Finite Group Databasebeta

← #27182912

Subgroups — #27182912

Order 12928.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265992C211yesmaximal, characteristic, complemented
2#20265984C211yesmaximal, characteristic, complemented
3#20265980C211yesmaximal, characteristic, complemented
4#20265558C211yesmaximal, characteristic, complemented
5#20265450C211yesmaximal, characteristic, complemented
6#20265445C211yesmaximal, characteristic, complemented
7#20265440C211yesmaximal, characteristic, complemented
8#20265437C211yesmaximal, characteristic, complemented
9#20265355C211yesmaximal, characteristic, complemented
10#20265353C211yesmaximal, characteristic, complemented
11#20265350C211yesmaximal, characteristic, complemented
12#20265348C211yesmaximal, characteristic, complemented
13#20265294C211yesmaximal, characteristic, complemented
14#20265292C211yesmaximal, characteristic, complemented
15#577×C101C211yesmaximal, characteristic, complemented
16#15692302C2211yescharacteristic, complemented
17#15692301C2211yescharacteristic
18#15692287—21no—
19#15692286—21no—
20#15692286C2211yescharacteristic, complemented
21#15692285—21no—
22#15692285C2211yescharacteristic
23#15692285C2211yescharacteristic, complemented
24#15692281—21no—
25#15692281C2211yescharacteristic
26#15692280—21no—
27#15692280C2211yescharacteristic
28#15692280C2211yescharacteristic, complemented
29#15692279—21no—
30#15692276—21no—
31#15692275—21no—
32#15692275—21no—
33#15692275C2211yescharacteristic
34C2×#13108064—21no—
35C2×#13108064C2211yescharacteristic, complemented
36C2×#13108063C2211yescharacteristic, complemented
37SD16×D101—21no—
38C2×#13108059—21no—
39C2×#13108059C2211yescharacteristic
40#15692262C2211yescharacteristic
41#15692262C2211yescharacteristic, complemented
42#15692261C2211yescharacteristic, complemented
43#15692244C2211yescharacteristic, complemented
44#15692242C2211yescharacteristic, complemented
45#15692191C2211yescharacteristic, complemented
46#15692190C2211yescharacteristic, complemented
47#15692182C2211yescharacteristic
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, complemented
57H(5,𝔽2)×C101C2211yescharacteristic
58#137×C101—21no—
59#137×C101C2211yescharacteristic
60AGL(1,ℤ/8)×C101—21no—
61AGL(1,ℤ/8)×C101C2211yescharacteristic, complemented
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
78D4×D101—21no—
79D4×D101—41no—
80#13108064—21no—
81#13108064—21no—
82#13108064—21no—
83#13108064—21no—
84#13108064—21no—
85#13108064D411yescharacteristic
86#13108063—21no—
87#13108063—21no—
88#13108062—21no—
89#13108062—21no—
90#13108062—21no—
91#13108062D411yescharacteristic
92#13108061—21no—
93#13108061—21no—
94#13108061—21no—
95#13108061D411yescharacteristic
96#13108059—21no—
97#13108059—21no—
98#13108059—21no—
99#13108059—21no—
100#13108059—21no—
101#13108059D411yescharacteristic
102#13108058—21no—
103#13108058—21no—
104#13108058C2311yescharacteristic
105#13108058C2311yescharacteristic
106#13108058C2311yescharacteristic
107#13108057—21no—
108C2×#11717579—21no—
109C2×#11717579—21no—
110C2×#11717579C2311yescharacteristic
111C8×D101—21no—
112C22⋊Dic101—41no—
113#13108051—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
132SD16×C101—21no—
133SD16×C101—21no—
134SD16×C101—21no—
135SD16×C101—21no—
136SD16×C101D411yescharacteristic
137SD16×C101D411yescharacteristic
