Finite Group Databasebeta

← #29836631

Subgroups — #29836631

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840201C211yesmaximal, characteristic, complemented
2Q16×C22⋊D61C211yesmaximal, characteristic, complemented
3#21457535C211yesmaximal, characteristic
4#21448070C211yesmaximal, characteristic, complemented
5#21447927C211yesmaximal, characteristic, complemented
6#21447924C211yesmaximal, characteristic, complemented
7#21447867C211yesmaximal, characteristic, complemented
8#21447862C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447756C211yesmaximal, characteristic, complemented
11#21447753C211yesmaximal, characteristic, complemented
12#21447751C211yesmaximal, characteristic, complemented
13SD32×Dic61C211yesmaximal, characteristic, complemented
14#21447649C211yesmaximal, characteristic, complemented
15C2×C61×#510C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18Q8×C22⋊D61—21no—
19#16840411—21no—
20#16840201—24no—
21C2×Dic488C2211yescharacteristic
22C2×#14519723C2211yescharacteristic
23C2×#14519723C2211yescharacteristic
24C2×#14519723C2211yescharacteristic
25C2×#14519721C2211yescharacteristic
26C2×#14519716C2211yescharacteristic
27#16840140C2211yescharacteristic
28#16840140C2211yescharacteristic
29#16840003C2211yescharacteristic
30#16840003C2211yescharacteristic
31#16840000C2211yescharacteristic
32#16840000C2211yescharacteristic
33#16839970—21no—
34#16839965—21no—
35#16839924—21no—
36#16839921—21no—
37#16839918—21no—
38#16839916—21no—
39Q16×Dic61C2211yescharacteristic
40D8×Dic61C2211yescharacteristic
41C8×C22⋊D61C2211yescharacteristic
42#16839590C2211yescharacteristic
43#16839580C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839567C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839566C2211yescharacteristic
48#16839566C2211yescharacteristic
49#16839563C2211yescharacteristic
50C22⋊#11758185C2211yescharacteristic
51#16839541C2211yescharacteristic
52C16×Dic61C2211yescharacteristic
53C2×#135×C61C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C61×#510—24no—
56C2×Q32×C61C2211yescharacteristic
57C2×Q32×C61C2211yescharacteristic
58C2×SD32×C61C2211yescharacteristic
59C2×SD32×C61C2211yescharacteristic
60C2×M32×C61C2211yescharacteristic
61C2×#11758224—21no—
62C2×Q8×D61—21no—
63#14519796—22no—
64#14519796—41no—
65C2×Dic244C2311yescharacteristic
66C2×Dic244C2311yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70Q16×D61—22no—
71Q16×D61—41no—
72C2×#11758200—21no—
73C2×C8×D61C2311yescharacteristic
74#14519775—21no—
75#14519775—21no—
76#14519775—41no—
77#14519760—41no—
78#14519758—41no—
79#14519756—21no—
80#14519756—21no—
81#14519756—41no—
82#14519753—21no—
83#14519753—21no—
84#14519753—41no—
85#14519749—41no—
86Q8×Dic61—21no—
87D4×Dic61—21no—
88C4×C22⋊D61—41no—
89C4×C22⋊D61D411yescharacteristic
90C2×#11758196—41no—
91Dic488—22no—
92#14519723—22no—
93#14519723—41no—
94#14519723—41no—
95#14519721—22no—
96#14519716—22no—
97C2×#11758185C2311yescharacteristic
98C2×#11758185C2311yescharacteristic
99#14519711C2311yescharacteristic
100#14519711C2311yescharacteristic
101#14519706D411yescharacteristic
102#14519702—21no—
103#14519702—21no—
104#14519701—21no—
105#14519701—21no—
106#14519701—21no—
107#14519698—21no—
108C22⋊#92418D411yescharacteristic
109#14519689D411yescharacteristic
110C8×Dic61C2311yescharacteristic
111C2×#41×C61—21no—
112C22×Q8×C61—21no—
113#135×C61—22no—
114#135×C61—41no—
115C2×Q16×C61—22no—
116C2×Q16×C61—21no—
117C2×Q16×C61—41no—
118C2×Q16×C61C2311yescharacteristic
119C2×Q16×C61C2311yescharacteristic
120C2×Q16×C61C2311yescharacteristic
121C2×SD16×C61—21no—
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124Q32×C61—22no—
125Q32×C61—22no—
126SD32×C61—22no—
127SD32×C61D414yescomplemented
128M32×C61—22no—
129C2×C16×C61C2311yescharacteristic
130C2×C16×C61C2311yescharacteristic
131#41×C61—42no—
132#41×C61—81no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—42no—
137C2×Q8×C61—41no—
138C2×Q8×C61—81no—
139C2×D4×C61—21no—
140C2×D4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61C2×D411yescharacteristic
