Finite Group Databasebeta

← #49133025

Subgroups — #49133025

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#31827272C211yesmaximal, characteristic, complemented
3#31827268C211yesmaximal, characteristic, complemented
4#31826853C211yesmaximal, characteristic, complemented
5#31826733C211yesmaximal, characteristic, complemented
6#31826732C211yesmaximal, characteristic, complemented
7#31826728C211yesmaximal, characteristic, complemented
8#31826726C211yesmaximal, characteristic, complemented
9#31826643C211yesmaximal, characteristic, complemented
10#31826636C211yesmaximal, characteristic, complemented
11#31826633C211yesmaximal, characteristic, complemented
12#31826631C211yesmaximal, characteristic, complemented
13#31826587C211yesmaximal, characteristic, complemented
14#31826585C211yesmaximal, characteristic, complemented
15#576×C283C211yesmaximal, characteristic, complemented
16#22583657C2211yescharacteristic, complemented
17#22583656C2211yescharacteristic, complemented
18#22583642—21no—
19#22583640—21no—
20#22583640C2211yescharacteristic, complemented
21#22583640C2211yescharacteristic, complemented
22#22583636—21no—
23#22583636—21no—
24#22583636C2211yescharacteristic, complemented
25#22583636C2211yescharacteristic, complemented
26#22583635—21no—
27#22583635C2211yescharacteristic
28#22583635C2211yescharacteristic, complemented
29#22583634—21no—
30#22583631—21no—
31#22583630—21no—
32#22583630—21no—
33#22583630C2211yescharacteristic
34C2×#17985381C2211yescharacteristic, complemented
35C2×#17985380—21no—
36C2×#17985380C2211yescharacteristic, complemented
37SD16×D283—21no—
38C2×#17985377—21no—
39C2×#17985377C2211yescharacteristic, complemented
40#22583617C2211yescharacteristic, complemented
41#22583617C2211yescharacteristic, complemented
42#22583616C2211yescharacteristic, complemented
43#22583606C2211yescharacteristic, complemented
44#22583597C2211yescharacteristic, complemented
45#22583546C2211yescharacteristic, complemented
46#22583545C2211yescharacteristic, complemented
47#22583537C2211yescharacteristic, complemented
48#22583537C2211yescharacteristic, complemented
49#22583537C2211yescharacteristic, complemented
50#22583536C2211yescharacteristic, complemented
51#22583528C2211yescharacteristic, complemented
52#22583527C2211yescharacteristic, complemented
53#22583527C2211yescharacteristic, complemented
54#22583526C2211yescharacteristic, complemented
55#22583520C2211yescharacteristic, complemented
56H(5,𝔽2)×C283C2211yescharacteristic
57H(5,𝔽2)×C283C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C283—21no—
59AGL(1,ℤ/8)×C283—21no—
60AGL(1,ℤ/8)×C283C2211yescharacteristic
61AGL(1,ℤ/8)×C283C2211yescharacteristic, complemented
62#135×C283—21no—
63#135×C283C2211yescharacteristic
64C2×D8×C283—21no—
65C2×D8×C283C2211yescharacteristic, complemented
66#131×C283C2211yescharacteristic, complemented
67#17985393—21no—
68#17985392—21no—
69#17985392—21no—
70#17985392C2311yescharacteristic
71#17985391—21no—
72#17985391—41no—
73C2×Dic566—21no—
74C2×Dic566C2311yescharacteristic
75Q8×D283—21no—
76C2×D1132—21no—
77C2×D1132C2311yescharacteristic
78C2×C22⋊D283—41no—
79D4×D283—21no—
80D4×D283—41no—
81#17985382—21no—
82#17985382—21no—
83#17985382D411yescharacteristic, complemented
84#17985381—21no—
85#17985381—21no—
86#17985380—21no—
87#17985380—21no—
88#17985380—21no—
89#17985380—21no—
90#17985380—21no—
91#17985380—21no—
92#17985380D411yescharacteristic, complemented
93#17985379—21no—
94#17985379—21no—
95#17985379D411yescharacteristic, complemented
96#17985377—21no—
97#17985377—21no—
98#17985377—21no—
99#17985377—21no—
100#17985377—21no—
101#17985377—21no—
102#17985377D411yescharacteristic, complemented
103#17985376—21no—
104#17985376—21no—
105#17985376C2311yescharacteristic
106#17985376C2311yescharacteristic
107#17985376C2311yescharacteristic
108#17985375—21no—
109C2×C283⋊C8—21no—
110C2×C283⋊C8—21no—
111C2×C283⋊C8C2311yescharacteristic
112C8×D283—21no—
113C22⋊Dic283—41no—
114C4×Dic283C2311yescharacteristic
115#41×C283—21no—
116#41×C283—21no—
