Finite Group Databasebeta

← #29665212

Subgroups — #29665212

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#21455819C211yesmaximal, characteristic
3#21454843C211yesmaximal, characteristic, complemented
4#21445258C212yesmaximal, complemented
5#21445244C212yesmaximal, complemented
6#21445182C212yesmaximal, complemented
7#21445162C212yesmaximal, complemented
8#21444600C211yesmaximal, characteristic, complemented
9#21444525C211yesmaximal, characteristic, complemented
10#21444484C211yesmaximal, characteristic, complemented
11C61×#3110C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840141C2211yescharacteristic, complemented
14#16840136C2211yescharacteristic, complemented
15#16840114C2211yescharacteristic, complemented
16#16840110C2211yescharacteristic, complemented
17#16840075—21no—
18#16840042C2212yes—
19#16839984—22no—
20#16839957C2212yes—
21#16839922—22no—
22#16839896—21no—
23C4×#11758199—22no—
24D4×#92418—22no—
25D4×#92418C2212yes—
26D4×#92418C2212yes—
27#16839815C2211yescharacteristic
28#16839815C2211yescharacteristic
29#16839815C2211yescharacteristic
30C2×C22⋊#92418—22no—
31#16839514C2212yescomplemented
32#16839512C2212yescomplemented
33#16839508C2212yescomplemented
34#16839508C2212yescomplemented
35#16839507C2212yescomplemented
36#16839505C2212yescomplemented
37C61×#533C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C61×#484—21no—
40C61×#474C2212yes—
41C61×#461C2212yes—
42C61×#450—21no—
43C61×#440—22no—
44C61×#432C2211yescharacteristic
45C2×#11758222—21no—
46C2×#11758202—22no—
47C2×#11758201—22no—
48C2×#11758199—22no—
49C22×#92418—22no—
50C22×#92418—22no—
51#14519777C2311yescharacteristic
52#14519774C2311yescharacteristic
53#14519765—21no—
54#14519751—21no—
55#14519751—21no—
56C4×D244—21no—
57C4×D244—21no—
58#14519710—22no—
59C4⋊#92418D412yes—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519700—22no—
63#14519698—22no—
64#14519697—22no—
65#14519697—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—22no—
69C22⋊#92418—22no—
70C22⋊#92418D412yes—
71C4×#92418—21no—
72C4×#92418—21no—
73C4×#92418C2311yescharacteristic
74C4×#92418C2311yescharacteristic
75C22×D4×C61D411yescharacteristic
76C23×C4×C61—21no—
77C2×SD16×C61—22no—
78C4⋊Q8×C61C2311yescharacteristic
79C4⋊Q8×C61C2311yescharacteristic
80#125×C61—21no—
81C22⋊Q8×C61—21no—
82C22⋊Q8×C61—21no—
83C4×Q8×C61—21no—
84C4×Q8×C61—21no—
85C4×D4×C61—21no—
86C4×D4×C61—21no—
87C4×D4×C61D412yes—
88C4×D4×C61C2312yes—
89#117×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C42×C61C2311yescharacteristic
93#107×C61—22no—
94C4⋊C8×C61C2312yes—
95#103×C61—22no—
96#102×C61—22no—
97C22⋊C8×C61—21no—
98#97×C61—21no—
99C24×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×D4×C61—24no—
103C2×D4×C61—21no—
104C2×D4×C61C2×D412yes—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—82no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—41no—
113#11758222—82no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×C4×D61—41no—
118C2×Dic122—21no—
119SD16×C61—44no—
120C2×C8×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61C2×D412yes—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—22no—
131C22⋊C4×C61—41no—
132C42×C61—41no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137#11758202—82no—
138#11758201—82no—
139#11758199—44no—
140#11758198—41no—
141#11758198—41no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C2×#92418—22no—
145C2×#92418—22no—
146C2×#92418—82no—
147C2×#92418—82no—
148C2×#92418C2×D412yes—
149C2×#92418C2×D412yes—
150C23×C61—24no—
151C23×C61—22no—
152C23×C61—81no—
153C23×C61—81no—
154C23×C61#12111yescharacteristic
155C22×D61—41no—
156Q8×C61—42no—
157Q8×C61—42no—
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—41no—
168C2×C4×C61—41no—
169C2×C4×C61—82no—
170C2×C4×C61—81no—
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C22×D411yescharacteristic
176C2×C4×C61C2×#4111yescharacteristic
177C22⋊D61—82no—
178C2×Dic61—41no—
179D244—42no—
180D244—42no—
181D244—81no—
182C4×D61—82no—
183Dic122—42no—
184C8×C61—42no—
185#92418—22no—
186#92418—22no—
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×C61—81no—
207C4×C61—81no—
208C4×C61C2×#13512yes—
209C4×C61#57612yes—
210Dic61—81no—
211#3210—611no—
212#3110D6111yescharacteristic, complemented
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#1544—611no—