144C2×C22⋊D61—41no—
145C2×C22⋊D61D811yescharacteristic
146C22×Dic61—41no—
147C22×Dic61—41no—
148Q8×D61—42no—
149Q8×D61—81no—
150#11758224—42no—
151#11758224—81no—
152C2×C4×D61—41no—
153C2×C4×D61C2×D411yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157Q16×C61—21no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—42no—
163Q16×C61C2×D412yes—
164SD16×C61—42no—
165D8×C61—21no—
166D8×C61C2×D412yes—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2411yescharacteristic
171C22⋊Dic61—41no—
172C22⋊Dic61—41no—
173C22⋊Dic61—41no—
174C22⋊Dic61D811yescharacteristic
175#11758202—42no—
176#11758202—81no—
177#11758202—81no—
178#11758200—42no—
179#11758198—41no—
180#11758198—41no—
181#11758198D811yescharacteristic
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—41no—
185#11758196—41no—
186#11758196—41no—
187#11758196—41no—
188#11758196—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—
200Dic244—21no—
201Dic244—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C23×C61C2×D811yescharacteristic
210C22×D61C2×D811yescharacteristic
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—82no—
218D4×C61—22no—
219D4×C61—41no—
220D4×C61—82no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—82no—
227C2×C4×C61—82no—
228C2×C4×C61C2×D811yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C22⋊D61—22no—
231C22⋊D61—82no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—82no—
237C2×Dic61—82no—
238C2×Dic61C2×D811yescharacteristic
239C2×Dic61C2×D811yescharacteristic
240C2×Dic61C2×D811yescharacteristic
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—82no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—82no—
249C8×C61—21no—
250C8×C61C22×D411yescharacteristic
251C8×C61H(5,𝔽2)11yescharacteristic
252#92418—21no—
253#92418—21no—
254#33842—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—42no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61C22×D811yescharacteristic
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C4×C61—21no—
266C4×C61—42no—
267C4×C61—81no—
268C4×C61—81no—
269C4×C61D4211yescharacteristic
270C4×C61#57611yescharacteristic
271Dic61—21no—
272Dic61—21no—
273Dic61—81no—
274Dic61—81no—
275Dic61—81no—
276Dic61—81no—
277Dic61#51012yescomplemented
278C2×#510—611no—
279C2×#510D6111yescharacteristic, complemented
280D4×Q16—611no—
281#3283—611no—
282#2247—611no—
283#2219—611no—
284#2219—611no—
285#2209—611no—
286#2208—611no—
287#2189—611no—
288#2189—611no—
289#2188—611no—
290#2188—611no—
291C4×SD32—611no—
292#2169—611no—
293C2×C61—41no—
294C2×C61—82no—
295C2×C61D4×D811yescharacteristic
296C2×C61C2×#51011yescharacteristic
297C2×C61#341711yescharacteristic
298D61—41no—
299C2×#135—611no—
300C2×#135C2×D6111yescharacteristic
301C22×Q16—611no—
302C22×Q16C2×D6111yescharacteristic
303D4×Q8—1221no—
304#547—1221no—
305#510—24no—
306#510—1224no—
307C2×Q32—611no—
308C2×Q32—611no—
309C2×Q32—611no—
310C2×Q32—611no—
311C2×Q32C2×D6111yescharacteristic
312C2×Q32C2×D6111yescharacteristic
313C2×SD32—611no—
314C2×SD32C2×D6111yescharacteristic
315C2×SD32C2×D6111yescharacteristic
316C2×M32—611no—
317C2×M32C2×D6111yescharacteristic
318#494—611no—
319#494—611no—
320#467—611no—
321#467—611no—
322#467—611no—
323#467—611no—
324#462—1221no—
325#461—1221no—
326#452—1221no—
327#452—1221no—
328#451—1221no—
329#451—1221no—
330C4×Q16—611no—
331C4×D8—611no—
332C8×D4—611no—
333#367—611no—
334C4⋊C16—611no—
335#358—611no—
336#358—611no—
337#358—611no—
338#358—611no—
339#358—611no—
340#357—611no—
341C22⋊C16—611no—
342C22⋊C16—611no—
343C4×C16—611no—
344C61#3384211yescharacteristic, complemented, Sylow
345C2×#41—21no—
346C2×#41—1221no—
347C22×Q8—21no—
348C22×Q8—1221no—
349#135—22no—
350#135—41no—
351#135—1222no—
352#135—2441no—
353C2×Q16—22no—
354C2×Q16—21no—
355C2×Q16—41no—
356C2×Q16—611no—
357C2×Q16—611no—
358C2×Q16—1221no—
359C2×Q16—1221no—
360C2×Q16—1221no—
361C2×Q16—1222no—
362C2×Q16—2441no—
363C2×Q16C22×D6111yescharacteristic
364C2×Q16C22×D6111yescharacteristic
365C2×Q16C22×D6111yescharacteristic
366C2×SD16—21no—
