Finite Group Databasebeta

← #29663121

Subgroups — #29663121

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456743C212yesmaximal
2#500×D61C211yesmaximal, characteristic, complemented
3#21454528C211yesmaximal, characteristic
4#21445333C211yesmaximal, characteristic, complemented
5#21445332C211yesmaximal, characteristic, complemented
6#21445329C211yesmaximal, characteristic, complemented
7#21445207C212yesmaximal, complemented
8#21445206C212yesmaximal, complemented
9#21444564C211yesmaximal, characteristic, complemented
10#21444502C212yesmaximal, complemented
11C61×#1693C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840309C2211yescharacteristic
14C4⋊Q8×D61C2212yes—
15#16840167C2211yescharacteristic
16#16840166—21no—
17#16840166—21no—
18#16840165C2212yes—
19#16840133—22no—
20#16840111—22no—
21#16839968C2212yes—
22#16839967C2212yes—
23#16839964C2212yes—
24#16839902—22no—
25#16839899—22no—
26#16839841C2211yescharacteristic
27#16839838—21no—
28D4×#92418C2211yescharacteristic
29#16839827C2211yescharacteristic
30#16839825—21no—
31#16839814C2212yes—
32#107×D61—21no—
33#107×D61—21no—
34#16839784C2211yescharacteristic
35#16839727—21no—
36C4×C8×D61C2211yescharacteristic
37#16839514C2212yes—
38#16839513C2211yescharacteristic
39#16839509C2212yes—
40#16839508C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#461C2212yes—
44C61×#436C2211yescharacteristic
45C61×#334C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758222—21no—
48C2×Q8×D61—22no—
49C2×#11758190—22no—
50C2×#11758200—22no—
51C2×#11758201—22no—
52#14519785—44no—
53C2×#11758189—21no—
54C2×#11758189—21no—
55C22×#92418—21no—
56C2×C8×D61—21no—
57C2×C8×D61D411yescharacteristic
58C2×C8×D61D411yescharacteristic
59#14519777C2312yes—
60#14519776—22no—
61#14519776—22no—
62#14519776D411yescharacteristic
63#14519775—22no—
64#14519773D411yescharacteristic
65#14519763—21no—
66#14519752D411yescharacteristic
67#14519750—21no—
68#14519748D411yescharacteristic
69C4×D244C2311yescharacteristic
70C4×C22⋊D61—21no—
71C4⋊C4×D61—22no—
72C4⋊C4×D61—22no—
73C42×D61C2311yescharacteristic
74#14519711—21no—
75#14519711—21no—
76#14519710—22no—
77#14519710—22no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2311yescharacteristic
80#14519706—21no—
81#14519702—22no—
82#14519701—22no—
83#14519700—22no—
84C22⋊#92418—21no—
85#14519689—22no—
86#14519689—21no—
87#14519689—21no—
88C4×#92418C2311yescharacteristic
89C8×Dic61—22no—
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C2×SD16×C61—22no—
93C2×M16×C61—21no—
94C4⋊Q8×C61C2312yes—
95C4×D4×C61C2311yescharacteristic
96#107×C61—21no—
97#107×C61—21no—
98C4⋊C8×C61C2312yes—
99C4⋊C8×C61C2311yescharacteristic
100#103×C61—22no—
101C22⋊C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C2×Q8×C61—22no—
104C2×D4×C61C2×D411yescharacteristic
105C22×C4×C61—21no—
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109Q8×D61—82no—
110Q8×D61—82no—
111#11758222—42no—
112#11758222—42no—
113C2×D244—21no—
114C2×D244C2×D411yescharacteristic
115C2×C4×D61—21no—
116C2×C4×D61—42no—
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61C2×D411yescharacteristic
119C2×C4×D61#4111yescharacteristic
120C2×Dic122—21no—
121C2×Dic122—22no—
122C2×Dic122—42no—
123SD16×C61—82no—
124M16×C61—42no—
125C2×C8×C61—21no—
126C2×C8×C61—22no—
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61C2×D411yescharacteristic
132C22⋊C4×C61—21no—
133C42×C61C2411yescharacteristic
