Finite Group Databasebeta

← #29663939

Subgroups — #29663939

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D61C211yesmaximal, characteristic, complemented
2#21455303C212yesmaximal
3C8×#11758226C211yesmaximal, characteristic
4#21445213C211yesmaximal, characteristic, complemented
5#21445211C212yesmaximal, complemented
6#21445209C211yesmaximal, characteristic, complemented
7#21445207C212yesmaximal, complemented
8#21445205C211yesmaximal, characteristic, complemented
9#21444576C211yesmaximal, characteristic, complemented
10#21444489C212yesmaximal, complemented
11C61×#1674C211yesmaximal, characteristic, complemented
12C2×#14519809—22no—
13C4⋊Q8×D61C2212yes—
14C4×#11758226C2211yescharacteristic
15#16840167C2211yescharacteristic
16#16840166—21no—
17#16840166—21no—
18#16840165C2212yes—
19#16839972C2212yes—
20#16839970—22no—
21#16839968C2212yes—
22#16839966—22no—
23#16839964C2212yes—
24Q8×#92418C2211yescharacteristic
25C8×D244—21no—
26C8×D244C2211yescharacteristic
27#16839827—21no—
28#16839827C2211yescharacteristic
29#16839814—21no—
30#16839814C2211yescharacteristic
31#107×D61—21no—
32#107×D61—21no—
33#16839784C2212yes—
34#16839769—22no—
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839514C2212yes—
39#16839509C2212yes—
40#16839508C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#462C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758226D411yescharacteristic
48C2×Q8×D61—22no—
49#14519809—48no—
50C2×#11758202—22no—
51C2×#11758190—22no—
52C2×#11758201—22no—
53C2×#11758189—22no—
54C2×C8×D61—21no—
55C2×C8×D61—21no—
56C2×C8×D61D411yescharacteristic
57C2×C8×D61D411yescharacteristic
58#14519777C2312yes—
59#14519776—22no—
60#14519776—22no—
61#14519775—22no—
62Q8×Dic61D411yescharacteristic
63C4×D244—21no—
64C4×D244C2311yescharacteristic
65#14519739—21no—
66#14519739D411yescharacteristic
67#14519735—21no—
68#14519735D411yescharacteristic
69C4⋊C4×D61—22no—
70C4⋊C4×D61—22no—
71C42×D61—21no—
72C42×D61C2311yescharacteristic
73#14519711—21no—
74#14519711—21no—
75#14519710—22no—
76#14519710—22no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2311yescharacteristic
80#14519706—22no—
81#14519702—22no—
82#14519701—22no—
83#14519700—22no—
84#14519689—21no—
85#14519689—21no—
86#14519689—21no—
87C4×#92418—21no—
88C4×#92418C2311yescharacteristic
89C8×Dic61—21no—
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C2×Q16×C61—22no—
93C4⋊Q8×C61C2312yes—
94C4×Q8×C61C2311yescharacteristic
95#107×C61—21no—
96#107×C61—21no—
97C4⋊C8×C61—21no—
98C4⋊C8×C61C2312yes—
99C4⋊C8×C61C2311yescharacteristic
100#103×C61—22no—
101C4×C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C2×Q8×C61—22no—
104C2×Q8×C61C2×D411yescharacteristic
105#11758226—22no—
106#11758226D814yes—
107Q8×D61—44no—
108Q8×D61—44no—
109C2×D244—21no—
110C2×D244C2×D411yescharacteristic
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61—42no—
115C2×C4×D61C2×D411yescharacteristic
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61#4111yescharacteristic
118C2×Dic122—22no—
119C2×Dic122—42no—
120Q16×C61—44no—
121C2×C8×C61—21no—
122C2×C8×C61—22no—
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61C2×D411yescharacteristic
129C42×C61—21no—
130C42×C61C2411yescharacteristic
131#11758202—44no—
132#11758201—44no—
133#11758198—21no—
134#11758198—21no—
135#11758198—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—22no—
