Finite Group Databasebeta

← #29665278

Subgroups — #29665278

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456980C211yesmaximal, characteristic, complemented
2#21455918C211yesmaximal, characteristic
3#21455827C211yesmaximal, characteristic
4#21445270C212yesmaximal, complemented
5#21445242C212yesmaximal, complemented
6#21445208C212yesmaximal, complemented
7#21445194C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444527C211yesmaximal, characteristic, complemented
10#21444483C211yesmaximal, characteristic, complemented
11C61×#2972C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840319C2211yescharacteristic
14#16840167C2211yescharacteristic, complemented
15#16840160C2211yescharacteristic, complemented
16#16840113C2211yescharacteristic, complemented
17#16840113C2211yescharacteristic, complemented
18#16840104—22no—
19#16840097—21no—
20#16840092—21no—
21#16840070—22no—
22#16840052C2212yes—
23#16840042C2212yes—
24#16839967C2212yes—
25#16839967C2212yes—
26#16839913—21no—
27#16839913—21no—
28#16839862—22no—
29C4×#11758200—22no—
30#16839819C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839515C2212yescomplemented
33#16839513C2212yescomplemented
34#16839510C2212yescomplemented
35#16839509C2212yescomplemented
36#16839507C2212yescomplemented
37#16839507C2212yescomplemented
38C2×C4×D4×C61C2211yescharacteristic
39C8×D4×C61—22no—
40C8×D4×C61C2212yes—
41C8×D4×C61C2212yes—
42C61×#431C2211yescharacteristic
43C61×#431C2211yescharacteristic
44C2×C22⋊C8×C61—22no—
45C2×#11758200—22no—
46C2×#11758200—22no—
47#14519777C2311yescharacteristic
48#14519777C2311yescharacteristic
49#14519777C2311yescharacteristic
50#14519777C2311yescharacteristic
51#14519769—21no—
52#14519769—21no—
53#14519756—21no—
54#14519756—21no—
55#14519756—21no—
56#14519756—21no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61#14519736—21no—
62#14519736—21no—
63#14519711—22no—
64#14519710—22no—
65#14519709—22no—
66#14519708—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519702—22no—
70#14519702—22no—
71#14519700—22no—
72#14519700—22no—
73#14519698—22no—
74#14519698—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×Dic122—21no—
111C2×Dic122—21no—
112C2×Dic122—21no—
113C2×Dic122—21no—
114C2×C8×C61—22no—
115C2×C8×C61—82no—
116C2×C8×C61—82no—
117C2×C8×C61C2×D412yes—
118C2×C8×C61C2×D412yes—
119C4⋊C4×C61—21no—
120C4⋊C4×C61C2×D412yes—
121C22⋊C4×C61—22no—
122C22⋊C4×C61—22no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129#11758200—44no—
130#11758200—44no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139#11758196—41no—
140#11758196—41no—
141#11758196—41no—
142#11758196—41no—
143C4×Dic61—41no—
144C4×Dic61—41no—
145C2×#92418—22no—
146C2×#92418—22no—
147C23×C61—24no—
148C23×C61—22no—
149C23×C61—81no—
150C23×C61—81no—
151C23×C61#12111yescharacteristic
152D4×C61—24no—
153D4×C61—22no—
154D4×C61#13514yes—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—82no—
162C2×C4×C61—81no—
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173Dic122—42no—
174Dic122—42no—
175Dic122—42no—
176Dic122—81no—
177C8×C61—22no—
178C8×C61—22no—
179C8×C61—42no—
180#92418—42no—
181#92418—42no—
182#19395—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—24no—
186C22×C61—21no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61—84no—
190C22×C61C2×#12111yescharacteristic
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—42no—
196C4×C61C2×#13512yes—
197C4×C61#57712yes—
198Dic61—81no—
199Dic61—81no—
200Dic61—81no—
201Dic61—81no—
202#3231—611no—
203#3110—611no—
204#3101—611no—
205#2972D6111yescharacteristic
206#1683—612no—
207#1679—612no—
208#1673—612no—
209#1671—612no—
210#1572—611no—
211#1554—611no—
212#1544—611no—
213C2×C61—41no—
214C2×C61—44no—
215C2×C61#304211yescharacteristic
216C2×C61#305011yescharacteristic
217C2×C61#306911yescharacteristic
218#533—611no—
219#533—611no—
220C2×C4×D4C2×D6111yescharacteristic
221#500—611no—
222#498—611no—
223#487—611no—
224#487—611no—
225#485—1222no—
226#484—1221no—
227#483—1221no—
228#479—1222no—
229#476—612no—
