Finite Group Databasebeta

← #27182896

Subgroups — #27182896

Order 12928.

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#20265988C211yesmaximal, characteristic, complemented
2#20265984C211yesmaximal, characteristic, complemented
3#20265980C211yesmaximal, characteristic, complemented
4#20265565C211yesmaximal, characteristic, complemented
5#20265445C211yesmaximal, characteristic, complemented
6#20265444C211yesmaximal, characteristic, complemented
7#20265440C211yesmaximal, characteristic, complemented
8#20265438C211yesmaximal, characteristic, complemented
9#20265355C211yesmaximal, characteristic, complemented
10#20265348C211yesmaximal, characteristic, complemented
11#20265345C211yesmaximal, characteristic, complemented
12#20265343C211yesmaximal, characteristic, complemented
13#20265299C211yesmaximal, characteristic, complemented
14#20265297C211yesmaximal, characteristic, complemented
15#576×C101C211yesmaximal, characteristic, complemented
16#15692302C2211yescharacteristic, complemented
17#15692301C2211yescharacteristic, complemented
18#15692287—21no—
19#15692285—21no—
20#15692285C2211yescharacteristic, complemented
21#15692285C2211yescharacteristic, complemented
22#15692281—21no—
23#15692281—21no—
24#15692281C2211yescharacteristic, complemented
25#15692281C2211yescharacteristic, complemented
26#15692280—21no—
27#15692280C2211yescharacteristic
28#15692280C2211yescharacteristic, complemented
29#15692279—21no—
30#15692276—21no—
31#15692275—21no—
32#15692275—21no—
33#15692275C2211yescharacteristic
34C2×#13108063C2211yescharacteristic, complemented
35C2×#13108062—21no—
36C2×#13108062C2211yescharacteristic, complemented
37SD16×D101—21no—
38C2×#13108059—21no—
39C2×#13108059C2211yescharacteristic, complemented
40#15692262C2211yescharacteristic, complemented
41#15692262C2211yescharacteristic, complemented
42#15692261C2211yescharacteristic, complemented
43#15692251C2211yescharacteristic, complemented
44#15692242C2211yescharacteristic, complemented
45#15692191C2211yescharacteristic, complemented
46#15692190C2211yescharacteristic, complemented
47#15692182C2211yescharacteristic, complemented
48#15692182C2211yescharacteristic, complemented
49#15692182C2211yescharacteristic, complemented
50#15692181C2211yescharacteristic, complemented
51#15692173C2211yescharacteristic, complemented
52#15692172C2211yescharacteristic, complemented
53#15692172C2211yescharacteristic, complemented
54#15692171C2211yescharacteristic, complemented
55#15692165C2211yescharacteristic, complemented
56H(5,𝔽2)×C101C2211yescharacteristic
57H(5,𝔽2)×C101C2211yescharacteristic, complemented
58AGL(1,ℤ/8)×C101—21no—
59AGL(1,ℤ/8)×C101—21no—
60AGL(1,ℤ/8)×C101C2211yescharacteristic
61AGL(1,ℤ/8)×C101C2211yescharacteristic, complemented
62#135×C101—21no—
63#135×C101C2211yescharacteristic
64C2×D8×C101—21no—
65C2×D8×C101C2211yescharacteristic, complemented
66#131×C101C2211yescharacteristic, complemented
67#13108075—21no—
68#13108074—21no—
69#13108074—21no—
70#13108074C2311yescharacteristic
71#13108073—21no—
72#13108073—41no—
73C2×Dic202—21no—
74C2×Dic202C2311yescharacteristic
75Q8×D101—21no—
76C2×D404—21no—
77C2×D404C2311yescharacteristic
78C2×C22⋊D101—41no—
79D4×D101—21no—
