Finite Group Databasebeta

← #19146362

Subgroups — #19146362

Order 5504.

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#14583711C211yesmaximal, characteristic, complemented
2#14583706C211yesmaximal, characteristic, complemented
3#14583704C211yesmaximal, characteristic, complemented
4#14583701C211yesmaximal, characteristic, complemented
5#14583288C211yesmaximal, characteristic, complemented
6#14583170C211yesmaximal, characteristic, complemented
7#14583169C211yesmaximal, characteristic, complemented
8#14583163C211yesmaximal, characteristic, complemented
9#14583162C211yesmaximal, characteristic, complemented
10#14583078C211yesmaximal, characteristic, complemented
11#14583073C211yesmaximal, characteristic, complemented
12#14583068C211yesmaximal, characteristic, complemented
13#14583068C211yesmaximal, characteristic, complemented
14#14583022C211yesmaximal, characteristic, complemented
15C43×#444C211yesmaximal, characteristic, complemented
16#13063372C2211yescharacteristic, complemented
17#13063371C2211yescharacteristic
18#13063371C2211yescharacteristic, complemented
19#13063371C2211yescharacteristic, complemented
20#13063358—21no—
21#13063358C2211yescharacteristic, complemented
22#13063357—21no—
23#13063355C2211yescharacteristic, complemented
24#13063353—21no—
25#13063353—21no—
26#13063353—21no—
27#13063353C2211yescharacteristic
28#13063353C2211yescharacteristic, complemented
29#13063353C2211yescharacteristic, complemented
30#13063352—21no—
31#13063352—21no—
32#13063350C2211yescharacteristic
33#13063350C2211yescharacteristic, complemented
34#13063350C2211yescharacteristic, complemented
35#13063349—21no—
36#13063348—21no—
37#13063347—21no—
38#13063347—21no—
39#13063347C2211yescharacteristic
40#13063346—21no—
41C2×#10621873—21no—
42C2×#10621873C2211yescharacteristic, complemented
43C2×#10621873C2211yescharacteristic, complemented
44SD16×D43—21no—
45C2×D344—21no—
46C2×D344C2211yescharacteristic, complemented
47C2×D344C2211yescharacteristic, complemented
48D8×D43—21no—
49#13063331C2211yescharacteristic, complemented
50#13063331C2211yescharacteristic, complemented
51#13063322C2211yescharacteristic, complemented
52#13063313C2211yescharacteristic, complemented
53#13063261C2211yescharacteristic, complemented
54#13063253C2211yescharacteristic, complemented
55#13063253C2211yescharacteristic, complemented
56#13063243C2211yescharacteristic, complemented
57#13063241C2211yescharacteristic, complemented
58#13063241C2211yescharacteristic, complemented
59#13063241C2211yescharacteristic, complemented
60#135×C43C2211yescharacteristic
61#131×C43C2211yescharacteristic, complemented
62#131×C43C2211yescharacteristic, complemented
63#108×C43C2211yescharacteristic, complemented
64#104×C43C2211yescharacteristic, complemented
65#104×C43C2211yescharacteristic, complemented
66#97×C43C2211yescharacteristic, complemented
67#41×C43C2311yescharacteristic
68#41×C43C2311yescharacteristic
69#10621902—21no—
70#10621902—21no—
71#10621902—21no—
72Q8×D43—21no—
73#10621900—21no—
74#10621900—41no—
75D4×D43—21no—
76D4×D43—21no—
77D4×D43—21no—
78D4×D43—41no—
79D4×D43—41no—