138C4⋊Dic61—22no—
139C4⋊Dic61C2×D411yescharacteristic
140#11758196—42no—
141#11758196—42no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C4×Dic61C2×D411yescharacteristic
145C4×Dic61C2×D411yescharacteristic
146C4×Dic61#4111yescharacteristic
147C2×#92418—21no—
148C2×#92418—22no—
149C2×#92418C2×D411yescharacteristic
150C2×#92418C2×D411yescharacteristic
151#11758190—44no—
152#11758189—44no—
153C8×D61—22no—
154C8×D61—22no—
155C8×D61—42no—
156C8×D61—42no—
157C22×D61—21no—
158C22×D61#12111yescharacteristic
159Q8×C61—21no—
160Q8×C61—42no—
161Q8×C61—42no—
162Q8×C61C2×D812yes—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—42no—
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C2×Dic61—21no—
170C2×Dic61—42no—
171C2×Dic61#12111yescharacteristic
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12111yescharacteristic
174D244—24no—
175D244—21no—
176D244C2×D812yes—
177C4×D61—21no—
178C4×D61—21no—
179C4×D61—24no—
180C4×D61—22no—
181C4×D61—41no—
182C4×D61—41no—
183C4×D61—42no—
184C4×D61—84no—
185C4×D61C2×D812yes—
186C4×D61#13512yes—
187Dic122—42no—
188Dic122—42no—
189Dic122—84no—
190C8×C61—21no—
191C8×C61—42no—
192C8×C61—41no—
193C8×C61#13712yescomplemented
194#92418—21no—
195#92418—42no—
196#92418—41no—
197#92418#13712yescomplemented
198#19271—611nomaximal, Sylow
199C22×C61C2×#12111yescharacteristic
200C2×D61—21no—
201C2×D61—21no—
202C2×D61—22no—
203C2×D61—41no—
204C2×D61—41no—
205C2×D61#46612yes—
206C4×C61—22no—
207C4×C61—21no—
208C4×C61—41no—
209C4×C61—82no—
210C4×C61C22×D811yescharacteristic
211C4×C61C2×#13511yescharacteristic
212C4×C61C2×#13711yescharacteristic
213C4×C61C2×#13711yescharacteristic
214Dic61—21no—
215Dic61—41no—
216Dic61—82no—
217Dic61#46612yes—
218Dic61#47312yes—
219C2×#500—611no—
220#3039—612no—
221C8×#41—611no—
222#1674—612no—
223#1674—611no—
224#1674D6111yescharacteristic, complemented
225#1673—611no—
226#1673—611no—
227#1673—612no—
228#1567—611no—
229#1546—612no—
230C2×C61#303911yescharacteristic
231C2×C61C2×#46611yescharacteristic
232C2×C61C2×#47311yescharacteristic
233D61—22no—
234D61—42no—
235C2×#137—1222no—
236C2×C4⋊Q8—612no—
237C4×#41—611no—
238#500—612no—
239#500—611no—
240#500—1221no—
241#500—1221no—
242#500C2×D6111yescharacteristic
243#462—612no—
244#462—1222no—
245#462C2×D6112yes—
246#461—612no—
247#461—612no—
248#461—1222no—
249C8×Q8—611no—
250C8×Q8C2×D6111yescharacteristic
251C8×D4—611no—
252C8×D4—611no—
253C8×D4—1221no—
254C8×D4—1221no—
255#431—611no—
256#431—1221no—
257C2×#107—1221no—
258C2×#107—1221no—
259#423—612no—
260#419—1222no—
261C2×C4×C8—611no—
262C2×C4×C8—1221no—
263#335—612no—
264#335—611no—
265#335C2×D6111yescharacteristic
266#332—612no—
267#332—612no—
268#332—612no—
269#332C2×D6112yes—
270C61#1927111yescharacteristic, complemented, Sylow
271C2×#41—611no—
272C22×Q8—1222no—
273#137—2448no—
274C2×Q16—22no—
275C2×Q16—1222no—
276C2×SD16—1222no—
277C2×SD16—1222no—
278C2×M16—1222no—
279C22×C8—611no—
280C22×C8—611no—
281C22×C8—1221no—
282C22×C8—1221no—
283C4⋊Q8—612no—
284C4⋊Q8—1222no—
285C4⋊Q8—1222no—
286C4⋊Q8—1222no—
287C4⋊Q8C22×D6112yes—
288C4×Q8—611no—
289C4×Q8C22×D6111yescharacteristic
290C4×D4—611no—
291C4×D4—611no—
292C4×D4—1221no—
293C4×D4—1221no—
294#117—611no—
295#117—1221no—
296C2×C4⋊C4—1222no—
297C2×C4⋊C4—1222no—
298C2×C42—611no—
299C2×C42—1221no—
300#107—21no—
301#107—21no—
302#107—1222no—
303#107—1222no—
304#107—1221no—
305#107—1221no—