80D4×D101—41no—
81#13108064—21no—
82#13108064—21no—
83#13108064D411yescharacteristic, complemented
84#13108063—21no—
85#13108063—21no—
86#13108062—21no—
87#13108062—21no—
88#13108062—21no—
89#13108062—21no—
90#13108062—21no—
91#13108062—21no—
92#13108062D411yescharacteristic, complemented
93#13108061—21no—
94#13108061—21no—
95#13108061D411yescharacteristic, complemented
96#13108059—21no—
97#13108059—21no—
98#13108059—21no—
99#13108059—21no—
100#13108059—21no—
101#13108059—21no—
102#13108059D411yescharacteristic, complemented
103#13108058—21no—
104#13108058—21no—
105#13108058C2311yescharacteristic
106#13108058C2311yescharacteristic
107#13108058C2311yescharacteristic
108#13108057—21no—
109C2×#11717579—21no—
110C2×#11717579—21no—
111C2×#11717579C2311yescharacteristic
112C8×D101—21no—
113C22⋊Dic101—41no—
114C4×Dic101C2311yescharacteristic
115#41×C101—21no—
116#41×C101—21no—
117#41×C101—21no—
118#41×C101—21no—
119#41×C101C2311yescharacteristic
120#41×C101C2311yescharacteristic
121#41×C101C2311yescharacteristic
122C2×D4×C101—21no—
123C2×D4×C101—21no—
124C2×D4×C101—21no—
125C2×D4×C101—21no—
126C2×D4×C101—41no—
127C2×D4×C101—41no—
128C2×D4×C101C2311yescharacteristic
129C2×D4×C101C2311yescharacteristic
130Q16×C101D411yescharacteristic, complemented
131SD16×C101—21no—
132SD16×C101—21no—
133SD16×C101D411yescharacteristic, complemented
134SD16×C101D411yescharacteristic, complemented
135D8×C101—21no—
136D8×C101—21no—
137D8×C101—21no—
138D8×C101—21no—
139D8×C101D411yescharacteristic, complemented
140M16×C101—21no—
141M16×C101C2311yescharacteristic
142C2×C8×C101—21no—
143C2×C8×C101C2311yescharacteristic
144C23×C101—21no—
145C23×C101—21no—
146C23×C101—41no—
147C23×C101—41no—
148C22×D101—21no—
149C22×D101—41no—
150Q8×C101C2×D411yescharacteristic
151Q8×C101C2×D411yescharacteristic
152Q8×C101C2×D411yescharacteristic
153D4×C101—21no—
154D4×C101—21no—
155D4×C101—21no—
156D4×C101—21no—
157D4×C101—21no—
158D4×C101—21no—
159D4×C101—21no—
160D4×C101—41no—
161D4×C101—41no—
162D4×C101—41no—
163D4×C101—41no—
164D4×C101C2×D411yescharacteristic
165D4×C101C2×D411yescharacteristic
166D4×C101C2×D411yescharacteristic
167C2×C4×C101—21no—
168C2×C4×C101—21no—
169C2×C4×C101—21no—
170C2×C4×C101—41no—
171C2×C4×C101—41no—
172C2×C4×C101C2411yescharacteristic
173C22⋊D101—21no—
174C22⋊D101—41no—
175C22⋊D101—41no—
176C22⋊D101—82no—
177C2×Dic101—21no—
178C2×Dic101—21no—
179C2×Dic101—41no—
180D404—21no—
181D404—21no—
182D404—41no—
183D404C2×D411yescharacteristic
184C4×D101—21no—
185C4×D101—41no—
186C4×D101—41no—
187Dic202—21no—
188Dic202—21no—
189Dic202—41no—
190Dic202C2×D411yescharacteristic
191C8×C101—21no—
192C8×C101C2×D411yescharacteristic
193C8×C101C2×D411yescharacteristic
194#11717579—21no—
195#11717579—21no—
196#11717579—21no—
197#11717579C2×D411yescharacteristic
198#11717579C2×D411yescharacteristic
199C22×C101—21no—
200C22×C101—21no—
201C22×C101—21no—
202C22×C101—41no—
203C22×C101—41no—
204C22×C101—81no—
205C22×C101—81no—
206C22×C101—161no—
207C22×C101—161no—
208C22×C101H(5,𝔽2)11yescharacteristic
209C2×D101—21no—
210C2×D101—41no—
211C2×D101—81no—
212C2×D101—161no—
213C4×C101—21no—
214C4×C101—21no—
215C4×C101—21no—
216C4×C101C22×D411yescharacteristic
