Finite Group Databasebeta

← #49133022

Subgroups — #49133022

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#31827276C211yesmaximal, characteristic, complemented
2#31827275C211yesmaximal, characteristic, complemented
3#31827269C211yesmaximal, characteristic, complemented
4#31827268C211yesmaximal, characteristic, complemented
5#31826852C211yesmaximal, characteristic, complemented
6#31826734C211yesmaximal, characteristic, complemented
7#31826733C211yesmaximal, characteristic, complemented
8#31826727C211yesmaximal, characteristic, complemented
9#31826726C211yesmaximal, characteristic, complemented
10#31826644C211yesmaximal, characteristic, complemented
11#31826639C211yesmaximal, characteristic, complemented
12#31826632C211yesmaximal, characteristic, complemented
13#31826584C211yesmaximal, characteristic, complemented
14#31826584C211yesmaximal, characteristic, complemented
15#453×C283C211yesmaximal, characteristic, complemented
16#22583657C2211yescharacteristic, complemented
17#22583656C2211yescharacteristic
18#22583655C2211yescharacteristic, complemented
19#22583643C2211yescharacteristic, complemented
20#22583640—21no—
21#22583640C2211yescharacteristic, complemented
22#22583639—21no—
23#22583638C2211yescharacteristic, complemented
24#22583637—21no—
25#22583636—21no—
26#22583636—21no—
27#22583636C2211yescharacteristic
28#22583636C2211yescharacteristic, complemented
29#22583635—21no—
30#22583635C2211yescharacteristic, complemented
31#22583634—21no—
32#22583634—21no—
33#22583631—21no—
34#22583630—21no—
35#22583630—21no—
36#22583630C2211yescharacteristic
37#22583630C2211yescharacteristic
38#22583629—21no—
39C2×#17985380—21no—
40C2×#17985380C2211yescharacteristic, complemented
41C2×#17985380C2211yescharacteristic, complemented
42SD16×D283—21no—
43C2×#17985377—21no—
44C2×#17985377C2211yescharacteristic, complemented
45C2×#17985377C2211yescharacteristic, complemented
46D8×D283—21no—
47#22583618C2211yescharacteristic, complemented
48#22583618C2211yescharacteristic, complemented
49#22583617C2211yescharacteristic, complemented
50#22583617C2211yescharacteristic, complemented
51#22583598C2211yescharacteristic, complemented
52#22583545C2211yescharacteristic, complemented
53#22583545C2211yescharacteristic, complemented
54#22583538C2211yescharacteristic, complemented
55#22583538C2211yescharacteristic, complemented
56#22583528C2211yescharacteristic, complemented
57#22583527C2211yescharacteristic, complemented
58#22583526C2211yescharacteristic, complemented
59#22583521C2211yescharacteristic, complemented
60H(5,𝔽2)×C283C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C283C2211yescharacteristic
62AGL(1,ℤ/8)×C283C2211yescharacteristic, complemented
63#127×C283C2211yescharacteristic, complemented
64#104×C283C2211yescharacteristic, complemented
65#104×C283C2211yescharacteristic, complemented
66#100×C283C2211yescharacteristic, complemented
67#17985393—21no—
68#17985392—21no—
69#17985392—21no—
70#17985392C2311yescharacteristic
71#17985392C2311yescharacteristic
72#17985391—21no—
73#17985391—21no—
74#17985391—21no—
75#17985391—41no—
76C2×Dic566—21no—
77C2×Dic566C2311yescharacteristic
78Q8×D283—21no—
79C2×D1132—21no—
80C2×D1132C2311yescharacteristic
81C2×C22⋊D283—41no—
82D4×D283—21no—
83D4×D283—21no—
84D4×D283—21no—
85D4×D283—41no—
86Dic1132—21no—
87#17985382—21no—
88#17985382D411yescharacteristic, complemented
89#17985382D411yescharacteristic, complemented
90#17985381—21no—
91#17985381—21no—
92#17985380—21no—
93#17985380—21no—
94#17985380—21no—
95#17985380—21no—
96#17985380—21no—
97#17985380—21no—
98#17985380—21no—
99#17985380D411yescharacteristic, complemented
100#17985380D411yescharacteristic, complemented
101#17985379—21no—
102#17985379D411yescharacteristic, complemented
103#17985379D411yescharacteristic, complemented
104D2264—21no—
105#17985377—21no—
106#17985377—21no—
107#17985377—21no—
