Finite Group Databasebeta

← #28402341

Subgroups — #28402341

Order 14464.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20299410C211yesmaximal, characteristic, complemented
2#20299405C211yesmaximal, characteristic, complemented
3#20299403C211yesmaximal, characteristic, complemented
4#20299400C211yesmaximal, characteristic, complemented
5#20298987C211yesmaximal, characteristic, complemented
6#20298869C211yesmaximal, characteristic, complemented
7#20298868C211yesmaximal, characteristic, complemented
8#20298862C211yesmaximal, characteristic, complemented
9#20298861C211yesmaximal, characteristic, complemented
10#20298777C211yesmaximal, characteristic, complemented
11#20298772C211yesmaximal, characteristic, complemented
12#20298767C211yesmaximal, characteristic, complemented
13#20298767C211yesmaximal, characteristic, complemented
14#20298721C211yesmaximal, characteristic, complemented
15#444×C113C211yesmaximal, characteristic, complemented
16#16813669C2211yescharacteristic, complemented
17#16813668C2211yescharacteristic
18#16813668C2211yescharacteristic, complemented
19#16813668C2211yescharacteristic, complemented
20#16813655—21no—
21#16813655C2211yescharacteristic, complemented
22#16813654—21no—
23#16813652C2211yescharacteristic, complemented
24#16813650—21no—
25#16813650—21no—
26#16813650—21no—
27#16813650C2211yescharacteristic
28#16813650C2211yescharacteristic, complemented
29#16813650C2211yescharacteristic, complemented
30#16813649—21no—
31#16813649—21no—
32#16813647C2211yescharacteristic
33#16813647C2211yescharacteristic, complemented
34#16813647C2211yescharacteristic, complemented
35#16813646—21no—
36#16813645—21no—
37#16813644—21no—
38#16813644—21no—
39#16813644C2211yescharacteristic
40#16813643—21no—
41C2×#14268636—21no—
42C2×#14268636C2211yescharacteristic, complemented
43C2×#14268636C2211yescharacteristic, complemented
44SD16×D113—21no—
45C2×D904—21no—
46C2×D904C2211yescharacteristic, complemented
47C2×D904C2211yescharacteristic, complemented
48D8×D113—21no—
49#16813628C2211yescharacteristic, complemented
50#16813628C2211yescharacteristic, complemented
51#16813619C2211yescharacteristic, complemented
52#16813610C2211yescharacteristic, complemented
53#16813558C2211yescharacteristic, complemented
54#16813550C2211yescharacteristic, complemented
55#16813550C2211yescharacteristic, complemented
56#16813540C2211yescharacteristic, complemented
57#16813538C2211yescharacteristic, complemented
58#16813538C2211yescharacteristic, complemented
59#16813538C2211yescharacteristic, complemented
60#135×C113C2211yescharacteristic
61#131×C113C2211yescharacteristic, complemented
62#131×C113C2211yescharacteristic, complemented
63#108×C113C2211yescharacteristic, complemented
64#104×C113C2211yescharacteristic, complemented
65#104×C113C2211yescharacteristic, complemented
66#97×C113C2211yescharacteristic, complemented
67#14268648—21no—
68#14268648—21no—
69#14268648—21no—
70#14268647—21no—
71#14268647—21no—
72#14268647—21no—
73#14268647—21no—
74#14268647C2311yescharacteristic
75#14268647C2311yescharacteristic
76#14268646—21no—
77#14268646—41no—
78C2×Dic226—21no—
79C2×Dic226C2311yescharacteristic
80Q8×D113—21no—
81C2×D452—21no—
82C2×D452—21no—
83C2×D452—21no—
84C2×D452—41no—
