Finite Group Databasebeta

← #29837716

Subgroups — #29837716

Order 15616.

484 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458828C211yesmaximal, characteristic
2#21458707C211yesmaximal, characteristic, complemented
3#21448052C211yesmaximal, characteristic, complemented
4#21448041C211yesmaximal, characteristic
5#21447978C211yesmaximal, characteristic, complemented
6#21447973C211yesmaximal, characteristic, complemented
7#21447959C211yesmaximal, characteristic, complemented
8#21447884C211yesmaximal, characteristic, complemented
9#21447878C211yesmaximal, characteristic, complemented
10#21447858C211yesmaximal, characteristic
11C16×Dic122C211yesmaximal, characteristic, complemented
12#21447675C211yesmaximal, characteristic, complemented
13#21447674C211yesmaximal, characteristic, complemented
14C4×#14519722C211yesmaximal, characteristic, complemented
15C61×#2242C211yesmaximal, characteristic, complemented
16#16840494—21no—
17#16840489—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519722C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21#16840166C2211yescharacteristic
22#16840163C2211yescharacteristic
23#16840147C2211yescharacteristic
24#16840137C2211yescharacteristic
25#16840072—21no—
26#16840066—21no—
27#16840046—21no—
28#16839979—21no—
29#16839974—21no—
30#16839960—21no—
31C8×Dic122C2211yescharacteristic
32C4×Dic244C2211yescharacteristic
33Q16×Dic61C2211yescharacteristic
34#16839856C2211yescharacteristic
35#16839856C2211yescharacteristic
36C4×D488C2211yescharacteristic
37D8×Dic61C2211yescharacteristic
38#16839590C2211yescharacteristic
39#16839589C2211yescharacteristic
40#16839587C2211yescharacteristic
41C4⋊#11758185C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839580C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839562C2211yescharacteristic
50C16×Dic61C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C61×#499C2211yescharacteristic
53C61×#495C2211yescharacteristic
54C61×#358C2211yescharacteristic
55C61×#358C2211yescharacteristic
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C4×C16×C61C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758201—21no—
61C2×D488C2311yescharacteristic
62C2×#11758199—21no—
63#14519776—21no—
64#14519776—41no—
65#14519773—21no—
66#14519768—21no—
67#14519767—41no—
68#14519767—41no—
69#14519766—21no—
70#14519763—41no—
71#14519761—41no—
72#14519759—41no—
73Q8×Dic61—21no—
74C4×Dic122—21no—
75C4×Dic122D411yescharacteristic
76#14519742—21no—
77#14519742—21no—
78#14519742—41no—
79C4×D244—21no—
80D4×Dic61—21no—
81#14519737—41no—
82#14519723—22no—
83#14519722—22no—
84#14519721—22no—
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519710—21no—
90#14519708—21no—
91C4⋊#92418D411yescharacteristic
92#14519706D411yescharacteristic
93#14519706D411yescharacteristic
94#14519702—21no—
95#14519701—21no—
96#14519700—21no—
97#14519700—21no—
98#14519699—21no—
99#14519698—21no—
100C8×Dic61C2311yescharacteristic
101C8×Dic61C2311yescharacteristic
102C2×Q16×C61C2311yescharacteristic
103C2×SD16×C61—21no—
104C2×D8×C61C2311yescharacteristic
105#125×C61—21no—
106#124×C61—21no—
107C2×C16×C61C2311yescharacteristic
108C2×C16×C61C2311yescharacteristic
109#107×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#106×C61—21no—
112C4×C8×C61C2311yescharacteristic
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×C22⋊D61—41no—
116C2×C22⋊D61—41no—
117C22×Dic61—41no—
118C2×D244—21no—
119C2×C4×D61—41no—
120C2×Dic122—21no—
121C2×Dic122—41no—
122C2×Dic122D811yescharacteristic
123Q16×C61—21no—
124Q16×C61—21no—
125SD16×C61—81no—
126D8×C61—21no—
127D8×C61—21no—