230#474—612no—
231#461—612no—
232#461—612no—
233#450—1221no—
234#450—1221no—
235#440—1222no—
236C4×SD16—1222no—
237C8×D4—22no—
238C8×D4C2×D6112yes—
239C8×D4C2×D6112yes—
240#432—611no—
241#431—611no—
242#431C2×D6111yescharacteristic
243#431C2×D6111yescharacteristic
244C2×C22⋊C8—22no—
245#335—612no—
246#334—612no—
247#333—612no—
248#332—612no—
249#332—612no—
250#332—612no—
251C61#1939511yescharacteristic, complemented, Sylow
252C22×D4D24411yescharacteristic
253C23×C4—21no—
254C2×SD16—1222no—
255C2×SD16—1222no—
256C22×C8—22no—
257C22×C8—22no—
258C4⋊Q8—611no—
259C4⋊Q8—611no—
260C4⋊Q8—611no—
261C4⋊Q8—611no—
262#125—1221no—
263#125—1221no—
264C22⋊Q8—1221no—
265C22⋊Q8—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268C4×Q8—1221no—
269C4×Q8—1221no—
270C4×Q8—1221no—
271C4×Q8—1221no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4D24412yes—
275C4×D4C22×D6112yes—
276#117—1221no—
277#117—1221no—
278C2×C4⋊C4D24411yescharacteristic
279C2×C22⋊C4—21no—
280C2×C42C22×D6111yescharacteristic
281#107—1222no—
282#107—1222no—
283#106—1222no—
284#106—1222no—
285C4⋊C8—612no—
286C4⋊C8—612no—
287C4⋊C8C22⋊D6112yes—
288C4⋊C8C22×D6112yes—
289#103—1222no—
290#103—1222no—
291#103—1222no—
292#103—1222no—
293#102—1222no—
294#102—1222no—
295C22⋊C8—21no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300C22⋊C8C22⋊D6112yes—
301#97—1221no—
302C4×C8—21no—
303C4×C8—1221no—
304C4×C8C22×D6111yescharacteristic
305C4×C8C22×D6111yescharacteristic
306C24—21no—
307C2×Q8—1221no—
308C2×Q8—1221no—
309C2×Q8—1221no—
310C2×Q8—1221no—
311C2×D4—24no—
312C2×D4—21no—
313C2×D4C2×D24412yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—82no—
318C22×C4C2×D24411yescharacteristic
319C22×C4#1175822211yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321SD16—2444no—
322SD16—2444no—
323C2×C8—22no—
324C2×C8—82no—
325C2×C8—82no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8C2×C22⋊D6112yes—
329C2×C8C2×C22⋊D6112yes—
330C4⋊C4—21no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C4⋊C4C2×D24412yes—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C42—2441no—
349C42—2441no—
350C42C2×D24411yescharacteristic
351C42#1175822211yescharacteristic
352C42C2×C22⋊D6111yescharacteristic
353C42C23×D6111yescharacteristic
354C23—24no—
355C23—22no—
356C23—81no—
357C23—81no—
358C23#1451975111yescharacteristic
359Q8—4881no—
360Q8—2442no—
361Q8—2442no—
362Q8—2442no—
363D4—24no—
364D4—22no—
365D4#1451979914yes—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4#1451975111yescharacteristic
379C2×C4#1451975111yescharacteristic
380C2×C4#1451975111yescharacteristic
381C2×C4C22×C22⋊D6111yescharacteristic
382C2×C4C22×D24411yescharacteristic
383C2×C4C2×#1175822211yescharacteristic
384C8—22no—
385C8—22no—
386C8—42no—
387C8—2442no—
388C8—2442no—
389C22—21no—
390C22—24no—
391C22—21no—
392C22—21no—
393C22—42no—
394C22—42no—
395C22—84no—
396C22C2×#1451975111yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—4881no—
403C4—4881no—
404C4—4881no—
405C4—4881no—
406C4C2×#1451979912yes—
407C4#1684057312yes—
408C2—41no—
409C2—44no—
410C2#2145533111yescharacteristic
411C2#2145540011yescharacteristic
412C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2972C2
socleC22×C61C2×#121
prime radicalC22×D4×C61D4
hypercenter#2972D61
hypocenterC61#19395
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. #21456980C2semidirect
4. #21445270C2semidirect
5. #21445242C2semidirect
6. #21445208C2semidirect
7. #21445194C2semidirect
8. #21444599C2semidirect
9. #21444527C2semidirect
10. #21444483C2semidirect
14. #16840167C22semidirect
15. #16840160C22semidirect
16. #16840113C22semidirect
17. #16840113C22semidirect
32. #16839515C22semidirect
33. #16839513C22semidirect
34. #16839510C22semidirect
35. #16839509C22semidirect
36. #16839507C22semidirect
37. #16839507C22semidirect
251. C61#19395semidirect

Maximal subgroups

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

subgroupindex
#214569802
#214559182
#214558272
#214452702
#214452422
#214452082
#214451942
#214445992
#214445272
#214444832
C61×#29722
#1939561