80D4×D43—41no—
81#10621898—21no—
82#10621898—21no—
83#10621898—21no—
84#10621898—21no—
85#10621898C2311yescharacteristic
86#10621898C2311yescharacteristic
87C2×D172—21no—
88C2×D172—21no—
89C2×D172—21no—
90C2×D172—41no—
91C2×D172C2311yescharacteristic
92C2×D172C2311yescharacteristic
93C2×D172C2311yescharacteristic
94C2×Dic86—21no—
95C2×Dic86C2311yescharacteristic
96Q16×C43D411yescharacteristic, complemented
97SD16×C43D411yescharacteristic, complemented
98SD16×C43D411yescharacteristic, complemented
99D8×C43D411yescharacteristic, complemented
100M16×C43—21no—
101M16×C43—21no—
102M16×C43C2311yescharacteristic
103M16×C43C2311yescharacteristic
104C2×C8×C43—21no—
105C2×C8×C43—21no—
106C2×C8×C43C2311yescharacteristic
107C2×C8×C43C2311yescharacteristic
108C42×C43C2311yescharacteristic
109#10621885—21no—
110#10621884—21no—
111#10621884—21no—
112#10621884—21no—
113#10621883—21no—
114#10621882—21no—
115#10621882—21no—
116#10621882—21no—
117(C2×D43)⋊C4—41no—
118#10621877C2311yescharacteristic
119#10621877C2311yescharacteristic
120Dic172—21no—
121Dic172D411yescharacteristic, complemented
122D344—21no—
123D344—21no—
124D344—21no—
125D344—21no—
126D344—21no—
127D344—21no—
128D344—21no—
129D344D411yescharacteristic, complemented
130#10621873—21no—
131#10621873—21no—
132#10621873—21no—
133#10621873—21no—
134#10621873—21no—
135#10621873—21no—
136#10621873—21no—
137#10621873—21no—
138#10621873D411yescharacteristic, complemented
139#10621873D411yescharacteristic, complemented
140#10621872—21no—
141#10621872—21no—
142C8×D43—21no—
143C8×D43—21no—
144C22×D43—21no—
145C22×D43—21no—
146C22×D43—21no—
147C22×D43—41no—
148C22×D43—41no—
149C22×D43—41no—
150Q8×C43C2×D411yescharacteristic
151Q8×C43C2×D411yescharacteristic
152D4×C43—21no—
153D4×C43—21no—
154D4×C43C2×D411yescharacteristic
155D4×C43C2×D411yescharacteristic
156C2×C4×C43—21no—
157C2×C4×C43—21no—
158C2×C4×C43—21no—
159C2×C4×C43C2411yescharacteristic
160C22⋊D43—21no—
161C22⋊D43—21no—
162C22⋊D43—41no—
163C22⋊D43—41no—
164C22⋊D43—41no—
165C22⋊D43—41no—
166C2×Dic43—21no—
167C2×Dic43—41no—
168D172—21no—
169D172—21no—
170D172—21no—
171D172—21no—
172D172—21no—
173D172—21no—
174D172—41no—
175D172—41no—
176D172—41no—
177D172—82no—
178D172C2×D411yescharacteristic
179D172C2×D411yescharacteristic
180C4×D43—21no—
181C4×D43—21no—
182C4×D43—41no—
183C4×D43—41no—
184C4×D43—41no—
185C4×D43—41no—
186Dic86—21no—
187Dic86—21no—
188Dic86—41no—
189Dic86C2×D411yescharacteristic
190Dic86C2×D411yescharacteristic
191C8×C43—21no—
192C8×C43—21no—
193C8×C43C2×D411yescharacteristic
194C8×C43C2×D411yescharacteristic
195C8×C43C2×D411yescharacteristic
196C8×C43C2×D411yescharacteristic
197C43⋊C8—21no—
198C43⋊C8—21no—
199C22×C43—21no—
200C22×C43—21no—
201C22×C43H(5,𝔽2)11yescharacteristic
202C2×D43—21no—
203C2×D43—21no—
204C2×D43—41no—
205C2×D43—41no—
206C2×D43—41no—
207C2×D43—81no—
208C2×D43—81no—
209C2×D43—81no—
210C2×D43—161no—
211C2×D43—161no—
212C2×D43—161no—
213C4×C43—21no—
214C4×C43—21no—
215C4×C43—41no—
216C4×C43C22×D411yescharacteristic
217C4×C43C22×D411yescharacteristic
218Dic43—21no—
219Dic43—21no—
220Dic43—41no—
221#3288—431nomaximal, Sylow
