Finite Group Databasebeta

← #27153498

Subgroups — #27153498

Order 12416.

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#20261543C211yesmaximal, characteristic, complemented
2#20261538C211yesmaximal, characteristic, complemented
3#20261537C211yesmaximal, characteristic, complemented
4#20261121C211yesmaximal, characteristic, complemented
5#20261005C211yesmaximal, characteristic, complemented
6#20261003C211yesmaximal, characteristic, complemented
7#20261000C211yesmaximal, characteristic, complemented
8#20260995C211yesmaximal, characteristic, complemented
9#20260911C211yesmaximal, characteristic, complemented
10#20260908C211yesmaximal, characteristic, complemented
11#20260903C211yesmaximal, characteristic, complemented
12#20260901C211yesmaximal, characteristic, complemented
13#20260857C211yesmaximal, characteristic, complemented
14#20260855C211yesmaximal, characteristic, complemented
15#577×C97C211yesmaximal, characteristic, complemented
16#15690856C2211yescharacteristic
17#15690856C2211yescharacteristic, complemented
18#15690841—21no—
19#15690841C2211yescharacteristic, complemented
20#15690837—21no—
21#15690837—21no—
22#15690836—21no—
23#15690836C2211yescharacteristic
24#15690835—21no—
25#15690835—21no—
26#15690835C2211yescharacteristic
27#15690835C2211yescharacteristic, complemented
28#15690835C2211yescharacteristic, complemented
29#15690835C2211yescharacteristic, complemented
30#15690833—21no—
31#15690830—21no—
32#15690830—21no—
33#15690830C2211yescharacteristic
34C2×#13095498—21no—
35C2×#13095498C2211yescharacteristic, complemented
36C2×D776C2211yescharacteristic, complemented
37C2×#13095496—21no—
38C2×#13095496C2211yescharacteristic, complemented
39D8×D97—21no—
40#15690817C2211yescharacteristic, complemented
41#15690817C2211yescharacteristic, complemented
42#15690816C2211yescharacteristic, complemented
43#15690806C2211yescharacteristic, complemented
44#15690799C2211yescharacteristic, complemented
45#15690746C2211yescharacteristic, complemented
46#15690745C2211yescharacteristic, complemented
47#15690737C2211yescharacteristic, complemented
48#15690737C2211yescharacteristic, complemented
49#15690737C2211yescharacteristic, complemented
50#15690736C2211yescharacteristic, complemented
51#15690729C2211yescharacteristic, complemented
52#15690727C2211yescharacteristic, complemented
53#15690726C2211yescharacteristic, complemented
54#15690726C2211yescharacteristic, complemented
55#15690720C2211yescharacteristic, complemented
562−1+4×C97C2211yescharacteristic, complemented
57H(5,𝔽2)×C97C2211yescharacteristic, complemented
58#137×C97—21no—
59#137×C97C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C97—21no—
61AGL(1,ℤ/8)×C97C2211yescharacteristic, complemented
62#135×C97—21no—
63#135×C97C2211yescharacteristic
64C2×SD16×C97—21no—
65C2×SD16×C97C2211yescharacteristic, complemented
66#131×C97C2211yescharacteristic, complemented
67#13095512—21no—
68#13095512—21no—
69#13095511—21no—
70#13095511—21no—
71#13095511C2311yescharacteristic
72C2×D388—21no—
73C2×D388—21no—
74C2×D388C2311yescharacteristic
75C2×D388C2311yescharacteristic
76C2×C22⋊D97—41no—
77D4×D97—21no—
78D4×D97—21no—
79D4×D97—41no—