217C4×C101C22×D411yescharacteristic
218Dic101—21no—
219Dic101—41no—
220Dic101—41no—
221C2×C101—21no—
222C2×C101—41no—
223C2×C101—41no—
224C2×C101—41no—
225C2×C101—81no—
226C2×C101—81no—
227C2×C101D4211yescharacteristic
228D101—41no—
229D101—81no—
230#3288—1011nomaximal, Sylow
231C101#328811yescharacteristic, complemented, Sylow
232#577—1011no—
233#577—1011no—
234#576—1011no—
235#576D10111yescharacteristic, complemented
236#496—1011no—
237#472—1011no—
238#472—1011no—
239#471—1011no—
240#471—1011no—
241#455—1011no—
242#454—1011no—
243#453—1011no—
244#453—1011no—
245#444—1011no—
246#444—1011no—
2472−1+4—1011no—
248H(5,𝔽2)—1011no—
249H(5,𝔽2)C2×D10111yescharacteristic
250H(5,𝔽2)C2×D10111yescharacteristic, complemented
251#137—1011no—
252#137—1011no—
253#137—2021no—
254#137—2021no—
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—1011no—
258AGL(1,ℤ/8)—1011no—
259AGL(1,ℤ/8)—1011no—
260AGL(1,ℤ/8)—1011no—
261AGL(1,ℤ/8)—2021no—
262AGL(1,ℤ/8)—2021no—
263AGL(1,ℤ/8)—2021no—
264AGL(1,ℤ/8)—2021no—
265AGL(1,ℤ/8)C2×D10111yescharacteristic
266AGL(1,ℤ/8)C2×D10111yescharacteristic, complemented
267#135—21no—
268#135—1011no—
269#135—2021no—
270#135—2021no—
271#135—2021no—
272#135C2×D10111yescharacteristic
273C2×SD16—1011no—
274C2×SD16—1011no—
275C2×SD16—2021no—
276C2×SD16—2021no—
277C2×D8—21no—
278C2×D8—1011no—
279C2×D8—2021no—
280C2×D8C2×D10111yescharacteristic, complemented
281#131—1011no—
282#131—1011no—
283#131—1011no—
284#131C2×D10111yescharacteristic, complemented
285#127—1011no—
286#124—1011no—
287#108—1011no—
288#108—1011no—
289#104—1011no—
290#104—1011no—
291#104—1011no—
292#104—1011no—
293#101—1011no—
294#100—1011no—
295#100—1011no—
296#100—1011no—
297#97—1011no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—1011no—
303#41—4041no—
304#41—2021no—
305#41—2021no—
306#41—2021no—
307#41—2021no—
308#41C22×D10111yescharacteristic
309#41C22×D10111yescharacteristic
310#41C22×D10111yescharacteristic
311C2×Q8—1011no—
312C2×Q8—2021no—
313C2×Q8—2021no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—1011no—
321C2×D4—4041no—
322C2×D4—4041no—
323C2×D4—2021no—
324C2×D4—2021no—
325C2×D4C22×D10111yescharacteristic
326C2×D4C22×D10111yescharacteristic
327Q16—1011no—
328Q16—2021no—
329Q16—2021no—
330Q16C22⋊D10111yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—1011no—
334SD16—1011no—
335SD16—2021no—
336SD16—2021no—
337SD16—2021no—
338SD16—2021no—
339SD16—2021no—
340SD16—2021no—
341SD16—2021no—
342SD16—2021no—
343SD16—2021no—
344SD16—2021no—
345SD16C22⋊D10111yescharacteristic, complemented
346SD16C22⋊D10111yescharacteristic, complemented
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—1011no—
352D8—2021no—
353D8—2021no—
354D8—2021no—
355D8—2021no—
356D8—2021no—
357D8—2021no—
358D8C22⋊D10111yescharacteristic, complemented
359M16—21no—
360M16—1011no—
361M16—1011no—
362M16—1011no—
363M16—2021no—
364M16—2021no—
365M16—2021no—
366M16C22×D10111yescharacteristic
367C2×C8—21no—
368C2×C8—1011no—
369C2×C8—2021no—
370C2×C8—2021no—
371C2×C8—2021no—
372C2×C8C22×D10111yescharacteristic
373C22⋊C4—4041no—
374C42—1011no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—4041no—
380C23—2021no—
381Q8—1011no—
382Q8—4041no—
383Q8—2021no—
384Q8—2021no—
