Finite Group Databasebeta

← #29665209

Subgroups — #29665209

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#21456777C211yesmaximal, characteristic, complemented
2#21455918C211yesmaximal, characteristic, complemented
3#21455819C211yesmaximal, characteristic
4#21445256C212yesmaximal, complemented
5#21445242C212yesmaximal, complemented
6#21445184C212yesmaximal, complemented
7#21445160C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444522C211yesmaximal, characteristic, complemented
10#21444483C211yesmaximal, characteristic, complemented
11C61×#2972C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840142C2211yescharacteristic, complemented
15#16840135C2211yescharacteristic, complemented
16#16840113C2211yescharacteristic, complemented
17#16840109C2211yescharacteristic, complemented
18#16840097—21no—
19#16840075—21no—
20#16840042C2212yes—
21#16840042C2212yes—
22#16839985—22no—
23#16839967C2212yes—
24#16839957C2212yes—
25#16839926—22no—
26#16839913—21no—
27#16839896—21no—
28#16839862—22no—
29C4×#11758199—22no—
30#16839819C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839515C2212yescomplemented
33#16839513C2212yescomplemented
34#16839509C2212yescomplemented
35#16839509C2212yescomplemented
36#16839507C2212yescomplemented
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×Dic244—22no—
47C2×#11758190—22no—
48C2×#11758200—22no—
49C2×#11758199—22no—
50#14519777C2311yescharacteristic
51#14519777C2311yescharacteristic
52#14519777C2311yescharacteristic
53#14519774C2311yescharacteristic
54#14519769—21no—
55#14519765—21no—
56#14519756—21no—
57#14519756—21no—
58#14519751—21no—
59#14519751—21no—
60C4×Dic122—21no—
61C4×Dic122—21no—
62C4×D244—21no—
63C4×D244—21no—
64#14519736—21no—
65#14519710—22no—
66#14519710—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519702—22no—
70#14519700—22no—
71#14519699—22no—
72#14519698—22no—
73#14519698—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—
111#11758222—82no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×C4×D61—41no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×C8×C61—22no—
120C2×C8×C61—82no—
121C2×C8×C61—82no—
122C2×C8×C61C2×D412yes—
123C2×C8×C61C2×D412yes—
124C4⋊C4×C61—21no—
125C4⋊C4×C61C2×D412yes—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—22no—
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61#4111yescharacteristic
131C42×C61C2411yescharacteristic
132C22⋊Dic61—41no—
133#11758200—44no—
134#11758199—44no—
135#11758198—41no—
136#11758198—41no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143#11758196—41no—
144#11758196—41no—
145C4×Dic61—41no—
146C2×#92418—22no—
147C2×#92418—22no—
148Dic244—82no—
149#11758190—82no—
150C23×C61—24no—
151C23×C61—22no—
152C23×C61—81no—
153C23×C61—81no—
154C23×C61#12111yescharacteristic
155C22×D61—41no—
156D4×C61—24no—
157D4×C61—22no—
158D4×C61#13514yes—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—82no—
166C2×C4×C61—81no—
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C22⋊D61—82no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177D244—42no—
178D244—42no—
179D244—81no—
180C4×D61—82no—
181Dic122—42no—
182Dic122—42no—
183Dic122—42no—
184C8×C61—22no—
185C8×C61—22no—
186C8×C61—42no—
187#92418—42no—
188#92418—42no—
189#19388—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—24no—
193C22×C61—21no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61—84no—
197C22×C61C2×#12111yescharacteristic
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—42no—
206C4×C61C2×#13512yes—
207C4×C61#57612yes—
208Dic61—81no—
209Dic61—81no—
210Dic61—81no—
211#3210—611no—
212#3110—611no—
213#3100—611no—
214#2972D6111yescharacteristic, complemented
215#1681—612no—
216#1679—612no—
217#1669—612no—
218#1665—612no—
219#1572—611no—
220#1553—611no—
221#1544—611no—
222C2×C61—41no—
223C2×C61—44no—
224C2×C61#304111yescharacteristic
225C2×C61#305011yescharacteristic
226C2×C61#306911yescharacteristic
227D61—161no—
228#533—611no—
229#532—611no—
230C2×C4×D4C2×D6111yescharacteristic
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—22no—
248C8×D4C2×D6112yes—
249C8×D4C2×D6112yes—