134C22⋊Dic61—21no—
135#11758201—82no—
136#11758200—82no—
137#11758198—21no—
138#11758198—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61C2×D411yescharacteristic
142#11758196—21no—
143#11758196—42no—
144#11758196—42no—
145C4×Dic61C2×D411yescharacteristic
146C4×Dic61C2×D411yescharacteristic
147C4×Dic61#4111yescharacteristic
148C2×#92418—22no—
149C2×#92418—21no—
150C2×#92418—42no—
151C2×#92418C2×D411yescharacteristic
152C2×#92418C2×D411yescharacteristic
153#11758190—82no—
154#11758189—42no—
155#11758189—42no—
156C8×D61—22no—
157C8×D61—42no—
158C8×D61SD1614yescomplemented
159C23×C61—21no—
160C22×D61—21no—
161C22×D61#12111yescharacteristic
162Q8×C61—42no—
163Q8×C61—42no—
164D4×C61—22no—
165C2×C4×C61—21no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C22⋊D61—42no—
173C22⋊D61—42no—
174C22⋊D61—42no—
175C2×Dic61—21no—
176C2×Dic61—42no—
177C2×Dic61#12111yescharacteristic
178C2×Dic61#12111yescharacteristic
179C2×Dic61#12111yescharacteristic
180D244—22no—
181D244—42no—
182C4×D61—22no—
183C4×D61—22no—
184C4×D61—41no—
185C4×D61—41no—
186C4×D61—84no—
187C4×D61C2×SD1612yes—
188C4×D61#13712yes—
189Dic122—42no—
190Dic122—42no—
191Dic122—42no—
192Dic122—84no—
193C8×C61—21no—
194C8×C61—41no—
195C8×C61—42no—
196C8×C61C2×SD1612yes—
197#92418—21no—
198#92418—41no—
199#92418—42no—
200#92418C2×SD1612yes—
201#19192—611nomaximal, Sylow
202C22×C61—41no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61C2×#12111yescharacteristic
206C2×D61—21no—
207C2×D61—21no—
208C2×D61—41no—
209C2×D61—41no—
210C2×D61—41no—
211C2×D61#46112yes—
212C4×C61—21no—
213C4×C61—41no—
214C4×C61—82no—
215C4×C61C22×SD1611yescharacteristic
216C4×C61C2×#13711yescharacteristic
217C4×C61#57612yes—
218Dic61—21no—
219Dic61—41no—
220Dic61—82no—
221Dic61#46112yes—
222Dic61#47312yes—
223#3206—612no—
224C2×#500—611no—
225#2931—611no—
226#1693—611no—
227#1693—611no—
228#1693—611no—
229#1693D6111yescharacteristic, complemented
230#1673—612no—
231#1673—612no—
232#1564—611no—
233#1549—612no—
234C2×C61—81no—
235C2×C61C2×#46111yescharacteristic
236C2×C61#305811yescharacteristic
237C2×C61C2×#47311yescharacteristic
238D61—22no—
239D61—81no—
240C2×#131—1221no—
241#532—611no—
242C2×C4⋊Q8—612no—
243#500—612no—
244#500—611no—
245#500—1221no—
246#500—1221no—
247#500C2×D6111yescharacteristic
248#492—1222no—
249#487—1222no—
250#461—612no—
251#461—612no—
252#461—612no—
253#461C2×D6112yes—
254#448—1222no—
255#448—1222no—
256#436—611no—
257#436—1221no—
258#436C2×D6111yescharacteristic
259C8×D4—611no—
260C8×D4—611no—
261C8×D4—1221no—
262#431—612no—
263C2×#107—1221no—
264C2×#107—1221no—
265#423—611no—
266#408—1221no—
267C2×C4×C8—611no—
268#335—612no—
269#334—611no—
270#334C2×D6111yescharacteristic
271#332—612no—
272#332—612no—
273#332—612no—
274#332C2×D6112yes—
275C61#1919211yescharacteristic, complemented, Sylow
276C2×#41—1221no—
277C22×Q8—1222no—
278C2×SD16—22no—
279C2×SD16—1222no—
280C2×SD16—1222no—
281C2×SD16—1222no—
282#131—2444no—
283C2×M16—21no—
284C2×M16—1221no—
285C2×M16—1221no—
286C22×C8—611no—
287C22×C8—611no—
288C22×C8—1221no—
289C22×C8—1221no—
290C4⋊Q8—611no—
291C4⋊Q8—612no—
292C4⋊Q8—1222no—
293C4⋊Q8—1222no—
294C4⋊Q8—1222no—
295C4⋊Q8C22×D6112yes—
296#127—611no—
297#124—1221no—
298#121—611no—
299#121—611no—
300#121—1221no—
301C4×D4—611no—
302C4×D4—1221no—
303C4×D4C22×D6111yescharacteristic
304C2×C4⋊C4—1222no—
305C2×C4⋊C4—1222no—
306C2×C42—611no—
307#107—21no—
308#107—21no—
309#107—1222no—
