Finite Group Databasebeta

← #29665220

Subgroups — #29665220

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456754C211yesmaximal, characteristic, complemented
2#21455910C211yesmaximal, characteristic, complemented
3#21455819C211yesmaximal, characteristic, complemented
4#21445270C212yesmaximal, complemented
5#21445228C212yesmaximal, complemented
6#21445174C212yesmaximal, complemented
7#21445160C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444522C211yesmaximal, characteristic, complemented
10#21444479C211yesmaximal, characteristic, complemented
11C61×#2972C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840139C2211yescharacteristic, complemented
15#16840135C2211yescharacteristic, complemented
16#16840113C2211yescharacteristic, complemented
17#16840109C2211yescharacteristic, complemented
18#16840080—21no—
19#16840075—21no—
20#16840052C2212yes—
21#16840042C2212yes—
22#16839978—22no—
23#16839957C2212yes—
24#16839957C2212yes—
25#16839909—22no—
26#16839896—21no—
27#16839896—21no—
28#16839872—22no—
29C4×#11758199—22no—
30#16839819C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839515C2212yescomplemented
33#16839513C2212yescomplemented
34#16839510C2212yescomplemented
35#16839509C2212yescomplemented
36#16839505C2212yescomplemented
37#16839505C2212yescomplemented
38C2×C4×D4×C61C2211yescharacteristic
39C8×D4×C61—22no—
40C8×D4×C61C2212yes—
41C8×D4×C61C2212yes—
42C61×#431C2211yescharacteristic
43C61×#431C2211yescharacteristic
44C2×C22⋊C8×C61—22no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758190—22no—
48C2×D488—22no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51#14519777C2311yescharacteristic
52#14519777C2311yescharacteristic
53#14519774C2311yescharacteristic
54#14519774C2311yescharacteristic
55#14519765—21no—
56#14519765—21no—
57#14519751—21no—
58#14519751—21no—
59#14519751—21no—
60#14519751—21no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244—21no—
64C4×D244—21no—
65#14519710—22no—
66#14519708—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519702—22no—
70#14519699—22no—
71#14519698—22no—
72#14519698—22no—
73#14519697—22no—
74#14519697—22no—
75C22⋊#92418—21no—
76C22⋊#92418—21no—
77#14519688—21no—
78C4×#92418—21no—
79C22×D4×C61D411yescharacteristic
80C23×C4×C61—21no—
81C22×C8×C61—22no—
82C22×C8×C61—22no—
83C4×D4×C61—21no—
84C4×D4×C61—21no—
85C4×D4×C61D412yes—
86C4×D4×C61C2312yes—
87C2×C4⋊C4×C61D411yescharacteristic
88C2×C22⋊C4×C61—21no—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61D412yes—
91C4⋊C8×C61C2312yes—
92C22⋊C8×C61—21no—
93C22⋊C8×C61—22no—
94C22⋊C8×C61—22no—
95C22⋊C8×C61D412yes—
96C4×C8×C61—21no—
97C4×C8×C61C2311yescharacteristic
98C4×C8×C61C2311yescharacteristic
99C24×C61—21no—
100C2×D4×C61—24no—
101C2×D4×C61—21no—
102C2×D4×C61C2×D412yes—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—82no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C2×C22⋊D61—41no—
111C2×C22⋊D61—41no—
112#11758222—82no—
113#11758222—82no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×C4×D61—41no—
121C2×C4×D61—41no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×C8×C61—22no—
125C2×C8×C61—82no—
126C2×C8×C61—82no—
127C2×C8×C61C2×D412yes—
128C2×C8×C61C2×D412yes—
129C4⋊C4×C61—21no—