222C2×C43—21no—
223C2×C43—41no—
224C2×C43—41no—
225C2×C43D4211yescharacteristic
226D43—41no—
227D43—41no—
228D43—81no—
229D43—81no—
230D43—81no—
231#577—431no—
232#577—431no—
233#576—431no—
234#576—431no—
235#496—431no—
236#472—431no—
237#472—431no—
238#471—431no—
239#471—431no—
240#455—431no—
241#454—431no—
242#453—431no—
243#453—431no—
244#444—431no—
245#444D4311yescharacteristic, complemented
246C43#328811yescharacteristic, complemented, Sylow
2472−1+4—431no—
248H(5,𝔽2)—431no—
249H(5,𝔽2)—431no—
250H(5,𝔽2)—431no—
251#137—431no—
252#137—431no—
253#137—861no—
254#137—861no—
255AGL(1,ℤ/8)—431no—
256AGL(1,ℤ/8)—431no—
257AGL(1,ℤ/8)—431no—
258AGL(1,ℤ/8)—431no—
259AGL(1,ℤ/8)—431no—
260AGL(1,ℤ/8)—431no—
261AGL(1,ℤ/8)—861no—
262AGL(1,ℤ/8)—861no—
263AGL(1,ℤ/8)—861no—
264AGL(1,ℤ/8)—861no—
265AGL(1,ℤ/8)—861no—
266AGL(1,ℤ/8)—861no—
267#135—431no—
268#135—861no—
269#135—861no—
270#135—861no—
271#135—861no—
272#135C2×D4311yescharacteristic
273C2×SD16—431no—
274C2×SD16—431no—
275C2×SD16—861no—
276C2×SD16—861no—
277C2×D8—431no—
278C2×D8—431no—
279C2×D8—861no—
280C2×D8—861no—
281#131—431no—
282#131—431no—
283#131C2×D4311yescharacteristic, complemented
284#131C2×D4311yescharacteristic, complemented
285#127—431no—
286#124—431no—
287#108—431no—
288#108C2×D4311yescharacteristic, complemented
289#104—431no—
290#104—431no—
291#104C2×D4311yescharacteristic, complemented
292#104C2×D4311yescharacteristic, complemented
293#101—431no—
294#100—431no—
295#100—431no—
296#100—431no—
297#97C2×D4311yescharacteristic, complemented
298#41—431no—
299#41—431no—
300#41—861no—
301#41—861no—
302#41—861no—
303#41—861no—
304#41—861no—
305#41—861no—
306#41—861no—
307#41—861no—
308#41—1721no—
309#41C22×D4311yescharacteristic
310#41C22×D4311yescharacteristic
311C2×Q8—431no—
312C2×Q8—861no—
313C2×Q8—861no—
314C2×D4—431no—
315C2×D4—431no—
316C2×D4—431no—
317C2×D4—861no—
318C2×D4—861no—
319C2×D4—861no—
320C2×D4—861no—
321C2×D4—861no—
322C2×D4—861no—
323C2×D4—1721no—
324C2×D4—1721no—
325C2×D4—1721no—
326C2×D4—1721no—
327Q16—431no—
328Q16—861no—
329Q16—861no—
330Q16D17211yescharacteristic, complemented
331SD16—431no—
332SD16—431no—
333SD16—861no—
334SD16—861no—
335SD16—861no—
336SD16—861no—
337SD16—861no—
338SD16—861no—
339SD16—861no—
340SD16—861no—
341SD16—861no—
342SD16—861no—
343SD16—861no—
344SD16—861no—
345SD16D17211yescharacteristic, complemented
346SD16D17211yescharacteristic, complemented
347D8—431no—
348D8—861no—
349D8—861no—
350D8—861no—
351D8—861no—
352D8—861no—
353D8—861no—
354D8—861no—
355D8—861no—
356D8—861no—
357D8—861no—
358D8D17211yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—431no—
362M16—431no—
363M16—861no—
364M16—861no—
365M16C22×D4311yescharacteristic
366M16C22×D4311yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—861no—
370C2×C8—861no—
371C2×C8C22×D4311yescharacteristic
372C2×C8C22×D4311yescharacteristic
373C22⋊C4—1721no—
374C42C22×D4311yescharacteristic
375C23—861no—
376C23—861no—
377C23—861no—
378C23—1721no—
379C23—1721no—
380C23—1721no—
381Q8—431no—
382Q8—431no—
383Q8—861no—
384Q8—861no—
385Q8—1721no—
386Q8C2×D17211yescharacteristic