108#17985377—21no—
109#17985377—21no—
110#17985377—21no—
111#17985377—21no—
112#17985377D411yescharacteristic, complemented
113#17985377D411yescharacteristic, complemented
114#17985376—21no—
115#17985376—21no—
116#17985376C2311yescharacteristic
117#17985376C2311yescharacteristic
118#17985375—21no—
119#17985375—21no—
120C2×C283⋊C8—21no—
121C2×C283⋊C8—21no—
122C2×C283⋊C8C2311yescharacteristic
123C2×C283⋊C8C2311yescharacteristic
124C8×D283—21no—
125C8×D283—21no—
126(C2×D283)⋊C4—41no—
127#41×C283—21no—
128#41×C283—21no—
129#41×C283C2311yescharacteristic
130#41×C283C2311yescharacteristic
131C2×D4×C283—21no—
132C2×D4×C283—21no—
133C2×D4×C283—41no—
134C2×D4×C283—41no—
135C2×D4×C283C2311yescharacteristic
136C2×D4×C283C2311yescharacteristic
137SD16×C283—21no—
138SD16×C283—21no—
139D8×C283—21no—
140D8×C283—21no—
141M16×C283C2311yescharacteristic
142M16×C283C2311yescharacteristic
143C42×C283C2311yescharacteristic
144C22×D283—21no—
145C22×D283—41no—
146C22×D283—41no—
147Dic566—21no—
148Dic566—21no—
149Dic566—41no—
150Dic566C2×D411yescharacteristic
151Dic566C2×D411yescharacteristic
152D1132—21no—
153D1132—21no—
154D1132—41no—
155D1132C2×D411yescharacteristic
156D1132C2×D411yescharacteristic
157C22⋊D283—21no—
158C22⋊D283—21no—
159C22⋊D283—41no—
160C22⋊D283—41no—
161C22⋊D283—41no—
162C22⋊D283—41no—
163C2×Dic283—21no—
164C2×Dic283—41no—
165C2×Dic283—41no—
166C4×D283—21no—
167C4×D283—21no—
168C4×D283—41no—
169C4×D283—41no—
170C283⋊C8—21no—
171C283⋊C8—21no—
172C283⋊C8C2×D411yescharacteristic
173C283⋊C8C2×D411yescharacteristic
174C283⋊C8C2×D411yescharacteristic
175C283⋊C8C2×D411yescharacteristic
176C23×C283—21no—
177C23×C283—21no—
178C23×C283—41no—
179Q8×C283C2×D411yescharacteristic
180Q8×C283C2×D411yescharacteristic
181D4×C283—21no—
182D4×C283—21no—
183D4×C283—21no—
184D4×C283—21no—
185D4×C283—21no—
186D4×C283—21no—
187D4×C283—41no—
188D4×C283—41no—
189D4×C283—82no—
190D4×C283C2×D411yescharacteristic
191D4×C283C2×D411yescharacteristic
192C2×C4×C283—21no—
193C2×C4×C283—21no—
194C2×C4×C283—21no—
195C2×C4×C283—41no—
196C2×C4×C283C2411yescharacteristic
197C8×C283—21no—
198C8×C283—21no—
199C22×C283—21no—
200C22×C283—21no—
201C22×C283—41no—
202C22×C283—41no—
203C22×C283—81no—
204C22×C283—81no—
205C22×C283—161no—
206C22×C283H(5,𝔽2)11yescharacteristic
207C2×D283—21no—
208C2×D283—21no—
209C2×D283—41no—
210C2×D283—81no—
211C2×D283—161no—
212C2×D283—161no—
213C4×C283—21no—
214C4×C283—21no—
215C4×C283—41no—
216C4×C283C22×D411yescharacteristic
217C4×C283C22×D411yescharacteristic
218Dic283—21no—
219Dic283—21no—
220Dic283—41no—
221C2×C283—21no—
222C2×C283—41no—
223C2×C283—41no—
224C2×C283—81no—
225C2×C283—81no—
226C2×C283D4211yescharacteristic
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#472—2831no—
239#471—2831no—
240#471—2831no—
241#455—2831no—
242#454—2831no—
243#453—2831no—
244#453D28311yescharacteristic, complemented
245#444—2831no—
246#444—2831no—
2472−1+4—2831no—
248H(5,𝔽2)—2831no—
249H(5,𝔽2)—2831no—
250H(5,𝔽2)C2×D28311yescharacteristic, complemented
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
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—2831no—
279C2×D8—5661no—
280C2×D8—5661no—
281#131—2831no—
282#131—2831no—
283#131—2831no—
284#131—2831no—
285#127C2×D28311yescharacteristic, complemented
286#124—2831no—
287#108—2831no—
288#108—2831no—
289#104—2831no—
290#104—2831no—
291#104C2×D28311yescharacteristic, complemented
292#104C2×D28311yescharacteristic, complemented
293#101—2831no—
294#100—2831no—
295#100—2831no—
296#100C2×D28311yescharacteristic, complemented
297#97—2831no—
298#41—21no—
299#41—21no—
300#41—2831no—
301#41—2831no—
302#41—5661no—
303#41—5661no—
304#41—5661no—
305#41—5661no—
306#41—5661no—
307#41—5661no—
308#41—11321no—
309#41C22×D28311yescharacteristic
310#41C22×D28311yescharacteristic
311C2×Q8—2831no—
312C2×Q8—5661no—