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2411yescharacteristic
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C42×C61C2×D411yescharacteristic
137C22⋊Dic61—41no—
138#11758201—81no—
139#11758199—81no—
140#11758198—41no—
141#11758198—41no—
142#11758198—41no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—41no—
146C4⋊Dic61—41no—
147C4⋊Dic61D811yescharacteristic
148#11758196—41no—
149#11758196—41no—
150#11758196—41no—
151#11758196—41no—
152#11758196—41no—
153#11758196D811yescharacteristic
154#11758196D811yescharacteristic
155C4×Dic61—41no—
156C4×Dic61—41no—
157C4×Dic61—41no—
158C4×Dic61—41no—
159C4×Dic61C2×D411yescharacteristic
160C4×Dic61C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163Dic244—21no—
164Dic244#4112yes—
165D488—21no—
166D488—21no—
167C16×C61—21no—
168C16×C61—21no—
169#11758185—21no—
170#11758185—21no—
171C23×C61—41no—
172C22×D61—41no—
173Q8×C61—41no—
174Q8×C61—41no—
175D4×C61—41no—
176D4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61C2×D811yescharacteristic
181C2×C4×C61C2×D811yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C22⋊D61—82no—
184C22⋊D61—161no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—161no—
189C2×Dic61C2×D811yescharacteristic
190C2×Dic61C2×D811yescharacteristic
191C2×Dic61C2×D811yescharacteristic
192C2×Dic61C2×D811yescharacteristic
193D244—41no—
194D244—41no—
195C4×D61—161no—
196Dic122—22no—
197Dic122—22no—
198Dic122—41no—
199Dic122—82no—
200C8×C61—21no—
201C8×C61C2×#4111yescharacteristic
202C8×C612−1+411yescharacteristic
203#92418—21no—
204#92418—21no—
205#33932—611nomaximal, Sylow
206C22×C61—81no—
207C22×C61—81no—
208C22×C61—81no—
209C22×C61C22×D811yescharacteristic
210C2×D61—81no—
211C2×D61—81no—
212C2×D61—81no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—81no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61#54811yescharacteristic
219C4×C61#57711yescharacteristic
220Dic61—21no—
221Dic61—21no—
222Dic61—21no—
223Dic61—42no—
224Dic61—81no—
225Dic61—81no—
226Dic61#50812yes—
227#3396—611no—
228#3383—611no—
229#2242—611no—
230#2242D6111yescharacteristic, complemented
231#2240—611no—
232#2228—611no—
233#2228—611no—
234#2225—611no—
235#2211—611no—
236#2211—611no—
237#2207—611no—
238Q8×C16—611no—
239C4×Q32—611no—
240C4×SD32—611no—
241C4×SD32—611no—
242C2×C61—161no—
243C2×C61#335811yescharacteristic
244C2×C61C2×#50811yescharacteristic
245C2×C61#341611yescharacteristic
246D61—161no—
247#559—1221no—
248#557—1221no—
249C2×Q32—611no—
250C2×SD32—611no—
251C2×SD32—611no—
252#500—611no—
253#499—611no—
254#499C2×D6111yescharacteristic
255#495—611no—
256#495C2×D6111yescharacteristic
257#493—611no—
258#479—1221no—
259#479—1221no—
260#475—1221no—
261#463—1221no—
262#463—1221no—
263#460—1221no—
264C8×Q8—611no—
265C4×Q16—611no—
266C4×Q16—611no—
267C4×Q16—611no—
268C4×Q16—611no—
269C4×D8—611no—
270C4×D8—611no—
271#367—611no—
272#367—611no—
273#366—611no—
274C4⋊C16—611no—
275C4⋊C16—611no—
276C4⋊C16—611no—
277#358—611no—
278#358—611no—
279#358C2×D6111yescharacteristic
280#358C2×D6111yescharacteristic
281#357—611no—
282#357—611no—
283#357—611no—
284#357—611no—
285#357C2×D6111yescharacteristic
286#357C2×D6111yescharacteristic
287C4×C16—611no—
288C4×C16—611no—
289C4×C16C2×D6111yescharacteristic
290C61#3393211yescharacteristic, complemented, Sylow
291C2×Q16—611no—
292C2×Q16C22×D6111yescharacteristic
293C2×SD16—21no—
294C2×SD16—1221no—
295C2×D8—611no—
296C2×D8—1221no—
297C2×D8C22×D6111yescharacteristic
298C4⋊Q8—1221no—
299C4⋊Q8—2441no—
300#127—1221no—
301#125—21no—
302#125—1221no—
303#125—2441no—
304#125—2441no—
305#124—21no—
306#124—1221no—
307#124—2441no—
