Finite Group Databasebeta

← #44560091

Subgroups — #44560091

Order 32128.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632483C211yesmaximal, characteristic, complemented
2C2×#22548739C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D251C211yesmaximal, characteristic, complemented
4#30632452C211yesmaximal, characteristic, complemented
5#30631956C211yesmaximal, characteristic, complemented
6#30631955C211yesmaximal, characteristic, complemented
7#30631954C211yesmaximal, characteristic, complemented
8#30631951C211yesmaximal, characteristic, complemented
9#30631950C211yesmaximal, characteristic, complemented
10#30631949C211yesmaximal, characteristic, complemented
11#30631948C211yesmaximal, characteristic, complemented
12#30631734C211yesmaximal, characteristic, complemented
13#30631668C211yesmaximal, characteristic, complemented
14#100×D251C211yesmaximal, characteristic, complemented
15#472×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17#41×D251C2211yescharacteristic
18C2×D4×D251C2211yescharacteristic
19#22548748C2211yescharacteristic, complemented
20#22548745C2211yescharacteristic, complemented
21#22548744—21no—
22#22548743C2211yescharacteristic
23#22548742—21no—
24#22548741C2211yescharacteristic
25#22548741C2211yescharacteristic, complemented
26#22548740C2211yescharacteristic
27#22548739—21no—
28#22548739—21no—
29#22548739—21no—
30#22548739—21no—
31#22548739—21no—
32#22548739—21no—
33#22548736—21no—
34#22548734—21no—
35C2×#16860799C2211yescharacteristic, complemented
36C2×#16860798C2211yescharacteristic, complemented
37SD16×D251—21no—
38C2×#16860795C2211yescharacteristic, complemented
39D8×D251—21no—
40#22548723—21no—
41#22548723C2211yescharacteristic, complemented
42#22548722C2211yescharacteristic, complemented
43#22548721—21no—
44#22548721C2211yescharacteristic, complemented
45C2×#16860793—21no—
46C2×#16860793C2211yescharacteristic
47M16×D251—21no—
48M16×D251C2211yescharacteristic
49M16×D251C2211yescharacteristic
50M16×D251C2211yescharacteristic
51#22548651C2211yescharacteristic, complemented
52#22548650C2211yescharacteristic, complemented
53#22548650C2211yescharacteristic, complemented
54#22548633C2211yescharacteristic, complemented
55#22548633C2211yescharacteristic, complemented
56#22548632C2211yescharacteristic, complemented
57#22548632C2211yescharacteristic, complemented
58#22548631C2211yescharacteristic, complemented
59#22548631C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C251C2211yescharacteristic
61AGL(1,ℤ/8)×C251C2211yescharacteristic, complemented
62C2×D8×C251C2211yescharacteristic, complemented
63#131×C251C2211yescharacteristic, complemented
64#108×C251C2211yescharacteristic, complemented
65#100×C251C2211yescharacteristic, complemented
66#100×C251C2211yescharacteristic, complemented
67C23×D251—21no—
68#16860811D411yescharacteristic
69#16860810—21no—
70#16860810C2311yescharacteristic
71#16860809—21no—
72#16860809—21no—
73#16860809—21no—
74#16860809—21no—
75#16860809—21no—
76#16860809D411yescharacteristic
77C2×Dic502C2311yescharacteristic
78Q8×D251D411yescharacteristic
