Finite Group Databasebeta

← #28402338

Subgroups — #28402338

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#20299409C211yesmaximal, characteristic, complemented
2#20299405C211yesmaximal, characteristic, complemented
3#20299401C211yesmaximal, characteristic, complemented
4#20298986C211yesmaximal, characteristic, complemented
5#20298866C211yesmaximal, characteristic, complemented
6#20298865C211yesmaximal, characteristic, complemented
7#20298861C211yesmaximal, characteristic, complemented
8#20298859C211yesmaximal, characteristic, complemented
9#20298776C211yesmaximal, characteristic, complemented
10#20298769C211yesmaximal, characteristic, complemented
11#20298766C211yesmaximal, characteristic, complemented
12#20298764C211yesmaximal, characteristic, complemented
13#20298720C211yesmaximal, characteristic, complemented
14#20298718C211yesmaximal, characteristic, complemented
15#576×C113C211yesmaximal, characteristic, complemented
16#16813669C2211yescharacteristic, complemented
17#16813668C2211yescharacteristic, complemented
18#16813654—21no—
19#16813652—21no—
20#16813652C2211yescharacteristic, complemented
21#16813652C2211yescharacteristic, complemented
22#16813648—21no—
23#16813648—21no—
24#16813648C2211yescharacteristic, complemented
25#16813648C2211yescharacteristic, complemented
26#16813647—21no—
27#16813647C2211yescharacteristic
28#16813647C2211yescharacteristic, complemented
29#16813646—21no—
30#16813643—21no—
31#16813642—21no—
32#16813642—21no—
33#16813642C2211yescharacteristic
34C2×#14268636C2211yescharacteristic, complemented
35C2×#14268635—21no—
36C2×#14268635C2211yescharacteristic, complemented
37SD16×D113—21no—
38C2×#14268632—21no—
39C2×#14268632C2211yescharacteristic, complemented
40#16813629C2211yescharacteristic, complemented
41#16813629C2211yescharacteristic, complemented
42#16813628C2211yescharacteristic, complemented
43#16813618C2211yescharacteristic, complemented
44#16813609C2211yescharacteristic, complemented
45#16813558C2211yescharacteristic, complemented
46#16813557C2211yescharacteristic, complemented
47#16813549C2211yescharacteristic, complemented
48#16813549C2211yescharacteristic, complemented
49#16813549C2211yescharacteristic, complemented
50#16813548C2211yescharacteristic, complemented
51#16813540C2211yescharacteristic, complemented
52#16813539C2211yescharacteristic, complemented
53#16813539C2211yescharacteristic, complemented
54#16813538C2211yescharacteristic, complemented
55#16813532C2211yescharacteristic, complemented
56H(5,𝔽2)×C113C2211yescharacteristic
57H(5,𝔽2)×C113C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C113—21no—
59AGL(1,ℤ/8)×C113—21no—
60AGL(1,ℤ/8)×C113C2211yescharacteristic
61AGL(1,ℤ/8)×C113C2211yescharacteristic, complemented
62#135×C113—21no—
63#135×C113C2211yescharacteristic
64C2×D8×C113—21no—
65C2×D8×C113C2211yescharacteristic, complemented
66#131×C113C2211yescharacteristic, complemented
67#14268648—21no—
68#14268647—21no—
69#14268647—21no—
70#14268647C2311yescharacteristic
71#14268646—21no—
72#14268646—41no—
73C2×Dic226—21no—
74C2×Dic226C2311yescharacteristic
