Finite Group Databasebeta

← #27156512

Subgroups — #27156512

Order 12416.

1464 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261571C211yesmaximal, characteristic, complemented
2C2×#15690855C211yesmaximal, characteristic, complemented
3#20261545C214yesmaximal, complemented
4#20261541C214yesmaximal, complemented
5#20261539C214yesmaximal, complemented
6#20261536C214yesmaximal, complemented
7#20261535C211yesmaximal, characteristic, complemented
8#20261533C211yesmaximal, characteristic, complemented
9#20261531C214yesmaximal, complemented
10C2×#15690836C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×D97C214yesmaximal, complemented
12#20261501C211yesmaximal, characteristic, complemented
13C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
14#15690858C2212yescomplemented
15#15690857—21no—
16#15690857C2212yescomplemented
17#15690856—21no—
18#15690856C2212yescomplemented
19#15690855—22no—
20#15690855—22no—
21#15690855—22no—
22#15690855C2214yescomplemented
23#15690855C2214yescomplemented
24#15690855C2212yescomplemented
25C2×#13095511—22no—
26C2×#13095511C2211yescharacteristic
27C2×#13095511C2211yescharacteristic, complemented
28C2×#13095511C2211yescharacteristic, complemented
29C2×#13095510—21no—
30C2×#13095510C2212yescomplemented
31#41×D97—22no—
32#41×D97C2214yescomplemented
33C22×C22⋊D97—22no—
34C2×D4×D97—21no—
35C2×D4×D97C2212yescomplemented
36#15690843C2212yescomplemented
37#15690843C2212yescomplemented
38#15690841C2212yescomplemented
39#15690840C2212yescomplemented
40#15690839C2218yescomplemented
41#15690838C2212yescomplemented
42#15690838C2212yescomplemented
43#15690837C2218yescomplemented
44#15690836C2214yescomplemented
45#15690836C2214yescomplemented
46#15690836C2212yescomplemented
47#15690835C2212yescomplemented
48#15690834C2218yescomplemented
49#15690834C2218yescomplemented
50#15690832C2214yescomplemented
51#15690831C2218yescomplemented
52#15690830C2214yescomplemented
53#15690829C2218yescomplemented
54C2×#13095499C2212yescomplemented
55SD16×D97C2218yescomplemented
56C2×#13095496C2212yescomplemented
57D8×D97C2218yescomplemented
58#15690818C2214yescomplemented
59#15690816C2214yescomplemented
60C2×#13095495C2211yescharacteristic, complemented
61M16×D97C2214yescomplemented
62C2×#41×C97C2211yescharacteristic, complemented
63C22×D4×C97C2211yescharacteristic, complemented
64AGL(1,ℤ/8)×C97C2214yescomplemented
65AGL(1,ℤ/8)×C97C2214yescomplemented
66C2×SD16×C97C2212yescomplemented
67C2×D8×C97C2212yescomplemented
68C2×M16×C97C2211yescharacteristic, complemented
69C23×D97—22no—
70#13095512—22no—
71#13095512C2314yescomplemented
72#13095511—21no—
73#13095511—21no—
74#13095511—21no—
75#13095511—21no—
76#13095511—24no—
77#13095511—22no—