222C2×C61—41no—
223C2×C61—44no—
224C2×C61#304111yescharacteristic
225C2×C61#305011yescharacteristic
226C2×C61#306911yescharacteristic
227D61—161no—
228#533C2×D6111yescharacteristic
229#532—611no—
230C2×C4×D4C2×D6111yescharacteristic
231#494—611no—
232#492—611no—
233#487—611no—
234#486—611no—
235#484—21no—
236#480—1221no—
237#474—612no—
238#474C2×D6112yes—
239#464—1222no—
240#461C2×D6112yes—
241#459—612no—
242#452—1222no—
243#450—21no—
244#447—1221no—
245#440—22no—
246C4×D8—1222no—
247C8×D4—612no—
248C8×D4—612no—
249C8×D4—1222no—
250#432C2×D6111yescharacteristic
251#431—611no—
252#431—611no—
253#431—611no—
254C2×C22⋊C8—1222no—
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×D4C22⋊D6111yescharacteristic
264C23×C4—21no—
265C2×Q16—1222no—
266C2×SD16—22no—
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—21no—
284C4×D4—21no—
285C4×D4—1221no—
286C4×D4—1221no—
287C4×D4C22⋊D6112yes—
288C4×D4C22×D6112yes—
289#117—21no—
290C2×C4⋊C4C22⋊D6111yescharacteristic
291C2×C22⋊C4—21no—
292C2×C42C22×D6111yescharacteristic
293#107—22no—
294#107—1222no—
295C4⋊C8—612no—
296C4⋊C8—612no—
297C4⋊C8—612no—
298C4⋊C8C22×D6112yes—
299#103—22no—
300#103—1222no—
301#102—22no—
302#102—1222no—
303#102—1222no—
304#102—1222no—
305C22⋊C8—21no—
306C22⋊C8—612no—
307C22⋊C8—1222no—
308C22⋊C8—1222no—
309C22⋊C8—1221no—
310C22⋊C8—1221no—
311#97—21no—
312C4×C8—611no—
313C4×C8—611no—
314C4×C8—1221no—
315C4×C8—1221no—
316C24—21no—
317#41—4882no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—1221no—
321C2×D4—24no—
322C2×D4—21no—
323C2×D4—1221no—
324C2×D4—1221no—
325C2×D4—1221no—
326C2×D4—2441no—
327C2×D4C2×C22⋊D6112yes—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—82no—
332C22×C4—2441no—
333C22×C4D4×D6111yescharacteristic
334C22×C4#1175822411yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336Q16—4882no—
337SD16—44no—
338SD16—4882no—
339D8—2444no—
340C2×C8—22no—
341C2×C8—612no—
342C2×C8—612no—
343C2×C8—1222no—
344C2×C8—1222no—
345C2×C8—4882no—
346C2×C8—4882no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—1221no—
355C4⋊C4—1221no—
356C4⋊C4C2×C22⋊D6112yes—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—41no—
360C22⋊C4—2441no—
361C22⋊C4—2441no—
362C42—41no—
363C42D4×D6111yescharacteristic
364C42#1175822411yescharacteristic
365C42C2×C22⋊D6111yescharacteristic
366C42C23×D6111yescharacteristic
367C23—24no—
368C23—22no—
369C23—81no—
370C23—81no—
371C23—2441no—
372C23#1451974911yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—2442no—
376D4—24no—
377D4—22no—
378D4—4881no—
379D4—4882no—
380D4—2442no—
381D4—2442no—
382D4#1451979714yes—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—4882no—
394C2×C4—2441no—
395C2×C4#1451974911yescharacteristic
396C2×C4#1451974911yescharacteristic
397C2×C4#1451974911yescharacteristic
398C2×C4C2×D4×D6111yescharacteristic
399C2×C4C22×C22⋊D6111yescharacteristic
400C2×C4C2×#1175822411yescharacteristic
401C8—42no—
402C8—1222no—
403C8—1222no—
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×#1451974911yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—42no—
422C4—81no—
423C4—81no—
424C4—4881no—
425C4C2×#1451979712yes—
426C4#1684056612yes—
427C2—41no—
428C2—44no—
429C2—9761no—
430C2#2145532411yescharacteristic
431C2#2145539711yescharacteristic
432C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3110C2
socleC22×C61C2×#121
prime radical#21455819C2
hypercenter#3110D61
hypocenterC61#19388
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #21456771C2semidirect
3. #21454843C2semidirect
4. #21445258C2semidirect
5. #21445244C2semidirect
6. #21445182C2semidirect
7. #21445162C2semidirect
8. #21444600C2semidirect
9. #21444525C2semidirect
10. #21444484C2semidirect
11. C61×#3110C2semidirect
13. #16840141C22semidirect
14. #16840136C22semidirect
15. #16840114C22semidirect
16. #16840110C22semidirect
31. #16839514C22semidirect
32. #16839512C22semidirect
33. #16839508C22semidirect
34. #16839508C22semidirect
35. #16839507C22semidirect
36. #16839505C22semidirect
212. #3110D61semidirect
261. C61#19388semidirect

Maximal subgroups

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

subgroupindex
#214567712
#214558192
#214548432
#214452582
#214452442
#214451822
#214451622
#214446002
#214445252
#214444842
C61×#31102
#1938861