Finite Group Databasebeta

← #30613296

Subgroups — #30613296

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462753C211yesmaximal, characteristic, complemented
2#21462702C211yesmaximal, characteristic, complemented
3#21461451C212yesmaximal, complemented
4#21461446C212yesmaximal, complemented
5#21461439C212yesmaximal, complemented
6#21461403C212yesmaximal, complemented
7#21461397C211yesmaximal, characteristic, complemented
8#21461166C211yesmaximal, characteristic, complemented
9#21461113C211yesmaximal, characteristic, complemented
10#21461025C211yesmaximal, characteristic, complemented
11#21455882C211yesmaximal, characteristic
12Q8×#92420C2212yes—
13#16840796—21no—
14#16840796—22no—
15#16840796C2212yes—
16#16840796C2212yes—
17#16840790—21no—
18#16840790C2212yes—
19#16840781—21no—
20#16840781C2211yescharacteristic
21#16840781C2211yescharacteristic
22#16840766—21no—
23#16840766C2211yescharacteristic
24#16840737—22no—
25C4×D61⋊C8C2211yescharacteristic
26#16840625C2212yes—
27#16840624C2212yes—
28Q8⋊#92420C2211yescharacteristic
29Q8⋊#92420C2211yescharacteristic
30#16840610C2211yescharacteristic
31#16840610C2211yescharacteristic
32#16840610C2212yes—
33#16840610C2212yes—
34#92420⋊C8C2212yes—
35C8×#92420C2212yes—
36#16840313C2211yescharacteristic
37C4×#11758226C2211yescharacteristic
38#16840089C412yes—
39#16840049C411yescharacteristic
40#16839962C411yescharacteristic
41#16839958C411yescharacteristic
42#16839910C412yes—
43#16839847C2212yes—
44#16839844C2212yes—
45#16839814C2211yescharacteristic
46C61×#477C411yescharacteristic
47C2×#11758213—21no—
48C2×#11758213—22no—
49C2×#11758213—22no—
50C2×D61⋊C8—21no—
51C2×D61⋊C8—22no—
52C4⋊#92420—22no—
53C4⋊#92420—21no—
54C4⋊#92420D411yescharacteristic
55C4⋊#92420D411yescharacteristic
56C4⋊#92420C2312yes—
57C4⋊#92420C2311yescharacteristic
58C4⋊#92420C2311yescharacteristic
59#14519834—21no—
60#14519834—22no—
61#14519834—22no—
62#14519834—22no—
63#14519834D412yes—
64C4×#92420—22no—
65C4×#92420—21no—
66C4×#92420—21no—
67C4×#92420C2311yescharacteristic
68C4×#92420C2311yescharacteristic
69C4×#92420C2312yes—
70C2×#11758226—21no—
71C2×#11758226C2×C411yescharacteristic
72C2×#11758201C2×C412yes—
73C2×D488C2×C412yes—
74#14519774C2×C411yescharacteristic
75#14519766—21no—
76#14519752—21no—
77#14519752C2×C412yes—
78Q8×Dic61C2311yescharacteristic
79C4×D244—21no—
80C4×D244C2×C411yescharacteristic
81#14519739—21no—
82#14519739—21no—
83#14519739C2311yescharacteristic
84#14519739C2312yes—
85#14519735—21no—
86#14519735C2×C411yescharacteristic
87C42×D61—21no—
88C42×D61C2311yescharacteristic
89#14519708C2×C412yes—
90C4⋊#92418C2×C411yescharacteristic
91#14519699C2×C412yes—
92#14519697C2×C412yes—
93#14519697C2×C412yes—
94#14519689C2×C412yes—
95C8×Dic61C2312yes—
96C4⋊Q8×C61C2×C411yescharacteristic
97C4×Q8×C61C2×C411yescharacteristic
98#107×C61C2×C412yes—
99C4⋊C8×C61C2×C411yescharacteristic
100#103×C61C2×C412yes—
101C2×Q8×C61—21no—
102C2×Q8×C61C22×C411yescharacteristic
103#11758226—22no—
104#11758226—41no—
105#11758226—81no—
106#11758226—81no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—22no—
110C2×D244—21no—
111C2×D244C22×C411yescharacteristic
112C2×D244C22×C412yes—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—22no—
116C2×C4×D61—21no—
117C2×C4×D61C22×C411yescharacteristic
118C2×C4×D61C2×D411yescharacteristic
119C2×C4×D61#4111yescharacteristic
120C2×#92420—21no—
121C2×#92420—21no—
122C2×#92420—21no—
123C2×#92420—21no—
