Finite Group Databasebeta

← #29667872

Subgroups — #29667872

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21456374C211yesmaximal, characteristic, complemented
4#21456365C211yesmaximal, characteristic, complemented
5C2×#16840108C211yesmaximal, characteristic, complemented
6#21445445C212yesmaximal, complemented
7#21445438C212yesmaximal, complemented
8#21445418C212yesmaximal, complemented
9#21445408C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3151C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840171—22no—
15#16840167C2212yescomplemented
16#16840160C2212yescomplemented
17#16840158C2211yescharacteristic, complemented
18#16840134C2212yes—
19#16840134C2212yescomplemented
20#16840134C2211yescharacteristic, complemented
21#16840114C2211yescharacteristic, complemented
22#16840114C2212yescomplemented
23#16840108—22no—
24#16840108C2212yes—
25#16840108C2212yes—
26#16840108C2212yes—
27#16840108C2211yescharacteristic, complemented
28#16839989—22no—
29#16839967—22no—
30#16839959—22no—
31#16839913—22no—
32#16839900—22no—
33C4×#11758200—22no—
34#16839815C2211yescharacteristic
35#16839804—22no—
36C2×#14519698—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yescomplemented
39C8×#92418C2212yescomplemented
40C2×AGL(1,ℤ/8)×C61—22no—
41C61×#532C2211yescharacteristic
42C2×#127×C61C2211yescharacteristic
43C61×#496—22no—
44C61×#493C2212yes—
45C61×#492C2212yes—
46C4×M16×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758200—22no—
49C2×#11758200—24no—
50C2×#11758200—24no—
51C22×#92418D412yes—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66#14519711—24no—
67#14519709—24no—
68#14519708—22no—
69C4⋊#92418D412yes—
70#14519700—24no—
71#14519700—22no—
72#14519698—24no—
73#14519698—22no—
74#14519698—24no—
75#14519698—24no—
76C22⋊#92418D412yes—
77C4×#92418D412yes—
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85AGL(1,ℤ/8)×C61—48no—
86C2×SD16×C61—24no—
87C2×D8×C61—24no—
88C2×M16×C61D412yes—
89C4⋊Q8×C61C2311yescharacteristic
90#127×C61—21no—
91#127×C61—21no—
92#127×C61—21no—
93#127×C61C2312yes—
94#127×C61C2311yescharacteristic
95#124×C61—21no—
96#121×C61—21no—
97#121×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103C24×C61—41no—
104#41×C61—44no—
105C2×Q8×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—44no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—44no—
118C2×D4×C61—82no—
119C22×C4×C61—41no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×Dic61—41no—
124C2×Dic122—22no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—41no—
128C2×Dic122—82no—
129SD16×C61—48no—
130D8×C61—48no—
131M16×C61SD1618yescomplemented
132C2×C8×C61C2×D414yes—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C22⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2411yescharacteristic
141C22⋊Dic61—41no—
142#11758200—44no—
143#11758200—84no—
144#11758200—84no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153#11758196—41no—
154#11758196—41no—
155C4×Dic61—41no—
156C2×#92418—42no—
157C2×#92418C2×D412yes—
158C2×#92418C2×D412yes—
159C2×#92418C2×D414yes—
160C23×C61—41no—
161C23×C61—42no—
162C23×C61—82no—
163C23×C61—82no—
164C23×C61—82no—
165C23×C61#12711yescharacteristic
166Q8×C61—42no—
167D4×C61—42no—
168D4×C61—44no—
169D4×C61—42no—
170D4×C61—44no—
171D4×C61—42no—
172D4×C61—44no—
173D4×C61—82no—
174D4×C61—82no—
175D4×C61—82no—
176D4×C61—82no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—82no—
181C2×C4×C61#12711yescharacteristic
182C2×C4×C61#12711yescharacteristic
183C2×C4×C61#12711yescharacteristic
184C2×C4×C61C2×SD1614yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—82no—
192Dic122—42no—
193Dic122—42no—
194Dic122—44no—
195Dic122—82no—
196C8×C61C2×SD1618yes—
197#92418—22no—
198#92418—22no—
199#92418—24no—
200#19623—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—81no—
204C22×C61—82no—
205C22×C61—82no—
206C22×C61—81no—
207C22×C61—82no—
208C22×C61—81no—
209C22×C61—164no—
210C22×C61#49212yes—
211C22×C61C2×#12711yescharacteristic
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—81no—
215C4×C61#49212yes—
216C4×C61C22×SD1612yes—
217C4×C61#57512yes—
218C4×C61#57612yes—
219Dic61—81no—
220Dic61—82no—
221Dic61—81no—
222#3208—611no—
223#3206—611no—
224#3167—611no—
225#3165—611no—
226#3151D6111yescharacteristic
227C2×#486—611no—
228#1715—612no—
229#1714—612no—
230#1708—612no—
231#1706—612no—
232C8×M16—611no—
233C2×C61—22no—
234C2×C61—161no—
235C2×C61—162no—
236C2×C61C2×#49211yescharacteristic
237C2×C61#315311yescharacteristic
238C2×C61#315511yescharacteristic
239C2×AGL(1,ℤ/8)—22no—
240#533—611no—
241#532C2×D6111yescharacteristic
242C2×C4⋊Q8—611no—
243C2×#127C2×D6111yescharacteristic
244#501—1222no—
