Finite Group Databasebeta

← #29665267

Subgroups — #29665267

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#21456957C211yesmaximal, characteristic, complemented
2#21455910C211yesmaximal, characteristic
3#21455827C211yesmaximal, characteristic, complemented
4#21445256C212yesmaximal, complemented
5#21445228C212yesmaximal, complemented
6#21445218C212yesmaximal, complemented
7#21445194C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444527C211yesmaximal, characteristic, complemented
10#21444479C211yesmaximal, characteristic, complemented
11C61×#2972C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840167C2211yescharacteristic, complemented
15#16840160C2211yescharacteristic, complemented
16#16840109C2211yescharacteristic, complemented
17#16840109C2211yescharacteristic, complemented
18#16840092—21no—
19#16840087—22no—
20#16840080—21no—
21#16840063—22no—
22#16840052C2212yes—
23#16840052C2212yes—
24#16839967C2212yes—
25#16839957C2212yes—
26#16839913—21no—
27#16839896—21no—
28#16839872—22no—
29C4×#11758200—22no—
30#16839819C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839515C2212yescomplemented
33#16839513C2212yescomplemented
34#16839510C2212yescomplemented
35#16839510C2212yescomplemented
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×#11758200—22no—
47C2×#11758199—22no—
48#14519777C2311yescharacteristic
49#14519777C2311yescharacteristic
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519769—21no—
53#14519765—21no—
54#14519756—21no—
55#14519756—21no—
56#14519751—21no—
57#14519751—21no—
58C4×Dic122—21no—
59C4×Dic122—21no—
60C4×D244—21no—
61C4×D244—21no—
62#14519736—21no—
63#14519711—22no—
64#14519709—22no—
65#14519708—22no—
66#14519708—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519700—22no—
70#14519699—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—
148C23×C61—24no—
149C23×C61—22no—
150C23×C61—81no—
151C23×C61—81no—
152C23×C61#12111yescharacteristic
153C22×D61—41no—
154D4×C61—24no—
155D4×C61—22no—
156D4×C61#13514yes—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—82no—
164C2×C4×C61—81no—
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C22⋊D61—82no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175D244—42no—
176D244—42no—
177D244—81no—
178C4×D61—82no—
179Dic122—42no—
180Dic122—81no—
181Dic122—81no—
182C8×C61—22no—
183C8×C61—22no—
184C8×C61—42no—
185#92418—42no—
186#92418—42no—
187#19394—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—24no—
191C22×C61—21no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—84no—
195C22×C61C2×#12111yescharacteristic
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—42no—
204C4×C61C2×#13512yes—
205C4×C61#57712yes—
206Dic61—81no—
207Dic61—81no—
208Dic61—81no—
209#3228—611no—
210#3109—611no—
211#3101—611no—
212#2972D6111yescharacteristic, complemented
213#1681—612no—
214#1677—612no—
215#1675—612no—
216#1671—612no—
217#1572—611no—
218#1554—611no—
219#1543—611no—
220C2×C61—41no—
221C2×C61—44no—
222C2×C61#304211yescharacteristic
223C2×C61#305011yescharacteristic
224C2×C61#306811yescharacteristic
225D61—161no—
226#533—611no—
227#532—611no—
228C2×C4×D4C2×D6111yescharacteristic
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—22no—
246C8×D4C2×D6112yes—
247C8×D4C2×D6112yes—
248#432—611no—
249#431—611no—
250#431C2×D6111yescharacteristic
251#431C2×D6111yescharacteristic
252C2×C22⋊C8—22no—
253#335—612no—
254#334—612no—
255#333—612no—
256#333—612no—
257#332—612no—
258#331—612no—
259C61#1939411yescharacteristic, complemented, Sylow
260C2×#41—1221no—
261C22×D4D24411yescharacteristic
262C23×C4—21no—
263C2×SD16—1222no—