85C2×D452C2311yescharacteristic
86C2×D452C2311yescharacteristic
87C2×D452C2311yescharacteristic
88D4×D113—21no—
89D4×D113—21no—
90D4×D113—21no—
91D4×D113—41no—
92D4×D113—41no—
93D4×D113—41no—
94Dic452—21no—
95Dic452D411yescharacteristic, complemented
96#14268637—21no—
97#14268636—21no—
98#14268636—21no—
99#14268636—21no—
100#14268636—21no—
101#14268636—21no—
102#14268636—21no—
103#14268636—21no—
104#14268636—21no—
105#14268636D411yescharacteristic, complemented
106#14268636D411yescharacteristic, complemented
107#14268635—21no—
108#14268634—21no—
109#14268634—21no—
110#14268634—21no—
111D904—21no—
112D904—21no—
113D904—21no—
114D904—21no—
115D904—21no—
116D904—21no—
117D904—21no—
118D904D411yescharacteristic, complemented
119#14268632—21no—
120#14268632—21no—
121#14268632—21no—
122#14268631C2311yescharacteristic
123#14268631C2311yescharacteristic
124#14268630—21no—
125#14268630—21no—
126C8×D113—21no—
127C8×D113—21no—
128#14268624—41no—
129#41×C113C2311yescharacteristic
130#41×C113C2311yescharacteristic
131Q16×C113D411yescharacteristic, complemented
132SD16×C113D411yescharacteristic, complemented
133SD16×C113D411yescharacteristic, complemented
134D8×C113D411yescharacteristic, complemented
135M16×C113—21no—
136M16×C113—21no—
137M16×C113C2311yescharacteristic
138M16×C113C2311yescharacteristic
139C2×C8×C113—21no—
140C2×C8×C113—21no—
141C2×C8×C113C2311yescharacteristic
142C2×C8×C113C2311yescharacteristic
143C42×C113C2311yescharacteristic
144C22×D113—21no—
145C22×D113—21no—
146C22×D113—21no—
147C22×D113—41no—
148C22×D113—41no—
149C22×D113—41no—
150Q8×C113C2×D411yescharacteristic
151Q8×C113C2×D411yescharacteristic
152D4×C113—21no—
153D4×C113—21no—
154D4×C113C2×D411yescharacteristic
155D4×C113C2×D411yescharacteristic
156C2×C4×C113—21no—
157C2×C4×C113—21no—
158C2×C4×C113—21no—
159C2×C4×C113C2411yescharacteristic
160C22⋊D113—21no—
161C22⋊D113—21no—
162C22⋊D113—41no—
163C22⋊D113—41no—
164C22⋊D113—41no—
165C22⋊D113—41no—
166C2×Dic113—21no—
167C2×Dic113—41no—
168D452—21no—
169D452—21no—
170D452—21no—
171D452—21no—
172D452—21no—
173D452—21no—
174D452—41no—
175D452—41no—
176D452—41no—
177D452—82no—
178D452C2×D411yescharacteristic
179D452C2×D411yescharacteristic
180C4×D113—21no—
181C4×D113—21no—
182C4×D113—41no—
183C4×D113—41no—
184C4×D113—41no—
185C4×D113—41no—
186Dic226—21no—
187Dic226—21no—
188Dic226—41no—
189Dic226C2×D411yescharacteristic
190Dic226C2×D411yescharacteristic
191C8×C113—21no—
192C8×C113—21no—
193C8×C113C2×D411yescharacteristic
194C8×C113C2×D411yescharacteristic
195C8×C113C2×D411yescharacteristic
196C8×C113C2×D411yescharacteristic
197#11744751—21no—
198#11744751—21no—
199C22×C113—21no—
200C22×C113—21no—
201C22×C113H(5,𝔽2)11yescharacteristic
202C2×D113—21no—
203C2×D113—21no—
204C2×D113—41no—
205C2×D113—41no—
206C2×D113—41no—
207C2×D113—81no—
208C2×D113—81no—
209C2×D113—81no—
210C2×D113—161no—
211C2×D113—161no—
212C2×D113—161no—
213C4×C113—21no—
214C4×C113—21no—
215C4×C113—41no—
216C4×C113C22×D411yescharacteristic
217C4×C113C22×D411yescharacteristic
218Dic113—21no—
219Dic113—21no—
220Dic113—41no—
221C2×C113—21no—
222C2×C113—41no—
223C2×C113—41no—
224C2×C113D4211yescharacteristic
225D113—41no—
226D113—41no—
227D113—81no—
228D113—81no—
229D113—81no—
230#3288—1131nomaximal, Sylow