245#500—612no—
246#498—611no—
247#498—612no—
248#496—22no—
249#493C2×D6112yes—
250#492—611no—
251#492—612no—
252#492—612no—
253#492C2×D6112yes—
254#487—611no—
255#487—612no—
256#486—612no—
257#486—612no—
258#486—611no—
259#486—612no—
260#486—1222no—
261#465—1222no—
262#461—1222no—
263#460—1222no—
264#450—1222no—
265#448—1222no—
266C4×SD16—1222no—
267#431—611no—
268#428—1222no—
269C2×#102—1222no—
270C4×M16C2×D6111yescharacteristic
271C2×C4×C8—611no—
272#322—612no—
273C82—612no—
274C61#1962311yescharacteristic, complemented, Sylow
275C2×#41—21no—
276C22×Q8—1221no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280AGL(1,ℤ/8)—48no—
281C2×SD16—24no—
282C2×SD16—1224no—
283C2×SD16—1224no—
284C2×SD16—1222no—
285C2×D8—24no—
286C2×M16C22⋊D6112yes—
287C22×C8—612no—
288C4⋊Q8—612no—
289C4⋊Q8—611no—
290C4⋊Q8—611no—
291C4⋊Q8—1221no—
292C4⋊Q8—1221no—
293C4⋊Q8—1221no—
294C4⋊Q8C22×D6111yescharacteristic
295#127—21no—
296#127—21no—
297#127—21no—
298#127C22×D6112yes—
299#127C22×D6111yescharacteristic
300#125—1221no—
301#124—21no—
302C22⋊Q8—1221no—
303C22⋊Q8—1221no—
304#121—21no—
305#121—21no—
306C4×Q8—1221no—
307C4×Q8—1221no—
308C4×D4—21no—
309C4×D4—21no—
310#117—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1224no—
315#106—1222no—
316#106—1224no—
317C4⋊C8—612no—
318#103—1222no—
319#103—1224no—
320#102—1224no—
321#102—1224no—
322#102—1222no—
323#102—1224no—
324C22⋊C8—612no—
325#97C22⋊D6112yes—
326C4×C8—612no—
327C4×C8—611no—
328C4×C8—612no—
329C4×C8—611no—
330C4×C8C22×D6112yes—
331C24—41no—
332#41—44no—
333C2×Q8—21no—
334C2×Q8—1221no—
335C2×Q8—1222no—
336C2×Q8—1221no—
337C2×Q8—4882no—
338C2×Q8—2441no—
339C2×D4—22no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—44no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—44no—
351C2×D4—82no—
352C22×C4—41no—
353C22×C4—2441no—
354C22×C4D4×D6111yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357SD16—48no—
358SD16—4884no—
359SD16—4884no—
360SD16—2444no—
361D8—48no—
362M16#1175820018yescomplemented
363C2×C8—612no—
364C2×C8—612no—
365C2×C8—614no—
366C2×C8—2442no—
367C2×C8C2×C22⋊D6114yes—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—1222no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1222no—
375C4⋊C4—1221no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C4⋊C4—2441no—
379C4⋊C4—2441no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—2441no—
383C42—2441no—
384C42D4×D6111yescharacteristic
385C42D4×D6111yescharacteristic
386C42C2×C22⋊D6111yescharacteristic
387C42C23×D6111yescharacteristic
388C23—41no—
389C23—42no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23#1451977311yescharacteristic
394Q8—42no—
395Q8—4882no—
396Q8—2442no—
397Q8—2442no—
398Q8—2444no—
399D4—42no—
400D4—44no—
401D4—42no—
402D4—44no—
403D4—42no—
404D4—44no—
405D4—82no—
406D4—82no—
407D4—82no—
408D4—82no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—82no—
413C2×C4—4882no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2442no—
417C2×C4#1451977311yescharacteristic
418C2×C4#1451977311yescharacteristic
419C2×C4#1451977311yescharacteristic
420C2×C4C2×#1175820014yes—
421C2×C4C2×D4×D6111yescharacteristic
422C2×C4C2×D4×D6111yescharacteristic
423C2×C4C22×C22⋊D6111yescharacteristic
424C8—1222no—
425C8—1222no—
426C8—1224no—
427C8C2×#1175820018yes—
428C22—21no—
429C22—21no—
430C22—81no—
431C22—82no—
432C22—82no—
433C22—81no—
434C22—82no—
435C22—81no—
436C22—164no—
437C22#1684013412yes—
438C22C2×#1451977311yescharacteristic
439C4—21no—
440C4—21no—
441C4—81no—
442C4—4881no—
443C4—4882no—
444C4—4881no—
445C4#1684013412yes—
446C4C22×#1175820012yes—
447C4#1684055812yes—
448C4#1684056612yes—
449C2—22no—
450C2—162no—
451C2—161no—
452C2C2×#1684013411yescharacteristic
453C2#2145628111yescharacteristic
454C2#2145630411yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3151C2
socleC22×C61C2×#127
hypercenter#3151D61
hypocenterC61#19623
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21456741C2semidirect
3. #21456374C2semidirect
4. #21456365C2semidirect
5. C2×#16840108C2semidirect
6. #21445445C2semidirect
7. #21445438C2semidirect
8. #21445418C2semidirect
9. #21445408C2semidirect
10. M16×#92418C2semidirect
15. #16840167C22semidirect
16. #16840160C22semidirect
17. #16840158C22semidirect
19. #16840134C22semidirect
20. #16840134C22semidirect
21. #16840114C22semidirect
22. #16840114C22semidirect
27. #16840108C22semidirect
38. #16839483C22semidirect
39. C8×#92418C22semidirect
131. M16×C61SD16semidirect
274. C61#19623semidirect
362. M16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214567412
#214563742
#214563652
C2×#168401082
#214454452
#214454382
#214454182
#214454082
M16×#924182
C61×#31512
#1962361