Finite Group Databasebeta

← #49133028

Subgroups — #49133028

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#31827272C211yesmaximal, characteristic, complemented
3#31827270C211yesmaximal, characteristic, complemented
4#31827267C211yesmaximal, characteristic, complemented
5#31826854C211yesmaximal, characteristic, complemented
6#31826736C211yesmaximal, characteristic, complemented
7#31826735C211yesmaximal, characteristic, complemented
8#31826729C211yesmaximal, characteristic, complemented
9#31826728C211yesmaximal, characteristic, complemented
10#31826644C211yesmaximal, characteristic, complemented
11#31826639C211yesmaximal, characteristic, complemented
12#31826634C211yesmaximal, characteristic, complemented
13#31826634C211yesmaximal, characteristic, complemented
14#31826588C211yesmaximal, characteristic, complemented
15#444×C283C211yesmaximal, characteristic, complemented
16#22583657C2211yescharacteristic, complemented
17#22583656C2211yescharacteristic
18#22583656C2211yescharacteristic, complemented
19#22583656C2211yescharacteristic, complemented
20#22583643—21no—
21#22583643C2211yescharacteristic, complemented
22#22583642—21no—
23#22583640C2211yescharacteristic, complemented
24#22583638—21no—
25#22583638—21no—
26#22583638—21no—
27#22583638C2211yescharacteristic
28#22583638C2211yescharacteristic, complemented
29#22583638C2211yescharacteristic, complemented
30#22583637—21no—
31#22583637—21no—
32#22583635C2211yescharacteristic
33#22583635C2211yescharacteristic, complemented
34#22583635C2211yescharacteristic, complemented
35#22583634—21no—
36#22583633—21no—
37#22583632—21no—
38#22583632—21no—
39#22583632C2211yescharacteristic
40#22583631—21no—
41C2×#17985381—21no—
42C2×#17985381C2211yescharacteristic, complemented
43C2×#17985381C2211yescharacteristic, complemented
44SD16×D283—21no—
45C2×D2264—21no—
46C2×D2264C2211yescharacteristic, complemented
47C2×D2264C2211yescharacteristic, complemented
48D8×D283—21no—
49#22583616C2211yescharacteristic, complemented
50#22583616C2211yescharacteristic, complemented
51#22583607C2211yescharacteristic, complemented
52#22583598C2211yescharacteristic, complemented
53#22583546C2211yescharacteristic, complemented
54#22583538C2211yescharacteristic, complemented
55#22583538C2211yescharacteristic, complemented
56#22583528C2211yescharacteristic, complemented
57#22583526C2211yescharacteristic, complemented
58#22583526C2211yescharacteristic, complemented
59#22583526C2211yescharacteristic, complemented
60#135×C283C2211yescharacteristic
61#131×C283C2211yescharacteristic, complemented
62#131×C283C2211yescharacteristic, complemented
63#108×C283C2211yescharacteristic, complemented
64#104×C283C2211yescharacteristic, complemented
65#104×C283C2211yescharacteristic, complemented
66#97×C283C2211yescharacteristic, complemented
67#17985393—21no—
68#17985393—21no—
69#17985393—21no—
70#17985392—21no—
71#17985392—21no—
72#17985392—21no—
73#17985392—21no—
74#17985392C2311yescharacteristic
75#17985392C2311yescharacteristic
76#17985391—21no—
77#17985391—41no—
78C2×Dic566—21no—
79C2×Dic566C2311yescharacteristic
80Q8×D283—21no—
81C2×D1132—21no—
82C2×D1132—21no—
83C2×D1132—21no—
84C2×D1132—41no—
85C2×D1132C2311yescharacteristic
86C2×D1132C2311yescharacteristic
87C2×D1132C2311yescharacteristic
88D4×D283—21no—
89D4×D283—21no—
90D4×D283—21no—
91D4×D283—41no—
92D4×D283—41no—
93D4×D283—41no—
94Dic1132—21no—
95Dic1132D411yescharacteristic, complemented
96#17985382—21no—
97#17985381—21no—
98#17985381—21no—
99#17985381—21no—
100#17985381—21no—
101#17985381—21no—
102#17985381—21no—
103#17985381—21no—
104#17985381—21no—
105#17985381D411yescharacteristic, complemented
106#17985381D411yescharacteristic, complemented
107#17985380—21no—
108#17985379—21no—
109#17985379—21no—
110#17985379—21no—
111D2264—21no—
112D2264—21no—
113D2264—21no—
114D2264—21no—
115D2264—21no—
116D2264—21no—
117D2264—21no—
118D2264D411yescharacteristic, complemented
119#17985377—21no—
120#17985377—21no—