79C2×D1004C2311yescharacteristic
80C2×C22⋊D251—21no—
81C2×C22⋊D251—21no—
82D4×D251—21no—
83D4×D251—21no—
84D4×D251—21no—
85D4×D251—21no—
86D4×D251—21no—
87D4×D251D411yescharacteristic
88C22×Dic251—21no—
89C2×C4×D251—21no—
90C2×C4×D251C2311yescharacteristic
91Dic1004—21no—
92#16860800—21no—
93#16860799—21no—
94#16860799—21no—
95#16860799—21no—
96#16860799—21no—
97#16860798—21no—
98#16860798—21no—
99#16860798—21no—
100#16860798—21no—
101#16860798—21no—
102#16860797—21no—
103D2008—21no—
104#16860795—21no—
105#16860795—21no—
106#16860795—21no—
107#16860795—21no—
108#16860795—21no—
109#16860794—21no—
110#16860794C2311yescharacteristic
111#16860794C2311yescharacteristic
112#16860794C2311yescharacteristic
113#16860793—21no—
114#16860793—21no—
115#16860793—21no—
116#16860793—21no—
117#16860793—21no—
118#16860793D411yescharacteristic
119#16860793D411yescharacteristic
120#16860793D411yescharacteristic
121#16860793D411yescharacteristic
122C2×C251⋊C8—21no—
123C2×C251⋊C8C2311yescharacteristic
124C8×D251—21no—
125C8×D251—21no—
126C8×D251—21no—
127C8×D251—21no—
128C8×D251—21no—
129#41×C251C2311yescharacteristic
130C2×D4×C251C2311yescharacteristic
131C2×D4×C251C2311yescharacteristic
132SD16×C251—21no—
133SD16×C251—21no—
134D8×C251—21no—
135D8×C251—21no—
136D8×C251—21no—
137D8×C251—21no—
138M16×C251—21no—
139M16×C251C2311yescharacteristic
140M16×C251C2311yescharacteristic
141M16×C251C2311yescharacteristic
142C2×C8×C251—21no—
143C2×C8×C251C2311yescharacteristic
144C22×D251—21no—
145C22×D251—21no—
146C22×D251—41no—
147C22×D251—41no—
148C22×D251—82no—
149C22×D251#4111yescharacteristic
150Dic502—21no—
151Dic502—21no—
152Dic502—21no—
153Dic502C2×D411yescharacteristic
154D1004—21no—
155D1004—21no—
156D1004—21no—
157D1004C2×D411yescharacteristic
158C22⋊D251—21no—
159C22⋊D251—21no—
160C22⋊D251—21no—
161C22⋊D251—41no—
162C22⋊D251—41no—
163C22⋊D251—41no—
164C22⋊D251—41no—
165C2×Dic251—21no—
166C2×Dic251—21no—
167C2×Dic251—41no—
168C2×Dic251—41no—
169C2×Dic251#4111yescharacteristic
170C4×D251—21no—
171C4×D251—21no—
172C4×D251—21no—
173C4×D251C2×D411yescharacteristic
174C4×D251C2×D411yescharacteristic
175C4×D251C2×D411yescharacteristic
176C4×D251C2×D411yescharacteristic
177C251⋊C8—21no—
178C251⋊C8—21no—
179C251⋊C8—21no—
180C251⋊C8C2×D411yescharacteristic
181C251⋊C8C2×D411yescharacteristic
182C23×C251—21no—
183C23×C251—21no—
184Q8×C251C2×D411yescharacteristic
185D4×C251—21no—
186D4×C251—21no—
187D4×C251—21no—
188D4×C251—21no—
189D4×C251—21no—
190D4×C251C2×D411yescharacteristic
191C2×C4×C251—21no—
192C2×C4×C251C2411yescharacteristic
193C8×C251—21no—
194C8×C251—21no—
195C8×C251—21no—
196C8×C251C2×D411yescharacteristic
197C8×C251C2×D411yescharacteristic
198C22×C251—21no—
199C22×C251—41no—
200C22×C251—41no—
201C22×C251—81no—
202C22×C251—81no—
203C22×C251C2×#4111yescharacteristic
204C2×D251—21no—
205C2×D251—41no—
206C2×D251—41no—
207C2×D251—81no—
208C2×D251—82no—
209C2×D251—82no—
210C2×D251—81no—
211C2×D251#12111yescharacteristic
212C2×D251#12111yescharacteristic
213C4×C251—21no—
214C4×C251C22×D411yescharacteristic
