Finite Group Databasebeta

← #49133053

Subgroups — #49133053

Order 36224.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 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#31827274C211yesmaximal, characteristic, complemented
4#31827271C211yesmaximal, characteristic, complemented
5#31826836C211yesmaximal, characteristic, complemented
6#31826731C211yesmaximal, characteristic, complemented
7#31826729C211yesmaximal, characteristic, complemented
8#31826727C211yesmaximal, characteristic, complemented
9#31826650C211yesmaximal, characteristic, complemented
10#31826646C211yesmaximal, characteristic, complemented
11#31826635C211yesmaximal, characteristic, complemented
12#31826631C211yesmaximal, characteristic, complemented
13#31826581C211yesmaximal, characteristic, complemented
14#31826580C211yesmaximal, characteristic, complemented
15#471×C283C211yesmaximal, characteristic, complemented
16#22583658C2211yescharacteristic
17#22583657C2211yescharacteristic
18#22583656C2211yescharacteristic, complemented
19#22583655C2211yescharacteristic, complemented
20#22583643—21no—
21#22583643C2211yescharacteristic, complemented
22#22583643C2211yescharacteristic, complemented
23#22583642—21no—
24#22583641C2211yescharacteristic, complemented
25#22583639—21no—
26#22583639—21no—
27#22583638—21no—
28#22583638C2211yescharacteristic
29#22583638C2211yescharacteristic
30#22583637—21no—
31#22583637—21no—
32#22583636C2211yescharacteristic
33#22583634—21no—
34#22583633—21no—
35#22583632—21no—
36#22583632C2211yescharacteristic
37#22583631—21no—
38#22583630C2211yescharacteristic
39#22583629—21no—
40C2×#17985381—21no—
41C2×#17985381C2211yescharacteristic, complemented
42C2×#17985380C2211yescharacteristic, complemented
43C2×#17985379C2211yescharacteristic, complemented
44SD16×D283—21no—
45SD16×D283—21no—
46SD16×D283—21no—
47#22583618C2211yescharacteristic, complemented
48#22583618C2211yescharacteristic, complemented
49#22583616C2211yescharacteristic, complemented
50#22583608C2211yescharacteristic, complemented
51#22583606C2211yescharacteristic
52#22583545C2211yescharacteristic, complemented
53#22583537C2211yescharacteristic, complemented
54#22583536C2211yescharacteristic, complemented
55#22583536C2211yescharacteristic, complemented
56#22583536C2211yescharacteristic, complemented
57#22583528C2211yescharacteristic, complemented
58#22583526C2211yescharacteristic, complemented
59C8×Dic283C2211yescharacteristic, complemented
60#137×C283C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C283C2211yescharacteristic
62C2×SD16×C283C2211yescharacteristic, complemented
63#131×C283C2211yescharacteristic, complemented
64#108×C283C2211yescharacteristic, complemented
65#101×C283C2211yescharacteristic, complemented
66#100×C283C2211yescharacteristic, complemented
67#17985393—21no—
68#17985393—21no—
69#17985393—21no—
70#17985392—21no—
71#17985392—21no—
72#17985392—41no—
73#17985392C2311yescharacteristic
74#17985392C2311yescharacteristic
75#17985392C2311yescharacteristic
76#17985391—21no—
77#17985391—21no—
78#17985391—21no—
79#17985391—41no—
80C2×Dic566—21no—
81C2×Dic566C2311yescharacteristic
82Q8×D283—21no—
83Q8×D283—21no—
84Q8×D283—21no—
85C2×D1132—21no—
86C2×D1132C2311yescharacteristic
87C2×C22⋊D283—41no—
88C2×C22⋊D283—41no—
89D4×D283—21no—
90D4×D283—21no—
91D4×D283—21no—
92D4×D283—41no—
93Dic1132—21no—
94Dic1132—21no—
95Dic1132D411yescharacteristic, complemented
96#17985382—21no—
97#17985382D411yescharacteristic, complemented
98#17985381—21no—
99#17985381—21no—
100#17985381—21no—
101#17985381—21no—
102#17985381—21no—
103#17985381—21no—
104#17985381D411yescharacteristic, complemented
105#17985381D411yescharacteristic, complemented
106#17985380—21no—
107#17985380—21no—
108#17985380—21no—
109#17985380D411yescharacteristic, complemented
110#17985379—21no—
111#17985379—21no—
112#17985379—21no—
113#17985379D411yescharacteristic, complemented
114D2264—21no—
115D2264—21no—
