Finite Group Databasebeta

← #29665272

Subgroups — #29665272

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456951C211yesmaximal, characteristic, complemented
2#21455906C211yesmaximal, characteristic
3#21455828C211yesmaximal, characteristic, complemented
4#21454842C211yesmaximal, characteristic, complemented
5#21445255C212yesmaximal, complemented
6#21445227C212yesmaximal, complemented
7#21445217C212yesmaximal, complemented
8#21445197C212yesmaximal, complemented
9#21444597C211yesmaximal, characteristic, complemented
10#21444527C211yesmaximal, characteristic, complemented
11C61×#1543C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840165C2211yescharacteristic, complemented
15#16840158C2211yescharacteristic, complemented
16#16840108C2211yescharacteristic, complemented
17#16840108C2211yescharacteristic, complemented
18#16840093—21no—
19#16840083—22no—
20#16840078—21no—
21#16840062—22no—
22#16840053C2212yes—
23#16840049C2212yes—
24#16839968C2212yes—
25#16839954C2212yes—
26#16839915—21no—
27#16839895—21no—
28#16839872—22no—
29C4×#11758190—22no—
30C8×D244—22no—
31#16839827C2212yes—
32#16839827C2212yes—
33#16839819C2211yescharacteristic
34#16839815C2211yescharacteristic
35C22⋊C8×D61—22no—
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839510C2212yescomplemented
39#16839507C2212yescomplemented
40C61×#532C2211yescharacteristic
41C61×#431C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C61×#333C2212yescomplemented
44C61×#331C2212yescomplemented
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C2×#11758190—22no—
48C2×#11758199—22no—
49C2×C8×D61—22no—
50C2×C8×D61—22no—
51#14519777C2311yescharacteristic
52#14519777C2311yescharacteristic
53#14519769—21no—
54#14519756—21no—
55#14519756—21no—
56C4×Dic122—21no—
57C4×Dic122—21no—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244D412yes—
61C4×D244C2312yes—
62#14519736—21no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×#11758198—21no—
65#14519710—22no—
66#14519709—22no—
67#14519708—22no—
68#14519708—22no—
69C4⋊#92418D412yes—
70C4⋊#92418C2312yes—
71#14519702—22no—
72#14519699—22no—
73#14519698—22no—
74#14519698—22no—
75#14519698—22no—
76#14519697—22no—
77C22⋊#92418—21no—
78C22⋊#92418D412yes—
79#14519689—22no—
80#14519689—22no—
81#14519688—21no—
82C4×#92418C2311yescharacteristic
83C4×#92418C2311yescharacteristic
84C2×#41×C61—21no—
85C4⋊Q8×C61C2311yescharacteristic
86#127×C61C2311yescharacteristic
87#124×C61—21no—
88#121×C61—21no—
89#121×C61—21no—
90C4×D4×C61—21no—
91C4×D4×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—
100#41×C61—82no—
101C2×Q8×C61—21no—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C22×Dic61—21no—
111C2×D244—21no—
112C2×D244—24no—
113C2×D244C2×D412yes—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—82no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×C8×C61—22no—
120C2×C8×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C22⋊C4×C61—41no—
124C22⋊C4×C61—41no—
125C42×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61#4111yescharacteristic
128C42×C61C2411yescharacteristic
129C22⋊Dic61—41no—
130#11758199—44no—
131#11758198—22no—
132#11758198—22no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61C2×D412yes—
139#11758196—41no—
140#11758196—41no—
141C4×Dic61—41no—
142C2×#92418—22no—
143C2×#92418C2×D412yes—
144C2×#92418C2×D412yes—
145#11758190—44no—
146C8×D61—82no—
147C8×D61—82no—
148C23×C61—41no—
149C23×C61#12111yescharacteristic
150C22×D61—22no—
151C22×D61—24no—
152C22×D61—81no—
153C22×D61—81no—
154Q8×C61—81no—
155D4×C61—42no—
156D4×C61—42no—
157D4×C61—81no—
158D4×C61—82no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—82no—
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C2×#4111yescharacteristic
170C2×Dic61—22no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—81no—
174D244—24no—
175D244—22no—
176D244#13514yes—
177C4×D61—42no—
178C4×D61—42no—
179C4×D61—82no—
180Dic122—42no—
181Dic122—81no—
182C8×C61—42no—
183C8×C61—42no—
184#92418—22no—
185#92418—22no—
186#92418—42no—
187#19394—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61C2×#12111yescharacteristic
195C2×D61—24no—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61—84no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—81no—
204C4×C61C2×#13512yes—
205C4×C61#57712yes—
206Dic61—42no—
207Dic61—81no—
208Dic61—81no—
209#3228—611no—
210#3109—611no—
211#3101—611no—
212#2972—611no—
213#1681—612no—
214#1677—612no—
215#1675—612no—
216#1671—612no—
217#1572—611no—
218#1554—611no—
219#1543D6111yescharacteristic, complemented
220C2×C61—41no—
221C2×C61—161no—
222C2×C61#304211yescharacteristic
223C2×C61#305011yescharacteristic
224C2×C61#306811yescharacteristic
225D61—44no—