121#17985377—21no—
122#17985376C2311yescharacteristic
123#17985376C2311yescharacteristic
124#17985375—21no—
125#17985375—21no—
126C8×D283—21no—
127C8×D283—21no—
128(C2×D283)⋊C4—41no—
129#41×C283C2311yescharacteristic
130#41×C283C2311yescharacteristic
131Q16×C283D411yescharacteristic, complemented
132SD16×C283D411yescharacteristic, complemented
133SD16×C283D411yescharacteristic, complemented
134D8×C283D411yescharacteristic, complemented
135M16×C283—21no—
136M16×C283—21no—
137M16×C283C2311yescharacteristic
138M16×C283C2311yescharacteristic
139C2×C8×C283—21no—
140C2×C8×C283—21no—
141C2×C8×C283C2311yescharacteristic
142C2×C8×C283C2311yescharacteristic
143C42×C283C2311yescharacteristic
144C22×D283—21no—
145C22×D283—21no—
146C22×D283—21no—
147C22×D283—41no—
148C22×D283—41no—
149C22×D283—41no—
150Dic566—21no—
151Dic566—21no—
152Dic566—41no—
153Dic566C2×D411yescharacteristic
154Dic566C2×D411yescharacteristic
155D1132—21no—
156D1132—21no—
157D1132—21no—
158D1132—21no—
159D1132—21no—
160D1132—21no—
161D1132—41no—
162D1132—41no—
163D1132—41no—
164D1132—82no—
165D1132C2×D411yescharacteristic
166D1132C2×D411yescharacteristic
167C22⋊D283—21no—
168C22⋊D283—21no—
169C22⋊D283—41no—
170C22⋊D283—41no—
171C22⋊D283—41no—
172C22⋊D283—41no—
173C2×Dic283—21no—
174C2×Dic283—41no—
175C4×D283—21no—
176C4×D283—21no—
177C4×D283—41no—
178C4×D283—41no—
179C4×D283—41no—
180C4×D283—41no—
181C283⋊C8—21no—
182C283⋊C8—21no—
183Q8×C283C2×D411yescharacteristic
184Q8×C283C2×D411yescharacteristic
185D4×C283—21no—
186D4×C283—21no—
187D4×C283C2×D411yescharacteristic
188D4×C283C2×D411yescharacteristic
189C2×C4×C283—21no—
190C2×C4×C283—21no—
191C2×C4×C283—21no—
192C2×C4×C283C2411yescharacteristic
193C8×C283—21no—
194C8×C283—21no—
195C8×C283C2×D411yescharacteristic
196C8×C283C2×D411yescharacteristic
197C8×C283C2×D411yescharacteristic
198C8×C283C2×D411yescharacteristic
199C22×C283—21no—
200C22×C283—21no—
201C22×C283H(5,𝔽2)11yescharacteristic
202C2×D283—21no—
203C2×D283—21no—
204C2×D283—41no—
205C2×D283—41no—
206C2×D283—41no—
207C2×D283—81no—
208C2×D283—81no—
209C2×D283—81no—
210C2×D283—161no—
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×C283D4211yescharacteristic
225D283—41no—
226D283—41no—
227D283—81no—
228D283—81no—
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#453—2831no—
245#444—2831no—
246#444D28311yescharacteristic, complemented
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)—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)—5661no—
266AGL(1,ℤ/8)—5661no—
267#135—2831no—
268#135—5661no—
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—2831no—
278C2×D8—2831no—
279C2×D8—5661no—
280C2×D8—5661no—
281#131—2831no—
282#131—2831no—
283#131C2×D28311yescharacteristic, complemented
284#131C2×D28311yescharacteristic, complemented
285#127—2831no—
286#124—2831no—
287#108—2831no—
288#108C2×D28311yescharacteristic, complemented
289#104—2831no—
290#104—2831no—
291#104C2×D28311yescharacteristic, complemented
292#104C2×D28311yescharacteristic, complemented
293#101—2831no—
294#100—2831no—
295#100—2831no—
296#100—2831no—
297#97C2×D28311yescharacteristic, complemented
298#41—2831no—
299#41—2831no—
300#41—5661no—
301#41—5661no—
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—2831no—
315C2×D4—2831no—
316C2×D4—2831no—
317C2×D4—5661no—
318C2×D4—5661no—
319C2×D4—5661no—
320C2×D4—5661no—
321C2×D4—5661no—
322C2×D4—5661no—
323C2×D4—11321no—
324C2×D4—11321no—
325C2×D4—11321no—
326C2×D4—11321no—
327Q16—2831no—
328Q16—5661no—
329Q16—5661no—
330Q16D113211yescharacteristic, complemented
331SD16—2831no—
332SD16—2831no—
333SD16—5661no—
334SD16—5661no—
335SD16—5661no—
336SD16—5661no—
337SD16—5661no—
338SD16—5661no—
339SD16—5661no—
340SD16—5661no—
341SD16—5661no—
342SD16—5661no—
343SD16—5661no—
344SD16—5661no—
345SD16D113211yescharacteristic, complemented
346SD16D113211yescharacteristic, complemented
347D8—2831no—
348D8—5661no—
349D8—5661no—
350D8—5661no—
351D8—5661no—
352D8—5661no—