78#13095511—24no—
79#13095511—22no—
80#13095511D412yes—
81#13095511D412yes—
82#13095511C2312yes—
83#13095511C2312yes—
84#13095511C2312yes—
85#13095511C2312yes—
86#13095511C2312yes—
87#13095511C2312yescomplemented
88#13095510—24no—
89#13095510—24no—
90#13095510—24no—
91#13095510—22no—
92#13095510—24no—
93#13095510—24no—
94#13095510C2318yes—
95#13095510C2314yescomplemented
96C2×Dic194—22no—
97C2×Dic194D411yescharacteristic
98C2×Dic194C2311yescharacteristic
99C2×Dic194C2311yescharacteristic, complemented
100Q8×D97—22no—
101Q8×D97C2314yescomplemented
102C2×D388—22no—
103C2×D388D411yescharacteristic
104C2×D388C2311yescharacteristic
105C2×D388C2311yescharacteristic, complemented
106C2×C22⋊D97—21no—
107C2×C22⋊D97—21no—
108C2×C22⋊D97—22no—
109C2×C22⋊D97—28no—
110C2×C22⋊D97—28no—
111C2×C22⋊D97—24no—
112C2×C22⋊D97—22no—
113C2×C22⋊D97D412yes—
114D4×D97—24no—
115D4×D97—24no—
116D4×D97—24no—
117D4×D97—24no—
118D4×D97—22no—
119D4×D97—24no—
120D4×D97C2318yes—
121D4×D97C2314yescomplemented
122C22×Dic97—22no—
123C2×C4×D97—21no—
124C2×C4×D97—21no—
125C2×C4×D97—24no—
126C2×C4×D97C2312yes—
127Dic388C2314yescomplemented
128#13095501C2314yescomplemented
129#13095500C2314yescomplemented
130#13095500C2314yescomplemented
131#13095499C2318yescomplemented
132#13095499C2314yescomplemented
133#13095498C2314yescomplemented
134D776C2314yescomplemented
135#13095496C2314yescomplemented
136#13095496C2318yescomplemented
137#13095495C2312yes—
138#13095495C2312yescomplemented
139#13095494C2318yescomplemented
140C2×#11715895C2312yescomplemented
141C8×D97C2318yescomplemented
142C24×C97—21no—
143#41×C97—21no—
144#41×C97—21no—
145#41×C97C2312yes—
146#41×C97C2312yescomplemented
147C2×Q8×C97C2311yescharacteristic, complemented
148C2×D4×C97—21no—
149C2×D4×C97—21no—
150C2×D4×C97—21no—
151C2×D4×C97—21no—
152C2×D4×C97C2312yes—
153C2×D4×C97C2312yes—
154C2×D4×C97C2312yes—
155C2×D4×C97C2311yescharacteristic, complemented
156C22×C4×C97—21no—
157C22×C4×C97C2311yescharacteristic
158SD16×C97C2318yescomplemented
159D8×C97C2318yescomplemented
160M16×C97C2312yes—
161M16×C97C2312yescomplemented
162C2×C8×C97C2312yescomplemented
163C23×C97—22no—
164C23×C97—21no—
165C23×C97—22no—
166C23×C97—22no—
167C23×C97—82no—
168C23×C97C2×D411yescharacteristic
169C22×D97—21no—
170C22×D97—21no—
171C22×D97—24no—
172C22×D97—28no—
173C22×D97—84no—
174C22×D97C2×D412yes—
175Q8×C97—21no—
176Q8×C97C2412yes—
177D4×C97—22no—
178D4×C97—22no—
179D4×C97—22no—
180D4×C97—22no—
181D4×C97—22no—
182D4×C97—21no—
183D4×C97C2412yes—
184D4×C97C2414yes—
185C2×C4×C97—21no—
186C2×C4×C97—22no—
187C2×C4×C97—22no—
188C2×C4×C97C2×D411yescharacteristic
189C2×C4×C97C2×D411yescharacteristic
190C2×C4×C97C2×D411yescharacteristic
191C2×C4×C97C2411yescharacteristic
