Finite Group Databasebeta

← #29665283

Subgroups — #29665283

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#21456974C211yesmaximal, characteristic, complemented
2#21455914C211yesmaximal, characteristic
3#21455828C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445269C212yesmaximal, complemented
6#21445241C212yesmaximal, complemented
7#21445207C212yesmaximal, complemented
8#21445193C212yesmaximal, complemented
9#21444597C211yesmaximal, characteristic, complemented
10#21444527C211yesmaximal, characteristic, complemented
11C61×#1544C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840165C2211yescharacteristic, complemented
15#16840158C2211yescharacteristic, complemented
16#16840112C2211yescharacteristic, complemented
17#16840112C2211yescharacteristic, complemented
18#16840100—22no—
19#16840095—21no—
20#16840093—21no—
21#16840069—22no—
22#16840053C2212yes—
23#16840039C2212yes—
24#16839968C2212yes—
25#16839964C2212yes—
26#16839915—21no—
27#16839912—21no—
28#16839860—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×#533C2211yescharacteristic
41C61×#431C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C61×#332C2212yescomplemented
44C61×#332C2212yescomplemented
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C2×#11758190—22no—
48C2×#11758201—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#14519710—22no—
67#14519709—22no—
68#14519708—22no—
69C4⋊#92418D412yes—
70C4⋊#92418C2312yes—
71#14519702—22no—
72#14519701—22no—
73#14519701—22no—
74#14519701—22no—
75#14519699—22no—
76#14519697—22no—
77C22⋊#92418—21no—
78C22⋊#92418D412yes—
79#14519689—22no—
80#14519689—22no—
81#14519688—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—
97C4×C8×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
105C22×Dic61—21no—
106C2×D244—21no—
107C2×D244—24no—
108C2×D244C2×D412yes—
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—82no—
112C2×Dic122—21no—
113C2×Dic122—21no—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C42×C61—41no—
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C42×C61C2411yescharacteristic
128C22⋊Dic61—41no—
129#11758201—44no—
130#11758198—22no—
131#11758198—22no—
132C4⋊Dic61—21no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61C2×D412yes—
138#11758196—41no—
139#11758196—41no—
140C4×Dic61—41no—
141C2×#92418—22no—
142C2×#92418C2×D412yes—
143C2×#92418C2×D412yes—
144#11758190—44no—
145C8×D61—82no—
146C8×D61—82no—
147C23×C61#12111yescharacteristic
148C22×D61—22no—
149C22×D61—24no—
150C22×D61—81no—
151C22×D61—81no—
152Q8×C61—42no—
153Q8×C61—42no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C2×Dic61—22no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—81no—
169D244—24no—
170D244—22no—
171D244#13514yes—
172C4×D61—42no—
173C4×D61—42no—
174C4×D61—82no—
175Dic122—42no—
176Dic122—81no—
177C8×C61—42no—
178C8×C61—42no—
179#92418—22no—
180#92418—22no—
181#92418—42no—
182#19395—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61C2×#12111yescharacteristic
187C2×D61—24no—
188C2×D61—42no—
189C2×D61—42no—
190C2×D61—84no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—81no—
196C4×C61—81no—
197C4×C61C2×#13512yes—
198C4×C61#57712yes—
199Dic61—42no—
200Dic61—81no—
201Dic61—81no—
202#3231—611no—
203#3110—611no—
204#3101—611no—
205#2972—611no—
206#1683—612no—
207#1679—612no—
208#1673—612no—
209#1671—612no—
210#1572—611no—
211#1554—611no—
212#1544D6111yescharacteristic, complemented
213C2×C61—41no—
214C2×C61#304211yescharacteristic
215C2×C61#305011yescharacteristic
216C2×C61#306911yescharacteristic
217D61—44no—
218#533—611no—
219#533C2×D6111yescharacteristic
220C2×C4×D4—611no—
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—612no—
238C8×D4—612no—
239C8×D4—1222no—
240#432—611no—
241#431—611no—
242#431C2×D6111yescharacteristic
