Finite Group Databasebeta

← #49133024

Subgroups — #49133024

Order 36224.

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#31827277C211yesmaximal, characteristic, complemented
2#31827275C211yesmaximal, characteristic, complemented
3#31827269C211yesmaximal, characteristic, complemented
4#31827266C211yesmaximal, characteristic, complemented
5#31826853C211yesmaximal, characteristic, complemented
6#31826734C211yesmaximal, characteristic, complemented
7#31826729C211yesmaximal, characteristic, complemented
8#31826727C211yesmaximal, characteristic, complemented
9#31826645C211yesmaximal, characteristic, complemented
10#31826636C211yesmaximal, characteristic, complemented
11#31826635C211yesmaximal, characteristic, complemented
12#31826631C211yesmaximal, characteristic, complemented
13#31826586C211yesmaximal, characteristic, complemented
14#31826585C211yesmaximal, characteristic, complemented
15#472×C283C211yesmaximal, characteristic, complemented
16#22583657C2211yescharacteristic, complemented
17#22583656C2211yescharacteristic, complemented
18#22583655C2211yescharacteristic
19#22583655C2211yescharacteristic, complemented
20#22583643—21no—
21#22583643C2211yescharacteristic, complemented
22#22583643C2211yescharacteristic, complemented
23#22583639—21no—
24#22583638—21no—
25#22583638C2211yescharacteristic
26#22583638C2211yescharacteristic, complemented
27#22583637—21no—
28#22583636C2211yescharacteristic
29#22583636C2211yescharacteristic, complemented
30#22583634—21no—
31#22583634—21no—
32#22583634—21no—
33#22583634—21no—
34#22583632—21no—
35#22583632C2211yescharacteristic
36#22583631—21no—
37#22583630C2211yescharacteristic
38#22583629—21no—
39#22583629—21no—
40C2×#17985381—21no—
41C2×#17985381C2211yescharacteristic, complemented
42C2×#17985380C2211yescharacteristic, complemented
43SD16×D283—21no—
44C2×#17985377C2211yescharacteristic, complemented
45D8×D283—21no—
46D8×D283—21no—
47#22583618C2211yescharacteristic, complemented
48#22583618C2211yescharacteristic, complemented
49#22583616C2211yescharacteristic, complemented
50#22583606C2211yescharacteristic, complemented
51#22583597C2211yescharacteristic, complemented
52#22583545C2211yescharacteristic, complemented
53#22583537C2211yescharacteristic, complemented
54#22583536C2211yescharacteristic, complemented
55#22583536C2211yescharacteristic, complemented
56#22583536C2211yescharacteristic, complemented
57#22583528C2211yescharacteristic, complemented
58#22583526C2211yescharacteristic, complemented
59#22583520C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C283C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C283C2211yescharacteristic, complemented
62C2×D8×C283C2211yescharacteristic, complemented
63#131×C283C2211yescharacteristic, complemented
64#108×C283C2211yescharacteristic, complemented
65#100×C283C2211yescharacteristic, complemented
66#100×C283C2211yescharacteristic, complemented
67#17985393—21no—
68#17985392—21no—
69#17985392—21no—
70#17985392C2311yescharacteristic
71#17985392C2311yescharacteristic
72#17985392C2311yescharacteristic
73#17985391—21no—
74#17985391—21no—
75#17985391—21no—
76#17985391—21no—
77#17985391—21no—
78#17985391—41no—
79C2×Dic566—21no—