226#533—611no—
227#532C2×D6111yescharacteristic
228C2×C4×D4—611no—
229#500—611no—
230#498—611no—
231#486—611no—
232#486—611no—
233#483—1221no—
234#482—1222no—
235#481—1221no—
236#478—1222no—
237#476—612no—
238#476—612no—
239#461—612no—
240#459—612no—
241#450—1221no—
242#447—1221no—
243#442—1222no—
244C4×SD16—1222no—
245C8×D4—612no—
246C8×D4—612no—
247C8×D4—1222no—
248#432—611no—
249#431—611no—
250#431C2×D6111yescharacteristic
251#431C2×D6111yescharacteristic
252C2×C22⋊C8—1222no—
253#335—612no—
254#334—612no—
255#333—612no—
256#333C2×D6112yescomplemented
257#332—612no—
258#331C2×D6112yescomplemented
259C61#1939411yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C22×D4—611no—
262C23×C4—1221no—
263C2×SD16—1222no—
264C2×D8—1222no—
265C22×C8—1222no—
266C22×C8—1222no—
267C4⋊Q8—611no—
268C4⋊Q8—611no—
269C4⋊Q8C22×D6111yescharacteristic
270#127C22×D6111yescharacteristic
271#125—1221no—
272#124—21no—
273C22⋊Q8—1221no—
274C22⋊Q8—1221no—
275#121—21no—
276#121—21no—
277C4×Q8—1221no—
278C4×Q8—1221no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—612no—
282C4×D4—612no—
283C4×D4—1221no—
284C4×D4—1221no—
285#117—1221no—
286C2×C4⋊C4—611no—
287C2×C22⋊C4—1221no—
288C2×C42C22×D6111yescharacteristic
289#107—1222no—
290#106—1222no—
291#106—1222no—
292#106—1222no—
293C4⋊C8—612no—
294C4⋊C8—612no—
295C4⋊C8C22×D6112yes—
296C4⋊C8C22×D6112yes—
297#103—1222no—
298#102—1222no—
299#102—1222no—
300#102—1222no—
301#102—1222no—
302#102—1222no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—612no—
306C22⋊C8—1222no—
307C22⋊C8—1222no—
308C22⋊C8—1221no—
309#97—1221no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—611no—
313C4×C8—611no—
314C24—1221no—
315#41—82no—
316C2×Q8—21no—
317C2×Q8—1221no—
318C2×Q8—1221no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—612no—
324C2×D4—1224no—
325C2×D4—1221no—
326C22×C4—41no—
327C22×C4—1221no—
328C22×C4—1222no—
329C22×C4—1222no—
330C22×C4—4882no—
331C22×C4#1175822211yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4D4×D6111yescharacteristic
334SD16—2444no—
335D8—2444no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—612no—
339C2×C8—612no—
340C2×C8—1222no—
341C2×C8—4882no—
342C2×C8—4882no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—612no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—1221no—
351C4⋊C4—2441no—
352C4⋊C4—2441no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—1222no—
356C22⋊C4—1222no—
357C22⋊C4—2441no—
358C42—2441no—
359C42#1175822211yescharacteristic
360C42D4×D6111yescharacteristic
361C42D4×D6111yescharacteristic
362C42C23×D6111yescharacteristic
363C23—41no—
364C23—1224no—
365C23—1222no—
366C23—4881no—
367C23—4881no—
368C23#1451975011yescharacteristic
369Q8—81no—
370Q8—4881no—
371Q8—2442no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—82no—
376D4—614no—
377D4—1224no—
378D4—1222no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—82no—
384C2×C4—1222no—
385C2×C4—4882no—
386C2×C4—4881no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4#1451975011yescharacteristic
392C2×C4#1451975011yescharacteristic
393C2×C4#1451975011yescharacteristic
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4C2×D4×D6111yescharacteristic
396C2×C4C2×#1175822211yescharacteristic
397C8—42no—
398C8—42no—
399C8—1222no—
400C8—1222no—
401C8—2442no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—1224no—
409C22—4884no—
410C22—2442no—
411C22—2442no—
412C22C2×#1451975011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—4881no—
419C4—4881no—
420C4—2442no—
421C4#135×D6112yes—
422C4#1684056912yes—
423C2—41no—
424C2—161no—
425C2—2444no—
426C2#2145533411yescharacteristic
427C2#2145539911yescharacteristic
428C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1543C2
socleC22×C61C2×#121
prime radical#21455906C2
hypercenter#1543D61
hypocenterC61#19394
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
1. #21456951C2semidirect
3. #21455828C2semidirect
4. #21454842C2semidirect
5. #21445255C2semidirect
6. #21445227C2semidirect
7. #21445217C2semidirect
8. #21445197C2semidirect
9. #21444597C2semidirect
10. #21444527C2semidirect
11. C61×#1543C2semidirect
14. #16840165C22semidirect
15. #16840158C22semidirect
16. #16840108C22semidirect
17. #16840108C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
38. #16839510C22semidirect
39. #16839507C22semidirect
43. C61×#333C22semidirect
44. C61×#331C22semidirect
219. #1543D61semidirect
256. #333C2×D61semidirect
258. #331C2×D61semidirect
259. C61#19394semidirect

Maximal subgroups

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

subgroupindex
#214569512
#214559062
#214558282
#214548422
#214452552
#214452272
#214452172
#214451972
#214445972
#214445272
C61×#15432
#1939461