Q8—1221no—
394C4×D4—611no—
395C4×D4—1221no—
396C4×D4—2441no—
397C2×C4⋊C4—2441no—
398Q32—22no—
399Q32—1222no—
400SD32—22no—
401SD32—1222no—
402SD32—2441no—
403SD32—2441no—
404SD32C22⋊D6114yescomplemented
405D16—22no—
406D16—1222no—
407C2×C16—22no—
408C2×C16—611no—
409C2×C16—611no—
410C2×C16—1222no—
411C2×C16C22×D6111yescharacteristic
412C2×C16C22×D6111yescharacteristic
413#107—611no—
414#107—611no—
415C4⋊C8—611no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—611no—
423C22⋊C8—611no—
424C4×C8—611no—
425#41—42no—
426#41—42no—
427#41—81no—
428#41—81no—
429#41—4881no—
430#41—4881no—
431#41—2442no—
432#41—2442no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—1221no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—611no—
441C2×D4—1221no—
442C2×D4—2441no—
443C2×D4—2441no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4—2441no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—611no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4—2441no—
454C22×C4D4×D6111yescharacteristic
455Q16—21no—
456Q16—21no—
457Q16—21no—
458Q16—1221no—
459Q16—1221no—
460Q16—4881no—
461Q16C2×C22⋊D6112yes—
462SD16—42no—
463SD16—42no—
464SD16—2442no—
465SD16—2442no—
466D8—21no—
467D8—21no—
468D8—21no—
469D8—1221no—
470D8—1221no—
471D8—4881no—
472D8C2×C22⋊D6112yes—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8D4×D6111yescharacteristic
480C2×C8C23×D6111yescharacteristic
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C4⋊C4—2441no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C22⋊C4—611no—
490C22⋊C4—611no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C42—611no—
497C42—2441no—
498C16—21no—
499C16—1221no—
500C16—1221no—
501C16C2×C22⋊D6112yes—
502C23—41no—
503C23—41no—
504C23—611no—
505C23—2441no—
506C23D8×D6111yescharacteristic
507Q8—22no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—2441no—
512Q8—2441no—
513D4—22no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—82no—
518D4—82no—
519D4—1222no—
520D4—9761no—
521D4—9761no—
522D4—9761no—
523D4—4882no—
524D4—4882no—
525D4—4882no—
526D4—2441no—
527D4—2441no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—4882no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4D8×D6111yescharacteristic
546C2×C4C2×D4×D6111yescharacteristic
547C8—21no—
548C8—1221no—
549C8—1221no—
550C8C22×C22⋊D6111yescharacteristic
551C8#1451982211yescharacteristic
552C22—21no—
553C22—21no—
554C22—21no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22C2×D8×D6111yescharacteristic
568C4—21no—
569C4—42no—
570C4—81no—
571C4—612no—
572C4—1221no—
573C4—1221no—
574C4—4881no—
575C4—4881no—
576C4—4881no—
577C4D4×C22⋊D6111yescharacteristic
578C4#1684056311yescharacteristic
579C2—41no—
580C2—82no—
581C2—161no—
582C2—9761no—
583C2—2441no—
584C2D8×C22⋊D6111yescharacteristic
585C2#508×D6111yescharacteristic
586C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#508C2
socleC22×C61C22×D8
hypercenterC2×#508D61
hypocenterC61#33850
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. C2×#16840193C2semidirect
2. #21457573C2semidirect
3. #21457535C2semidirect
4. #21448073C2semidirect
5. #21447915C2semidirect
6. #21447912C2semidirect
7. #21447867C2semidirect
8. #21447852C2semidirect
9. #21447759C2semidirect
10. #21447758C2semidirect
11. #21447742C2semidirect
12. #21447739C2semidirect
13. SD32×Dic61C2semidirect
14. C16×C22⋊D61C2semidirect
15. C2×C61×#508C2semidirect
126. SD32×C61D4semidirect
285. Dic61#508semidirect
287. C2×#508D61semidirect
354. C61#33850semidirect
404. SD32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401932
#214575732
#214575352
#214480732
#214479152
#214479122
#214478672
#214478522
#214477592
#214477582
#214477422
#214477392
SD32×Dic612
C16×C22⋊D612
C2×C61×#5082
#3385061