80D4×D97—41no—
81#13095501—21no—
82#13095501D411yescharacteristic, complemented
83#13095499—21no—
84#13095499D411yescharacteristic, complemented
85#13095498—21no—
86#13095498—21no—
87#13095498—21no—
88#13095498—21no—
89#13095498—21no—
90#13095498—21no—
91#13095498—21no—
92#13095498D411yescharacteristic, complemented
93D776—21no—
94D776—21no—
95#13095496—21no—
96#13095496—21no—
97#13095496—21no—
98#13095496—21no—
99#13095496—21no—
100#13095496—21no—
101#13095496—21no—
102#13095496D411yescharacteristic, complemented
103#13095495—21no—
104#13095495—21no—
105#13095495C2311yescharacteristic
106#13095495C2311yescharacteristic
107#13095495C2311yescharacteristic
108#13095494—21no—
109C2×#11715895—21no—
110C2×#11715895—21no—
111C2×#11715895C2311yescharacteristic
112C8×D97—21no—
113#13095488—41no—
114C4×Dic97C2311yescharacteristic
115#41×C97—21no—
116#41×C97—21no—
117#41×C97—21no—
118#41×C97—21no—
119#41×C97—41no—
120#41×C97C2311yescharacteristic
121#41×C97C2311yescharacteristic
122#41×C97C2311yescharacteristic
123C2×Q8×C97—21no—
124C2×Q8×C97—21no—
125C2×Q8×C97C2311yescharacteristic
126C2×D4×C97—21no—
127C2×D4×C97—21no—
128C2×D4×C97—41no—
129C2×D4×C97C2311yescharacteristic
130Q16×C97—21no—
131Q16×C97D411yescharacteristic, complemented
132SD16×C97—21no—
133SD16×C97—21no—
134SD16×C97—21no—
135SD16×C97—21no—
136SD16×C97D411yescharacteristic, complemented
137SD16×C97D411yescharacteristic, complemented
138D8×C97—21no—
139D8×C97D411yescharacteristic, complemented
140M16×C97—21no—
141M16×C97C2311yescharacteristic
142C2×C8×C97—21no—
143C2×C8×C97C2311yescharacteristic
144C23×C97—21no—
145C23×C97—41no—
146C22×D97—21no—
147C22×D97—21no—
148C22×D97—41no—
149C22×D97—41no—
150Q8×C97—21no—
151Q8×C97—21no—
152Q8×C97—41no—
153Q8×C97C2×D411yescharacteristic
154Q8×C97C2×D411yescharacteristic
155Q8×C97C2×D411yescharacteristic
156D4×C97—21no—
157D4×C97—21no—
158D4×C97—21no—
159D4×C97—21no—
160D4×C97—21no—
161D4×C97—41no—
162D4×C97—41no—
163D4×C97—41no—
164D4×C97C2×D411yescharacteristic
165D4×C97C2×D411yescharacteristic
166D4×C97C2×D411yescharacteristic
167C2×C4×C97—21no—
168C2×C4×C97—21no—
169C2×C4×C97—21no—
170C2×C4×C97—21no—
171C2×C4×C97—41no—
172C2×C4×C97—41no—
173C2×C4×C97—41no—
174C2×C4×C97C2411yescharacteristic
175C22⋊D97—21no—
176C22⋊D97—41no—
177C22⋊D97—41no—
178C22⋊D97—82no—
179C2×Dic97—21no—
180D388—21no—
181D388—21no—
182D388—21no—
183D388—21no—
184D388—41no—
185D388—41no—
186D388C2×D411yescharacteristic
187C4×D97—21no—
188C4×D97—41no—
189C4×D97—41no—
190Dic194C2×D411yescharacteristic
191C8×C97—21no—
192C8×C97C2×D411yescharacteristic
193C8×C97C2×D411yescharacteristic
194#11715895—21no—
195#11715895—21no—
196#11715895—21no—
197#11715895C2×D411yescharacteristic
198#11715895C2×D411yescharacteristic
199C22×C97—21no—
200C22×C97—21no—
201C22×C97—21no—
202C22×C97—41no—
203C22×C97—81no—
204C22×C97—161no—
205C22×C97H(5,𝔽2)11yescharacteristic
206C2×D97—21no—
207C2×D97—41no—
208C2×D97—41no—
209C2×D97—81no—
210C2×D97—81no—
211C2×D97—161no—
212C2×D97—161no—
213C4×C97—21no—
214C4×C97—21no—
215C4×C97—21no—
216C4×C97—41no—
217C4×C97C22×D411yescharacteristic
218C4×C97C22×D411yescharacteristic
219Dic97—21no—
220Dic97—41no—