264C2×D8—1222no—
265C22×C8—22no—
266C22×C8—22no—
267C4⋊Q8—611no—
268C4⋊Q8—611no—
269C4⋊Q8—611no—
270#127—611no—
271#125—1221no—
272#124—1221no—
273C22⋊Q8—1221no—
274C22⋊Q8—1221no—
275#121—1221no—
276#121—1221no—
277C4×Q8—1221no—
278C4×Q8—1221no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—1221no—
282C4×D4—1221no—
283C4×D4D24412yes—
284C4×D4C22×D6112yes—
285#117—1221no—
286C2×C4⋊C4D24411yescharacteristic
287C2×C22⋊C4—21no—
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—22no—
305C22⋊C8—22no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308C22⋊C8C22⋊D6112yes—
309#97—1221no—
310C4×C8—21no—
311C4×C8—1221no—
312C4×C8C22×D6111yescharacteristic
313C4×C8C22×D6111yescharacteristic
314C24—21no—
315#41—4882no—
316C2×Q8—1221no—
317C2×Q8—1221no—
318C2×Q8—1221no—
319C2×D4—24no—
320C2×D4—21no—
321C2×D4—1221no—
322C2×D4—1221no—
323C2×D4—1221no—
324C2×D4—2441no—
325C2×D4C2×D24412yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—82no—
330C22×C4—2441no—
331C22×C4C2×D24411yescharacteristic
332C22×C4#1175822211yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—2444no—
335D8—2444no—
336C2×C8—22no—
337C2×C8—82no—
338C2×C8—82no—
339C2×C8—1222no—
340C2×C8—1222no—
341C2×C8C2×C22⋊D6112yes—
342C2×C8C2×C22⋊D6112yes—
343C4⋊C4—21no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—2441no—
351C4⋊C4—2441no—
352C4⋊C4C2×D24412yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C42—2441no—
359C42C2×D24411yescharacteristic
360C42#1175822211yescharacteristic
361C42C2×C22⋊D6111yescharacteristic
362C42C23×D6111yescharacteristic
363C23—24no—
364C23—22no—
365C23—81no—
366C23—81no—
367C23—2441no—
368C23#1451975111yescharacteristic
369Q8—4881no—
370Q8—4881no—
371Q8—2442no—
372D4—24no—
373D4—22no—
374D4—4881no—
375D4—4882no—
376D4—2442no—
377D4—2442no—
378D4#1451979914yes—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—4882no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4#1451975111yescharacteristic
392C2×C4#1451975111yescharacteristic
393C2×C4#1451975111yescharacteristic
394C2×C4C22×C22⋊D6111yescharacteristic
395C2×C4C22×D24411yescharacteristic
396C2×C4C2×#1175822211yescharacteristic
397C8—22no—
398C8—22no—
399C8—42no—
400C8—2442no—
401C8—2442no—
402C22—21no—
403C22—24no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—84no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22C2×#1451975111yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—4881no—
419C4—4881no—
420C4—4881no—
421C4C2×#1451979912yes—
422C4#1684057312yes—
423C2—41no—
424C2—44no—
425C2—9761no—
426C2#2145533111yescharacteristic
427C2#2145540011yescharacteristic
428C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2972C2
socleC22×C61C2×#121
prime radical#21455910C2
hypercenter#2972D61
hypocenterC61#19394
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. #21456957C2semidirect
3. #21455827C2semidirect
4. #21445256C2semidirect
5. #21445228C2semidirect
6. #21445218C2semidirect
7. #21445194C2semidirect
8. #21444599C2semidirect
9. #21444527C2semidirect
10. #21444479C2semidirect
11. C61×#2972C2semidirect
14. #16840167C22semidirect
15. #16840160C22semidirect
16. #16840109C22semidirect
17. #16840109C22semidirect
32. #16839515C22semidirect
33. #16839513C22semidirect
34. #16839510C22semidirect
35. #16839510C22semidirect
36. #16839507C22semidirect
37. #16839505C22semidirect
212. #2972D61semidirect
259. C61#19394semidirect

Maximal subgroups

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

subgroupindex
#214569572
#214559102
#214558272
#214452562
#214452282
#214452182
#214451942
#214445992
#214445272
#214444792
C61×#29722
#1939461