75Q8×D113—21no—
76C2×D452—21no—
77C2×D452C2311yescharacteristic
78C2×C22⋊D113—41no—
79D4×D113—21no—
80D4×D113—41no—
81#14268637—21no—
82#14268637—21no—
83#14268637D411yescharacteristic, complemented
84#14268636—21no—
85#14268636—21no—
86#14268635—21no—
87#14268635—21no—
88#14268635—21no—
89#14268635—21no—
90#14268635—21no—
91#14268635—21no—
92#14268635D411yescharacteristic, complemented
93#14268634—21no—
94#14268634—21no—
95#14268634D411yescharacteristic, complemented
96#14268632—21no—
97#14268632—21no—
98#14268632—21no—
99#14268632—21no—
100#14268632—21no—
101#14268632—21no—
102#14268632D411yescharacteristic, complemented
103#14268631—21no—
104#14268631—21no—
105#14268631C2311yescharacteristic
106#14268631C2311yescharacteristic
107#14268631C2311yescharacteristic
108#14268630—21no—
109C2×#11744751—21no—
110C2×#11744751—21no—
111C2×#11744751C2311yescharacteristic
112C8×D113—21no—
113C22⋊Dic113—41no—
114C4×Dic113C2311yescharacteristic
115#41×C113—21no—
116#41×C113—21no—
117#41×C113—21no—
118#41×C113—21no—
119#41×C113C2311yescharacteristic
120#41×C113C2311yescharacteristic
121#41×C113C2311yescharacteristic
122C2×D4×C113—21no—
123C2×D4×C113—21no—
124C2×D4×C113—21no—
125C2×D4×C113—21no—
126C2×D4×C113—41no—
127C2×D4×C113—41no—
128C2×D4×C113C2311yescharacteristic
129C2×D4×C113C2311yescharacteristic
130Q16×C113D411yescharacteristic, complemented
131SD16×C113—21no—
132SD16×C113—21no—
133SD16×C113D411yescharacteristic, complemented
134SD16×C113D411yescharacteristic, complemented
135D8×C113—21no—
136D8×C113—21no—
137D8×C113—21no—
138D8×C113—21no—
139D8×C113D411yescharacteristic, complemented
140M16×C113—21no—
141M16×C113C2311yescharacteristic
142C2×C8×C113—21no—
143C2×C8×C113C2311yescharacteristic
144C23×C113—21no—
145C23×C113—21no—
146C23×C113—41no—
147C23×C113—41no—
148C22×D113—21no—
149C22×D113—41no—
150Q8×C113C2×D411yescharacteristic
151Q8×C113C2×D411yescharacteristic
152Q8×C113C2×D411yescharacteristic
153D4×C113—21no—
154D4×C113—21no—
155D4×C113—21no—
156D4×C113—21no—
157D4×C113—21no—
158D4×C113—21no—
159D4×C113—21no—
160D4×C113—41no—
161D4×C113—41no—
162D4×C113—41no—
163D4×C113—41no—
164D4×C113C2×D411yescharacteristic
165D4×C113C2×D411yescharacteristic
166D4×C113C2×D411yescharacteristic
167C2×C4×C113—21no—
168C2×C4×C113—21no—
169C2×C4×C113—21no—
170C2×C4×C113—41no—
171C2×C4×C113—41no—
172C2×C4×C113C2411yescharacteristic
173C22⋊D113—21no—
174C22⋊D113—41no—
175C22⋊D113—41no—
176C22⋊D113—82no—
177C2×Dic113—21no—
178C2×Dic113—21no—
179C2×Dic113—41no—
180D452—21no—
181D452—21no—
182D452—41no—
183D452C2×D411yescharacteristic
184C4×D113—21no—
185C4×D113—41no—
186C4×D113—41no—
187Dic226—21no—
188Dic226—21no—
189Dic226—41no—
190Dic226C2×D411yescharacteristic
191C8×C113—21no—
192C8×C113C2×D411yescharacteristic
193C8×C113C2×D411yescharacteristic
194#11744751—21no—
195#11744751—21no—
196#11744751—21no—
197#11744751C2×D411yescharacteristic
198#11744751C2×D411yescharacteristic
199C22×C113—21no—
200C22×C113—21no—
201C22×C113—21no—
202C22×C113—41no—
203C22×C113—41no—
204C22×C113—81no—
205C22×C113—81no—
206C22×C113—161no—