221C2×C97—21no—
222C2×C97—41no—
223C2×C97—41no—
224C2×C97—41no—
225C2×C97—81no—
226C2×C97D4211yescharacteristic
227D97—41no—
228D97—81no—
229D97—81no—
230#3288—971nomaximal, Sylow
231C97#328811yescharacteristic, complemented, Sylow
232#577—971no—
233#577D9711yescharacteristic, complemented
234#576—971no—
235#576—971no—
236#496—971no—
237#472—971no—
238#472—971no—
239#471—971no—
240#471—971no—
241#455—971no—
242#454—971no—
243#453—971no—
244#453—971no—
245#444—971no—
246#444—971no—
2472−1+4C2×D9711yescharacteristic, complemented
248H(5,𝔽2)—971no—
249H(5,𝔽2)—971no—
250H(5,𝔽2)C2×D9711yescharacteristic, complemented
251#137—21no—
252#137—971no—
253#137—1941no—
254#137C2×D9711yescharacteristic, complemented
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—971no—
257AGL(1,ℤ/8)—971no—
258AGL(1,ℤ/8)—971no—
259AGL(1,ℤ/8)—971no—
260AGL(1,ℤ/8)—971no—
261AGL(1,ℤ/8)—1941no—
262AGL(1,ℤ/8)—1941no—
263AGL(1,ℤ/8)—1941no—
264AGL(1,ℤ/8)—1941no—
265AGL(1,ℤ/8)—1941no—
266AGL(1,ℤ/8)C2×D9711yescharacteristic, complemented
267#135—21no—
268#135—971no—
269#135—1941no—
270#135—1941no—
271#135—1941no—
272#135C2×D9711yescharacteristic
273C2×SD16—21no—
274C2×SD16—971no—
275C2×SD16—1941no—
276C2×SD16C2×D9711yescharacteristic, complemented
277C2×D8—971no—
278C2×D8—971no—
279C2×D8—1941no—
280C2×D8—1941no—
281#131—971no—
282#131—971no—
283#131—971no—
284#131C2×D9711yescharacteristic, complemented
285#127—971no—
286#124—971no—
287#108—971no—
288#108—971no—
289#104—971no—
290#104—971no—
291#104—971no—
292#104—971no—
293#101—971no—
294#100—971no—
295#100—971no—
296#100—971no—
297#97—971no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—41no—
303#41—971no—
304#41—1941no—
305#41—1941no—
306#41—1941no—
307#41—1941no—
308#41C22×D9711yescharacteristic
309#41C22×D9711yescharacteristic
310#41C22×D9711yescharacteristic
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8C22×D9711yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—971no—
318C2×D4—971no—
319C2×D4—3881no—
320C2×D4—3881no—
321C2×D4—3881no—
322C2×D4—1941no—
323C2×D4—1941no—
324C2×D4—1941no—
325C2×D4—1941no—
326C2×D4C22×D9711yescharacteristic
327Q16—21no—
328Q16—971no—
329Q16—1941no—
330Q16C22⋊D9711yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—971no—
336SD16—971no—
337SD16—1941no—
338SD16—1941no—
339SD16—1941no—
340SD16—1941no—
341SD16—1941no—
342SD16—1941no—
343SD16—1941no—
344SD16—1941no—
345SD16C22⋊D9711yescharacteristic, complemented
346SD16C22⋊D9711yescharacteristic, complemented
347D8—21no—
348D8—971no—
349D8—1941no—
350D8—1941no—
351D8—1941no—
352D8—1941no—
353D8—1941no—
354D8—1941no—
355D8—1941no—
356D8—1941no—
357D8—1941no—
358D8C22⋊D9711yescharacteristic, complemented
359M16—21no—
360M16—971no—
361M16—971no—
362M16—971no—
363M16—1941no—
364M16—1941no—
365M16—1941no—
366M16C22×D9711yescharacteristic
367C2×C8—21no—
368C2×C8—971no—
369C2×C8—1941no—
370C2×C8—1941no—
371C2×C8—1941no—
372C2×C8C22×D9711yescharacteristic
373C22⋊C4—3881no—
374C42—971no—
375C23—21no—
376C23—41no—
377C23—3881no—
378C23—3881no—
379C23—1941no—
380C23—1941no—
381Q8—21no—
382Q8—21no—
383Q8—41no—
384Q8—971no—
385Q8D4×D9711yescharacteristic
386Q8C2×C22⋊D9711yescharacteristic
