Finite Group Databasebeta

← #29837727

Subgroups — #29837727

Order 15616.

532 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458792C211yesmaximal, characteristic, complemented
2#21458683C211yesmaximal, characteristic, complemented
3#21448059C211yesmaximal, characteristic, complemented
4#21448041C211yesmaximal, characteristic, complemented
5#21447978C211yesmaximal, characteristic, complemented
6#21447973C211yesmaximal, characteristic, complemented
7#21447964C211yesmaximal, characteristic, complemented
8#21447853C211yesmaximal, characteristic, complemented
9#21447853C211yesmaximal, characteristic, complemented
10#21447851C211yesmaximal, characteristic, complemented
11#21447725C211yesmaximal, characteristic, complemented
12#21447696C211yesmaximal, characteristic, complemented
13#21447692C211yesmaximal, characteristic, complemented
14#21447663C211yesmaximal, characteristic, complemented
15C61×#2244C211yesmaximal, characteristic, complemented
16#16840468—21no—
17#16840432—21no—
18C2×D976C2211yescharacteristic
19C2×#14519718C2211yescharacteristic
20C2×#14519718C2211yescharacteristic
21#16840166C2211yescharacteristic
22#16840163C2211yescharacteristic
23#16840137C2211yescharacteristic
24#16840137C2211yescharacteristic
25#16840061—21no—
26#16840059—21no—
27#16840051—21no—
28#16839955—21no—
29#16839955—21no—
30#16839954—21no—
31C8×Dic122C2211yescharacteristic
32C4×D488C2211yescharacteristic
33C4×D488C2211yescharacteristic
34D8×Dic61C2211yescharacteristic
35D8×Dic61C2211yescharacteristic
36#16839844C2211yescharacteristic
37#16839844C2211yescharacteristic
38#16839590C2211yescharacteristic
39#16839589C2211yescharacteristic
40#16839587C2211yescharacteristic
41C4⋊#11758185C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839580C2211yescharacteristic
44#16839564C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839562C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C61×#499C2211yescharacteristic
53C61×#493C2211yescharacteristic
54C61×#357C2211yescharacteristic
55C61×#357C2211yescharacteristic
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×#346C2211yescharacteristic
59C2×#11758224—41no—
60C2×D488C2311yescharacteristic
61C2×D488C2311yescharacteristic
62C2×#11758199—21no—
63C2×#11758199—21no—
64#14519776—21no—
65#14519773—21no—
66#14519773—21no—
67#14519768—21no—
68#14519757—41no—
69#14519755—41no—
70#14519755—41no—
71#14519749—41no—
72#14519749—41no—
73#14519748—41no—
74C4×Dic122D411yescharacteristic
75C4×D244—21no—
76C4×D244—21no—
77D4×Dic61—21no—
78D4×Dic61—21no—
79#14519739—21no—
80#14519739—21no—
81C4⋊C4×D61—41no—
82D976—22no—
83#14519718—22no—
84#14519718—22no—
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519710—21no—
90#14519708—21no—
91C4⋊#92418D411yescharacteristic
92#14519706D411yescharacteristic
93#14519706D411yescharacteristic
94#14519699—21no—
95#14519699—21no—
96#14519698—21no—
97#14519698—21no—
98#14519697—21no—
99#14519697—21no—
100C8×Dic61C2311yescharacteristic
101C8×Dic61C2311yescharacteristic
102C2×D8×C61—21no—
103C2×D8×C61C2311yescharacteristic
104C2×D8×C61C2311yescharacteristic
105#127×C61—21no—
106#125×C61—21no—
107C2×C16×C61C2311yescharacteristic
108C2×C16×C61C2311yescharacteristic
109#107×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#106×C61—21no—
112C4×C8×C61C2311yescharacteristic
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×C22⋊D61—41no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C22×Dic61—41no—
122C22×Dic61—41no—
123#11758224—84no—
124C2×D244—21no—
125C2×D244—21no—
126C2×C4×D61—41no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×Dic122—41no—
131C2×Dic122D811yescharacteristic
132D8×C61—21no—
133D8×C61—21no—