207C22×C113—161no—
208C22×C113H(5,𝔽2)11yescharacteristic
209C2×D113—21no—
210C2×D113—41no—
211C2×D113—81no—
212C2×D113—161no—
213C4×C113—21no—
214C4×C113—21no—
215C4×C113—21no—
216C4×C113C22×D411yescharacteristic
217C4×C113C22×D411yescharacteristic
218Dic113—21no—
219Dic113—41no—
220Dic113—41no—
221C2×C113—21no—
222C2×C113—41no—
223C2×C113—41no—
224C2×C113—41no—
225C2×C113—81no—
226C2×C113—81no—
227C2×C113D4211yescharacteristic
228D113—41no—
229D113—81no—
230#3288—1131nomaximal, Sylow
231C113#328811yescharacteristic, complemented, Sylow
232#577—1131no—
233#577—1131no—
234#576—1131no—
235#576D11311yescharacteristic, complemented
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#444—1131no—
2472−1+4—1131no—
248H(5,𝔽2)—1131no—
249H(5,𝔽2)C2×D11311yescharacteristic
250H(5,𝔽2)C2×D11311yescharacteristic, complemented
251#137—1131no—
252#137—1131no—
253#137—2261no—
254#137—2261no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
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)C2×D11311yescharacteristic
266AGL(1,ℤ/8)C2×D11311yescharacteristic, complemented
267#135—21no—
268#135—1131no—
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—21no—
278C2×D8—1131no—
279C2×D8—2261no—
280C2×D8C2×D11311yescharacteristic, complemented
281#131—1131no—
282#131—1131no—
283#131—1131no—
284#131C2×D11311yescharacteristic, complemented
285#127—1131no—
286#124—1131no—
287#108—1131no—
288#108—1131no—
289#104—1131no—
290#104—1131no—
291#104—1131no—
292#104—1131no—
293#101—1131no—
294#100—1131no—
295#100—1131no—
296#100—1131no—
297#97—1131no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—1131no—
303#41—4521no—
304#41—2261no—
305#41—2261no—
306#41—2261no—
307#41—2261no—
308#41C22×D11311yescharacteristic
309#41C22×D11311yescharacteristic
310#41C22×D11311yescharacteristic
311C2×Q8—1131no—
312C2×Q8—2261no—
313C2×Q8—2261no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—1131no—
321C2×D4—4521no—
322C2×D4—4521no—
323C2×D4—2261no—
324C2×D4—2261no—
325C2×D4C22×D11311yescharacteristic
326C2×D4C22×D11311yescharacteristic
327Q16—1131no—
328Q16—2261no—
329Q16—2261no—
330Q16C22⋊D11311yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—1131no—
334SD16—1131no—
335SD16—2261no—
336SD16—2261no—
337SD16—2261no—
338SD16—2261no—
339SD16—2261no—
340SD16—2261no—
341SD16—2261no—
342SD16—2261no—
343SD16—2261no—
344SD16—2261no—
345SD16C22⋊D11311yescharacteristic, complemented
346SD16C22⋊D11311yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—1131no—
352D8—2261no—
353D8—2261no—
354D8—2261no—
355D8—2261no—
356D8—2261no—
357D8—2261no—
358D8C22⋊D11311yescharacteristic, complemented
359M16—21no—
360M16—1131no—
361M16—1131no—
362M16—1131no—
363M16—2261no—
364M16—2261no—
365M16—2261no—
366M16C22×D11311yescharacteristic
367C2×C8—21no—
368C2×C8—1131no—
369C2×C8—2261no—
370C2×C8—2261no—
371C2×C8—2261no—
372C2×C8C22×D11311yescharacteristic
373C22⋊C4—4521no—
374C42—1131no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—4521no—
380C23—2261no—
381Q8—1131no—
382Q8—4521no—
383Q8—2261no—
384Q8—2261no—
385Q8D4×D11311yescharacteristic