117#41×C283—21no—
118#41×C283—21no—
119#41×C283C2311yescharacteristic
120#41×C283C2311yescharacteristic
121#41×C283C2311yescharacteristic
122C2×D4×C283—21no—
123C2×D4×C283—21no—
124C2×D4×C283—21no—
125C2×D4×C283—21no—
126C2×D4×C283—41no—
127C2×D4×C283—41no—
128C2×D4×C283C2311yescharacteristic
129C2×D4×C283C2311yescharacteristic
130Q16×C283D411yescharacteristic, complemented
131SD16×C283—21no—
132SD16×C283—21no—
133SD16×C283D411yescharacteristic, complemented
134SD16×C283D411yescharacteristic, complemented
135D8×C283—21no—
136D8×C283—21no—
137D8×C283—21no—
138D8×C283—21no—
139D8×C283D411yescharacteristic, complemented
140M16×C283—21no—
141M16×C283C2311yescharacteristic
142C2×C8×C283—21no—
143C2×C8×C283C2311yescharacteristic
144C22×D283—21no—
145C22×D283—41no—
146Dic566—21no—
147Dic566—21no—
148Dic566—41no—
149Dic566C2×D411yescharacteristic
150D1132—21no—
151D1132—21no—
152D1132—41no—
153D1132C2×D411yescharacteristic
154C22⋊D283—21no—
155C22⋊D283—41no—
156C22⋊D283—41no—
157C22⋊D283—82no—
158C2×Dic283—21no—
159C2×Dic283—21no—
160C2×Dic283—41no—
161C4×D283—21no—
162C4×D283—41no—
163C4×D283—41no—
164C283⋊C8—21no—
165C283⋊C8—21no—
166C283⋊C8—21no—
167C283⋊C8C2×D411yescharacteristic
168C283⋊C8C2×D411yescharacteristic
169C23×C283—21no—
170C23×C283—21no—
171C23×C283—41no—
172C23×C283—41no—
173Q8×C283C2×D411yescharacteristic
174Q8×C283C2×D411yescharacteristic
175Q8×C283C2×D411yescharacteristic
176D4×C283—21no—
177D4×C283—21no—
178D4×C283—21no—
179D4×C283—21no—
180D4×C283—21no—
181D4×C283—21no—
182D4×C283—21no—
183D4×C283—41no—
184D4×C283—41no—
185D4×C283—41no—
186D4×C283—41no—
187D4×C283C2×D411yescharacteristic
188D4×C283C2×D411yescharacteristic
189D4×C283C2×D411yescharacteristic
190C2×C4×C283—21no—
191C2×C4×C283—21no—
192C2×C4×C283—21no—
193C2×C4×C283—41no—
194C2×C4×C283—41no—
195C2×C4×C283C2411yescharacteristic
196C8×C283—21no—
197C8×C283C2×D411yescharacteristic
198C8×C283C2×D411yescharacteristic
199C22×C283—21no—
200C22×C283—21no—
201C22×C283—21no—
202C22×C283—41no—
203C22×C283—41no—
204C22×C283—81no—
205C22×C283—81no—
206C22×C283—161no—
207C22×C283—161no—
208C22×C283H(5,𝔽2)11yescharacteristic
209C2×D283—21no—
210C2×D283—41no—
211C2×D283—81no—
212C2×D283—161no—
213C4×C283—21no—
214C4×C283—21no—
215C4×C283—21no—
216C4×C283C22×D411yescharacteristic
217C4×C283C22×D411yescharacteristic
218Dic283—21no—
219Dic283—41no—
220Dic283—41no—
221C2×C283—21no—
222C2×C283—41no—
223C2×C283—41no—
224C2×C283—41no—
225C2×C283—81no—
226C2×C283—81no—
227C2×C283D4211yescharacteristic
228D283—41no—
229D283—81no—
230C283#328811yescharacteristic, complemented, Sylow
231#3288—2831nomaximal, Sylow
232#577—2831no—
233#577—2831no—
234#576—2831no—
235#576D28311yescharacteristic, complemented
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#453—2831no—
245#444—2831no—
246#444—2831no—
2472−1+4—2831no—
248H(5,𝔽2)—2831no—
249H(5,𝔽2)C2×D28311yescharacteristic
250H(5,𝔽2)C2×D28311yescharacteristic, complemented
251#137—2831no—
252#137—2831no—
253#137—5661no—
254#137—5661no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—2831no—
258AGL(1,ℤ/8)—2831no—
259AGL(1,ℤ/8)—2831no—
260AGL(1,ℤ/8)—2831no—
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—21no—
268#135—2831no—
269#135—5661no—
270#135—5661no—
271#135—5661no—
272#135C2×D28311yescharacteristic
273C2×SD16—2831no—
274C2×SD16—2831no—
275C2×SD16—5661no—
276C2×SD16—5661no—
277C2×D8—21no—
278C2×D8—2831no—
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#108—2831no—
289#104—2831no—
290#104—2831no—
291#104—2831no—
292#104—2831no—
293#101—2831no—
294#100—2831no—
295#100—2831no—
296#100—2831no—
297#97—2831no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—2831no—
303#41—5661no—
304#41—5661no—
305#41—5661no—
306#41—5661no—
307#41—11321no—
308#41C22×D28311yescharacteristic
309#41C22×D28311yescharacteristic
310#41C22×D28311yescharacteristic
311C2×Q8—2831no—
312C2×Q8—5661no—
313C2×Q8—5661no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—2831no—
321C2×D4—5661no—
322C2×D4—5661no—
323C2×D4—11321no—
324C2×D4—11321no—