308#123—2441no—
309#123—2441no—
310C4×Q8—611no—
311C4×Q8—1221no—
312C4×Q8—1221no—
313C4×Q8—1221no—
314C4×Q8—1221no—
315C4×Q8—2441no—
316C4×D4—1221no—
317C4×D4—1221no—
318C4×D4—2441no—
319Q32—1222no—
320SD32—1222no—
321SD32—1222no—
322C2×C16—611no—
323C2×C16—611no—
324C2×C16C22×D6111yescharacteristic
325C2×C16C22×D6111yescharacteristic
326#107—611no—
327#107—611no—
328#107—1221no—
329#107C22×D6111yescharacteristic
330#107C22×D6111yescharacteristic
331#106—21no—
332#106—1221no—
333C4⋊C8—611no—
334C4⋊C8—611no—
335C4⋊C8—611no—
336#103—1221no—
337#103—1221no—
338#103—1221no—
339#103—1221no—
340#102—1221no—
341#102—1221no—
342C4×C8—611no—
343C4×C8—611no—
344C4×C8C22×D6111yescharacteristic
345C2×Q8—21no—
346C2×Q8—611no—
347C2×Q8—1221no—
348C2×Q8—2441no—
349C2×D4—21no—
350C2×D4—1221no—
351C2×D4—2441no—
352C2×D4—2441no—
353C22×C4—2441no—
354C22×C4—2441no—
355Q16—21no—
356Q16—21no—
357Q16—612no—
358Q16—1221no—
359SD16—81no—
360SD16—4881no—
361D8—21no—
362D8—21no—
363D8—1221no—
364D8—1221no—
365D8—4881no—
366C2×C8—611no—
367C2×C8—611no—
368C2×C8D4×D6111yescharacteristic
369C2×C8C23×D6111yescharacteristic
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—611no—
375C4⋊C4—611no—
376C4⋊C4—611no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—2441no—
380C4⋊C4—2441no—
381C4⋊C4—2441no—
382C4⋊C4—2441no—
383C4⋊C4—2441no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—2441no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C22⋊C4—2441no—
392C42—611no—
393C42—611no—
394C42—2441no—
395C42—2441no—
396C42—2441no—
397C42—2441no—
398C42D4×D6111yescharacteristic
399C16—21no—
400C16—21no—
401C16—1221no—
402C16—1221no—
403C23—41no—
404C23—2441no—
405Q8—41no—
406Q8—41no—
407Q8—1222no—
408Q8—1222no—
409Q8—4882no—
410Q8—2441no—
411D4—41no—
412D4—41no—
413D4—9761no—
414D4—4882no—
415D4—2441no—
416D4—2441no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—611no—
421C2×C4—611no—
422C2×C4—611no—
423C2×C4—611no—
424C2×C4—9761no—
425C2×C4—9761no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4D8×D6111yescharacteristic
430C2×C4D8×D6111yescharacteristic
431C2×C4C2×D4×D6111yescharacteristic
432C8—21no—
433C8—1221no—
434C8—1221no—
435C8#41×D6111yescharacteristic
436C8#1451982411yescharacteristic
437C22—81no—
438C22—81no—
439C22—81no—
440C22—4881no—
441C22—4881no—
442C22—4881no—
443C22C2×D8×D6111yescharacteristic
444C4—21no—
445C4—21no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—612no—
450C4—1221no—
451C4—1221no—
452C4—1221no—
453C4—4881no—
454C4—4881no—
455C4—2442no—
456C4#1684042011yescharacteristic
457C4#1684056811yescharacteristic
458C2—161no—
459C2—9761no—
460C2#2145845111yescharacteristic
461C2#508×D6111yescharacteristic
462C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2242C2
socleC22×C61C22×D8
prime radical#16840137C22
hypercenter#2242D61
hypocenterC61#33932
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2242

Complements

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

NH ≅ G/Ntype
2. #21458707C2semidirect
3. #21448052C2semidirect
5. #21447978C2semidirect
6. #21447973C2semidirect
7. #21447959C2semidirect
8. #21447884C2semidirect
9. #21447878C2semidirect
11. C16×Dic122C2semidirect
12. #21447675C2semidirect
13. #21447674C2semidirect
14. C4×#14519722C2semidirect
15. C61×#2242C2semidirect
230. #2242D61semidirect
290. C61#33932semidirect

Maximal subgroups

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

subgroupindex
#214588282
#214587072
#214480522
#214480412
#214479782
#214479732
#214479592
#214478842
#214478782
#214478582
C16×Dic1222
#214476752
#214476742
C4×#145197222
C61×#22422
#3393261