215C4×C251C22×D411yescharacteristic
216Dic251—21no—
217Dic251—41no—
218Dic251#12111yescharacteristic
219Dic251#12111yescharacteristic
220C2×C251—21no—
221C2×C251—41no—
222C2×C251—81no—
223C2×C251—81no—
224C2×C251C2×#12111yescharacteristic
225D251—21no—
226D251—21no—
227D251—41no—
228D251—81no—
229C251#306811yescharacteristic, complemented, Sylow
230#3068—2511nomaximal, Sylow
231#575—2511no—
232C2×AGL(1,ℤ/8)—2511no—
233C2×AGL(1,ℤ/8)—2511no—
234#568—2511no—
235#472—2511no—
236#472—2511no—
237#472—2511no—
238#472D25111yescharacteristic, complemented
239#471—2511no—
240#471—2511no—
241#471—2511no—
242#471—2511no—
243#430—2511no—
244#413—2511no—
245C2×#100—2511no—
246C2×#41—2511no—
247C2×#41—2511no—
248C22×D4—2511no—
249#137—5021no—
250#137—2511no—
251#137—2511no—
252AGL(1,ℤ/8)—5021no—
253AGL(1,ℤ/8)—5021no—
254AGL(1,ℤ/8)—5021no—
255AGL(1,ℤ/8)—5021no—
256AGL(1,ℤ/8)—5021no—
257AGL(1,ℤ/8)—5021no—
258AGL(1,ℤ/8)—5021no—
259AGL(1,ℤ/8)—2511no—
260AGL(1,ℤ/8)—2511no—
261AGL(1,ℤ/8)—2511no—
262AGL(1,ℤ/8)—2511no—
263AGL(1,ℤ/8)C2×D25111yescharacteristic
264AGL(1,ℤ/8)C2×D25111yescharacteristic, complemented
265#135—5021no—
266#135—5021no—
267C2×SD16—5021no—
268C2×SD16—2511no—
269C2×SD16—2511no—
270C2×D8—5021no—
271C2×D8—2511no—
272C2×D8C2×D25111yescharacteristic, complemented
273#131—5021no—
274#131—5021no—
275#131—2511no—
276#131—2511no—
277#131—2511no—
278#131C2×D25111yescharacteristic, complemented
279C2×M16—5021no—
280C2×M16—5021no—
281C2×M16—2511no—
282C2×M16—2511no—
283C2×M16—2511no—
284C2×M16—2511no—
285#108—2511no—
286#108—2511no—
287#108—2511no—
288#108C2×D25111yescharacteristic, complemented
289#101—2511no—
290#101—2511no—
291#100—2511no—
292#100—2511no—
293#100—2511no—
294#100—2511no—
295#100C2×D25111yescharacteristic, complemented
296#100C2×D25111yescharacteristic, complemented
297C24—5021no—
298#41—5021no—
299#41—5021no—
300#41—5021no—
301#41—5021no—
302#41—5021no—
303#41—5021no—
304#41—2511no—
305#41—2511no—
306#41—2511no—
307#41C22×D25111yescharacteristic
308C2×Q8—2511no—
309C2×Q8—2511no—
310C2×D4—5021no—
311C2×D4—5021no—
312C2×D4—5021no—
313C2×D4—5021no—
314C2×D4—5021no—
315C2×D4—5021no—
316C2×D4—5021no—
317C2×D4—2511no—
318C2×D4—2511no—
319C2×D4C22×D25111yescharacteristic
320C2×D4C22×D25111yescharacteristic
321C22×C4—5021no—
322C22×C4—5021no—
323C22×C4—2511no—
324Q16—5021no—
325Q16—5021no—
326SD16—21no—
327SD16—21no—
328SD16—5021no—
329SD16—5021no—
330SD16—5021no—
331SD16—5021no—
332SD16—5021no—
333SD16—5021no—
334SD16—5021no—
335SD16—5021no—
336SD16—5021no—
337SD16—5021no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—5021no—
343D8—5021no—
344D8—5021no—
345D8—5021no—
346D8—5021no—
347D8—5021no—
348M16—21no—
349M16—5021no—
350M16—5021no—
351M16—5021no—
352M16—5021no—
353M16—5021no—
354M16—5021no—
355M16—2511no—
356M16—2511no—
357M16—2511no—
358M16—2511no—
359M16—2511no—
360M16—2511no—
361M16—2511no—
362M16C22×D25111yescharacteristic
363M16C22×D25111yescharacteristic
364M16C22×D25111yescharacteristic
365C2×C8—21no—
366C2×C8—5021no—
367C2×C8—5021no—
368C2×C8—5021no—
369C2×C8—5021no—
370C2×C8—5021no—
371C2×C8—5021no—