367C2×SD16—1221no—
368C2×D8C22×D6111yescharacteristic
369C22×C8—611no—
370C22×C8C22×D6111yescharacteristic
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8—2441no—
374#123—2441no—
375#123—2441no—
376C22⋊Q8—1221no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379C22⋊Q8—1221no—
380C22⋊Q8—2441no—
381C22⋊Q8—2441no—
382#121—2441no—
383C4×Q8—1221no—
384C4×D4—611no—
385C4×D4—1221no—
386C4×D4—2441no—
387C2×C4⋊C4—2441no—
388Q32—22no—
389Q32—22no—
390Q32—1222no—
391Q32—1222no—
392Q32—2441no—
393Q32—2441no—
394SD32—22no—
395SD32—1222no—
396SD32C22⋊D6114yescomplemented
397M32—22no—
398M32—1222no—
399C2×C16—611no—
400C2×C16—611no—
401C2×C16C22×D6111yescharacteristic
402C2×C16C22×D6111yescharacteristic
403#107—611no—
404#107—611no—
405C4⋊C8—611no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#102—1221no—
412C22⋊C8—611no—
413C22⋊C8—611no—
414C4×C8—611no—
415#41—42no—
416#41—81no—
417#41—4881no—
418#41—2442no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—42no—
423C2×Q8—41no—
424C2×Q8—81no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—4881no—
428C2×Q8—2441no—
429C2×Q8—2442no—
430C2×D4—21no—
431C2×D4—41no—
432C2×D4—611no—
433C2×D4—2441no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—611no—
437C22×C4—2441no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4D4×D6111yescharacteristic
441Q16—21no—
442Q16—21no—
443Q16—21no—
444Q16—21no—
445Q16—21no—
446Q16—42no—
447Q16—1221no—
448Q16—1221no—
449Q16—1221no—
450Q16—1221no—
451Q16—4881no—
452Q16—4881no—
453Q16—2442no—
454Q16C2×C22⋊D6112yes—
455SD16—42no—
456SD16—2442no—
457D8—21no—
458D8C2×C22⋊D6112yes—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8D4×D6111yescharacteristic
466C2×C8C23×D6111yescharacteristic
467C4⋊C4—611no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C22⋊C4—611no—
480C22⋊C4—611no—
481C22⋊C4—2441no—
482C22⋊C4—2441no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C42—611no—
487C42—2441no—
488C16—21no—
489C16—1221no—
490C16—1221no—
491C16C2×C22⋊D6112yes—
492C23—41no—
493C23—611no—
494C23D8×D6111yescharacteristic
495Q8—22no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502Q8—4882no—
503Q8—2441no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507D4—22no—
508D4—41no—
509D4—82no—
510D4—1222no—
511D4—4882no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—611no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—4882no—
525C2×C4—4882no—
526C2×C4—4882no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4D8×D6111yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C8—21no—
534C8—1221no—
535C8—1221no—
536C8C22×C22⋊D6111yescharacteristic
537C8#1451982211yescharacteristic
538C22—21no—
539C22—21no—
540C22—21no—
541C22—42no—
542C22—81no—
543C22—81no—
544C22—1221no—
545C22—1221no—
546C22—1221no—
547C22C2×D8×D6111yescharacteristic
548C4—21no—
549C4—42no—
550C4—81no—
551C4—81no—
552C4—612no—
553C4—1221no—
554C4—1221no—
555C4—4881no—
556C4—4881no—
557C4—4881no—
558C4—4881no—
559C4D4×C22⋊D6111yescharacteristic
560C4#1684056311yescharacteristic
561C2—41no—
562C2—82no—
563C2—2441no—
564C2D8×C22⋊D6111yescharacteristic
565C2#510×D6111yescharacteristic
566C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#510C2
socleC22×C61C22×D8
prime radical#21457535C2
hypercenterC2×#510D61
hypocenterC61#33842
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
1. C2×#16840201C2semidirect
2. Q16×C22⋊D61C2semidirect
4. #21448070C2semidirect
5. #21447927C2semidirect
6. #21447924C2semidirect
7. #21447867C2semidirect
8. #21447862C2semidirect
9. #21447759C2semidirect
10. #21447756C2semidirect
11. #21447753C2semidirect
12. #21447751C2semidirect
13. SD32×Dic61C2semidirect
14. #21447649C2semidirect
15. C2×C61×#510C2semidirect
127. SD32×C61D4semidirect
277. Dic61#510semidirect
279. C2×#510D61semidirect
344. C61#33842semidirect
396. SD32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
Q16×C22⋊D612
#214575352
#214480702
#214479272
#214479242
#214478672
#214478622
#214477592
#214477562
#214477532
#214477512
SD32×Dic612
#214476492
C2×C61×#5102
#3384261