Finite Group Databasebeta

← #29665266

Subgroups — #29665266

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#21456955C211yesmaximal, characteristic, complemented
2#21455911C211yesmaximal, characteristic
3#21455823C211yesmaximal, characteristic, complemented
4#21454842C211yesmaximal, characteristic, complemented
5#21445259C212yesmaximal, complemented
6#21445231C212yesmaximal, complemented
7#21445217C212yesmaximal, complemented
8#21445193C212yesmaximal, complemented
9#21444600C211yesmaximal, characteristic, complemented
10#21444479C211yesmaximal, characteristic, complemented
11C61×#1554C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840165C2211yescharacteristic, complemented
15#16840158C2211yescharacteristic, complemented
16#16840110C2211yescharacteristic, complemented
17#16840110C2211yescharacteristic, complemented
18#16840090—21no—
19#16840086—22no—
20#16840081—21no—
21#16840059—22no—
22#16840053C2212yes—
23#16840049C2212yes—
24#16839964C2212yes—
25#16839958C2212yes—
26#16839912—21no—
27#16839898—21no—
28#16839875—22no—
29C4×#11758201—22no—
30C8×D244—22no—
31#16839827C2212yes—
32#16839827C2212yes—
33#16839815C2211yescharacteristic
34#16839815C2211yescharacteristic
35C22⋊C8×D61—22no—
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839510C2212yescomplemented
39#16839505C2212yescomplemented
40C61×#533C2211yescharacteristic
41C61×#432C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C61×#333C2212yescomplemented
44C61×#332C2212yescomplemented
45C2×#11758222—21no—
46C22×D244D411yescharacteristic
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×D488—22no—
50C2×C8×D61—22no—
51C2×C8×D61—22no—
52#14519777C2311yescharacteristic
53#14519774C2311yescharacteristic
54#14519765—21no—
55#14519751—21no—
56#14519751—21no—
57C4×D244—21no—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244D412yes—
62C4×D244C2312yes—
63C2×C4⋊Dic61D411yescharacteristic
64C2×#11758198—21no—
65#14519710—22no—
66#14519709—22no—
67#14519708—22no—
68#14519708—22no—
69C4⋊#92418D412yes—
70C4⋊#92418C2312yes—
71#14519701—22no—
72#14519699—22no—
73#14519699—22no—
74#14519697—22no—
75#14519697—22no—
76#14519697—22no—
77C22⋊#92418—21no—
78C22⋊#92418D412yes—
79#14519689—22no—
80#14519689—22no—
81C4×#92418—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—
97#97×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#11758201—44no—
134#11758198—22no—
135#11758198—22no—
136#11758198—41no—
137#11758198—41no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61C2×D412yes—
142C2×#92418—22no—
143C2×#92418C2×D412yes—
144C2×#92418C2×D412yes—
145D488—44no—
146C8×D61—82no—
147C8×D61—82no—
148C23×C61#12111yescharacteristic
149C22×D61—22no—
150C22×D61—24no—
151C22×D61—41no—
152C22×D61—81no—
153C22×D61—81no—
154Q8×C61—42no—
155Q8×C61—81no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C22⋊D61—82no—
168C2×Dic61—22no—
169C2×Dic61—41no—
170C2×Dic61—81no—
171D244—24no—
172D244—22no—
173D244—42no—
174D244—42no—
175D244—81no—
176D244#13514yes—
177C4×D61—42no—
178C4×D61—42no—
179C4×D61—82no—
180C4×D61—82no—
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×C61C2×#12111yescharacteristic
192C2×D61—24no—
193C2×D61—42no—
194C2×D61—42no—
195C2×D61—84no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—81no—
204C4×C61—81no—
205C4×C61C2×#13512yes—
206C4×C61#57712yes—
207Dic61—42no—
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#1554D6111yescharacteristic, complemented
219#1543—611no—
220C2×C61—41no—
221C2×C61#304211yescharacteristic
222C2×C61#305011yescharacteristic
223C2×C61#306811yescharacteristic
224D61—44no—
225D61—161no—
226#533C2×D6111yescharacteristic