250#432—611no—
251#431—611no—
252#431C2×D6111yescharacteristic
253#431C2×D6111yescharacteristic
254C2×C22⋊C8—22no—
255#335—612no—
256#334—612no—
257#332—612no—
258#332—612no—
259#332—612no—
260#331—612no—
261C61#1938811yescharacteristic, complemented, Sylow
262C2×#41—1221no—
263C22×D4D24411yescharacteristic
264C23×C4—21no—
265C2×Q16—1222no—
266C2×SD16—1222no—
267C2×SD16—1222no—
268C2×D8—1222no—
269C22×C8—22no—
270C22×C8—22no—
271C4⋊Q8—611no—
272C4⋊Q8—611no—
273C4⋊Q8—611no—
274#127—611no—
275#125—1221no—
276#124—1221no—
277C22⋊Q8—1221no—
278C22⋊Q8—1221no—
279#121—1221no—
280#121—1221no—
281C4×Q8—1221no—
282C4×Q8—1221no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—1221no—
286C4×D4—1221no—
287C4×D4D24412yes—
288C4×D4C22×D6112yes—
289#117—1221no—
290C2×C4⋊C4D24411yescharacteristic
291C2×C22⋊C4—21no—
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—22no—
307C22⋊C8—22no—
308C22⋊C8—1221no—
309C22⋊C8—1221no—
310C22⋊C8C22⋊D6112yes—
311#97—1221no—
312C4×C8—21no—
313C4×C8—1221no—
314C4×C8C22×D6111yescharacteristic
315C4×C8C22×D6111yescharacteristic
316C24—21no—
317#41—4882no—
318C2×Q8—1221no—
319C2×Q8—1221no—
320C2×Q8—1221no—
321C2×D4—24no—
322C2×D4—21no—
323C2×D4—1221no—
324C2×D4—1221no—
325C2×D4—1221no—
326C2×D4—2441no—
327C2×D4C2×D24412yes—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—82no—
332C22×C4—2441no—
333C22×C4C2×D24411yescharacteristic
334C22×C4#1175822211yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336Q16—4882no—
337SD16—4882no—
338SD16—2444no—
339D8—2444no—
340C2×C8—22no—
341C2×C8—82no—
342C2×C8—82no—
343C2×C8—1222no—
344C2×C8—1222no—
345C2×C8C2×C22⋊D6112yes—
346C2×C8C2×C22⋊D6112yes—
347C4⋊C4—21no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—1221no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—2441no—
355C4⋊C4—2441no—
356C4⋊C4C2×D24412yes—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C22⋊C4—2441no—
362C42—2441no—
363C42C2×D24411yescharacteristic
364C42#1175822211yescharacteristic
365C42C2×C22⋊D6111yescharacteristic
366C42C23×D6111yescharacteristic
367C23—24no—
368C23—22no—
369C23—81no—
370C23—81no—
371C23—2441no—
372C23#1451975111yescharacteristic
373Q8—2442no—
374Q8—2442no—
375Q8—2442no—
376D4—24no—
377D4—22no—
378D4—4881no—
379D4—4882no—
380D4—2442no—
381D4—2442no—
382D4#1451979914yes—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—4882no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4#1451975111yescharacteristic
396C2×C4#1451975111yescharacteristic
397C2×C4#1451975111yescharacteristic
398C2×C4C22×C22⋊D6111yescharacteristic
399C2×C4C22×D24411yescharacteristic
400C2×C4C2×#1175822211yescharacteristic
401C8—22no—
402C8—22no—
403C8—42no—
404C8—2442no—
405C8—2442no—
406C22—21no—
407C22—24no—
408C22—21no—
409C22—21no—
410C22—42no—
411C22—42no—
412C22—84no—
413C22—4881no—
414C22—4881no—
415C22—4881no—
416C22C2×#1451975111yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—42no—
422C4—4881no—
423C4—4881no—
424C4—4881no—
425C4C2×#1451979912yes—
426C4#1684056512yes—
427C2—41no—
428C2—44no—
429C2—9761no—
430C2#2145532211yescharacteristic
431C2#2145540011yescharacteristic
432C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2972C2
socleC22×C61C2×#121
prime radical#21455819C2
hypercenter#2972D61
hypocenterC61#19388
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. #21456777C2semidirect
2. #21455918C2semidirect
4. #21445256C2semidirect
5. #21445242C2semidirect
6. #21445184C2semidirect
7. #21445160C2semidirect
8. #21444599C2semidirect
9. #21444522C2semidirect
10. #21444483C2semidirect
11. C61×#2972C2semidirect
14. #16840142C22semidirect
15. #16840135C22semidirect
16. #16840113C22semidirect
17. #16840109C22semidirect
32. #16839515C22semidirect
33. #16839513C22semidirect
34. #16839509C22semidirect
35. #16839509C22semidirect
36. #16839507C22semidirect
37. #16839505C22semidirect
214. #2972D61semidirect
261. C61#19388semidirect

Maximal subgroups

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

subgroupindex
#214567772
#214559182
#214558192
#214452562
#214452422
#214451842
#214451602
#214445992
#214445222
#214444832
C61×#29722
#1938861