386Q8C2×C22⋊D11311yescharacteristic
387Q8C2×C22⋊D11311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—1131no—
400D4—9042no—
401D4—4521no—
402D4—4521no—
403D4—4521no—
404D4—2261no—
405D4—2261no—
406D4—2261no—
407D4D4×D11311yescharacteristic
408D4C2×C22⋊D11311yescharacteristic
409D4C2×C22⋊D11311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—4521no—
416C2×C4—4521no—
417C2×C4—4521no—
418C2×C4—2261no—
419C2×C4—2261no—
420C2×C4—2261no—
421C2×C4C23×D11311yescharacteristic
422C8—21no—
423C8—1131no—
424C8—1131no—
425C8—2261no—
426C8—2261no—
427C8—2261no—
428C8C2×C22⋊D11311yescharacteristic
429C8C2×C22⋊D11311yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—18081no—
440C22—9041no—
441C22—4521no—
442C22—2261no—
443C22#1681366711yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—4521no—
448C4—4521no—
449C4—2261no—
450C4C2×D4×D11311yescharacteristic
451C4C22×C22⋊D11311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—9041no—
459C2—4521no—
460C2D4×C22⋊D11311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#576×C113C2
socleC2×C113D42
hypercenter#576D113
hypocenterC113#3288
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20299409C2semidirect
2. #20299405C2semidirect
3. #20299401C2semidirect
4. #20298986C2semidirect
5. #20298866C2semidirect
6. #20298865C2semidirect
7. #20298861C2semidirect
8. #20298859C2semidirect
9. #20298776C2semidirect
10. #20298769C2semidirect
11. #20298766C2semidirect
12. #20298764C2semidirect
13. #20298720C2semidirect
14. #20298718C2semidirect
15. #576×C113C2semidirect
16. #16813669C22semidirect
17. #16813668C22semidirect
20. #16813652C22semidirect
21. #16813652C22semidirect
24. #16813648C22semidirect
25. #16813648C22semidirect
28. #16813647C22semidirect
34. C2×#14268636C22semidirect
36. C2×#14268635C22semidirect
39. C2×#14268632C22semidirect
40. #16813629C22semidirect
41. #16813629C22semidirect
42. #16813628C22semidirect
43. #16813618C22semidirect
44. #16813609C22semidirect
45. #16813558C22semidirect
46. #16813557C22semidirect
47. #16813549C22semidirect
48. #16813549C22semidirect
49. #16813549C22semidirect
50. #16813548C22semidirect
51. #16813540C22semidirect
52. #16813539C22semidirect
53. #16813539C22semidirect
54. #16813538C22semidirect
55. #16813532C22semidirect
57. H(5,𝔽2)×C113C22semidirect
61. AGL(1,ℤ/8)×C113C22semidirect
65. C2×D8×C113C22semidirect
66. #131×C113C22semidirect
83. #14268637D4semidirect
92. #14268635D4semidirect
95. #14268634D4semidirect
102. #14268632D4semidirect
130. Q16×C113D4semidirect
133. SD16×C113D4semidirect
134. SD16×C113D4semidirect
139. D8×C113D4semidirect
231. C113#3288semidirect
235. #576D113semidirect
250. H(5,𝔽2)C2×D113semidirect
266. AGL(1,ℤ/8)C2×D113semidirect
280. C2×D8C2×D113semidirect
284. #131C2×D113semidirect
330. Q16C22⋊D113semidirect
345. SD16C22⋊D113semidirect
346. SD16C22⋊D113semidirect
358. D8C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
#202994092
#202994052
#202994012
#202989862
#202988662
#202988652
#202988612
#202988592
#202987762
#202987692
#202987662
#202987642
#202987202
#202987182
#576×C1132
#3288113