306C4⋊C8—21no—
307C4⋊C8—611no—
308C4⋊C8—612no—
309C4⋊C8—1221no—
310C4⋊C8—1222no—
311C4⋊C8C22×D6111yescharacteristic
312C4⋊C8C22×D6112yes—
313#103—22no—
314#103—1222no—
315#103—1222no—
316#103—1222no—
317C22⋊C8—1221no—
318C22⋊C8—1221no—
319C22⋊C8—1221no—
320C4×C8—21no—
321C4×C8—611no—
322C4×C8—611no—
323C4×C8—611no—
324C4×C8—1221no—
325C4×C8—1221no—
326C4×C8C22×D6111yescharacteristic
327#41—614no—
328#41—1222no—
329C2×Q8—22no—
330C2×Q8—1222no—
331C2×Q8—2442no—
332C2×Q8—2444no—
333C2×Q8—2444no—
334C2×Q8D4×D6111yescharacteristic
335C2×D4—611no—
336C2×D4—1221no—
337C22×C4—611no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—2442no—
344Q16—44no—
345Q16—2444no—
346SD16—2444no—
347SD16—2444no—
348M16—2444no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—611no—
352C2×C8—611no—
353C2×C8—1222no—
354C2×C8—1222no—
355C2×C8—1221no—
356C2×C8—1222no—
357C2×C8—2442no—
358C2×C8—2442no—
359C2×C8D4×D6111yescharacteristic
360C2×C8D4×D6111yescharacteristic
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—611no—
365C4⋊C4—1221no—
366C4⋊C4—1222no—
367C4⋊C4—1222no—
368C4⋊C4—2442no—
369C4⋊C4—2442no—
370C4⋊C4D4×D6111yescharacteristic
371C22⋊C4—1221no—
372C22⋊C4—1221no—
373C22⋊C4—1221no—
374C42—21no—
375C42—611no—
376C42—611no—
377C42—611no—
378C42—1221no—
379C42—1221no—
380C42C23×D6111yescharacteristic
381C23—611no—
382C23—1221no—
383Q8—21no—
384Q8—42no—
385Q8—42no—
386Q8—4884no—
387Q8—2442no—
388Q8—2442no—
389Q8D8×D6112yes—
390D4—612no—
391D4—1224no—
392D4—1221no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—611no—
397C2×C4—611no—
398C2×C4—612no—
399C2×C4—612no—
400C2×C4—611no—
401C2×C4—1222no—
402C2×C4—1224no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—4884no—
407C2×C4—2442no—
408C2×C4—2442no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4C2×D4×D6111yescharacteristic
412C2×C4C2×D4×D6111yescharacteristic
413C2×C4#41×D6111yescharacteristic
414C8—21no—
415C8—42no—
416C8—41no—
417C8—612no—
418C8—1221no—
419C8—2442no—
420C8—2441no—
421C8#1451980912yescomplemented
422C22—612no—
423C22—1221no—
424C22—1221no—
425C22—1222no—
426C22—2441no—
427C22—2441no—
428C22#121×D6111yescharacteristic
429C4—22no—
430C4—21no—
431C4—41no—
432C4—82no—
433C4—612no—
434C4—612no—
435C4—1221no—
436C4—4882no—
437C4—2441no—
438C4C2×D8×D6111yescharacteristic
439C4#135×D6111yescharacteristic
440C4#137×D6111yescharacteristic
441C4C2×#1451980911yescharacteristic
442C2—1222no—
443C2—2442no—
444C2#2145530311yescharacteristic
445C2#466×D6111yescharacteristic
446C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1674C2
socleC22×C61C2×#121
prime radicalC2×#11758226D4
hypercenter#1674D61
hypocenterC61#19271
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1674

Complements

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

NH ≅ G/Ntype
1. #500×D61C2semidirect
4. #21445213C2semidirect
5. #21445211C2semidirect
6. #21445209C2semidirect
7. #21445207C2semidirect
8. #21445205C2semidirect
9. #21444576C2semidirect
10. #21444489C2semidirect
11. C61×#1674C2semidirect
193. C8×C61#137semidirect
197. #92418#137semidirect
224. #1674D61semidirect
270. C61#19271semidirect
421. C8#14519809semidirect

Maximal subgroups

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

subgroupindex
#500×D612
#214553032
C8×#117582262
#214452132
#214452112
#214452092
#214452072
#214452052
#214445762
#214444892
C61×#16742
#1927161