243#431C2×D6111yescharacteristic
244C2×C22⋊C8—1222no—
245#335—612no—
246#334—612no—
247#333—612no—
248#332—612no—
249#332C2×D6112yescomplemented
250#332C2×D6112yescomplemented
251C61#1939511yescharacteristic, complemented, Sylow
252C22×D4—611no—
253C23×C4—1221no—
254C2×SD16—1222no—
255C2×SD16—1222no—
256C22×C8—1222no—
257C22×C8—1222no—
258C4⋊Q8—611no—
259C4⋊Q8—611no—
260C4⋊Q8C22×D6111yescharacteristic
261C4⋊Q8C22×D6111yescharacteristic
262#125—21no—
263#125—1221no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—1221no—
271C4×Q8—1221no—
272C4×D4—612no—
273C4×D4—612no—
274C4×D4—1221no—
275C4×D4—1221no—
276#117—21no—
277#117—1221no—
278C2×C4⋊C4—611no—
279C2×C22⋊C4—1221no—
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—21no—
297C22⋊C8—612no—
298C22⋊C8—1222no—
299C22⋊C8—1222no—
300C22⋊C8—1221no—
301#97—1221no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—611no—
305C4×C8—611no—
306C24—1221no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—1221no—
310C2×Q8—1221no—
311C2×D4—612no—
312C2×D4—1224no—
313C2×D4—1221no—
314C22×C4—1221no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—4882no—
318C22×C4#1175822211yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4D4×D6111yescharacteristic
321SD16—2444no—
322SD16—2444no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—612no—
326C2×C8—612no—
327C2×C8—1222no—
328C2×C8—4882no—
329C2×C8—4882no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—612no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C4⋊C4—2441no—
343C4⋊C4—2441no—
344C22⋊C4—41no—
345C22⋊C4—1222no—
346C22⋊C4—1222no—
347C22⋊C4—2441no—
348C42—41no—
349C42—2441no—
350C42#1175822211yescharacteristic
351C42D4×D6111yescharacteristic
352C42D4×D6111yescharacteristic
353C42C23×D6111yescharacteristic
354C23—1224no—
355C23—1222no—
356C23—4881no—
357C23—4881no—
358C23#1451975011yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—4881no—
362Q8—2442no—
363D4—614no—
364D4—1224no—
365D4—1222no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—1222no—
372C2×C4—4882no—
373C2×C4—4881no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4#1451975011yescharacteristic
379C2×C4#1451975011yescharacteristic
380C2×C4#1451975011yescharacteristic
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C2×D4×D6111yescharacteristic
383C2×C4C2×#1175822211yescharacteristic
384C8—42no—
385C8—42no—
386C8—1222no—
387C8—1222no—
388C8—2442no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—1224no—
393C22—4884no—
394C22—2442no—
395C22—2442no—
396C22C2×#1451975011yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—4881no—
404C4—4881no—
405C4—2442no—
406C4#135×D6112yes—
407C4#1684056912yes—
408C2—41no—
409C2—2444no—
410C2#2145533411yescharacteristic
411C2#2145539911yescharacteristic
412C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1544C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1544D61
hypocenterC61#19395
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
1. #21456974C2semidirect
5. #21445269C2semidirect
6. #21445241C2semidirect
7. #21445207C2semidirect
8. #21445193C2semidirect
9. #21444597C2semidirect
10. #21444527C2semidirect
11. C61×#1544C2semidirect
14. #16840165C22semidirect
15. #16840158C22semidirect
16. #16840112C22semidirect
17. #16840112C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
38. #16839510C22semidirect
39. #16839507C22semidirect
43. C61×#332C22semidirect
44. C61×#332C22semidirect
212. #1544D61semidirect
249. #332C2×D61semidirect
250. #332C2×D61semidirect
251. C61#19395semidirect

Maximal subgroups

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

subgroupindex
#214569742
#214559142
#214558282
#214548422
#214452692
#214452412
#214452072
#214451932
#214445972
#214445272
C61×#15442
#1939561