80C2×Dic566C2311yescharacteristic
81Q8×D283—21no—
82C2×D1132—21no—
83C2×D1132C2311yescharacteristic
84C2×C22⋊D283—41no—
85C2×C22⋊D283—41no—
86C2×C22⋊D283—41no—
87D4×D283—21no—
88D4×D283—21no—
89D4×D283—21no—
90D4×D283—21no—
91D4×D283—21no—
92D4×D283—41no—
93Dic1132—21no—
94Dic1132—21no—
95Dic1132D411yescharacteristic, complemented
96#17985382D411yescharacteristic, complemented
97#17985381—21no—
98#17985381—21no—
99#17985381—21no—
100#17985381—21no—
101#17985381—21no—
102#17985381—21no—
103#17985381D411yescharacteristic, complemented
104#17985381D411yescharacteristic, complemented
105#17985380—21no—
106#17985380—21no—
107#17985380—21no—
108#17985380—21no—
109#17985380D411yescharacteristic, complemented
110#17985379D411yescharacteristic, complemented
111D2264—21no—
112D2264—21no—
113D2264D411yescharacteristic, complemented
114#17985377—21no—
115#17985377—21no—
116#17985377—21no—
117#17985377—21no—
118#17985377D411yescharacteristic, complemented
119#17985376C2311yescharacteristic
120#17985375—21no—
121#17985375—21no—
122#17985375—21no—
123C2×C283⋊C8C2311yescharacteristic
124C8×D283—21no—
125C8×D283—21no—
126C8×D283—21no—
127C22⋊Dic283—41no—
128C4×Dic283C2311yescharacteristic
129#41×C283C2311yescharacteristic
130C2×D4×C283C2311yescharacteristic
131C2×D4×C283C2311yescharacteristic
132SD16×C283—21no—
133SD16×C283—21no—
134D8×C283—21no—
135D8×C283—21no—
136D8×C283—21no—
137D8×C283—21no—
138M16×C283—21no—
139M16×C283C2311yescharacteristic
140M16×C283C2311yescharacteristic
141M16×C283C2311yescharacteristic
142C2×C8×C283—21no—
143C2×C8×C283C2311yescharacteristic
144C22×D283—21no—
145C22×D283—41no—
146C22×D283—41no—
147C22×D283—41no—
148Dic566—21no—
149Dic566—21no—
150Dic566—41no—
151Dic566C2×D411yescharacteristic
152Dic566C2×D411yescharacteristic
153Dic566C2×D411yescharacteristic
154D1132—21no—
155D1132—21no—
156D1132—41no—
157D1132C2×D411yescharacteristic
158D1132C2×D411yescharacteristic
159D1132C2×D411yescharacteristic
160C22⋊D283—21no—
161C22⋊D283—21no—
162C22⋊D283—21no—
163C22⋊D283—41no—
164C22⋊D283—41no—
165C22⋊D283—41no—
166C22⋊D283—41no—
167C22⋊D283—41no—
168C22⋊D283—41no—
169C22⋊D283—82no—
170C2×Dic283—21no—
171C2×Dic283—21no—
172C2×Dic283—41no—
173C2×Dic283—41no—
174C2×Dic283—41no—
175C4×D283—21no—
176C4×D283—21no—
177C4×D283—21no—
178C4×D283—41no—
179C4×D283—41no—
180C283⋊C8—21no—
181C283⋊C8C2×D411yescharacteristic
182C283⋊C8C2×D411yescharacteristic
183C23×C283—21no—
184C23×C283—21no—
185Q8×C283C2×D411yescharacteristic
186D4×C283—21no—
187D4×C283—21no—
188D4×C283—21no—
189D4×C283—21no—
190D4×C283—21no—
191D4×C283C2×D411yescharacteristic
192C2×C4×C283—21no—
193C2×C4×C283C2411yescharacteristic
194C8×C283—21no—
195C8×C283—21no—
196C8×C283—21no—
197C8×C283C2×D411yescharacteristic
198C8×C283C2×D411yescharacteristic
199C22×C283—21no—
200C22×C283—41no—
201C22×C283—41no—
202C22×C283—81no—
203C22×C283—81no—
204C22×C283H(5,𝔽2)11yescharacteristic
205C2×D283—21no—
206C2×D283—21no—
207C2×D283—21no—
208C2×D283—41no—
209C2×D283—81no—
210C2×D283—161no—
211C2×D283—161no—
212C2×D283—161no—
213C4×C283—21no—
214C4×C283C22×D411yescharacteristic
215C4×C283C22×D411yescharacteristic
216Dic283—21no—
217Dic283—21no—
218Dic283—21no—
219Dic283—41no—
220Dic283—41no—
