Finite Group Databasebeta

← #29667879

Subgroups — #29667879

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#21456757C211yesmaximal, characteristic, complemented
2#21456744C211yesmaximal, characteristic, complemented
3#21456379C211yesmaximal, characteristic, complemented
4#21456271C211yesmaximal, characteristic
5C2×#16840110C211yesmaximal, characteristic, complemented
6#21445448C212yesmaximal, complemented
7#21445442C212yesmaximal, complemented
8#21445421C212yesmaximal, complemented
9#21445412C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3165C211yesmaximal, characteristic, complemented
12C2×#14519805—22no—
13#16840312C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15#16840158C2211yescharacteristic, complemented
16#16840151—22no—
17#16840139C2212yes—
18#16840136C2212yes—
19#16840136C2212yescomplemented
20#16840136C2211yescharacteristic, complemented
21#16840135C2212yes—
22#16840112C2211yescharacteristic, complemented
23#16840112C2212yescomplemented
24#16840110—22no—
25#16840110C2212yes—
26#16840110C2212yes—
27#16840110C2212yes—
28#16840110C2211yescharacteristic, complemented
29#16839993—22no—
30#16839964—22no—
31#16839962—22no—
32#16839912—22no—
33#16839901—22no—
34C4×#11758201—22no—
35#16839815C2211yescharacteristic
36C2×#14519697—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#501—22no—
43C61×#500C2212yescomplemented
44C61×#498C2212yescomplemented
45C61×#428—22no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51#14519805—48no—
52C2×#11758190—24no—
53C2×#11758201—22no—
54C2×#11758201—24no—
55C2×#11758201—24no—
56C2×D488—24no—
57C22×#92418D412yes—
58#14519777C2311yescharacteristic
59#14519774—21no—
60#14519774—21no—
61#14519774—21no—
62#14519774C2312yes—
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519708—22no—
70C4⋊#92418D412yes—
71#14519701—24no—
72#14519701—22no—
73#14519697—24no—
74#14519697—22no—
75#14519697—24no—
76#14519697—24no—
77C22⋊#92418D412yes—
78C4×#92418D412yes—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2312yes—
82C22×Q8×C61—21no—
83C2×M16×C61D412yes—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2312yes—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95#117×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#107×C61—24no—
100#106×C61—24no—
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103C23×D61—41no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—41no—
108C2×Q8×C61—82no—
109C22×C4×C61—41no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—41no—
114#11758222—44no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—44no—
122C2×D244—44no—
123C2×D244—41no—
124C2×D244—41no—
125C2×D244—41no—
126C2×D244—82no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129M16×C61SD1618yescomplemented
130C2×C8×C61C2×D414yes—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C42×C61—41no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61C2411yescharacteristic
147#11758201—44no—
148#11758201—84no—
149#11758201—84no—
150#11758198—41no—
151#11758198—41no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C2×#92418—42no—
155C2×#92418C2×D412yes—
156C2×#92418C2×D412yes—
157C2×#92418C2×D414yes—
158D488—48no—
159#11758190—48no—
160C23×C61#12711yescharacteristic
161C22×D61—41no—
162C22×D61—42no—
163C22×D61—82no—
164C22×D61—82no—
165C22×D61—82no—
166Q8×C61—42no—
167Q8×C61—42no—
168Q8×C61—44no—
169Q8×C61—82no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—82no—
176C2×C4×C61#12711yescharacteristic
177C2×C4×C61#12711yescharacteristic
178C2×C4×C61#12711yescharacteristic
179C2×C4×C61C2×SD1614yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C22⋊D61—82no—
184C2×Dic61—41no—
185D244—44no—
186D244—42no—
187D244—42no—
188D244—44no—
189D244—42no—
190D244—44no—
191D244—82no—
192D244—82no—
193D244—82no—
194C4×D61—82no—
195Dic122—42no—
196C8×C61C2×SD1618yes—
197#92418—22no—
198#92418—22no—
199#92418—24no—
200#19623—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61#49212yes—
204C22×C61C2×#12711yescharacteristic
205C2×D61—81no—
206C2×D61—82no—
207C2×D61—82no—
208C2×D61—81no—
209C2×D61—82no—
210C2×D61—81no—
211C2×D61—164no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—81no—
215C4×C61—82no—
216C4×C61—81no—
217C4×C61#49212yes—
218C4×C61C22×SD1612yes—
219C4×C61#57512yes—
220C4×C61#57612yes—
221Dic61—81no—
222#3208—611no—
223#3206—611no—
224#3167—611no—
225#3165D6111yescharacteristic, complemented
226#3151—611no—
227C2×#486—611no—
228#1715—612no—
229#1714—612no—
230#1708—612no—
231#1706—612no—
232C8×M16—611no—
233C2×C61—22no—
234C2×C61C2×#49211yescharacteristic
235C2×C61#315311yescharacteristic
