Finite Group Databasebeta

← #27182923

Subgroups — #27182923

Order 12928.

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#20265988C211yesmaximal, characteristic, complemented
2#20265985C211yesmaximal, characteristic, complemented
3#20265984C211yesmaximal, characteristic, complemented
4#20265548C211yesmaximal, characteristic, complemented
5#20265442C211yesmaximal, characteristic, complemented
6#20265440C211yesmaximal, characteristic, complemented
7#20265438C211yesmaximal, characteristic, complemented
8#20265437C211yesmaximal, characteristic, complemented
9#20265360C211yesmaximal, characteristic, complemented
10#20265358C211yesmaximal, characteristic, complemented
11#20265345C211yesmaximal, characteristic, complemented
12#20265343C211yesmaximal, characteristic, complemented
13#20265294C211yesmaximal, characteristic, complemented
14#20265292C211yesmaximal, characteristic, complemented
15#577×C101C211yesmaximal, characteristic, complemented
16#15692302C2211yescharacteristic
17#15692301C2211yescharacteristic, complemented
18#15692287—21no—
19#15692286—21no—
20#15692286C2211yescharacteristic, complemented
21#15692285—21no—
22#15692285C2211yescharacteristic, complemented
23#15692285C2211yescharacteristic, complemented
24#15692281—21no—
25#15692281C2211yescharacteristic
26#15692280—21no—
27#15692280C2211yescharacteristic
28#15692280C2211yescharacteristic
29#15692279—21no—
30#15692276—21no—
31#15692275—21no—
32#15692275—21no—
33#15692275C2211yescharacteristic
34C2×#13108063C2211yescharacteristic, complemented
35C2×#13108062—21no—
36C2×#13108062C2211yescharacteristic, complemented
37C2×#13108061—21no—
38C2×#13108061C2211yescharacteristic, complemented
39SD16×D101—21no—
40#15692262C2211yescharacteristic, complemented
41#15692262C2211yescharacteristic, complemented
42#15692261C2211yescharacteristic, complemented
43#15692253C2211yescharacteristic, complemented
44#15692251C2211yescharacteristic
45#15692191C2211yescharacteristic, complemented
46#15692190C2211yescharacteristic, complemented
47#15692182C2211yescharacteristic, complemented
48#15692182C2211yescharacteristic, complemented
49#15692182C2211yescharacteristic, complemented
50#15692181C2211yescharacteristic, complemented
51#15692174C2211yescharacteristic, complemented
52#15692173C2211yescharacteristic, complemented
53#15692172C2211yescharacteristic, complemented
54#15692171C2211yescharacteristic, complemented
55C8×Dic101C2211yescharacteristic, complemented
562−1+4×C101C2211yescharacteristic
57H(5,𝔽2)×C101C2211yescharacteristic, complemented
58#137×C101—21no—
59#137×C101C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C101—21no—
61AGL(1,ℤ/8)×C101C2211yescharacteristic
62#135×C101—21no—
63#135×C101C2211yescharacteristic
64C2×SD16×C101—21no—
65C2×SD16×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#13108064—21no—
84#13108064D411yescharacteristic, complemented
85#13108063—21no—
86#13108063—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#13108061—21no—
96#13108061—21no—
97#13108061—21no—
98#13108061D411yescharacteristic, complemented
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—
113#13108053—41no—
114C4×Dic101C2311yescharacteristic
115#41×C101—21no—
116#41×C101—21no—
117#41×C101—21no—
118#41×C101—21no—
119#41×C101—41no—
120#41×C101C2311yescharacteristic
121#41×C101C2311yescharacteristic
122#41×C101C2311yescharacteristic
123C2×Q8×C101—21no—
124C2×Q8×C101—21no—
125C2×Q8×C101C2311yescharacteristic
126C2×D4×C101—21no—
127C2×D4×C101—21no—
128C2×D4×C101—41no—
129C2×D4×C101C2311yescharacteristic
130Q16×C101—21no—
131Q16×C101D411yescharacteristic, complemented
132SD16×C101—21no—
133SD16×C101—21no—
134SD16×C101—21no—