221C2×C283—21no—
222C2×C283—41no—
223C2×C283—81no—
224C2×C283—81no—
225C2×C283D4211yescharacteristic
226D283—41no—
227D283—41no—
228D283—41no—
229D283—81no—
230C283#328811yescharacteristic, complemented, Sylow
231#3288—2831nomaximal, Sylow
232#577—2831no—
233#577—2831no—
234#576—2831no—
235#576—2831no—
236#496—2831no—
237#472—2831no—
238#472D28311yescharacteristic, complemented
239#471—2831no—
240#471—2831no—
241#455—2831no—
242#454—2831no—
243#453—2831no—
244#453—2831no—
245#444—2831no—
246#444—2831no—
2472−1+4—2831no—
248H(5,𝔽2)—2831no—
249H(5,𝔽2)—2831no—
250H(5,𝔽2)—2831no—
251#137—2831no—
252#137—2831no—
253#137—5661no—
254#137—5661no—
255AGL(1,ℤ/8)—2831no—
256AGL(1,ℤ/8)—2831no—
257AGL(1,ℤ/8)—2831no—
258AGL(1,ℤ/8)—2831no—
259AGL(1,ℤ/8)—5661no—
260AGL(1,ℤ/8)—5661no—
261AGL(1,ℤ/8)—5661no—
262AGL(1,ℤ/8)—5661no—
263AGL(1,ℤ/8)—5661no—
264AGL(1,ℤ/8)—5661no—
265AGL(1,ℤ/8)C2×D28311yescharacteristic, complemented
266AGL(1,ℤ/8)C2×D28311yescharacteristic, complemented
267#135—2831no—
268#135—2831no—
269#135—5661no—
270#135—5661no—
271#135—5661no—
272#135—5661no—
273C2×SD16—2831no—
274C2×SD16—2831no—
275C2×SD16—5661no—
276C2×SD16—5661no—
277C2×D8—2831no—
278C2×D8—5661no—
279C2×D8—5661no—
280C2×D8C2×D28311yescharacteristic, complemented
281#131—2831no—
282#131—2831no—
283#131—2831no—
284#131C2×D28311yescharacteristic, complemented
285#127—2831no—
286#124—2831no—
287#108—2831no—
288#108C2×D28311yescharacteristic, complemented
289#104—2831no—
290#104—2831no—
291#104—2831no—
292#104—2831no—
293#101—2831no—
294#100—2831no—
295#100C2×D28311yescharacteristic, complemented
296#100C2×D28311yescharacteristic, complemented
297#97—2831no—
298#41—2831no—
299#41—2831no—
300#41—2831no—
301#41—5661no—
302#41—5661no—
303#41—5661no—
304#41—5661no—
305#41—5661no—
306#41—5661no—
307#41—5661no—
308#41—5661no—
309#41—11321no—
310#41C22×D28311yescharacteristic
311C2×Q8—2831no—
312C2×Q8—5661no—
313C2×Q8—5661no—
314C2×D4—2831no—
315C2×D4—5661no—
316C2×D4—5661no—
317C2×D4—5661no—
318C2×D4—5661no—
319C2×D4—5661no—
320C2×D4—5661no—
321C2×D4—11321no—
322C2×D4—11321no—
323C2×D4—11321no—
324C2×D4—11321no—
325C2×D4C22×D28311yescharacteristic
326C2×D4C22×D28311yescharacteristic
327Q16—2831no—
328Q16—2831no—
329Q16—5661no—
330Q16—5661no—
331SD16—21no—
332SD16—21no—
333SD16—2831no—
334SD16—2831no—
335SD16—2831no—
336SD16—2831no—
337SD16—5661no—
338SD16—5661no—
339SD16—5661no—
340SD16—5661no—
341SD16—5661no—
342SD16—5661no—
343SD16—5661no—
344SD16—5661no—
345SD16—5661no—
346SD16—5661no—
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—2831no—
352D8—2831no—
353D8—5661no—
354D8—5661no—
355D8—5661no—
356D8—5661no—
357D8—5661no—
358D8—5661no—
359M16—21no—
360M16—2831no—
361M16—5661no—
362M16—5661no—
363M16—5661no—
364M16C22×D28311yescharacteristic
365M16C22×D28311yescharacteristic
366M16C22×D28311yescharacteristic
367C2×C8—21no—
368C2×C8—2831no—
369C2×C8—5661no—
370C2×C8—5661no—
371C2×C8—5661no—
372C2×C8C22×D28311yescharacteristic
373C22⋊C4—11321no—
374C42—2831no—
375C23—21no—
376C23—21no—
377C23—5661no—
378C23—11321no—
379C23—11321no—
380C23—11321no—
381Q8—2831no—
382Q8—2831no—
383Q8—2831no—
384Q8—5661no—
385Q8—5661no—
386Q8—11321no—
387Q8D4×D28311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—2831no—