116D2264D411yescharacteristic, complemented
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—
127Dic283⋊C4—41no—
128C4×Dic283C2311yescharacteristic
129#41×C283C2311yescharacteristic
130C2×Q8×C283C2311yescharacteristic
131C2×D4×C283C2311yescharacteristic
132Q16×C283—21no—
133SD16×C283—21no—
134SD16×C283—21no—
135SD16×C283—21no—
136SD16×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—
147Dic566—21no—
148Dic566—21no—
149Dic566—41no—
150Dic566—41no—
151Dic566C2×D411yescharacteristic
152Dic566C2×D411yescharacteristic
153Dic566C2×D411yescharacteristic
154D1132—21no—
155D1132—21no—
156D1132—41no—
157D1132—41no—
158D1132C2×D411yescharacteristic
159D1132C2×D411yescharacteristic
160D1132C2×D411yescharacteristic
161C22⋊D283—21no—
162C22⋊D283—21no—
163C22⋊D283—21no—
164C22⋊D283—41no—
165C22⋊D283—41no—
166C22⋊D283—41no—
167C22⋊D283—41no—
168C22⋊D283—82no—
169C2×Dic283—21no—
170C2×Dic283—21no—
171C2×Dic283—41no—
172C2×Dic283—41no—
173C4×D283—21no—
174C4×D283—21no—
175C4×D283—21no—
176C4×D283—41no—
177C4×D283—41no—
178C4×D283—41no—
179C4×D283—41no—
180C283⋊C8—21no—
181C283⋊C8C2×D411yescharacteristic
182C283⋊C8C2×D411yescharacteristic
183C23×C283—21no—
184Q8×C283—21no—
185Q8×C283—21no—
186Q8×C283C2×D411yescharacteristic
187D4×C283—21no—
188D4×C283—21no—
189D4×C283—21no—
190D4×C283C2×D411yescharacteristic
191C2×C4×C283—21no—
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—81no—
202C22×C283H(5,𝔽2)11yescharacteristic
203C2×D283—21no—
204C2×D283—21no—
205C2×D283—21no—
206C2×D283—41no—
207C2×D283—81no—
208C2×D283—161no—
209C2×D283—161no—
210C4×C283—21no—
211C4×C283—41no—
212C4×C283C22×D411yescharacteristic
213C4×C283C22×D411yescharacteristic
214Dic283—22no—
215Dic283—21no—
216Dic283—21no—
217Dic283—21no—
218Dic283—41no—
219C2×C283—21no—
220C2×C283—41no—
221C2×C283—81no—
222C2×C283D4211yescharacteristic
223D283—41no—
224D283—41no—
225D283—41no—
226D283—81no—
227C283#329111yescharacteristic, complemented, Sylow
228#3291—2831nomaximal, Sylow
229#577—2831no—
230#577—2831no—
231#577—2831no—
232#577—2831no—
233#492—2831no—
234#471—2831no—
235#471—2831no—
236#471—2831no—
237#471D28311yescharacteristic, complemented
238#456—2831no—
239#456—2831no—
240#453—2831no—
241#453—2831no—
242#443—2831no—
243#443—2831no—
2442−1+4—2831no—
2452−1+4—2831no—
246H(5,𝔽2)—2831no—
247H(5,𝔽2)—2831no—
248#137—2831no—
249#137—2831no—
250#137—2831no—
251#137—5661no—
252#137—5661no—
253#137—5661no—
254#137—5661no—
255#137C2×D28311yescharacteristic, complemented
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)C2×D28311yescharacteristic
264#135—2831no—
265#135—2831no—
266#135—5661no—
267#135—5661no—
268#135—5661no—
269#135—5661no—
270C2×SD16—2831no—
271C2×SD16—2831no—
272C2×SD16—2831no—
273C2×SD16—5661no—
274C2×SD16—5661no—
275C2×SD16—5661no—
276C2×SD16—5661no—
277C2×SD16C2×D28311yescharacteristic, complemented
278#131—2831no—
279#131—2831no—
280#131—2831no—
281#131C2×D28311yescharacteristic, complemented
282C4⋊Q8—2831no—
283#127—2831no—
284#108—2831no—
285#108C2×D28311yescharacteristic, complemented
286#104—2831no—
287#104—2831no—
288#104—2831no—
289#104—2831no—
290#101—2831no—
291#101C2×D28311yescharacteristic, complemented
292#100—2831no—
293#100C2×D28311yescharacteristic, complemented
294C4×C8—2831no—
295#41—2831no—
296#41—2831no—
297#41—2831no—
298#41—5661no—
299#41—5661no—
300#41—5661no—
301#41—5661no—
302#41—5661no—
303#41—5661no—
304#41—5661no—
305#41—5661no—
306#41—11321no—
307#41—11321no—
308#41C22×D28311yescharacteristic
309C2×Q8—2831no—
310C2×Q8—5661no—
311C2×Q8—5661no—
312C2×Q8—5661no—
313C2×Q8—5661no—
314C2×Q8C22×D28311yescharacteristic
315C2×D4—2831no—
316C2×D4—5661no—
317C2×D4—5661no—
318C2×D4—5661no—
319C2×D4—5661no—
320C2×D4—11321no—
321C2×D4—11321no—
322C2×D4—11321no—
323C2×D4C22×D28311yescharacteristic
324Q16—21no—
325Q16—2831no—
326Q16—2831no—
327Q16—5661no—
328Q16—5661no—
329Q16—5661no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—2831no—
335SD16—2831no—