124C2×#92420—22no—
125C2×#92420C2×D412yes—
126C2×#92420C2×D412yes—
127#11758213—82no—
128#11758213—82no—
129#11758213—81no—
130D61⋊C8—24no—
131D61⋊C8—82no—
132C2×C8×C61C22×C412yes—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61C22×C412yes—
136C4⋊C4×C61C22×C411yescharacteristic
137C42×C61—21no—
138C42×C61C22×C411yescharacteristic
139#11758201—24no—
140#11758198—21no—
141#11758198—21no—
142#11758198—22no—
143#11758198—21no—
144#11758198—22no—
145C4⋊Dic61—21no—
146C4⋊Dic61C22×C411yescharacteristic
147C4×Dic61—21no—
148C4×Dic61—21no—
149C4×Dic61C2×D411yescharacteristic
150C4×Dic61#4111yescharacteristic
151C4×Dic61C2411yescharacteristic
152C2×#92418C22×C412yes—
153D488—42no—
154C22×D61—21no—
155C22×D61—22no—
156C22×D61C4×D411yescharacteristic
157Q8×C61—21no—
158Q8×C61—81no—
159Q8×C61#13112yes—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—22no—
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61C4×D411yescharacteristic
165C2×C4×C61C23×C411yescharacteristic
166C2×Dic61—21no—
167C2×Dic61C4×D411yescharacteristic
168C2×Dic61C22×D411yescharacteristic
169C2×Dic61C2×#4111yescharacteristic
170D244—22no—
171D244—21no—
172D244—21no—
173D244—22no—
174D244—42no—
175D244—42no—
176D244—81no—
177D244—82no—
178C4×D61—21no—
179C4×D61—21no—
180C4×D61—22no—
181C4×D61—21no—
182C4×D61—21no—
183C4×D61—42no—
184C4×D61—42no—
185C4×D61—41no—
186C4×D61—42no—
187C4×D61—42no—
188C4×D61—41no—
189#92420—22no—
190#92420—22no—
191#92420—22no—
192#92420—22no—
193#92420—41no—
194#92420—42no—
195#92420—41no—
196C8×C61—22no—
197#92418—22no—
198#18018—611nomaximal, Sylow
199C22×C61C2×C4×D411yescharacteristic
200C2×D61—21no—
201C2×D61—21no—
202C2×D61—21no—
203C2×D61—41no—
204C2×D61—42no—
205C2×D61—42no—
206C2×D61—41no—
207C2×D61—41no—
208C2×D61—42no—
209C4×C61—22no—
210C4×C61—21no—
211C4×C61—42no—
212C4×C61—41no—
213C4×C61#44312yes—
214C4×C61C2×#13111yescharacteristic
215C4×C61#56811yescharacteristic
216Dic61—21no—
217Dic61—21no—
218Dic61—41no—
219Dic61#57611yescharacteristic
220Dic61#57711yescharacteristic
221#3106—611no—
222#2991—611no—
223#2979—611no—
224#1589—612no—
225#1588—612no—
226#1586—612no—
227C8×SD16—612no—
228#1574—611no—
229#1492—611no—
230#1476—611no—
231#1450—611no—
232C2×C61#293311yescharacteristic
233C2×C61#294611yescharacteristic
234C2×C61C2×#44311yescharacteristic
235D61—41no—
236D61—81no—
237D61—82no—
238#532—611no—
239C4×#41—611no—
240#482—612no—
241#477#692311yescharacteristic
242#476—611no—
243#460—611no—
244#459—611no—
245#449—612no—
246C8×Q8—612no—
247C4×SD16—612no—
248C4×D8—612no—
249#436—612no—
250#436—612no—
251#436—1222no—
252#436—1221no—
253C8×D4—612no—
254C8×D4—1221no—
255#431—611no—
256#431—611no—
257#431—611no—
258#431—1221no—
259#423—611no—
260#423—1221no—
261#408—1222no—
262C2×C4×C8—611no—
263#335—612no—
264#334—612no—
265Q8⋊C8—611no—
266Q8⋊C8—611no—
267D4⋊C8—612no—
268D4⋊C8—611no—
269D4⋊C8—612no—
270D4⋊C8—611no—
271#322—612no—
272C82—612no—
273C61#1801811yescharacteristic, complemented, Sylow
274C2×#41—611no—
275C2×#41—1221no—
276C2×SD16—612no—
277C2×D8—612no—
278C2×M16—1222no—
279C2×M16—1222no—
280C2×M16—1221no—
281C22×C8—1222no—
282C22×C8—1221no—
283C4⋊Q8C2×#692311yescharacteristic
284#127—611no—
285#124—1221no—
286#121—612no—
287#121—1221no—
288C4×Q8—611no—
289C4×Q8C2×#692311yescharacteristic
290C4×D4—611no—
291C4×D4—612no—
292C4×D4—611no—
293C4×D4—1221no—
294C4×D4—1221no—
295C4×D4—1221no—
296#117—611no—
297#117—1221no—
298C2×C42—611no—