394D4—2831no—
395D4—2831no—
396D4—5661no—
397D4—5661no—
398D4—5661no—
399D4—5661no—
400D4—5661no—
401D4—11321no—
402D4—11321no—
403D4—11321no—
404D4—11321no—
405D4—11321no—
406D4—11321no—
407D4—11321no—
408D4—22642no—
409D4D4×D28311yescharacteristic
410C2×C4—21no—
411C2×C4—5661no—
412C2×C4—5661no—
413C2×C4—5661no—
414C2×C4—5661no—
415C2×C4—5661no—
416C2×C4—11321no—
417C2×C4—11321no—
418C2×C4—11321no—
419C2×C4—11321no—
420C2×C4—11321no—
421C2×C4C23×D28311yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—2831no—
426C8—2831no—
427C8—5661no—
428C8D4×D28311yescharacteristic
429C8D4×D28311yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—5661no—
436C22—5661no—
437C22—5661no—
438C22—11321no—
439C22—45281no—
440C22—45281no—
441C22—45281no—
442C22—22641no—
443C22#2258365611yescharacteristic
444C4—21no—
445C4—5661no—
446C4—5661no—
447C4—5661no—
448C4—11321no—
449C4—11321no—
450C4C2×D4×D28311yescharacteristic
451C4C2×D4×D28311yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—11321no—
457C2—11321no—
458C2—11321no—
459C2—22641no—
460C2#3182709911yescharacteristic
subgroupG/H
centerC2#31827099
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#472×C283C2
socleC2×C283D42
hypercenter#472D283
hypocenterC283#3288
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #31827277C2semidirect
2. #31827275C2semidirect
3. #31827269C2semidirect
4. #31827266C2semidirect
5. #31826853C2semidirect
6. #31826734C2semidirect
7. #31826729C2semidirect
8. #31826727C2semidirect
9. #31826645C2semidirect
10. #31826636C2semidirect
11. #31826635C2semidirect
12. #31826631C2semidirect
13. #31826586C2semidirect
14. #31826585C2semidirect
15. #472×C283C2semidirect
16. #22583657C22semidirect
17. #22583656C22semidirect
19. #22583655C22semidirect
21. #22583643C22semidirect
22. #22583643C22semidirect
26. #22583638C22semidirect
29. #22583636C22semidirect
41. C2×#17985381C22semidirect
42. C2×#17985380C22semidirect
44. C2×#17985377C22semidirect
47. #22583618C22semidirect
48. #22583618C22semidirect
49. #22583616C22semidirect
50. #22583606C22semidirect
51. #22583597C22semidirect
52. #22583545C22semidirect
53. #22583537C22semidirect
54. #22583536C22semidirect
55. #22583536C22semidirect
56. #22583536C22semidirect
57. #22583528C22semidirect
58. #22583526C22semidirect
59. #22583520C22semidirect
60. AGL(1,ℤ/8)×C283C22semidirect
61. AGL(1,ℤ/8)×C283C22semidirect
62. C2×D8×C283C22semidirect
63. #131×C283C22semidirect
64. #108×C283C22semidirect
65. #100×C283C22semidirect
66. #100×C283C22semidirect
95. Dic1132D4semidirect
96. #17985382D4semidirect
103. #17985381D4semidirect
104. #17985381D4semidirect
109. #17985380D4semidirect
110. #17985379D4semidirect
113. D2264D4semidirect
118. #17985377D4semidirect
230. C283#3288semidirect
238. #472D283semidirect
265. AGL(1,ℤ/8)C2×D283semidirect
266. AGL(1,ℤ/8)C2×D283semidirect
280. C2×D8C2×D283semidirect
284. #131C2×D283semidirect
288. #108C2×D283semidirect
295. #100C2×D283semidirect
296. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272772
#318272752
#318272692
#318272662
#318268532
#318267342
#318267292
#318267272
#318266452
#318266362
#318266352
#318266312
#318265862
#318265852
#472×C2832
#3288283