387Q8C2×D17211yescharacteristic
388D4—21no—
389D4—21no—
390D4—431no—
391D4—431no—
392D4—861no—
393D4—861no—
394D4—861no—
395D4—861no—
396D4—861no—
397D4—861no—
398D4—861no—
399D4—861no—
400D4—3442no—
401D4—1721no—
402D4—1721no—
403D4—1721no—
404D4—1721no—
405D4—1721no—
406D4—1721no—
407D4—1721no—
408D4C2×D17211yescharacteristic
409D4C2×D17211yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—861no—
414C2×C4—861no—
415C2×C4—861no—
416C2×C4—1721no—
417C2×C4—1721no—
418C2×C4—1721no—
419C2×C4—1721no—
420C2×C4—1721no—
421C2×C4C23×D4311yescharacteristic
422C8—21no—
423C8—21no—
424C8—861no—
425C8—861no—
426C8C2×D17211yescharacteristic
427C8C2×D17211yescharacteristic
428C8D4×D4311yescharacteristic
429C8D4×D4311yescharacteristic
430C22—21no—
431C22—21no—
432C22—861no—
433C22—861no—
434C22—3441no—
435C22—3441no—
436C22—3441no—
437C22—1721no—
438C22—1721no—
439C22—1721no—
440C22—6881no—
441C22—6881no—
442C22—6881no—
443C22#1306337111yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—861no—
448C4—861no—
449C4—1721no—
450C4C2×D4×D4311yescharacteristic
451C4C22×D17211yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—3441no—
456C2—3441no—
457C2—3441no—
458C2—1721no—
459C2—1721no—
460C2D4×D17211yescharacteristic
subgroupG/H
centerC2D4×D172
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#444C2
socleC2×C43D42
hypercenter#444D43
hypocenterC43#3288
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #14583711C2semidirect
2. #14583706C2semidirect
3. #14583704C2semidirect
4. #14583701C2semidirect
5. #14583288C2semidirect
6. #14583170C2semidirect
7. #14583169C2semidirect
8. #14583163C2semidirect
9. #14583162C2semidirect
10. #14583078C2semidirect
11. #14583073C2semidirect
12. #14583068C2semidirect
13. #14583068C2semidirect
14. #14583022C2semidirect
15. C43×#444C2semidirect
16. #13063372C22semidirect
18. #13063371C22semidirect
19. #13063371C22semidirect
21. #13063358C22semidirect
23. #13063355C22semidirect
28. #13063353C22semidirect
29. #13063353C22semidirect
33. #13063350C22semidirect
34. #13063350C22semidirect
42. C2×#10621873C22semidirect
43. C2×#10621873C22semidirect
46. C2×D344C22semidirect
47. C2×D344C22semidirect
49. #13063331C22semidirect
50. #13063331C22semidirect
51. #13063322C22semidirect
52. #13063313C22semidirect
53. #13063261C22semidirect
54. #13063253C22semidirect
55. #13063253C22semidirect
56. #13063243C22semidirect
57. #13063241C22semidirect
58. #13063241C22semidirect
59. #13063241C22semidirect
61. #131×C43C22semidirect
62. #131×C43C22semidirect
63. #108×C43C22semidirect
64. #104×C43C22semidirect
65. #104×C43C22semidirect
66. #97×C43C22semidirect
96. Q16×C43D4semidirect
97. SD16×C43D4semidirect
98. SD16×C43D4semidirect
99. D8×C43D4semidirect
121. Dic172D4semidirect
129. D344D4semidirect
138. #10621873D4semidirect
139. #10621873D4semidirect
245. #444D43semidirect
246. C43#3288semidirect
283. #131C2×D43semidirect
284. #131C2×D43semidirect
288. #108C2×D43semidirect
291. #104C2×D43semidirect
292. #104C2×D43semidirect
297. #97C2×D43semidirect
330. Q16D172semidirect
345. SD16D172semidirect
346. SD16D172semidirect
358. D8D172semidirect