134D8×C61—21no—
135D8×C61—21no—
136D8×C61—81no—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61—41no—
143C42×C61C2×D411yescharacteristic
144C22⋊Dic61—41no—
145C22⋊Dic61—41no—
146#11758199—81no—
147#11758199—81no—
148#11758198—41no—
149#11758198—41no—
150#11758198—41no—
151#11758198—41no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—41no—
155C4⋊Dic61D811yescharacteristic
156#11758196—41no—
157#11758196—41no—
158#11758196D811yescharacteristic
159#11758196D811yescharacteristic
160C4×Dic61C2×D411yescharacteristic
161C4×Dic61C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164D488—21no—
165D488—21no—
166D488—21no—
167D488#4112yescomplemented
168C16×C61—21no—
169C16×C61—21no—
170#11758185—21no—
171#11758185—21no—
172C23×C61—41no—
173C23×C61—41no—
174C22×D61—41no—
175C22×D61—41no—
176D4×C61—41no—
177D4×C61—41no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—82no—
181D4×C61—161no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61C2×D811yescharacteristic
185C2×C4×C61C2×D811yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C22⋊D61—82no—
188C22⋊D61—82no—
189C22⋊D61—161no—
190C22⋊D61—161no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—82no—
194C2×Dic61—82no—
195C2×Dic61C2×D811yescharacteristic
196C2×Dic61C2×D811yescharacteristic
197C2×Dic61C2×D811yescharacteristic
198C2×Dic61C2×D811yescharacteristic
199D244—22no—
200D244—41no—
201D244—41no—
202D244—41no—
203C4×D61—82no—
204C4×D61—82no—
205C4×D61—161no—
206C4×D61—161no—
207Dic122—22no—
208Dic122—161no—
209C8×C61—21no—
210C8×C61C2×#4111yescharacteristic
211C8×C612−1+411yescharacteristic
212#92418—21no—
213#92418—21no—
214#33933—611nomaximal, Sylow
215C22×C61—81no—
216C22×C61—81no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61C22×D811yescharacteristic
222C2×D61—42no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61#54811yescharacteristic
233C4×C61#57711yescharacteristic
234Dic61—21no—
235Dic61—21no—
236Dic61—21no—
237Dic61—81no—
238Dic61—81no—
239Dic61#50912yes—
240#3391—611no—
241#3380—611no—
242#2244—611no—
243#2244D6111yescharacteristic, complemented
244#2240—611no—
245#2228—611no—
246#2228—611no—
247#2226—611no—
248#2206—611no—
249#2206—611no—
250#2206—611no—
251#2183—611no—
252#2177—611no—
253#2177—611no—
254C4×D16—611no—
255C2×C61—161no—
256C2×C61—161no—
257C2×C61#335811yescharacteristic
258C2×C61C2×#50911yescharacteristic
259C2×C61#341511yescharacteristic
260D61—82no—
261D61—161no—
262#554—1221no—
263#550—1221no—
264C2×D16—611no—
265C2×D16—611no—
266C2×D16—611no—
267#500—611no—
268#499—611no—
269#499C2×D6111yescharacteristic
270#493—611no—
271#493—611no—
272#493C2×D6111yescharacteristic
273#478—1221no—
274#478—1221no—
275#476—1221no—
276#459—1221no—
277#459—1221no—
278#459—1221no—
279C8×Q8—611no—
280C4×D8—611no—
281C4×D8—611no—
282C4×D8—611no—
283C4×D8—611no—
284C4×D8—611no—
285C4×D8—611no—
286#367—611no—
287#367—611no—
288#366—611no—
289C4⋊C16—611no—
290C4⋊C16—611no—
291C4⋊C16—611no—
292#357—611no—
293#357—611no—
294#357—611no—
295#357—611no—
296#357—611no—
297#357—611no—
298#357C2×D6111yescharacteristic
299#357C2×D6111yescharacteristic
300#357C2×D6111yescharacteristic
301#357C2×D6111yescharacteristic
302#346—611no—
303#346C2×D6111yescharacteristic
304C4×C16—611no—
305C61#3393311yescharacteristic, complemented, Sylow
306C2×#41—2441no—
307C2×D8—21no—
308C2×D8—611no—
309C2×D8—611no—
310C2×D8—1221no—
311C2×D8—1221no—
312C2×D8C22×D6111yescharacteristic
313C2×D8C22×D6111yescharacteristic
314C4⋊Q8—1221no—
315#127—21no—
316#127—1221no—
317#127—1221no—
318#125—21no—
319#125—1221no—
320C22⋊Q8—2441no—
321C22⋊Q8—2441no—
322C22⋊Q8—2441no—
323#121—2441no—
324#121—2441no—
325#121—2441no—