130C4⋊C4×C61C2×D412yes—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137#11758199—44no—
138#11758199—44no—
139#11758198—41no—
140#11758198—41no—
141#11758198—41no—
142#11758198—41no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C2×#92418—22no—
148C2×#92418—22no—
149D488—82no—
150#11758190—82no—
151C23×C61—24no—
152C23×C61—22no—
153C23×C61—81no—
154C23×C61—81no—
155C23×C61#12111yescharacteristic
156C22×D61—41no—
157C22×D61—41no—
158D4×C61—24no—
159D4×C61—22no—
160D4×C61#13514yes—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—82no—
168C2×C4×C61—81no—
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C22⋊D61—82no—
176C22⋊D61—82no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179D244—42no—
180D244—42no—
181D244—42no—
182D244—42no—
183D244—81no—
184D244—81no—
185C4×D61—82no—
186C4×D61—82no—
187Dic122—42no—
188Dic122—81no—
189C8×C61—22no—
190C8×C61—22no—
191C8×C61—42no—
192#92418—42no—
193#92418—42no—
194#19389—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—24no—
198C22×C61—21no—
199C22×C61—42no—
200C22×C61—42no—
201C22×C61—84no—
202C22×C61C2×#12111yescharacteristic
203C2×D61—81no—
204C2×D61—81no—
205C2×D61—81no—
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—42no—
214C4×C61C2×#13512yes—
215C4×C61#57612yes—
216Dic61—81no—
217Dic61—81no—
218#3207—611no—
219#3109—611no—
220#3100—611no—
221#2972D6111yescharacteristic, complemented
222#1683—612no—
223#1677—612no—
224#1667—612no—
225#1665—612no—
226#1572—611no—
227#1553—611no—
228#1543—611no—
229C2×C61—41no—
230C2×C61—44no—
231C2×C61#304111yescharacteristic
232C2×C61#305011yescharacteristic
233C2×C61#306811yescharacteristic
234D61—161no—
235D61—161no—
236#532—611no—
237#532—611no—
238C2×C4×D4C2×D6111yescharacteristic
239#493—611no—
240#492—611no—
241#487—611no—
242#486—611no—
243#481—1221no—
244#480—1221no—
245#476—612no—
246#474—612no—
247#463—1222no—
248#459—612no—
249#459—612no—
250#449—1222no—
251#447—1221no—
252#447—1221no—
253#442—1222no—
254C4×D8—1222no—
255C8×D4—22no—
256C8×D4C2×D6112yes—
257C8×D4C2×D6112yes—
258#432—611no—
259#431—611no—
260#431C2×D6111yescharacteristic
261#431C2×D6111yescharacteristic
262C2×C22⋊C8—22no—
263#335—612no—
264#334—612no—
265#333—612no—
266#332—612no—
267#331—612no—
268#331—612no—
269C61#1938911yescharacteristic, complemented, Sylow
270C2×#41—1221no—
271C2×#41—1221no—
272C22×D4D24411yescharacteristic
273C23×C4—21no—
274C2×SD16—1222no—
275C2×D8—1222no—
276C2×D8—1222no—
277C2×D8—1222no—
278C22×C8—22no—
279C22×C8—22no—
280C4⋊Q8—611no—
281C4⋊Q8—611no—
282#127—611no—
283#127—611no—
284#124—1221no—
285#124—1221no—
286#121—1221no—
287#121—1221no—
288#121—1221no—
289#121—1221no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—1221no—
293C4×D4—1221no—
294C4×D4—1221no—
295C4×D4—1221no—
296C4×D4D24412yes—
297C4×D4C22×D6112yes—
298C2×C4⋊C4D24411yescharacteristic
299C2×C22⋊C4—21no—
300C2×C42C22×D6111yescharacteristic
301#107—1222no—
302#106—1222no—
303C4⋊C8—612no—
304C4⋊C8—612no—
305C4⋊C8C22⋊D6112yes—
306C4⋊C8C22×D6112yes—
307#103—1222no—
308#102—1222no—
309#102—1222no—
310#102—1222no—
311#102—1222no—
312#102—1222no—
313C22⋊C8—21no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—1221no—
317C22⋊C8—1221no—
318C22⋊C8C22⋊D6112yes—
319#97—1221no—
320C4×C8—21no—