192C2×C4×C97C2411yescharacteristic
193C2×C4×C97C2411yescharacteristic
194C22⋊D97—22no—
195C22⋊D97—22no—
196C22⋊D97—22no—
197C22⋊D97—22no—
198C22⋊D97—28no—
199C22⋊D97—24no—
200C22⋊D97—28no—
201C22⋊D97—24no—
202C22⋊D97C2×D414yes—
203C22⋊D97C2×D414yes—
204C2×Dic97—21no—
205C2×Dic97—21no—
206C2×Dic97—24no—
207C2×Dic97—28no—
208C2×Dic97C2×D412yes—
209D388—24no—
210D388—21no—
211D388—21no—
212D388C2×D412yes—
213D388C2×D412yes—
214D388C2412yes—
215D388C2412yes—
216C4×D97—24no—
217C4×D97—22no—
218C4×D97—22no—
219C4×D97—22no—
220C4×D97—22no—
221C4×D97—24no—
222C4×D97C2×D414yes—
223C4×D97C2414yes—
224Dic194—21no—
225Dic194—21no—
226Dic194—24no—
227Dic194C2×D412yes—
228Dic194C2×D412yes—
229Dic194C2412yes—
230Dic194C2412yes—
231C8×C97C2414yes—
232#11715895C2414yes—
233C22×C97—22no—
234C22×C97—24no—
235C22×C97—41no—
236C22×C97—44no—
237C22×C97—81no—
238C22×C97—82no—
239C22×C97—82no—
240C22×C97C22×D411yescharacteristic
241C22×C97C22×D411yescharacteristic
242C22×C97C22×D411yescharacteristic
243C2×D97—22no—
244C2×D97—22no—
245C2×D97—42no—
246C2×D97—42no—
247C2×D97—84no—
248C2×D97—88no—
249C2×D97—81no—
250C2×D97C22×D414yes—
251C4×C97—22no—
252C4×C97C22×D411yescharacteristic
253C4×C97C22×D411yescharacteristic
254C4×C97C22×D411yescharacteristic
255C4×C97C2511yescharacteristic
256Dic97—22no—
257Dic97—22no—
258Dic97C22×D414yes—
259C2×C97—21no—
260C2×C97—21no—
261C2×C97—21no—
262C2×C97—44no—
263C2×C97—42no—
264C2×C97C23×D411yescharacteristic
265D97—24no—
266D97—42no—
267D97—42no—
268#3585—971nomaximal, Sylow
269C97#358511yescharacteristic, complemented, Sylow
270#585—971no—
271C2×H(5,𝔽2)—971no—
272#577—974no—
273#577—974no—
274#576—974no—
275#576—974no—
276#575—971no—
277#575—974no—
278#575—971no—
279C2×AGL(1,ℤ/8)—971no—
280C2×AGL(1,ℤ/8)—974no—
281C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
282#568—971no—
2832−1+4—972no—
2842−1+4—972no—
2852−1+4—1941no—
286H(5,𝔽2)—972no—
287H(5,𝔽2)—974no—
288H(5,𝔽2)—974no—
289H(5,𝔽2)—972no—
290H(5,𝔽2)—1942no—
291H(5,𝔽2)—1942no—
292H(5,𝔽2)—1942no—
293H(5,𝔽2)—1941no—
294C2×#41—971no—
295C2×#41—974no—
296C2×#41—972no—
297C2×#41—971no—
298C2×#41—971no—
299C2×#41—1942no—
300C2×#41—1941no—
301C2×#41—1942no—
302C2×#41C2×D9711yescharacteristic, complemented
303C22×D4—972no—
304C22×D4—1941no—
305C22×D4—1942no—
306C22×D4C2×D9711yescharacteristic, complemented
307#137—972no—
308#137—972no—
309#137—978no—
310#137—972no—
311#137—972no—
312AGL(1,ℤ/8)—972no—
313AGL(1,ℤ/8)—974no—
314AGL(1,ℤ/8)—974no—
315AGL(1,ℤ/8)—972no—
316AGL(1,ℤ/8)—978no—
317AGL(1,ℤ/8)—978no—
318AGL(1,ℤ/8)—978no—
319AGL(1,ℤ/8)—972no—
320AGL(1,ℤ/8)—972no—
321AGL(1,ℤ/8)C2×D9714yescomplemented
322AGL(1,ℤ/8)C2×D9714yescomplemented
323#135—974no—
324#135—978no—