385Q8D4×D10111yescharacteristic
386Q8C2×C22⋊D10111yescharacteristic
387Q8C2×C22⋊D10111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—21no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—8082no—
400D4—1011no—
401D4—4041no—
402D4—4041no—
403D4—4041no—
404D4—2021no—
405D4—2021no—
406D4—2021no—
407D4D4×D10111yescharacteristic
408D4C2×C22⋊D10111yescharacteristic
409D4C2×C22⋊D10111yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—4041no—
416C2×C4—4041no—
417C2×C4—4041no—
418C2×C4—2021no—
419C2×C4—2021no—
420C2×C4—2021no—
421C2×C4C23×D10111yescharacteristic
422C8—21no—
423C8—1011no—
424C8—1011no—
425C8—2021no—
426C8—2021no—
427C8—2021no—
428C8C2×C22⋊D10111yescharacteristic
429C8C2×C22⋊D10111yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22—161no—
438C22—161no—
439C22—8081no—
440C22—16161no—
441C22—4041no—
442C22—2021no—
443C22#1569230011yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—4041no—
448C4—4041no—
449C4—2021no—
450C4C2×D4×D10111yescharacteristic
451C4C22×C22⋊D10111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—8081no—
459C2—4041no—
460C2D4×C22⋊D10111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#576×C101C2
socleC2×C101D42
hypercenter#576D101
hypocenterC101#3288
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #20265988C2semidirect
2. #20265984C2semidirect
3. #20265980C2semidirect
4. #20265565C2semidirect
5. #20265445C2semidirect
6. #20265444C2semidirect
7. #20265440C2semidirect
8. #20265438C2semidirect
9. #20265355C2semidirect
10. #20265348C2semidirect
11. #20265345C2semidirect
12. #20265343C2semidirect
13. #20265299C2semidirect
14. #20265297C2semidirect
15. #576×C101C2semidirect
16. #15692302C22semidirect
17. #15692301C22semidirect
20. #15692285C22semidirect
21. #15692285C22semidirect
24. #15692281C22semidirect
25. #15692281C22semidirect
28. #15692280C22semidirect
34. C2×#13108063C22semidirect
36. C2×#13108062C22semidirect
39. C2×#13108059C22semidirect
40. #15692262C22semidirect
41. #15692262C22semidirect
42. #15692261C22semidirect
43. #15692251C22semidirect
44. #15692242C22semidirect
45. #15692191C22semidirect
46. #15692190C22semidirect
47. #15692182C22semidirect
48. #15692182C22semidirect
49. #15692182C22semidirect
50. #15692181C22semidirect
51. #15692173C22semidirect
52. #15692172C22semidirect
53. #15692172C22semidirect
54. #15692171C22semidirect
55. #15692165C22semidirect
57. H(5,𝔽2)×C101C22semidirect
61. AGL(1,ℤ/8)×C101C22semidirect
65. C2×D8×C101C22semidirect
66. #131×C101C22semidirect
83. #13108064D4semidirect
92. #13108062D4semidirect
95. #13108061D4semidirect
102. #13108059D4semidirect
130. Q16×C101D4semidirect
133. SD16×C101D4semidirect
134. SD16×C101D4semidirect
139. D8×C101D4semidirect
231. C101#3288semidirect
235. #576D101semidirect
250. H(5,𝔽2)C2×D101semidirect
266. AGL(1,ℤ/8)C2×D101semidirect
280. C2×D8C2×D101semidirect
284. #131C2×D101semidirect
330. Q16C22⋊D101semidirect
345. SD16C22⋊D101semidirect
346. SD16C22⋊D101semidirect
358. D8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202659882
#202659842
#202659802
#202655652
#202654452
#202654442
#202654402
#202654382
#202653552
#202653482
#202653452
#202653432
#202652992
#202652972
#576×C1012
#3288101