Finite Group Databasebeta

← #29665214

Subgroups — #29665214

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456771C211yesmaximal, characteristic, complemented
2#21455914C211yesmaximal, characteristic, complemented
3#21455820C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic, complemented
5#21445259C212yesmaximal, complemented
6#21445245C212yesmaximal, complemented
7#21445183C212yesmaximal, complemented
8#21445163C212yesmaximal, complemented
9#21444597C211yesmaximal, characteristic, complemented
10#21444522C211yesmaximal, characteristic, complemented
11C61×#1544C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840141C2211yescharacteristic, complemented
15#16840136C2211yescharacteristic, complemented
16#16840114C2211yescharacteristic, complemented
17#16840110C2211yescharacteristic, complemented
18#16840095—21no—
19#16840076—21no—
20#16840043C2212yes—
21#16840039C2212yes—
22#16839984—22no—
23#16839964C2212yes—
24#16839958C2212yes—
25#16839922—22no—
26#16839912—21no—
27#16839898—21no—
28#16839860—22no—
29C4×D488—22no—
30C8×D244—22no—
31#16839827C2212yes—
32#16839827C2212yes—
33#16839819C2211yescharacteristic
34#16839815C2211yescharacteristic
35C22⋊C8×D61—22no—
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839509C2212yescomplemented
39#16839505C2212yescomplemented
40C61×#533C2211yescharacteristic
41C61×#431C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C61×#332C2212yescomplemented
44C61×#332C2212yescomplemented
45C2×#11758222—21no—
46C22×D244D411yescharacteristic
47C22×C4×D61—21no—
48C2×#11758202—22no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C2×D488—22no—
52C2×C8×D61—22no—
53C2×C8×D61—22no—
54#14519777C2311yescharacteristic
55#14519774C2311yescharacteristic
56#14519765—21no—
57#14519751—21no—
58#14519751—21no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244D412yes—
64C4×D244C2312yes—
65C2×C4⋊Dic61D411yescharacteristic
66C2×#11758198—21no—
67#14519711—22no—
68#14519710—22no—
69C4⋊#92418D412yes—
70C4⋊#92418C2312yes—
71#14519701—22no—
72#14519700—22no—
73#14519699—22no—
74#14519699—22no—
75#14519697—22no—
76#14519697—22no—
77C22⋊#92418—21no—
78C22⋊#92418D412yes—
79#14519689—22no—
80#14519689—22no—
81#14519688—21no—
82C4×#92418C2311yescharacteristic
83C4×#92418C2311yescharacteristic
84C4⋊Q8×C61C2311yescharacteristic
85C4⋊Q8×C61C2311yescharacteristic
86#125×C61—21no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91#117×C61—21no—
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61C2312yes—
94C4⋊C8×C61C2312yes—
95C22⋊C8×C61—21no—
96C22⋊C8×C61—21no—
97C4×C8×C61—21no—
98C4×C8×C61—21no—
99C23×D61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—41no—
106C22×Dic61—21no—
107#11758222—82no—
108C2×D244—21no—
109C2×D244—24no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244C2×D412yes—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—41no—
117C2×C4×D61—82no—
118C2×Dic122—21no—
119C2×C8×C61—22no—
120C2×C8×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C42×C61—41no—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61C2411yescharacteristic
133#11758202—82no—
134#11758201—44no—
135#11758201—82no—
136#11758198—22no—
137#11758198—22no—
138#11758198—41no—
139#11758198—41no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61C2×D412yes—
144C2×#92418—22no—
145C2×#92418C2×D412yes—
146C2×#92418C2×D412yes—
147D488—44no—
148C8×D61—82no—
149C8×D61—82no—
150C23×C61#12111yescharacteristic
151C22×D61—22no—
152C22×D61—24no—
153C22×D61—41no—
154C22×D61—81no—
155C22×D61—81no—
156Q8×C61—42no—
157Q8×C61—42no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C22⋊D61—82no—
170C2×Dic61—22no—
171C2×Dic61—41no—
172C2×Dic61—81no—
173D244—24no—
174D244—22no—
175D244—42no—
176D244—42no—
177D244—81no—
178D244#13514yes—
179C4×D61—42no—
180C4×D61—42no—
181C4×D61—82no—
182C4×D61—82no—
183Dic122—42no—
184C8×C61—42no—
185C8×C61—42no—
186#92418—22no—
187#92418—22no—
188#92418—42no—
189#19388—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61C2×#12111yescharacteristic
194C2×D61—24no—
195C2×D61—42no—
196C2×D61—42no—
197C2×D61—84no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—81no—
206C4×C61—81no—
207C4×C61C2×#13512yes—
208C4×C61#57612yes—
209Dic61—42no—
210Dic61—81no—
211#3210—611no—
212#3110—611no—
213#3100—611no—
214#2972—611no—
215#1681—612no—
216#1679—612no—
217#1669—612no—
218#1665—612no—
219#1572—611no—
220#1553—611no—
221#1544D6111yescharacteristic, complemented
222C2×C61—41no—
223C2×C61#304111yescharacteristic
224C2×C61#305011yescharacteristic
225C2×C61#306911yescharacteristic
226D61—44no—
227D61—161no—
228#533C2×D6111yescharacteristic
229#532—611no—
230C2×C4×D4—611no—
231#494—611no—
232#492—611no—
233#487—611no—
234#486—611no—