372C2×C8—2511no—
373C2×C8C22×D25111yescharacteristic
374C23—21no—
375C23—21no—
376C23—20082no—
377C23—10041no—
378C23—10041no—
379C23—5021no—
380C23—5021no—
381C23—2511no—
382Q8—5021no—
383Q8—5021no—
384Q8—5021no—
385Q8—2511no—
386Q8D4×D25111yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—10041no—
393D4—10041no—
394D4—10041no—
395D4—10041no—
396D4—5021no—
397D4—5021no—
398D4—5021no—
399D4—5021no—
400D4—5021no—
401D4—5021no—
402D4—2511no—
403D4D4×D25111yescharacteristic
404C2×C4—21no—
405C2×C4—10041no—
406C2×C4—10041no—
407C2×C4—5021no—
408C2×C4—5021no—
409C2×C4—5021no—
410C2×C4—5021no—
411C2×C4—5021no—
412C2×C4—2511no—
413C2×C4—2511no—
414C2×C4—2511no—
415C2×C4—2511no—
416C2×C4—2511no—
417C2×C4C23×D25111yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—5021no—
422C8—5021no—
423C8—5021no—
424C8—2511no—
425C8—2511no—
426C8D4×D25111yescharacteristic
427C8D4×D25111yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—20081no—
434C22—20081no—
435C22—20082no—
436C22—20082no—
437C22—10041no—
438C22—10041no—
439C22—5021no—
440C22—2511no—
441C22—2511no—
442C22#41×D25111yescharacteristic
443C4—21no—
444C4—10041no—
445C4—5021no—
446C4—2511no—
447C4—2511no—
448C4C2×D4×D25111yescharacteristic
449C4C2×D4×D25111yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—20081no—
455C2—10041no—
456C2—5021no—
457C2—5021no—
458C2#121×D25111yescharacteristic
subgroupG/H
centerC2#121×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#472×C251C2
socleC2×C251C2×#121
hypercenter#472D251
hypocenterC251#3068
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #30632483C2semidirect
2. C2×#22548739C2semidirect
3. AGL(1,ℤ/8)×D251C2semidirect
4. #30632452C2semidirect
5. #30631956C2semidirect
6. #30631955C2semidirect
7. #30631954C2semidirect
8. #30631951C2semidirect
9. #30631950C2semidirect
10. #30631949C2semidirect
11. #30631948C2semidirect
12. #30631734C2semidirect
13. #30631668C2semidirect
14. #100×D251C2semidirect
15. #472×C251C2semidirect
19. #22548748C22semidirect
20. #22548745C22semidirect
25. #22548741C22semidirect
35. C2×#16860799C22semidirect
36. C2×#16860798C22semidirect
38. C2×#16860795C22semidirect
41. #22548723C22semidirect
42. #22548722C22semidirect
44. #22548721C22semidirect
51. #22548651C22semidirect
52. #22548650C22semidirect
53. #22548650C22semidirect
54. #22548633C22semidirect
55. #22548633C22semidirect
56. #22548632C22semidirect
57. #22548632C22semidirect
58. #22548631C22semidirect
59. #22548631C22semidirect
61. AGL(1,ℤ/8)×C251C22semidirect
62. C2×D8×C251C22semidirect
63. #131×C251C22semidirect
64. #108×C251C22semidirect
65. #100×C251C22semidirect
66. #100×C251C22semidirect
229. C251#3068semidirect
238. #472D251semidirect
264. AGL(1,ℤ/8)C2×D251semidirect
272. C2×D8C2×D251semidirect
278. #131C2×D251semidirect
288. #108C2×D251semidirect
295. #100C2×D251semidirect
296. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324832
C2×#225487392
AGL(1,ℤ/8)×D2512
#306324522
#306319562
#306319552
#306319542
#306319512
#306319502
#306319492
#306319482
#306317342
#306316682
#100×D2512
#472×C2512
#3068251