227#532—611no—
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#432C2×D6111yescharacteristic
249#431—611no—
250#431—611no—
251#431C2×D6111yescharacteristic
252C2×C22⋊C8—1222no—
253#335—612no—
254#334—612no—
255#333—612no—
256#333C2×D6112yescomplemented
257#332C2×D6112yescomplemented
258#331—612no—
259C61#1939411yescharacteristic, complemented, Sylow
260C2×#41—1221no—
261C22×D4—611no—
262C23×C4—1221no—
263C2×SD16—1222no—
264C2×D8—1222no—
265C22×C8—1222no—
266C22×C8—1222no—
267C4⋊Q8—611no—
268C4⋊Q8C22×D6111yescharacteristic
269C4⋊Q8C22×D6111yescharacteristic
270#127—611no—
271#125—21no—
272#124—1221no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275#121—1221no—
276#121—1221no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×D4—612no—
280C4×D4—612no—
281C4×D4—1221no—
282C4×D4—1221no—
283C4×D4—1221no—
284C4×D4—1221no—
285#117—21no—
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—21no—
310C4×C8—21no—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—1221no—
314C24—1221no—
315#41—4882no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—1221no—
319C2×D4—612no—
320C2×D4—1224no—
321C2×D4—1221no—
322C2×D4—1221no—
323C2×D4—1221no—
324C2×D4—1221no—
325C2×D4—2441no—
326C22×C4—1221no—
327C22×C4—1222no—
328C22×C4—1222no—
329C22×C4—4882no—
330C22×C4—2441no—
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—21no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—612no—
350C4⋊C4—1221no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C22⋊C4—41no—
354C22⋊C4—1222no—
355C22⋊C4—1222no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C42—41no—
359C42#1175822211yescharacteristic
360C42D4×D6111yescharacteristic
361C42D4×D6111yescharacteristic
362C42C23×D6111yescharacteristic
363C23—1224no—
364C23—1222no—
365C23—4881no—
366C23—4881no—
367C23—2441no—
368C23#1451975011yescharacteristic
369Q8—42no—
370Q8—81no—
371Q8—4881no—
372D4—614no—
373D4—1224no—
374D4—1222no—
375D4—4881no—
376D4—4882no—
377D4—2442no—
378D4—2442no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—1222no—
385C2×C4—4882no—
386C2×C4—4882no—
387C2×C4—4881no—
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—1224no—
406C22—4884no—
407C22—4881no—
408C22—4881no—
409C22—4881no—
410C22—2442no—
411C22—2442no—
412C22C2×#1451975011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—4881no—
420C4—2442no—
421C4#135×D6112yes—
422C4#1684057112yes—
423C2—41no—
424C2—9761no—
425C2—2444no—
426C2#2145533011yescharacteristic
427C2#2145539911yescharacteristic
428C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1554C2
socleC22×C61C2×#121
prime radical#21455911C2
hypercenter#1554D61
hypocenterC61#19394
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1554

Complements

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

NH ≅ G/Ntype
1. #21456955C2semidirect
3. #21455823C2semidirect
4. #21454842C2semidirect
5. #21445259C2semidirect
6. #21445231C2semidirect
7. #21445217C2semidirect
8. #21445193C2semidirect
9. #21444600C2semidirect
10. #21444479C2semidirect
11. C61×#1554C2semidirect
14. #16840165C22semidirect
15. #16840158C22semidirect
16. #16840110C22semidirect
17. #16840110C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
38. #16839510C22semidirect
39. #16839505C22semidirect
43. C61×#333C22semidirect
44. C61×#332C22semidirect
218. #1554D61semidirect
256. #333C2×D61semidirect
257. #332C2×D61semidirect
259. C61#19394semidirect

Maximal subgroups

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

subgroupindex
#214569552
#214559112
#214558232
#214548422
#214452592
#214452312
#214452172
#214451932
#214446002
#214444792
C61×#15542
#1939461