313C2×Q8—5661no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—2831no—
319C2×D4—5661no—
320C2×D4—5661no—
321C2×D4—5661no—
322C2×D4—5661no—
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—2831no—
350D8—2831no—
351D8—5661no—
352D8—5661no—
353D8—5661no—
354D8—5661no—
355D8—5661no—
356D8—5661no—
357D8—5661no—
358D8—5661no—
359M16—2831no—
360M16—2831no—
361M16—5661no—
362M16—5661no—
363M16—5661no—
364M16—5661no—
365M16C22×D28311yescharacteristic
366M16C22×D28311yescharacteristic
367C2×C8—2831no—
368C2×C8—2831no—
369C2×C8—5661no—
370C2×C8—5661no—
371C2×C8—5661no—
372C2×C8—5661no—
373C22⋊C4—11321no—
374C42C22×D28311yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—5661no—
379C23—11321no—
380C23—11321no—
381Q8—2831no—
382Q8—2831no—
383Q8—5661no—
384Q8—5661no—
385Q8—11321no—
386Q8D4×D28311yescharacteristic
387Q8D4×D28311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—2831no—
398D4—2831no—
399D4—5661no—
400D4—5661no—
401D4—5661no—
402D4—5661no—
403D4—11321no—
404D4—11321no—
405D4—11321no—
406D4—11321no—
407D4—11321no—
408D4D4×D28311yescharacteristic
409D4D4×D28311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—5661no—
415C2×C4—5661no—
416C2×C4—5661no—
417C2×C4—11321no—
418C2×C4—11321no—
419C2×C4—11321no—
420C2×C4—11321no—
421C2×C4C23×D28311yescharacteristic
422C8—21no—
423C8—21no—
424C8—2831no—
425C8—2831no—
426C8—2831no—
427C8—2831no—
428C8—5661no—
429C8—5661no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—5661no—
438C22—5661no—
439C22—11321no—
440C22—45281no—
441C22—45281no—
442C22—22641no—
443C22#2258365511yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—5661no—
448C4—5661no—
449C4—11321no—
450C4C2×D4×D28311yescharacteristic
451C4C2×D4×D28311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—11321no—
458C2—11321no—
459C2—22641no—
460C2#3182709711yescharacteristic
subgroupG/H
centerC2#31827097
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#453×C283C2
socleC2×C283D42
hypercenter#453D283
hypocenterC283#3288
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #31827276C2semidirect
2. #31827275C2semidirect
3. #31827269C2semidirect
4. #31827268C2semidirect
5. #31826852C2semidirect
6. #31826734C2semidirect
7. #31826733C2semidirect
8. #31826727C2semidirect
9. #31826726C2semidirect
10. #31826644C2semidirect
11. #31826639C2semidirect
12. #31826632C2semidirect
13. #31826584C2semidirect
14. #31826584C2semidirect
15. #453×C283C2semidirect
16. #22583657C22semidirect
18. #22583655C22semidirect
19. #22583643C22semidirect
21. #22583640C22semidirect
23. #22583638C22semidirect
28. #22583636C22semidirect
30. #22583635C22semidirect
40. C2×#17985380C22semidirect
41. C2×#17985380C22semidirect
44. C2×#17985377C22semidirect
45. C2×#17985377C22semidirect
47. #22583618C22semidirect
48. #22583618C22semidirect
49. #22583617C22semidirect
50. #22583617C22semidirect
51. #22583598C22semidirect
52. #22583545C22semidirect
53. #22583545C22semidirect
54. #22583538C22semidirect
55. #22583538C22semidirect
56. #22583528C22semidirect
57. #22583527C22semidirect
58. #22583526C22semidirect
59. #22583521C22semidirect
60. H(5,𝔽2)×C283C22semidirect
62. AGL(1,ℤ/8)×C283C22semidirect
63. #127×C283C22semidirect
64. #104×C283C22semidirect
65. #104×C283C22semidirect
66. #100×C283C22semidirect
88. #17985382D4semidirect
89. #17985382D4semidirect
99. #17985380D4semidirect
100. #17985380D4semidirect
102. #17985379D4semidirect
103. #17985379D4semidirect
112. #17985377D4semidirect
113. #17985377D4semidirect
230. C283#3288semidirect
244. #453D283semidirect
250. H(5,𝔽2)C2×D283semidirect
266. AGL(1,ℤ/8)C2×D283semidirect
285. #127C2×D283semidirect
291. #104C2×D283semidirect
292. #104C2×D283semidirect
296. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272762
#318272752
#318272692
#318272682
#318268522
#318267342
#318267332
#318267272
#318267262
#318266442
#318266392
#318266322
#318265842
#453×C2832
#3288283