325C2×D4C22×D28311yescharacteristic
326C2×D4C22×D28311yescharacteristic
327Q16—2831no—
328Q16—5661no—
329Q16—5661no—
330Q16C22⋊D28311yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—2831no—
334SD16—2831no—
335SD16—5661no—
336SD16—5661no—
337SD16—5661no—
338SD16—5661no—
339SD16—5661no—
340SD16—5661no—
341SD16—5661no—
342SD16—5661no—
343SD16—5661no—
344SD16—5661no—
345SD16C22⋊D28311yescharacteristic, complemented
346SD16C22⋊D28311yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—2831no—
352D8—5661no—
353D8—5661no—
354D8—5661no—
355D8—5661no—
356D8—5661no—
357D8—5661no—
358D8C22⋊D28311yescharacteristic, complemented
359M16—21no—
360M16—2831no—
361M16—2831no—
362M16—2831no—
363M16—5661no—
364M16—5661no—
365M16—5661no—
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—41no—
378C23—41no—
379C23—5661no—
380C23—11321no—
381Q8—2831no—
382Q8—5661no—
383Q8—5661no—
384Q8—11321no—
385Q8D4×D28311yescharacteristic
386Q8C2×C22⋊D28311yescharacteristic
387Q8C2×C22⋊D28311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—2831no—
400D4—5661no—
401D4—5661no—
402D4—5661no—
403D4—11321no—
404D4—11321no—
405D4—11321no—
406D4—22642no—
407D4D4×D28311yescharacteristic
408D4C2×C22⋊D28311yescharacteristic
409D4C2×C22⋊D28311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—5661no—
416C2×C4—5661no—
417C2×C4—5661no—
418C2×C4—11321no—
419C2×C4—11321no—
420C2×C4—11321no—
421C2×C4C23×D28311yescharacteristic
422C8—21no—
423C8—2831no—
424C8—2831no—
425C8—5661no—
426C8—5661no—
427C8—5661no—
428C8C2×C22⋊D28311yescharacteristic
429C8C2×C22⋊D28311yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—5661no—
440C22—11321no—
441C22—45281no—
442C22—22641no—
443C22#2258365511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—5661no—
448C4—11321no—
449C4—11321no—
450C4C2×D4×D28311yescharacteristic
451C4C22×C22⋊D28311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—11321no—
459C2—22641no—
460C2D4×C22⋊D28311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#576×C283C2
socleC2×C283D42
hypercenter#576D283
hypocenterC283#3288
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #31827276C2semidirect
2. #31827272C2semidirect
3. #31827268C2semidirect
4. #31826853C2semidirect
5. #31826733C2semidirect
6. #31826732C2semidirect
7. #31826728C2semidirect
8. #31826726C2semidirect
9. #31826643C2semidirect
10. #31826636C2semidirect
11. #31826633C2semidirect
12. #31826631C2semidirect
13. #31826587C2semidirect
14. #31826585C2semidirect
15. #576×C283C2semidirect
16. #22583657C22semidirect
17. #22583656C22semidirect
20. #22583640C22semidirect
21. #22583640C22semidirect
24. #22583636C22semidirect
25. #22583636C22semidirect
28. #22583635C22semidirect
34. C2×#17985381C22semidirect
36. C2×#17985380C22semidirect
39. C2×#17985377C22semidirect
40. #22583617C22semidirect
41. #22583617C22semidirect
42. #22583616C22semidirect
43. #22583606C22semidirect
44. #22583597C22semidirect
45. #22583546C22semidirect
46. #22583545C22semidirect
47. #22583537C22semidirect
48. #22583537C22semidirect
49. #22583537C22semidirect
50. #22583536C22semidirect
51. #22583528C22semidirect
52. #22583527C22semidirect
53. #22583527C22semidirect
54. #22583526C22semidirect
55. #22583520C22semidirect
57. H(5,𝔽2)×C283C22semidirect
61. AGL(1,ℤ/8)×C283C22semidirect
65. C2×D8×C283C22semidirect
66. #131×C283C22semidirect
83. #17985382D4semidirect
92. #17985380D4semidirect
95. #17985379D4semidirect
102. #17985377D4semidirect
130. Q16×C283D4semidirect
133. SD16×C283D4semidirect
134. SD16×C283D4semidirect
139. D8×C283D4semidirect
230. C283#3288semidirect
235. #576D283semidirect
250. H(5,𝔽2)C2×D283semidirect
266. AGL(1,ℤ/8)C2×D283semidirect
280. C2×D8C2×D283semidirect
284. #131C2×D283semidirect
330. Q16C22⋊D283semidirect
345. SD16C22⋊D283semidirect
346. SD16C22⋊D283semidirect
358. D8C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
#318272762
#318272722
#318272682
#318268532
#318267332
#318267322
#318267282
#318267262
#318266432
#318266362
#318266332
#318266312
#318265872
#318265852
#576×C2832
#3288283