299C2×C42—1221no—
300#107C2×#692312yes—
301#106—612no—
302C4⋊C8—611no—
303C4⋊C8—611no—
304C4⋊C8—612no—
305C4⋊C8—611no—
306C4⋊C8—611no—
307C4⋊C8—611no—
308C4⋊C8—1221no—
309C4⋊C8—1222no—
310C4⋊C8C2×#692311yescharacteristic
311#103C2×#692312yes—
312#102—612no—
313#102—612no—
314#102—612no—
315C22⋊C8—612no—
316C22⋊C8—612no—
317C22⋊C8—1222no—
318C22⋊C8—1222no—
319C22⋊C8—1222no—
320C22⋊C8—1221no—
321C4×C8—612no—
322C4×C8—612no—
323C4×C8—611no—
324C4×C8—611no—
325C4×C8—1221no—
326C4×C8—1221no—
327C4×C8—1222no—
328#41—1222no—
329#41—4881no—
330#41—4881no—
331#41—2441no—
332C2×Q8—21no—
333C2×Q8C22×#692311yescharacteristic
334C2×D4—611no—
335C2×D4—612no—
336C2×D4—1222no—
337C2×D4—1221no—
338C2×D4—1221no—
339C2×D4—1221no—
340C22×C4—611no—
341C22×C4—611no—
342C22×C4—611no—
343C22×C4—1221no—
344C22×C4—1222no—
345C22×C4—1221no—
346C22×C4—1221no—
347SD16—1224no—
348D8—2442no—
349M16—4882no—
350M16—4882no—
351M16—4881no—
352C2×C8—612no—
353C2×C8—612no—
354C2×C8—612no—
355C2×C8—1221no—
356C2×C8—1221no—
357C2×C8—1224no—
358C2×C8—1221no—
359C2×C8—1221no—
360C2×C8—1222no—
361C2×C8—4882no—
362C2×C8C22×#692312yes—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—611no—
366C4⋊C4—1221no—
367C4⋊C4C22×#692311yescharacteristic
368C4⋊C4C22×#692312yes—
369C22⋊C4—1221no—
370C22⋊C4—1222no—
371C22⋊C4—1222no—
372C22⋊C4—1221no—
373C22⋊C4—1221no—
374C42—21no—
375C42—611no—
376C42—611no—
377C42—611no—
378C42—1221no—
379C42—1221no—
380C42C22×#692311yescharacteristic
381C23—611no—
382C23—1221no—
383C23—1222no—
384Q8—21no—
385Q8—81no—
386Q8#1451986512yes—
387D4—1222no—
388D4—1221no—
389D4—1221no—
390D4—1222no—
391D4—4881no—
392D4—4882no—
393D4—2442no—
394D4—2442no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—611no—
399C2×C4—611no—
400C2×C4—611no—
401C2×C4—1221no—
402C2×C4—1221no—
403C2×C4—1222no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—1221no—
407C2×C4—2442no—
408C2×C4—2442no—
409C2×C4—2441no—
410C2×C4—2442no—
411C2×C4—2441no—
412C2×C4—2442no—
413C2×C4D4×#692311yescharacteristic
414C2×C4D4×#692311yescharacteristic
415C2×C4C23×#692311yescharacteristic
416C8—22no—
417C8—1222no—
418C8—1222no—
419C8—1222no—
420C8—1222no—
421C8—1222no—
422C8—2441no—
423C8—2441no—
424C8—2442no—
425C22—1221no—
426C22—1221no—
427C22—1221no—
428C22—2441no—
429C22—2442no—
430C22—2442no—
431C22—2441no—
432C22—2441no—
433C22—2442no—
434C22C2×D4×#692311yescharacteristic
435C4—22no—
436C4—21no—
437C4—41no—
438C4—42no—
439C4—611no—
440C4—611no—
441C4—1221no—
442C4—1221no—
443C4—2441no—
444C4#1684080612yes—
445C4C2×#1451986511yescharacteristic
446C4#1684084111yescharacteristic
447C2—4881no—
448C2—4882no—
449C2—2441no—
450C2#2146255211yescharacteristic
451C2#2146257611yescharacteristic
452C2C2×#1684080611yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#477C4
socleC22×C61C2×C4×D4
prime radical#21455882C2
cosocleC4×Dic61C24
hypercenter#477#6923
hypocenterC61#18018
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #21462753C2semidirect
2. #21462702C2semidirect
3. #21461451C2semidirect
4. #21461446C2semidirect
5. #21461439C2semidirect
6. #21461403C2semidirect
7. #21461397C2semidirect
8. #21461166C2semidirect
9. #21461113C2semidirect
10. #21461025C2semidirect
273. C61#18018semidirect

Maximal subgroups

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

subgroupindex
#214627532
#214627022
#214614512
#214614462
#214614392
#214614032
#214613972
#214611662
#214611132
#214610252
#214558822
#1801861