Finite Group Databasebeta

← #29667896

Subgroups — #29667896

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456970C211yesmaximal, characteristic, complemented
2#21456737C211yesmaximal, characteristic
3#21456378C211yesmaximal, characteristic, complemented
4#21456215C211yesmaximal, characteristic, complemented
5#21456193C211yesmaximal, characteristic, complemented
6#21445470C212yesmaximal, complemented
7#21445456C212yesmaximal, complemented
8#21445418C212yesmaximal, complemented
9#21445409C212yesmaximal, complemented
10C8×#11758194C211yesmaximal, characteristic, complemented
11C2×C61×#492C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840165C2212yescomplemented
15#16840160C2211yescharacteristic, complemented
16#16840138C2212yes—
17#16840135C2211yescharacteristic, complemented
18#16840126—22no—
19#16840124—22no—
20#16840114C2212yescomplemented
21#16840114C2212yescomplemented
22#16840113C2211yescharacteristic, complemented
23#16840110C2212yescomplemented
24#16840109C2211yescharacteristic, complemented
25#16840108C2212yes—
26#16840108C2212yes—
27#16840108C2212yes—
28#16840097—22no—
29#16840054—22no—
30#16839991—22no—
31#16839954—22no—
32#16839896—22no—
33C4×#11758190—22no—
34#16839815C2211yescharacteristic
35#16839767—22no—
36#16839761—22no—
37C4×#11758194C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C22×SD16×C61—22no—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#127×C61C2211yescharacteristic
43C61×#492—22no—
44C61×#492C2212yes—
45C61×#492C2212yes—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222—21no—
48C2×#11758190—22no—
49C2×#11758199—24no—
50C2×#11758194D412yes—
51#14519777C2311yescharacteristic
52#14519777C2311yescharacteristic
53#14519777C2311yescharacteristic
54#14519774C2311yescharacteristic
55#14519769—21no—
56#14519765—21no—
57#14519756—21no—
58#14519756—21no—
59#14519751—21no—
60#14519751—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63C4×D244—21no—
64C4×D244—21no—
65#14519736—21no—
66#14519710—24no—
67#14519709—22no—
68C4⋊#92418D412yes—
69#14519702—22no—
70#14519700—24no—
71#14519700—24no—
72#14519699—22no—
73#14519698—24no—
74#14519698—24no—
75#14519698—24no—
76#14519697—24no—
77C22⋊#92418D412yes—
78#14519688D412yes—
79C4×#92418C2312yes—
80C4×#92418C2312yes—
81C22×Q8×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C2×SD16×C61—24no—
86C2×SD16×C61—24no—
87C2×SD16×C61—48no—
88C22×C8×C61D412yes—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61C2312yes—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61—21no—
96#127×C61C2312yes—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4×C8×C61D412yes—
101C4×C8×C61C2311yescharacteristic
102C4×C8×C61C2311yescharacteristic
103C24×C61—41no—
104C2×Q8×C61—22no—
105C2×Q8×C61—44no—
106C2×Q8×C61—41no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—44no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—41no—
114C2×D4×C61—44no—
115C2×D4×C61—82no—
116C22×C4×C61—41no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C2×C22⋊D61—41no—
121#11758222—44no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129SD16×C61—48no—
130SD16×C61—48no—
131C2×C8×C61—24no—
132C2×C8×C61C2×D412yes—
133C2×C8×C61C2×D412yes—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2411yescharacteristic
142C22⋊Dic61—41no—
143#11758199—84no—
144#11758198—41no—
145#11758198—41no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152#11758196—41no—
153#11758196—41no—
154C4×Dic61—41no—
155#11758194—24no—
156C2×#92418C2×D414yes—
157C2×#92418C2×D414yes—
158#11758190—44no—
159C23×C61—42no—
160C23×C61—82no—
161C23×C61—82no—
162C23×C61—82no—
163C23×C61#12711yescharacteristic
164C22×D61—41no—
165Q8×C61—44no—
166Q8×C61—82no—
167D4×C61—44no—
168D4×C61—44no—
169D4×C61—44no—
170D4×C61—82no—
171D4×C61—82no—
172D4×C61—82no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—42no—
176C2×C4×C61—82no—
177C2×C4×C61#12711yescharacteristic
178C2×C4×C61#12711yescharacteristic
179C2×C4×C61#12711yescharacteristic
180C2×C4×C61#13514yes—
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C22⋊D61—82no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188D244—42no—
189D244—42no—
190D244—42no—
191C4×D61—82no—
192Dic122—42no—
193Dic122—42no—
194Dic122—42no—
195C8×C61—22no—
196C8×C61AGL(1,ℤ/8)14yes—
197#92418—24no—
198#92418—24no—
199#19625—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—82no—
203C22×C61—82no—
204C22×C61—82no—
205C22×C61—164no—
206C22×C61#49512yes—
207C22×C61C2×#12711yescharacteristic
208C2×D61—81no—
209C2×D61—81no—
210C2×D61—81no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—82no—
214C4×C61#49512yes—
215C4×C61C2×#13512yes—
216C4×C61C2×AGL(1,ℤ/8)12yes—
217C4×C61#57712yes—
218Dic61—81no—
219Dic61—81no—
220Dic61—81no—
221#3230—611no—
222#3205—611no—
223#3167—611no—
224C2×#492D6111yescharacteristic, complemented
225#3141—611no—
226#3138—611no—
227#1719—612no—
228#1717—612no—
229#1708—612no—
230#1706—612no—