326C4×Q8—611no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—1221no—
331C4×D4—1221no—
332C4×D4—1221no—
333C2×C4⋊C4—2441no—
334D16—1222no—
335D16—1222no—
336D16—1222no—
337C2×C16—611no—
338C2×C16—611no—
339C2×C16C22×D6111yescharacteristic
340C2×C16C22×D6111yescharacteristic
341#107—611no—
342#107—611no—
343#107—1221no—
344#107C22×D6111yescharacteristic
345#107C22×D6111yescharacteristic
346#106—21no—
347#106—1221no—
348C4⋊C8—611no—
349C4⋊C8—611no—
350C4⋊C8—611no—
351#102—1221no—
352#102—1221no—
353#102—1221no—
354#102—1221no—
355#102—1221no—
356#102—1221no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8C22×D6111yescharacteristic
360#41—4884no—
361C2×Q8—611no—
362C2×Q8—2441no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—2441no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4—2441no—
377C22×C4—2441no—
378C22×C4—2441no—
379D8—21no—
380D8—21no—
381D8—21no—
382D8—21no—
383D8—81no—
384D8—612no—
385D8—1221no—
386D8—1221no—
387D8—1221no—
388D8—4881no—
389D8—4881no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8D4×D6111yescharacteristic
393C2×C8C23×D6111yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—611no—
399C4⋊C4—611no—
400C4⋊C4—611no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C22⋊C4—2441no—
407C22⋊C4—2441no—
408C22⋊C4—2441no—
409C22⋊C4—2441no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C42—611no—
413C42—611no—
414C42D4×D6111yescharacteristic
415C16—21no—
416C16—21no—
417C16—1221no—
418C16—1221no—
419C23—41no—
420C23—41no—
421C23—2441no—
422C23—2441no—
423Q8—1222no—
424Q8—9761no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—82no—
430D4—161no—
431D4—1222no—
432D4—9761no—
433D4—9761no—
434D4—4882no—
435D4—4882no—
436D4—2441no—
437D4—2441no—
438D4—2441no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—611no—
442C2×C4—611no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—9761no—
446C2×C4—9761no—
447C2×C4—4882no—
448C2×C4—4882no—
449C2×C4—4882no—
450C2×C4—4882no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4D8×D6111yescharacteristic
454C2×C4D8×D6111yescharacteristic
455C2×C4C2×D4×D6111yescharacteristic
456C8—21no—
457C8—1221no—
458C8—1221no—
459C8#41×D6111yescharacteristic
460C8#1451982411yescharacteristic
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—4881no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22—4881no—
472C22—2442no—
473C22C2×D8×D6111yescharacteristic
474C4—21no—
475C4—21no—
476C4—81no—
477C4—81no—
478C4—612no—
479C4—1221no—
480C4—1221no—
481C4—1221no—
482C4—4881no—
483C4—4881no—
484C4#1684042011yescharacteristic
485C4#1684056811yescharacteristic
486C2—161no—
487C2—161no—
488C2—9761no—
489C2—4882no—
490C2#2145845111yescharacteristic
491C2#509×D6111yescharacteristic
492C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2244C2
socleC22×C61C22×D8
hypercenter#2244D61
hypocenterC61#33933
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2244

Complements

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

NH ≅ G/Ntype
1. #21458792C2semidirect
2. #21458683C2semidirect
3. #21448059C2semidirect
4. #21448041C2semidirect
5. #21447978C2semidirect
6. #21447973C2semidirect
7. #21447964C2semidirect
8. #21447853C2semidirect
9. #21447853C2semidirect
10. #21447851C2semidirect
11. #21447725C2semidirect
12. #21447696C2semidirect
13. #21447692C2semidirect
14. #21447663C2semidirect
15. C61×#2244C2semidirect
167. D488#41semidirect
243. #2244D61semidirect
305. C61#33933semidirect

Maximal subgroups

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

subgroupindex
#214587922
#214586832
#214480592
#214480412
#214479782
#214479732
#214479642
#214478532
#214478512
#214477252
#214476962
#214476922
#214476632
C61×#22442
#3393361