236C2×C61#315511yescharacteristic
237D61—161no—
238D61—162no—
239C2×AGL(1,ℤ/8)—1222no—
240#533C2×D6111yescharacteristic
241#532—611no—
242C2×C4⋊Q8C2×D6111yescharacteristic
243C2×#127—611no—
244#501—22no—
245#500C2×D6112yescomplemented
246#498—611no—
247#498C2×D6112yescomplemented
248#496—1222no—
249#493—612no—
250#492—611no—
251#492—612no—
252#492—612no—
253#492—612no—
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—22no—
269C2×#102—1222no—
270C4×M16C2×D6111yescharacteristic
271C2×C4×C8—611no—
272#322—612no—
273C82—612no—
274C61#1962311yescharacteristic, complemented, Sylow
275C2×#41—1221no—
276C22×Q8—21no—
277C22×D4—1221no—
278C22×D4—1221no—
279C22×D4—1221no—
280AGL(1,ℤ/8)—2448no—
281C2×SD16—1224no—
282C2×SD16—1224no—
283C2×SD16—1222no—
284C2×SD16—1224no—
285C2×D8—1224no—
286C2×M16C22⋊D6112yes—
287C22×C8—612no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—611no—
292C4⋊Q8C22×D6112yes—
293C4⋊Q8C22×D6111yescharacteristic
294C4⋊Q8C22×D6111yescharacteristic
295#127—612no—
296#127—611no—
297#127—1221no—
298#127—1221no—
299#127—1221no—
300#125—21no—
301#124—1221no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—1221no—
305#121—1221no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×D4—1221no—
309C4×D4—1221no—
310#117—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C42C22×D6111yescharacteristic
314#107—24no—
315#106—24no—
316#106—1222no—
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—2441no—
332#41—2444no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—41no—
337C2×Q8—82no—
338C2×Q8—1221no—
339C2×D4—1222no—
340C2×D4—1221no—
341C2×D4—1222no—
342C2×D4—1221no—
343C2×D4—1221no—
344C2×D4—1222no—
345C2×D4—4882no—
346C2×D4—2444no—
347C2×D4—2441no—
348C2×D4—2441no—
349C2×D4—2441no—
350C2×D4—2441no—
351C2×D4—2444no—
352C22×C4—41no—
353C22×C4—2441no—
354C22×C4D4×D6111yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357SD16—4884no—
358SD16—4884no—
359SD16—2444no—
360SD16—2448no—
361D8—2448no—
362M16#1175820118yescomplemented
363C2×C8—612no—
364C2×C8—612no—
365C2×C8—614no—
366C2×C8—2442no—
367C2×C8C2×C22⋊D6114yes—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C22⋊C4—41no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C42—41no—
384C42D4×D6111yescharacteristic
385C42D4×D6111yescharacteristic
386C42C2×C22⋊D6111yescharacteristic
387C42C23×D6111yescharacteristic
388C23—4882no—
389C23—4882no—
390C23—4882no—
391C23—2441no—
392C23—2442no—
393C23#1451977311yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—44no—
397Q8—82no—
398Q8—2442no—
399D4—4882no—
400D4—4882no—
401D4—4882no—
402D4—4882no—
403D4—2442no—
404D4—2444no—
405D4—2442no—
406D4—2444no—
407D4—2442no—
408D4—2444no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—4882no—
416C2×C4—2441no—
417C2×C4#1451977311yescharacteristic
418C2×C4#1451977311yescharacteristic
419C2×C4#1451977311yescharacteristic
420C2×C4C2×#1175820114yes—
421C2×C4C2×D4×D6111yescharacteristic
422C2×C4C2×D4×D6111yescharacteristic
423C2×C4C22×C22⋊D6111yescharacteristic
424C8—1222no—
425C8—1222no—
426C8—1224no—
427C8C2×#1175820118yes—
428C22—21no—
429C22—21no—
430C22—9764no—
431C22—4881no—
432C22—4882no—
433C22—4882no—
434C22—4881no—
435C22—4882no—
436C22—4881no—
437C22#1684013612yes—
438C22C2×#1451977311yescharacteristic
439C4—21no—
440C4—21no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4—4881no—
445C4#1684013612yes—
446C4C22×#1175820112yes—
447C4#1684056112yes—
448C4#1684056612yes—
449C2—22no—
450C2—9762no—
451C2—9761no—
452C2C2×#1684013611yescharacteristic
453C2#2145628611yescharacteristic
454C2#2145630411yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3165C2
socleC22×C61C2×#127
prime radical#21456271C2
hypercenter#3165D61
hypocenterC61#19623
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
2. #21456744C2semidirect
3. #21456379C2semidirect
5. C2×#16840110C2semidirect
6. #21445448C2semidirect
7. #21445442C2semidirect
8. #21445421C2semidirect
9. #21445412C2semidirect
10. M16×#92418C2semidirect
11. C61×#3165C2semidirect
15. #16840158C22semidirect
19. #16840136C22semidirect
20. #16840136C22semidirect
22. #16840112C22semidirect
23. #16840112C22semidirect
28. #16840110C22semidirect
38. #16839483C22semidirect
39. C8×#92418C22semidirect
43. C61×#500C22semidirect
44. C61×#498C22semidirect
129. M16×C61SD16semidirect
225. #3165D61semidirect
245. #500C2×D61semidirect
247. #498C2×D61semidirect
274. C61#19623semidirect
362. M16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214567442
#214563792
#214562712
C2×#168401102
#214454482
#214454422
#214454212
#214454122
M16×#924182
C61×#31652
#1962361