231C113#328811yescharacteristic, complemented, Sylow
232#577—1131no—
233#577—1131no—
234#576—1131no—
235#576—1131no—
236#496—1131no—
237#472—1131no—
238#472—1131no—
239#471—1131no—
240#471—1131no—
241#455—1131no—
242#454—1131no—
243#453—1131no—
244#453—1131no—
245#444—1131no—
246#444D11311yescharacteristic, complemented
2472−1+4—1131no—
248H(5,𝔽2)—1131no—
249H(5,𝔽2)—1131no—
250H(5,𝔽2)—1131no—
251#137—1131no—
252#137—1131no—
253#137—2261no—
254#137—2261no—
255AGL(1,ℤ/8)—1131no—
256AGL(1,ℤ/8)—1131no—
257AGL(1,ℤ/8)—1131no—
258AGL(1,ℤ/8)—1131no—
259AGL(1,ℤ/8)—1131no—
260AGL(1,ℤ/8)—1131no—
261AGL(1,ℤ/8)—2261no—
262AGL(1,ℤ/8)—2261no—
263AGL(1,ℤ/8)—2261no—
264AGL(1,ℤ/8)—2261no—
265AGL(1,ℤ/8)—2261no—
266AGL(1,ℤ/8)—2261no—
267#135—1131no—
268#135—2261no—
269#135—2261no—
270#135—2261no—
271#135—2261no—
272#135C2×D11311yescharacteristic
273C2×SD16—1131no—
274C2×SD16—1131no—
275C2×SD16—2261no—
276C2×SD16—2261no—
277C2×D8—1131no—
278C2×D8—1131no—
279C2×D8—2261no—
280C2×D8—2261no—
281#131—1131no—
282#131—1131no—
283#131C2×D11311yescharacteristic, complemented
284#131C2×D11311yescharacteristic, complemented
285#127—1131no—
286#124—1131no—
287#108—1131no—
288#108C2×D11311yescharacteristic, complemented
289#104—1131no—
290#104—1131no—
291#104C2×D11311yescharacteristic, complemented
292#104C2×D11311yescharacteristic, complemented
293#101—1131no—
294#100—1131no—
295#100—1131no—
296#100—1131no—
297#97C2×D11311yescharacteristic, complemented
298#41—1131no—
299#41—1131no—
300#41—4521no—
301#41—2261no—
302#41—2261no—
303#41—2261no—
304#41—2261no—
305#41—2261no—
306#41—2261no—
307#41—2261no—
308#41—2261no—
309#41C22×D11311yescharacteristic
310#41C22×D11311yescharacteristic
311C2×Q8—1131no—
312C2×Q8—2261no—
313C2×Q8—2261no—
314C2×D4—1131no—
315C2×D4—1131no—
316C2×D4—1131no—
317C2×D4—4521no—
318C2×D4—4521no—
319C2×D4—4521no—
320C2×D4—4521no—
321C2×D4—2261no—
322C2×D4—2261no—
323C2×D4—2261no—
324C2×D4—2261no—
325C2×D4—2261no—
326C2×D4—2261no—
327Q16—1131no—
328Q16—2261no—
329Q16—2261no—
330Q16D45211yescharacteristic, complemented
331SD16—1131no—
332SD16—1131no—
333SD16—2261no—
334SD16—2261no—
335SD16—2261no—
336SD16—2261no—
337SD16—2261no—
338SD16—2261no—
339SD16—2261no—
340SD16—2261no—
341SD16—2261no—
342SD16—2261no—
343SD16—2261no—
344SD16—2261no—
345SD16D45211yescharacteristic, complemented
346SD16D45211yescharacteristic, complemented
347D8—1131no—
348D8—2261no—
349D8—2261no—
350D8—2261no—
351D8—2261no—
352D8—2261no—
353D8—2261no—
354D8—2261no—
355D8—2261no—
356D8—2261no—
357D8—2261no—
358D8D45211yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—1131no—
362M16—1131no—
363M16—2261no—
364M16—2261no—
365M16C22×D11311yescharacteristic
366M16C22×D11311yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—2261no—
370C2×C8—2261no—
371C2×C8C22×D11311yescharacteristic
372C2×C8C22×D11311yescharacteristic
373C22⋊C4—4521no—
374C42C22×D11311yescharacteristic
375C23—4521no—
376C23—4521no—
377C23—4521no—
378C23—2261no—
379C23—2261no—
380C23—2261no—
381Q8—1131no—
382Q8—1131no—
383Q8—4521no—
384Q8—2261no—
385Q8—2261no—
386Q8C2×D45211yescharacteristic
387Q8C2×D45211yescharacteristic
388D4—21no—
389D4—21no—
390D4—1131no—
391D4—1131no—
392D4—9042no—
393D4—4521no—
394D4—4521no—
395D4—4521no—
396D4—4521no—
397D4—4521no—