387Q8C2×C22⋊D9711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—7762no—
397D4—971no—
398D4—3881no—
399D4—3881no—
400D4—3881no—
401D4—3881no—
402D4—1941no—
403D4—1941no—
404D4—1941no—
405D4—1941no—
406D4—1941no—
407D4D4×D9711yescharacteristic
408D4C2×C22⋊D9711yescharacteristic
409D4C2×C22⋊D9711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—3881no—
418C2×C4—3881no—
419C2×C4—1941no—
420C2×C4—1941no—
421C2×C4C23×D9711yescharacteristic
422C8—21no—
423C8—971no—
424C8—971no—
425C8—1941no—
426C8—1941no—
427C8—1941no—
428C8C2×C22⋊D9711yescharacteristic
429C8C2×C22⋊D9711yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—81no—
435C22—7761no—
436C22—7761no—
437C22—161no—
438C22—15521no—
439C22—15521no—
440C22—3881no—
441C22—3881no—
442C22—1941no—
443C22#1569085511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—41no—
448C4—3881no—
449C4—1941no—
450C4C2×D4×D9711yescharacteristic
451C4C22×C22⋊D9711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—7761no—
458C2—7761no—
459C2—3881no—
460C2D4×C22⋊D9711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#577×C97C2
socleC2×C97D42
hypercenter#577D97
hypocenterC97#3288
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #20261543C2semidirect
2. #20261538C2semidirect
3. #20261537C2semidirect
4. #20261121C2semidirect
5. #20261005C2semidirect
6. #20261003C2semidirect
7. #20261000C2semidirect
8. #20260995C2semidirect
9. #20260911C2semidirect
10. #20260908C2semidirect
11. #20260903C2semidirect
12. #20260901C2semidirect
13. #20260857C2semidirect
14. #20260855C2semidirect
15. #577×C97C2semidirect
17. #15690856C22semidirect
19. #15690841C22semidirect
27. #15690835C22semidirect
28. #15690835C22semidirect
29. #15690835C22semidirect
35. C2×#13095498C22semidirect
36. C2×D776C22semidirect
38. C2×#13095496C22semidirect
40. #15690817C22semidirect
41. #15690817C22semidirect
42. #15690816C22semidirect
43. #15690806C22semidirect
44. #15690799C22semidirect
45. #15690746C22semidirect
46. #15690745C22semidirect
47. #15690737C22semidirect
48. #15690737C22semidirect
49. #15690737C22semidirect
50. #15690736C22semidirect
51. #15690729C22semidirect
52. #15690727C22semidirect
53. #15690726C22semidirect
54. #15690726C22semidirect
55. #15690720C22semidirect
56. 2−1+4×C97C22semidirect
57. H(5,𝔽2)×C97C22semidirect
59. #137×C97C22semidirect
61. AGL(1,ℤ/8)×C97C22semidirect
65. C2×SD16×C97C22semidirect
66. #131×C97C22semidirect
82. #13095501D4semidirect
84. #13095499D4semidirect
92. #13095498D4semidirect
102. #13095496D4semidirect
131. Q16×C97D4semidirect
136. SD16×C97D4semidirect
137. SD16×C97D4semidirect
139. D8×C97D4semidirect
231. C97#3288semidirect
233. #577D97semidirect
247. 2−1+4C2×D97semidirect
250. H(5,𝔽2)C2×D97semidirect
254. #137C2×D97semidirect
266. AGL(1,ℤ/8)C2×D97semidirect
276. C2×SD16C2×D97semidirect
284. #131C2×D97semidirect
330. Q16C22⋊D97semidirect
345. SD16C22⋊D97semidirect
346. SD16C22⋊D97semidirect
358. D8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202615432
#202615382
#202615372
#202611212
#202610052
#202610032
#202610002
#202609952
#202609112
#202609082
#202609032
#202609012
#202608572
#202608552
#577×C972
#328897