138D8×C101—21no—
139D8×C101D411yescharacteristic
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—
176C2×Dic101—21no—
177C2×Dic101—21no—
178C2×Dic101—41no—
179D404—21no—
180D404—21no—
181D404—41no—
182D404C2×D411yescharacteristic
183C4×D101—21no—
184C4×D101—41no—
185C4×D101—41no—
186Dic202—21no—
187Dic202—21no—
188Dic202—41no—
189Dic202C2×D411yescharacteristic
190C8×C101—21no—
191C8×C101C2×D411yescharacteristic
192C8×C101C2×D411yescharacteristic
193#11717579—21no—
194#11717579—21no—
195#11717579—21no—
196#11717579C2×D411yescharacteristic
197#11717579C2×D411yescharacteristic
198C22×C101—21no—
199C22×C101—21no—
200C22×C101—21no—
201C22×C101—41no—
202C22×C101—81no—
203C22×C101—161no—
204C22×C101H(5,𝔽2)11yescharacteristic
205C2×D101—21no—
206C2×D101—41no—
207C2×D101—81no—
208C2×D101—161no—
209C4×C101—21no—
210C4×C101—21no—
211C4×C101—21no—
212C4×C101—41no—
213C4×C101C22×D411yescharacteristic
214C4×C101C22×D411yescharacteristic
215Dic101—21no—
216Dic101—41no—
217Dic101—41no—
218C2×C101—21no—
219C2×C101—41no—
220C2×C101—41no—
221C2×C101—41no—
222C2×C101—81no—
223C2×C101D4211yescharacteristic
224D101—41no—
225D101—81no—
226#3290—1011nomaximal, Sylow
227C101#329011yescharacteristic, complemented, Sylow
228#578—1011no—
229#577—1011no—
230#577D10111yescharacteristic, complemented
231#576—1011no—
232#495—1011no—
233#473—1011no—
234#472—1011no—
235#471—1011no—
236#471—1011no—
237#455—1011no—
238#455—1011no—
239#454—1011no—
240#454—1011no—
241#443—1011no—
242#443—1011no—
2432−1+4—1011no—
2442−1+4C2×D10111yescharacteristic, complemented
245H(5,𝔽2)—1011no—
246H(5,𝔽2)C2×D10111yescharacteristic
247#137—21no—
248#137—1011no—
249#137—1011no—
250#137—1011no—
251#137—2021no—
252#137—2021no—
253#137—2021no—
254#137C2×D10111yescharacteristic
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—1011no—
257AGL(1,ℤ/8)—1011no—
258AGL(1,ℤ/8)—1011no—
259AGL(1,ℤ/8)—2021no—
260AGL(1,ℤ/8)—2021no—
261AGL(1,ℤ/8)—2021no—
262AGL(1,ℤ/8)C2×D10111yescharacteristic, complemented
263#135—21no—
264#135—1011no—
265#135—2021no—
266#135—2021no—
267#135—2021no—
268#135C2×D10111yescharacteristic
269C2×Q16—1011no—
270C2×Q16—2021no—
271C2×SD16—21no—
272C2×SD16—1011no—
273C2×SD16—2021no—
274C2×SD16C2×D10111yescharacteristic, complemented
275C2×D8—1011no—
276C2×D8—2021no—
277#131—1011no—
278#131—1011no—
279#131—1011no—
280#131C2×D10111yescharacteristic, complemented
281#124—1011no—
282#124—1011no—
283#108—1011no—
284#108—1011no—
285#104—1011no—
286#104—1011no—
287#104—1011no—
288#104—1011no—
289#101—1011no—
290#101—1011no—
291#100—1011no—
292#100—1011no—
293C4×C8—1011no—
294#41—21no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—41no—
299#41—1011no—
300#41—4041no—
301#41—2021no—
302#41—2021no—
303#41—2021no—
304#41—2021no—
305#41C22×D10111yescharacteristic
306#41C22×D10111yescharacteristic
307#41C22×D10111yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—1011no—
311C2×Q8—2021no—
312C2×Q8—2021no—
313C2×Q8C22×D10111yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—1011no—
318C2×D4—4041no—
319C2×D4—2021no—
320C2×D4—2021no—
321C2×D4C22×D10111yescharacteristic
322Q16—21no—
323Q16—1011no—
324Q16—2021no—
325Q16—2021no—
326Q16—2021no—
327Q16—2021no—
328Q16—2021no—
329Q16C22⋊D10111yescharacteristic
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—