353D8—5661no—
354D8—5661no—
355D8—5661no—
356D8—5661no—
357D8—5661no—
358D8D113211yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—2831no—
362M16—2831no—
363M16—5661no—
364M16—5661no—
365M16C22×D28311yescharacteristic
366M16C22×D28311yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—5661no—
370C2×C8—5661no—
371C2×C8C22×D28311yescharacteristic
372C2×C8C22×D28311yescharacteristic
373C22⋊C4—11321no—
374C42C22×D28311yescharacteristic
375C23—5661no—
376C23—5661no—
377C23—5661no—
378C23—11321no—
379C23—11321no—
380C23—11321no—
381Q8—2831no—
382Q8—2831no—
383Q8—5661no—
384Q8—5661no—
385Q8—11321no—
386Q8C2×D113211yescharacteristic
387Q8C2×D113211yescharacteristic
388D4—21no—
389D4—21no—
390D4—2831no—
391D4—2831no—
392D4—5661no—
393D4—5661no—
394D4—5661no—
395D4—5661no—
396D4—5661no—
397D4—5661no—
398D4—5661no—
399D4—5661no—
400D4—11321no—
401D4—11321no—
402D4—11321no—
403D4—11321no—
404D4—11321no—
405D4—11321no—
406D4—11321no—
407D4—22642no—
408D4C2×D113211yescharacteristic
409D4C2×D113211yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
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—5661no—
425C8—5661no—
426C8D4×D28311yescharacteristic
427C8D4×D28311yescharacteristic
428C8C2×D113211yescharacteristic
429C8C2×D113211yescharacteristic
430C22—21no—
431C22—21no—
432C22—5661no—
433C22—5661no—
434C22—11321no—
435C22—11321no—
436C22—11321no—
437C22—45281no—
438C22—45281no—
439C22—45281no—
440C22—22641no—
441C22—22641no—
442C22—22641no—
443C22#2258365611yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—5661no—
448C4—5661no—
449C4—11321no—
450C4C2×D4×D28311yescharacteristic
451C4C22×D113211yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—11321no—
456C2—11321no—
457C2—22641no—
458C2—22641no—
459C2—22641no—
460C2D4×D113211yescharacteristic
subgroupG/H
centerC2D4×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#444×C283C2
socleC2×C283D42
hypercenter#444D283
hypocenterC283#3288
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #31827277C2semidirect
2. #31827272C2semidirect
3. #31827270C2semidirect
4. #31827267C2semidirect
5. #31826854C2semidirect
6. #31826736C2semidirect
7. #31826735C2semidirect
8. #31826729C2semidirect
9. #31826728C2semidirect
10. #31826644C2semidirect
11. #31826639C2semidirect
12. #31826634C2semidirect
13. #31826634C2semidirect
14. #31826588C2semidirect
15. #444×C283C2semidirect
16. #22583657C22semidirect
18. #22583656C22semidirect
19. #22583656C22semidirect
21. #22583643C22semidirect
23. #22583640C22semidirect
28. #22583638C22semidirect
29. #22583638C22semidirect
33. #22583635C22semidirect
34. #22583635C22semidirect
42. C2×#17985381C22semidirect
43. C2×#17985381C22semidirect
46. C2×D2264C22semidirect
47. C2×D2264C22semidirect
49. #22583616C22semidirect
50. #22583616C22semidirect
51. #22583607C22semidirect
52. #22583598C22semidirect
53. #22583546C22semidirect
54. #22583538C22semidirect
55. #22583538C22semidirect
56. #22583528C22semidirect
57. #22583526C22semidirect
58. #22583526C22semidirect
59. #22583526C22semidirect
61. #131×C283C22semidirect
62. #131×C283C22semidirect
63. #108×C283C22semidirect
64. #104×C283C22semidirect
65. #104×C283C22semidirect
66. #97×C283C22semidirect
95. Dic1132D4semidirect
105. #17985381D4semidirect
106. #17985381D4semidirect
118. D2264D4semidirect
131. Q16×C283D4semidirect
132. SD16×C283D4semidirect
133. SD16×C283D4semidirect
134. D8×C283D4semidirect
230. C283#3288semidirect
246. #444D283semidirect
283. #131C2×D283semidirect
284. #131C2×D283semidirect
288. #108C2×D283semidirect
291. #104C2×D283semidirect
292. #104C2×D283semidirect
297. #97C2×D283semidirect
330. Q16D1132semidirect
345. SD16D1132semidirect
346. SD16D1132semidirect
358. D8D1132semidirect

Maximal subgroups

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

subgroupindex
#318272772
#318272722
#318272702
#318272672
#318268542
#318267362
#318267352
#318267292
#318267282
#318266442
#318266392
#318266342
#318265882
#444×C2832
#3288283