310#107—1222no—
311#107—1221no—
312#107—1221no—
313C4⋊C8—611no—
314C4⋊C8—612no—
315C4⋊C8—1221no—
316C4⋊C8C22×D6111yescharacteristic
317C4⋊C8C22×D6112yes—
318#103—22no—
319#103—1222no—
320#103—1222no—
321#103—1222no—
322C22⋊C8—21no—
323C22⋊C8—1221no—
324C22⋊C8—1222no—
325C22⋊C8—1221no—
326C22⋊C8—1221no—
327C4×C8—611no—
328C4×C8—611no—
329C4×C8—611no—
330C4×C8—1222no—
331C4×C8C22×D6111yescharacteristic
332#41—2442no—
333#41—2442no—
334C2×Q8—22no—
335C2×Q8—1221no—
336C2×Q8—1222no—
337C2×Q8—4882no—
338C2×Q8—4882no—
339C2×Q8—2442no—
340C2×D4—611no—
341C2×D4—1221no—
342C2×D4—1221no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4D4×D6111yescharacteristic
346C22×C4—21no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—611no—
350C22×C4—1221no—
351C22×C4—2442no—
352SD16—82no—
353SD16—4882no—
354SD16—4882no—
355SD16—4882no—
356M16—42no—
357M16—2442no—
358M16—2442no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—611no—
362C2×C8—614no—
363C2×C8—611no—
364C2×C8—1222no—
365C2×C8—1222no—
366C2×C8—1221no—
367C2×C8—2442no—
368C2×C8—2442no—
369C2×C8D4×D6111yescharacteristic
370C2×C8D4×D6111yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—611no—
374C4⋊C4—1221no—
375C4⋊C4—1222no—
376C4⋊C4—1222no—
377C4⋊C4—2442no—
378C4⋊C4—2442no—
379C4⋊C4D4×D6111yescharacteristic
380C22⋊C4—21no—
381C22⋊C4—1221no—
382C22⋊C4—1221no—
383C22⋊C4—1221no—
384C42—611no—
385C42—611no—
386C42—611no—
387C42C23×D6111yescharacteristic
388C23—21no—
389C23—611no—
390C23—1221no—
391Q8—42no—
392Q8—42no—
393Q8—4884no—
394Q8—2442no—
395Q8—2442no—
396Q8—2442no—
397D4—22no—
398D4—1222no—
399D4—2442no—
400D4—2442no—
401D4—2442no—
402D4—2442no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—611no—
408C2×C4—611no—
409C2×C4—612no—
410C2×C4—612no—
411C2×C4—611no—
412C2×C4—1222no—
413C2×C4—1222no—
414C2×C4—1221no—
415C2×C4—4884no—
416C2×C4—2442no—
417C2×C4—2441no—
418C2×C4—2441no—
419C2×C4C2×D4×D6111yescharacteristic
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4#41×D6111yescharacteristic
422C8—21no—
423C8—41no—
424C8—42no—
425C8—612no—
426C8—1221no—
427C8—2441no—
428C8—2442no—
429C8SD16×D6112yes—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—612no—
434C22—1221no—
435C22—1221no—
436C22—2441no—
437C22—2441no—
438C22—2441no—
439C22#121×D6111yescharacteristic
440C4—21no—
441C4—41no—
442C4—82no—
443C4—612no—
444C4—612no—
445C4—1221no—
446C4—4882no—
447C4—2441no—
448C4C2×SD16×D6111yescharacteristic
449C4#137×D6111yescharacteristic
450C4#1684056612yes—
451C2—81no—
452C2—1222no—
453C2—4881no—
454C2#461×D6111yescharacteristic
455C2#2145546511yescharacteristic
456C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1693C2
socleC22×C61C2×#121
prime radical#16840309C22
hypercenter#1693D61
hypocenterC61#19192
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1693

Complements

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

NH ≅ G/Ntype
2. #500×D61C2semidirect
4. #21445333C2semidirect
5. #21445332C2semidirect
6. #21445329C2semidirect
7. #21445207C2semidirect
8. #21445206C2semidirect
9. #21444564C2semidirect
10. #21444502C2semidirect
11. C61×#1693C2semidirect
158. C8×D61SD16semidirect
229. #1693D61semidirect
275. C61#19192semidirect

Maximal subgroups

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

subgroupindex
#214567432
#500×D612
#214545282
#214453332
#214453322
#214453292
#214452072
#214452062
#214445642
#214445022
C61×#16932
#1919261