325#135—978no—
326#135—974no—
327C2×SD16—972no—
328C2×SD16—978no—
329C2×SD16C2×D9712yescomplemented
330C2×D8—972no—
331C2×D8—978no—
332C2×D8C2×D9712yescomplemented
333#131—974no—
334#131—974no—
335C2×M16—971no—
336C2×M16—974no—
337C2×M16C2×D9711yescharacteristic, complemented
338C24—21no—
339C24—1942no—
340#41—21no—
341#41—21no—
342#41—972no—
343#41—972no—
344#41—972no—
345#41—972no—
346#41—974no—
347#41—978no—
348#41—974no—
349#41—972no—
350#41—972no—
351#41—972no—
352#41—972no—
353#41—1944no—
354#41—1944no—
355#41—1942no—
356#41—1942no—
357#41—1944no—
358#41—1944no—
359#41—1944no—
360#41—1944no—
361#41—1944no—
362#41—1942no—
363#41—1942no—
364#41—1941no—
365#41—1941no—
366#41—1941no—
367#41—1941no—
368#41C22×D9712yes—
369#41C22×D9712yescomplemented
370C2×Q8—971no—
371C2×Q8—974no—
372C2×Q8—971no—
373C2×Q8—971no—
374C2×Q8—1942no—
375C2×Q8—1942no—
376C2×Q8C22×D9711yescharacteristic, complemented
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—972no—
382C2×D4—978no—
383C2×D4—974no—
384C2×D4—971no—
385C2×D4—971no—
386C2×D4—971no—
387C2×D4—1942no—
388C2×D4—1944no—
389C2×D4—1942no—
390C2×D4—1944no—
391C2×D4—1944no—
392C2×D4—1944no—
393C2×D4—1942no—
394C2×D4—1944no—
395C2×D4—1948no—
396C2×D4—1948no—
397C2×D4—1944no—
398C2×D4—1942no—
399C2×D4—1941no—
400C2×D4—1941no—
401C2×D4C22×D9712yes—
402C2×D4C22×D9712yes—
403C2×D4C22×D9712yes—
404C2×D4C22×D9711yescharacteristic, complemented
405C22×C4—21no—
406C22×C4—972no—
407C22×C4—1944no—
408C22×C4—1942no—
409C22×C4—1941no—
410C22×C4—1941no—
411C22×C4C22×D9711yescharacteristic
412Q16—974no—
413Q16—974no—
414SD16—974no—
415SD16—978no—
416SD16—974no—
417SD16—974no—
418SD16—974no—
419SD16C22×D9718yescomplemented
420D8—978no—
421D8—974no—
422D8—974no—
423D8C22×D9718yescomplemented
424M16—972no—
425M16—972no—
426M16—978no—
427M16C22×D9712yes—
428M16C22×D9712yescomplemented
429C2×C8—972no—
430C2×C8—978no—
431C2×C8C22×D9712yescomplemented
432C23—22no—
433C23—21no—
434C23—22no—
435C23—22no—
436C23—82no—
437C23—7764no—
438C23—972no—
439C23—1944no—
440C23—1948no—
441C23—1941no—
442C23—1941no—
443C23D4×D9711yescharacteristic
444Q8—21no—
445Q8—972no—
446Q8—972no—
447Q8—972no—
448Q8—972no—
449Q8—1944no—
450Q8—1941no—
451Q8—1941no—
452Q8C23×D9712yes—
453D4—22no—
454D4—22no—
455D4—22no—
456D4—22no—
457D4—22no—
458D4—21no—
459D4—974no—
460D4—974no—
461D4—972no—
462D4—972no—
463D4—972no—
464D4—972no—
465D4—1944no—
466D4—1948no—
467D4—1944no—
468D4—1948no—
469D4—1944no—
470D4—1942no—
471D4—1942no—
472D4—1942no—
473D4—1942no—
474D4—1941no—
475D4—1941no—
476D4C23×D9712yes—
477D4C23×D9714yes—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—972no—
482C2×C4—974no—
483C2×C4—974no—
484C2×C4—1944no—
485C2×C4—1944no—