135SD16×C101—21no—
136SD16×C101D411yescharacteristic, complemented
137SD16×C101D411yescharacteristic, complemented
138D8×C101—21no—
139D8×C101D411yescharacteristic, complemented
140M16×C101—21no—
141M16×C101C2311yescharacteristic
142C2×C8×C101—21no—
143C2×C8×C101C2311yescharacteristic
144C23×C101—21no—
145C23×C101—41no—
146C22×D101—21no—
147C22×D101—41no—
148Q8×C101—21no—
149Q8×C101—21no—
150Q8×C101—41no—
151Q8×C101C2×D411yescharacteristic
152Q8×C101C2×D411yescharacteristic
153Q8×C101C2×D411yescharacteristic
154D4×C101—21no—
155D4×C101—21no—
156D4×C101—21no—
157D4×C101—21no—
158D4×C101—21no—
159D4×C101—41no—
160D4×C101—41no—
161D4×C101—41no—
162D4×C101C2×D411yescharacteristic
163D4×C101C2×D411yescharacteristic
164D4×C101C2×D411yescharacteristic
165C2×C4×C101—21no—
166C2×C4×C101—21no—
167C2×C4×C101—21no—
168C2×C4×C101—21no—
169C2×C4×C101—41no—
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—81no—
204C22×C101—161no—
205C22×C101H(5,𝔽2)11yescharacteristic
206C2×D101—21no—
207C2×D101—41no—
208C2×D101—81no—
209C2×D101—161no—
210C4×C101—21no—
211C4×C101—21no—
212C4×C101—21no—
213C4×C101—41no—
214C4×C101C22×D411yescharacteristic
215C4×C101C22×D411yescharacteristic
216Dic101—22no—
217Dic101—21no—
218Dic101—41no—
219C2×C101—21no—
220C2×C101—41no—
221C2×C101—41no—
222C2×C101—41no—
223C2×C101—81no—
224C2×C101D4211yescharacteristic
225D101—41no—
226D101—81no—
227#3291—1011nomaximal, Sylow
228C101#329111yescharacteristic, complemented, Sylow
229#577—1011no—
230#577—1011no—
231#577—1011no—
232#577D10111yescharacteristic, complemented
233#492—1011no—
234#471—1011no—
235#471—1011no—
236#471—1011no—
237#471—1011no—
238#456—1011no—
239#456—1011no—
240#453—1011no—
241#453—1011no—
242#443—1011no—
243#443—1011no—
2442−1+4—1011no—
2452−1+4C2×D10111yescharacteristic
246H(5,𝔽2)—1011no—
247H(5,𝔽2)C2×D10111yescharacteristic, complemented
248#137—21no—
249#137—1011no—
250#137—1011no—
251#137—1011no—
252#137—2021no—
253#137—2021no—
254#137—2021no—
255#137C2×D10111yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—1011no—
258AGL(1,ℤ/8)—1011no—
259AGL(1,ℤ/8)—1011no—
260AGL(1,ℤ/8)—2021no—
261AGL(1,ℤ/8)—2021no—
262AGL(1,ℤ/8)—2021no—
263AGL(1,ℤ/8)C2×D10111yescharacteristic
264#135—21no—
265#135—1011no—
266#135—2021no—
267#135—2021no—
268#135—2021no—
269#135C2×D10111yescharacteristic
270C2×SD16—21no—
271C2×SD16—1011no—
272C2×SD16—1011no—
273C2×SD16—1011no—
274C2×SD16—2021no—
275C2×SD16—2021no—
276C2×SD16—2021no—
277C2×SD16C2×D10111yescharacteristic, complemented
278#131—1011no—
279#131—1011no—
280#131—1011no—
281#131C2×D10111yescharacteristic, complemented
282C4⋊Q8—1011no—
283#127—1011no—
284#108—1011no—
285#108—1011no—
286#104—1011no—
287#104—1011no—
288#104—1011no—
289#104—1011no—
290#101—1011no—
291#101—1011no—
292#100—1011no—
293#100—1011no—
294C4×C8—1011no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—1011no—
301#41—4041no—
302#41—2021no—
303#41—2021no—
304#41—2021no—
305#41—2021no—
306#41C22×D10111yescharacteristic
307#41C22×D10111yescharacteristic
308#41C22×D10111yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—1011no—
312C2×Q8—2021no—
313C2×Q8—2021no—
314C2×Q8C22×D10111yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—1011no—
319C2×D4—4041no—
320C2×D4—4041no—
321C2×D4—2021no—
322C2×D4—2021no—
323C2×D4C22×D10111yescharacteristic
324Q16—21no—
325Q16—1011no—
326Q16—2021no—
327Q16—2021no—
328Q16—2021no—
329Q16C22⋊D10111yescharacteristic, complemented
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—1011no—
335SD16—1011no—
336SD16—2021no—
337SD16—2021no—
338SD16—2021no—
339SD16—2021no—
340SD16—2021no—
341SD16—2021no—
342SD16—2021no—
343SD16—2021no—
344SD16—2021no—
345SD16—2021no—
346SD16—2021no—