398D4—4521no—
399D4—4521no—
400D4—2261no—
401D4—2261no—
402D4—2261no—
403D4—2261no—
404D4—2261no—
405D4—2261no—
406D4—2261no—
407D4—2261no—
408D4C2×D45211yescharacteristic
409D4C2×D45211yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—4521no—
414C2×C4—4521no—
415C2×C4—4521no—
416C2×C4—4521no—
417C2×C4—4521no—
418C2×C4—2261no—
419C2×C4—2261no—
420C2×C4—2261no—
421C2×C4C23×D11311yescharacteristic
422C8—21no—
423C8—21no—
424C8—2261no—
425C8—2261no—
426C8D4×D11311yescharacteristic
427C8D4×D11311yescharacteristic
428C8C2×D45211yescharacteristic
429C8C2×D45211yescharacteristic
430C22—21no—
431C22—21no—
432C22—18081no—
433C22—18081no—
434C22—18081no—
435C22—9041no—
436C22—9041no—
437C22—9041no—
438C22—4521no—
439C22—4521no—
440C22—4521no—
441C22—2261no—
442C22—2261no—
443C22#1681366811yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—4521no—
448C4—2261no—
449C4—2261no—
450C4C2×D4×D11311yescharacteristic
451C4C22×D45211yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—9041no—
456C2—9041no—
457C2—9041no—
458C2—4521no—
459C2—4521no—
460C2D4×D45211yescharacteristic
subgroupG/H
centerC2D4×D452
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#444×C113C2
socleC2×C113D42
hypercenter#444D113
hypocenterC113#3288
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #20299410C2semidirect
2. #20299405C2semidirect
3. #20299403C2semidirect
4. #20299400C2semidirect
5. #20298987C2semidirect
6. #20298869C2semidirect
7. #20298868C2semidirect
8. #20298862C2semidirect
9. #20298861C2semidirect
10. #20298777C2semidirect
11. #20298772C2semidirect
12. #20298767C2semidirect
13. #20298767C2semidirect
14. #20298721C2semidirect
15. #444×C113C2semidirect
16. #16813669C22semidirect
18. #16813668C22semidirect
19. #16813668C22semidirect
21. #16813655C22semidirect
23. #16813652C22semidirect
28. #16813650C22semidirect
29. #16813650C22semidirect
33. #16813647C22semidirect
34. #16813647C22semidirect
42. C2×#14268636C22semidirect
43. C2×#14268636C22semidirect
46. C2×D904C22semidirect
47. C2×D904C22semidirect
49. #16813628C22semidirect
50. #16813628C22semidirect
51. #16813619C22semidirect
52. #16813610C22semidirect
53. #16813558C22semidirect
54. #16813550C22semidirect
55. #16813550C22semidirect
56. #16813540C22semidirect
57. #16813538C22semidirect
58. #16813538C22semidirect
59. #16813538C22semidirect
61. #131×C113C22semidirect
62. #131×C113C22semidirect
63. #108×C113C22semidirect
64. #104×C113C22semidirect
65. #104×C113C22semidirect
66. #97×C113C22semidirect
95. Dic452D4semidirect
105. #14268636D4semidirect
106. #14268636D4semidirect
118. D904D4semidirect
131. Q16×C113D4semidirect
132. SD16×C113D4semidirect
133. SD16×C113D4semidirect
134. D8×C113D4semidirect
231. C113#3288semidirect
246. #444D113semidirect
283. #131C2×D113semidirect
284. #131C2×D113semidirect
288. #108C2×D113semidirect
291. #104C2×D113semidirect
292. #104C2×D113semidirect
297. #97C2×D113semidirect
330. Q16D452semidirect
345. SD16D452semidirect
346. SD16D452semidirect
358. D8D452semidirect

Maximal subgroups

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

subgroupindex
#202994102
#202994052
#202994032
#202994002
#202989872
#202988692
#202988682
#202988622
#202988612
#202987772
#202987722
#202987672
#202987212
#444×C1132
#3288113