321C4×C8—1221no—
322C4×C8C22×D6111yescharacteristic
323C4×C8C22×D6111yescharacteristic
324C24—21no—
325#41—4882no—
326#41—4882no—
327C2×Q8—1221no—
328C2×Q8—1221no—
329C2×D4—24no—
330C2×D4—21no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4—1221no—
337C2×D4—2441no—
338C2×D4—2441no—
339C2×D4C2×D24412yes—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—82no—
344C22×C4—2441no—
345C22×C4—2441no—
346C22×C4C2×D24411yescharacteristic
347C22×C4#1175822211yescharacteristic
348C22×C4C2×C22⋊D6111yescharacteristic
349SD16—4882no—
350D8—4882no—
351D8—2444no—
352D8—2444no—
353C2×C8—22no—
354C2×C8—82no—
355C2×C8—82no—
356C2×C8—1222no—
357C2×C8—1222no—
358C2×C8C2×C22⋊D6112yes—
359C2×C8C2×C22⋊D6112yes—
360C4⋊C4—21no—
361C4⋊C4—1221no—
362C4⋊C4—1221no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4C2×D24412yes—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—2441no—
369C22⋊C4—2441no—
370C22⋊C4—2441no—
371C22⋊C4—2441no—
372C42C2×D24411yescharacteristic
373C42#1175822211yescharacteristic
374C42C2×C22⋊D6111yescharacteristic
375C42C23×D6111yescharacteristic
376C23—24no—
377C23—22no—
378C23—81no—
379C23—81no—
380C23—2441no—
381C23—2441no—
382C23#1451975111yescharacteristic
383Q8—4881no—
384Q8—2442no—
385D4—24no—
386D4—22no—
387D4—4881no—
388D4—4881no—
389D4—4882no—
390D4—4882no—
391D4—2442no—
392D4—2442no—
393D4—2442no—
394D4—2442no—
395D4#1451979914yes—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—4882no—
405C2×C4—4882no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4#1451975111yescharacteristic
409C2×C4#1451975111yescharacteristic
410C2×C4#1451975111yescharacteristic
411C2×C4C22×C22⋊D6111yescharacteristic
412C2×C4C22×D24411yescharacteristic
413C2×C4C2×#1175822211yescharacteristic
414C8—22no—
415C8—22no—
416C8—42no—
417C8—2442no—
418C8—2442no—
419C22—21no—
420C22—24no—
421C22—21no—
422C22—21no—
423C22—42no—
424C22—42no—
425C22—84no—
426C22—4881no—
427C22—4881no—
428C22—4881no—
429C22—4881no—
430C22—4881no—
431C22—4881no—
432C22C2×#1451975111yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—4881no—
439C4—4881no—
440C4C2×#1451979912yes—
441C4#1684056512yes—
442C2—41no—
443C2—44no—
444C2—9761no—
445C2—9761no—
446C2#2145532211yescharacteristic
447C2#2145540011yescharacteristic
448C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2972C2
socleC22×C61C2×#121
hypercenter#2972D61
hypocenterC61#19389
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #21456754C2semidirect
2. #21455910C2semidirect
3. #21455819C2semidirect
4. #21445270C2semidirect
5. #21445228C2semidirect
6. #21445174C2semidirect
7. #21445160C2semidirect
8. #21444599C2semidirect
9. #21444522C2semidirect
10. #21444479C2semidirect
11. C61×#2972C2semidirect
14. #16840139C22semidirect
15. #16840135C22semidirect
16. #16840113C22semidirect
17. #16840109C22semidirect
32. #16839515C22semidirect
33. #16839513C22semidirect
34. #16839510C22semidirect
35. #16839509C22semidirect
36. #16839505C22semidirect
37. #16839505C22semidirect
221. #2972D61semidirect
269. C61#19389semidirect

Maximal subgroups

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

subgroupindex
#214567542
#214559102
#214558192
#214452702
#214452282
#214451742
#214451602
#214445992
#214445222
#214444792
C61×#29722
#1938961