344SD16C22⋊D10111yescharacteristic
345SD16C22⋊D10111yescharacteristic
346D8—21no—
347D8—1011no—
348D8—2021no—
349D8—2021no—
350D8—2021no—
351D8—2021no—
352D8—2021no—
353D8C22⋊D10111yescharacteristic
354M16—21no—
355M16—1011no—
356M16—1011no—
357M16—1011no—
358M16—2021no—
359M16—2021no—
360M16—2021no—
361M16C22×D10111yescharacteristic
362C2×C8—21no—
363C2×C8—1011no—
364C2×C8—2021no—
365C2×C8—2021no—
366C2×C8—2021no—
367C2×C8C22×D10111yescharacteristic
368C22⋊C4—4041no—
369C22⋊C4—4041no—
370C42—1011no—
371C23—21no—
372C23—41no—
373C23—4041no—
374C23—2021no—
375Q8—21no—
376Q8—21no—
377Q8—41no—
378Q8—1011no—
379Q8—4041no—
380Q8—2021no—
381Q8—2021no—
382Q8D4×D10111yescharacteristic
383Q8C2×C22⋊D10111yescharacteristic
384Q8C2×C22⋊D10111yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—1011no—
394D4—4041no—
395D4—4041no—
396D4—4041no—
397D4—2021no—
398D4—2021no—
399D4—2021no—
400D4D4×D10111yescharacteristic
401D4C2×C22⋊D10111yescharacteristic
402D4C2×C22⋊D10111yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—4041no—
411C2×C4—4041no—
412C2×C4—4041no—
413C2×C4—2021no—
414C2×C4—2021no—
415C2×C4—2021no—
416C2×C4C23×D10111yescharacteristic
417C8—21no—
418C8—1011no—
419C8—1011no—
420C8—2021no—
421C8—2021no—
422C8—2021no—
423C8C2×C22⋊D10111yescharacteristic
424C8C2×C22⋊D10111yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—81no—
430C22—161no—
431C22—8081no—
432C22—16161no—
433C22—4041no—
434C22—2021no—
435C22#1569230011yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—4041no—
441C4—4041no—
442C4—2021no—
443C4C2×D4×D10111yescharacteristic
444C4C22×C22⋊D10111yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—81no—
450C2—8081no—
451C2—4041no—
452C2D4×C22⋊D10111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#577×C101C2
socleC2×C101D42
hypercenter#577D101
hypocenterC101#3290
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. #20265992C2semidirect
2. #20265984C2semidirect
3. #20265980C2semidirect
4. #20265558C2semidirect
5. #20265450C2semidirect
6. #20265445C2semidirect
7. #20265440C2semidirect
8. #20265437C2semidirect
9. #20265355C2semidirect
10. #20265353C2semidirect
11. #20265350C2semidirect
12. #20265348C2semidirect
13. #20265294C2semidirect
14. #20265292C2semidirect
15. #577×C101C2semidirect
16. #15692302C22semidirect
20. #15692286C22semidirect
23. #15692285C22semidirect
28. #15692280C22semidirect
35. C2×#13108064C22semidirect
36. C2×#13108063C22semidirect
41. #15692262C22semidirect
42. #15692261C22semidirect
43. #15692244C22semidirect
44. #15692242C22semidirect
45. #15692191C22semidirect
46. #15692190C22semidirect
48. #15692182C22semidirect
49. #15692182C22semidirect
50. #15692181C22semidirect
51. #15692174C22semidirect
52. #15692173C22semidirect
53. #15692172C22semidirect
54. #15692171C22semidirect
55. C8×Dic101C22semidirect
56. 2−1+4×C101C22semidirect
61. AGL(1,ℤ/8)×C101C22semidirect
65. C2×SD16×C101C22semidirect
66. #131×C101C22semidirect
227. C101#3290semidirect
230. #577D101semidirect
244. 2−1+4C2×D101semidirect
262. AGL(1,ℤ/8)C2×D101semidirect
274. C2×SD16C2×D101semidirect
280. #131C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659922
#202659842
#202659802
#202655582
#202654502
#202654452
#202654402
#202654372
#202653552
#202653532
#202653502
#202653482
#202652942
#202652922
#577×C1012
#3290101