235#484—1221no—
236#480—1221no—
237#474—612no—
238#474—612no—
239#464—1222no—
240#461—612no—
241#459—612no—
242#452—1222no—
243#450—1221no—
244#447—1221no—
245#440—1222no—
246C4×D8—1222no—
247C8×D4—612no—
248C8×D4—612no—
249C8×D4—1222no—
250#432—611no—
251#431—611no—
252#431C2×D6111yescharacteristic
253#431C2×D6111yescharacteristic
254C2×C22⋊C8—1222no—
255#335—612no—
256#334—612no—
257#332—612no—
258#332C2×D6112yescomplemented
259#332C2×D6112yescomplemented
260#331—612no—
261C61#1938811yescharacteristic, complemented, Sylow
262C2×#41—1221no—
263C22×D4—611no—
264C23×C4—1221no—
265C2×Q16—1222no—
266C2×SD16—1222no—
267C2×SD16—1222no—
268C2×D8—1222no—
269C22×C8—1222no—
270C22×C8—1222no—
271C4⋊Q8—611no—
272C4⋊Q8C22×D6111yescharacteristic
273C4⋊Q8C22×D6111yescharacteristic
274#127—611no—
275#125—21no—
276#124—1221no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279#121—1221no—
280#121—1221no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×D4—612no—
284C4×D4—612no—
285C4×D4—1221no—
286C4×D4—1221no—
287C4×D4—1221no—
288C4×D4—1221no—
289#117—21no—
290C2×C4⋊C4—611no—
291C2×C22⋊C4—1221no—
292C2×C42C22×D6111yescharacteristic
293#107—1222no—
294#107—1222no—
295C4⋊C8—612no—
296C4⋊C8—612no—
297C4⋊C8C22×D6112yes—
298C4⋊C8C22×D6112yes—
299#103—1222no—
300#103—1222no—
301#102—1222no—
302#102—1222no—
303#102—1222no—
304#102—1222no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—612no—
308C22⋊C8—1222no—
309C22⋊C8—1222no—
310C22⋊C8—1221no—
311#97—1221no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—611no—
315C4×C8—611no—
316C24—1221no—
317#41—4882no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—1221no—
321C2×D4—612no—
322C2×D4—1224no—
323C2×D4—1221no—
324C2×D4—1221no—
325C2×D4—1221no—
326C2×D4—1221no—
327C2×D4—2441no—
328C22×C4—1221no—
329C22×C4—1222no—
330C22×C4—1222no—
331C22×C4—4882no—
332C22×C4—2441no—
333C22×C4#1175822211yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4D4×D6111yescharacteristic
336Q16—4882no—
337SD16—4882no—
338SD16—2444no—
339D8—2444no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—612no—
343C2×C8—612no—
344C2×C8—1222no—
345C2×C8—4882no—
346C2×C8—4882no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—612no—
354C4⋊C4—1221no—
355C4⋊C4—1221no—
356C4⋊C4—1221no—
357C22⋊C4—41no—
358C22⋊C4—1222no—
359C22⋊C4—1222no—
360C22⋊C4—2441no—
361C22⋊C4—2441no—
362C42—41no—
363C42#1175822211yescharacteristic
364C42D4×D6111yescharacteristic
365C42D4×D6111yescharacteristic
366C42C23×D6111yescharacteristic
367C23—1224no—
368C23—1222no—
369C23—4881no—
370C23—4881no—
371C23—2441no—
372C23#1451975011yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—2442no—
376D4—614no—
377D4—1224no—
378D4—1222no—
379D4—4881no—
380D4—4882no—
381D4—2442no—
382D4—2442no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—1222no—
389C2×C4—4882no—
390C2×C4—4882no—
391C2×C4—4881no—
392C2×C4—2441no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4#1451975011yescharacteristic
396C2×C4#1451975011yescharacteristic
397C2×C4#1451975011yescharacteristic
398C2×C4C2×D4×D6111yescharacteristic
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C2×#1175822211yescharacteristic
401C8—42no—
402C8—42no—
403C8—1222no—
404C8—1222no—
405C8—2442no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—1224no—
410C22—4884no—
411C22—4881no—
412C22—4881no—
413C22—4881no—
414C22—2442no—
415C22—2442no—
416C22C2×#1451975011yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—81no—
422C4—81no—
423C4—4881no—
424C4—2442no—
425C4#135×D6112yes—
426C4#1684056412yes—
427C2—41no—
428C2—9761no—
429C2—2444no—
430C2#2145532511yescharacteristic
431C2#2145539911yescharacteristic
432C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1544C2
socleC22×C61C2×#121
prime radical#21455820C2
hypercenter#1544D61
hypocenterC61#19388
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
1. #21456771C2semidirect
2. #21455914C2semidirect
4. #21454842C2semidirect
5. #21445259C2semidirect
6. #21445245C2semidirect
7. #21445183C2semidirect
8. #21445163C2semidirect
9. #21444597C2semidirect
10. #21444522C2semidirect
11. C61×#1544C2semidirect
14. #16840141C22semidirect
15. #16840136C22semidirect
16. #16840114C22semidirect
17. #16840110C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
38. #16839509C22semidirect
39. #16839505C22semidirect
43. C61×#332C22semidirect
44. C61×#332C22semidirect
221. #1544D61semidirect
258. #332C2×D61semidirect
259. #332C2×D61semidirect
261. C61#19388semidirect

Maximal subgroups

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

subgroupindex
#214567712
#214559142
#214558202
#214548422
#214452592
#214452452
#214451832
#214451632
#214445972
#214445222
C61×#15442
#1938861