347SD16—2021no—
348SD16C22⋊D10111yescharacteristic, complemented
349SD16C22⋊D10111yescharacteristic, complemented
350D8—21no—
351D8—1011no—
352D8—2021no—
353D8—2021no—
354D8—2021no—
355D8C22⋊D10111yescharacteristic, complemented
356M16—21no—
357M16—1011no—
358M16—1011no—
359M16—1011no—
360M16—2021no—
361M16—2021no—
362M16—2021no—
363M16C22×D10111yescharacteristic
364C2×C8—21no—
365C2×C8—1011no—
366C2×C8—2021no—
367C2×C8—2021no—
368C2×C8—2021no—
369C2×C8C22×D10111yescharacteristic
370C4⋊C4—4041no—
371C42—1011no—
372C23—21no—
373C23—41no—
374C23—4041no—
375C23—2021no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—1011no—
380Q8—4041no—
381Q8—2021no—
382Q8—2021no—
383Q8D4×D10111yescharacteristic
384Q8C2×C22⋊D10111yescharacteristic
385Q8C2×C22⋊D10111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—8082no—
395D4—1011no—
396D4—4041no—
397D4—4041no—
398D4—4041no—
399D4—2021no—
400D4—2021no—
401D4—2021no—
402D4D4×D10111yescharacteristic
403D4C2×C22⋊D10111yescharacteristic
404D4C2×C22⋊D10111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—4041no—
413C2×C4—4041no—
414C2×C4—4041no—
415C2×C4—2021no—
416C2×C4—2021no—
417C2×C4—2021no—
418C2×C4C23×D10111yescharacteristic
419C8—21no—
420C8—1011no—
421C8—1011no—
422C8—2021no—
423C8—2021no—
424C8—2021no—
425C8C2×C22⋊D10111yescharacteristic
426C8C2×C22⋊D10111yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—8081no—
434C22—16161no—
435C22—4041no—
436C22—2021no—
437C22#1569230011yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—4041no—
443C4—2022no—
444C4—2021no—
445C4C2×D4×D10111yescharacteristic
446C4C22×C22⋊D10111yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—8081no—
453C2—4041no—
454C2D4×C22⋊D10111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#577×C101C2
socleC2×C101D42
hypercenter#577D101
hypocenterC101#3291
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #20265988C2semidirect
2. #20265985C2semidirect
3. #20265984C2semidirect
4. #20265548C2semidirect
5. #20265442C2semidirect
6. #20265440C2semidirect
7. #20265438C2semidirect
8. #20265437C2semidirect
9. #20265360C2semidirect
10. #20265358C2semidirect
11. #20265345C2semidirect
12. #20265343C2semidirect
13. #20265294C2semidirect
14. #20265292C2semidirect
15. #577×C101C2semidirect
17. #15692301C22semidirect
20. #15692286C22semidirect
22. #15692285C22semidirect
23. #15692285C22semidirect
34. C2×#13108063C22semidirect
36. C2×#13108062C22semidirect
38. C2×#13108061C22semidirect
40. #15692262C22semidirect
41. #15692262C22semidirect
42. #15692261C22semidirect
43. #15692253C22semidirect
45. #15692191C22semidirect
46. #15692190C22semidirect
47. #15692182C22semidirect
48. #15692182C22semidirect
49. #15692182C22semidirect
50. #15692181C22semidirect
51. #15692174C22semidirect
52. #15692173C22semidirect
53. #15692172C22semidirect
54. #15692171C22semidirect
55. C8×Dic101C22semidirect
57. H(5,𝔽2)×C101C22semidirect
59. #137×C101C22semidirect
65. C2×SD16×C101C22semidirect
66. #131×C101C22semidirect
84. #13108064D4semidirect
92. #13108062D4semidirect
98. #13108061D4semidirect
102. #13108059D4semidirect
131. Q16×C101D4semidirect
136. SD16×C101D4semidirect
137. SD16×C101D4semidirect
139. D8×C101D4semidirect
228. C101#3291semidirect
232. #577D101semidirect
247. H(5,𝔽2)C2×D101semidirect
255. #137C2×D101semidirect
277. C2×SD16C2×D101semidirect
281. #131C2×D101semidirect
329. Q16C22⋊D101semidirect
348. SD16C22⋊D101semidirect
349. SD16C22⋊D101semidirect
355. D8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202659882
#202659852
#202659842
#202655482
#202654422
#202654402
#202654382
#202654372
#202653602
#202653582
#202653452
#202653432
#202652942
#202652922
#577×C1012
#3291101