231C8×M16—611no—
232C2×C61—22no—
233C2×C61—162no—
234C2×C61C2×#49511yescharacteristic
235C2×C61#315111yescharacteristic
236C2×C61#315411yescharacteristic
237D61—161no—
238C22×SD16—22no—
239#533—611no—
240#532—611no—
241C2×C4⋊Q8C2×D6111yescharacteristic
242C2×#127C2×D6111yescharacteristic
243#500—612no—
244#498—611no—
245#493—612no—
246#492—22no—
247#492—611no—
248#492C2×D6112yes—
249#492C2×D6112yes—
250#490—1222no—
251#489—1222no—
252#487—612no—
253#487—612no—
254#487—611no—
255#486—612no—
256#486—612no—
257#486—612no—
258#486—612no—
259#486—611no—
260#484—1222no—
261#477—1222no—
262#465—1222no—
263#459—1222no—
264#447—1222no—
265C4×SD16—1222no—
266#431—611no—
267#418—1222no—
268#417—1222no—
269C4×M16—611no—
270C2×C4×C8C2×D6111yescharacteristic
271#322—612no—
272C82—612no—
273C61#1962511yescharacteristic, complemented, Sylow
274C2×#41—1221no—
275C22×Q8—21no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4—21no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—48no—
282C2×SD16—1222no—
283C2×D8—1224no—
284C2×M16—612no—
285C22×C8C22⋊D6112yes—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—611no—
290C4⋊Q8—611no—
291C4⋊Q8—611no—
292C4⋊Q8C22×D6112yes—
293#127—21no—
294#127—21no—
295#127—21no—
296#127—611no—
297#127C22×D6112yes—
298#125—1221no—
299#124—1221no—
300C22⋊Q8—1221no—
301C22⋊Q8—1221no—
302#121—1221no—
303#121—1221no—
304C4×Q8—1221no—
305C4×Q8—1221no—
306C4×D4—1221no—
307C4×D4—1221no—
308#117—1221no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C42C22×D6111yescharacteristic
312#107—1224no—
313#106—1222no—
314C4⋊C8—612no—
315#103—1224no—
316#103—1224no—
317#103—1222no—
318#102—1224no—
319#102—1224no—
320#102—1224no—
321#102—1224no—
322#102—1222no—
323C22⋊C8—612no—
324#97—612no—
325C4×C8—612no—
326C4×C8—612no—
327C4×C8C22⋊D6112yes—
328C4×C8C22×D6111yescharacteristic
329C4×C8C22×D6111yescharacteristic
330C24—41no—
331#41—2444no—
332C2×Q8—22no—
333C2×Q8—44no—
334C2×Q8—41no—
335C2×Q8—1221no—
336C2×Q8—1221no—
337C2×Q8—1221no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—44no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—44no—
346C2×D4—82no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—2441no—
351C22×C4—41no—
352C22×C4—2441no—
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356SD16—48no—
357SD16—48no—
358SD16—2444no—
359D8—4884no—
360M16—1224no—
361C2×C8—24no—
362C2×C8—614no—
363C2×C8—614no—
364C2×C8C2×C22⋊D6112yes—
365C2×C8C2×C22⋊D6112yes—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C22⋊C4—2441no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42—2441no—
382C42D4×D6111yescharacteristic
383C42D4×D6111yescharacteristic
384C42C2×C22⋊D6111yescharacteristic
385C42C23×D6111yescharacteristic
386C23—42no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—2441no—
391C23#1451977311yescharacteristic
392Q8—44no—
393Q8—82no—
394Q8—2442no—
395Q8—2442no—
396Q8—2442no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4—4882no—
404D4—2442no—
405D4—2442no—
406D4—2442no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—4882no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4#1451977311yescharacteristic
416C2×C4#1451977311yescharacteristic
417C2×C4#1451977311yescharacteristic
418C2×C4#1451979814yes—
419C2×C4C2×D4×D6111yescharacteristic
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4C22×C22⋊D6111yescharacteristic
422C8—22no—
423C8—1224no—
424C8—1224no—
425C8#1451980314yes—
426C22—21no—
427C22—21no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—164no—
432C22—4881no—
433C22—4881no—
434C22—4881no—
435C22#1684014312yes—
436C22C2×#1451977311yescharacteristic
437C4—21no—
438C4—21no—
439C4—82no—
440C4—4881no—
441C4—4881no—
442C4—4881no—
443C4#1684014312yes—
444C4C2×#1451979812yes—
445C4C2×#1451980312yes—
446C4#1684056812yes—
447C2—22no—
448C2—162no—
449C2—9761no—
450C2C2×#1684014311yescharacteristic
451C2#2145626911yescharacteristic
452C2#2145628911yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#492C2
socleC22×C61C2×#127
prime radical#21456737C2
hypercenterC2×#492D61
hypocenterC61#19625
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
1. #21456970C2semidirect
3. #21456378C2semidirect
4. #21456215C2semidirect
5. #21456193C2semidirect
6. #21445470C2semidirect
7. #21445456C2semidirect
8. #21445418C2semidirect
9. #21445409C2semidirect
10. C8×#11758194C2semidirect
11. C2×C61×#492C2semidirect
14. #16840165C22semidirect
15. #16840160C22semidirect
17. #16840135C22semidirect
20. #16840114C22semidirect
21. #16840114C22semidirect
22. #16840113C22semidirect
23. #16840110C22semidirect
24. #16840109C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
224. C2×#492D61semidirect
273. C61#19625semidirect

Maximal subgroups

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

subgroupindex
#214569702
#214567372
#214563782
#214562152
#214561932
#214454702
#214454562
#214454182
#214454092
C8×#117581942
C2×C61×#4922
#1962561