336SD16—2831no—
337SD16—2831no—
338SD16—5661no—
339SD16—5661no—
340SD16—5661no—
341SD16—5661no—
342SD16—5661no—
343SD16—5661no—
344SD16—5661no—
345SD16—5661no—
346SD16—5661no—
347SD16—5661no—
348SD16—5661no—
349SD16—5661no—
350D8—21no—
351D8—2831no—
352D8—2831no—
353D8—5661no—
354D8—5661no—
355D8—5661no—
356M16—21no—
357M16—2831no—
358M16—5661no—
359M16—5661no—
360M16—5661no—
361M16C22×D28311yescharacteristic
362M16C22×D28311yescharacteristic
363M16C22×D28311yescharacteristic
364C2×C8—21no—
365C2×C8—2831no—
366C2×C8—5661no—
367C2×C8—5661no—
368C2×C8—5661no—
369C2×C8C22×D28311yescharacteristic
370C4⋊C4—11321no—
371C42—2831no—
372C23—21no—
373C23—5661no—
374C23—11321no—
375C23—11321no—
376Q8—21no—
377Q8—21no—
378Q8—2831no—
379Q8—2831no—
380Q8—2831no—
381Q8—5661no—
382Q8—5661no—
383Q8—11321no—
384Q8—11321no—
385Q8D4×D28311yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—2831no—
390D4—2831no—
391D4—2831no—
392D4—5661no—
393D4—5661no—
394D4—5661no—
395D4—5661no—
396D4—5661no—
397D4—11321no—
398D4—11321no—
399D4—11321no—
400D4—11321no—
401D4—11321no—
402D4—11321no—
403D4—22642no—
404D4D4×D28311yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—5661no—
408C2×C4—5661no—
409C2×C4—5661no—
410C2×C4—5661no—
411C2×C4—5661no—
412C2×C4—11321no—
413C2×C4—11321no—
414C2×C4—11321no—
415C2×C4—11321no—
416C2×C4—11321no—
417C2×C4—11321no—
418C2×C4C23×D28311yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—2831no—
423C8—2831no—
424C8—5661no—
425C8D4×D28311yescharacteristic
426C8D4×D28311yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—5661no—
431C22—5661no—
432C22—5661no—
433C22—11321no—
434C22—45281no—
435C22—45281no—
436C22—22641no—
437C22#2258365611yescharacteristic
438C4—21no—
439C4—41no—
440C4—5662no—
441C4—5661no—
442C4—5661no—
443C4—5661no—
444C4—11321no—
445C4C2×D4×D28311yescharacteristic
446C4C2×D4×D28311yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—11321no—
451C2—11321no—
452C2—11321no—
453C2—22641no—
454C2#3182709911yescharacteristic
subgroupG/H
centerC2#31827099
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#471×C283C2
socleC2×C283D42
hypercenter#471D283
hypocenterC283#3291
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #31827277C2semidirect
2. #31827275C2semidirect
3. #31827274C2semidirect
4. #31827271C2semidirect
5. #31826836C2semidirect
6. #31826731C2semidirect
7. #31826729C2semidirect
8. #31826727C2semidirect
9. #31826650C2semidirect
10. #31826646C2semidirect
11. #31826635C2semidirect
12. #31826631C2semidirect
13. #31826581C2semidirect
14. #31826580C2semidirect
15. #471×C283C2semidirect
18. #22583656C22semidirect
19. #22583655C22semidirect
21. #22583643C22semidirect
22. #22583643C22semidirect
24. #22583641C22semidirect
41. C2×#17985381C22semidirect
42. C2×#17985380C22semidirect
43. C2×#17985379C22semidirect
47. #22583618C22semidirect
48. #22583618C22semidirect
49. #22583616C22semidirect
50. #22583608C22semidirect
52. #22583545C22semidirect
53. #22583537C22semidirect
54. #22583536C22semidirect
55. #22583536C22semidirect
56. #22583536C22semidirect
57. #22583528C22semidirect
58. #22583526C22semidirect
59. C8×Dic283C22semidirect
60. #137×C283C22semidirect
62. C2×SD16×C283C22semidirect
63. #131×C283C22semidirect
64. #108×C283C22semidirect
65. #101×C283C22semidirect
66. #100×C283C22semidirect
95. Dic1132D4semidirect
97. #17985382D4semidirect
104. #17985381D4semidirect
105. #17985381D4semidirect
109. #17985380D4semidirect
113. #17985379D4semidirect
116. D2264D4semidirect
118. #17985377D4semidirect
227. C283#3291semidirect
237. #471D283semidirect
255. #137C2×D283semidirect
277. C2×SD16C2×D283semidirect
281. #131C2×D283semidirect
285. #108C2×D283semidirect
291. #101C2×D283semidirect
293. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272772
#318272752
#318272742
#318272712
#318268362
#318267312
#318267292
#318267272
#318266502
#318266462
#318266352
#318266312
#318265812
#318265802
#471×C2832
#3291283