486C2×C4—1948no—
487C2×C4—1944no—
488C2×C4—1941no—
489C2×C4—1941no—
490C2×C4—1942no—
491C2×C4—1942no—
492C2×C4—1942no—
493C2×C4—1942no—
494C2×C4D4×D9711yescharacteristic
495C2×C4D4×D9711yescharacteristic
496C2×C4D4×D9711yescharacteristic
497C2×C4C23×D9711yescharacteristic
498C2×C4C23×D9711yescharacteristic
499C2×C4C23×D9711yescharacteristic
500C8—974no—
501C8C23×D9714yes—
502C22—22no—
503C22—24no—
504C22—41no—
505C22—44no—
506C22—82no—
507C22—82no—
508C22—81no—
509C22—7761no—
510C22—7768no—
511C22—7764no—
512C22—974no—
513C22—3882no—
514C22—3882no—
515C22—1942no—
516C22—1942no—
517C22C2×D4×D9711yescharacteristic
518C22C2×D4×D9711yescharacteristic
519C22C2×D4×D9711yescharacteristic
520C4—22no—
521C4—974no—
522C4—1942no—
523C4—1942no—
524C4C2×D4×D9711yescharacteristic
525C4C2×D4×D9711yescharacteristic
526C4C2×D4×D9711yescharacteristic
527C4C24×D9711yescharacteristic
528C2—21no—
529C2—21no—
530C2—21no—
531C2—44no—
532C2—42no—
533C2—3882no—
534C2—3882no—
535C2—1944no—
536C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC2C22×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC2×C97C23×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3585
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20261571C2semidirect
2. C2×#15690855C2semidirect
3. #20261545C2semidirect
4. #20261541C2semidirect
5. #20261539C2semidirect
6. #20261536C2semidirect
7. #20261535C2semidirect
8. #20261533C2semidirect
9. #20261531C2semidirect
10. C2×#15690836C2semidirect
11. AGL(1,ℤ/8)×D97C2semidirect
12. #20261501C2semidirect
13. C2×AGL(1,ℤ/8)×C97C2semidirect
14. #15690858C22semidirect
16. #15690857C22semidirect
18. #15690856C22semidirect
22. #15690855C22semidirect
23. #15690855C22semidirect
24. #15690855C22semidirect
27. C2×#13095511C22semidirect
28. C2×#13095511C22semidirect
30. C2×#13095510C22semidirect
32. #41×D97C22semidirect
35. C2×D4×D97C22semidirect
36. #15690843C22semidirect
37. #15690843C22semidirect
38. #15690841C22semidirect
39. #15690840C22semidirect
40. #15690839C22semidirect
41. #15690838C22semidirect
42. #15690838C22semidirect
43. #15690837C22semidirect
44. #15690836C22semidirect
45. #15690836C22semidirect
46. #15690836C22semidirect
47. #15690835C22semidirect
48. #15690834C22semidirect
49. #15690834C22semidirect
50. #15690832C22semidirect
51. #15690831C22semidirect
52. #15690830C22semidirect
53. #15690829C22semidirect
54. C2×#13095499C22semidirect
55. SD16×D97C22semidirect
56. C2×#13095496C22semidirect
57. D8×D97C22semidirect
58. #15690818C22semidirect
59. #15690816C22semidirect
60. C2×#13095495C22semidirect
61. M16×D97C22semidirect
62. C2×#41×C97C22semidirect
63. C22×D4×C97C22semidirect
64. AGL(1,ℤ/8)×C97C22semidirect
65. AGL(1,ℤ/8)×C97C22semidirect
66. C2×SD16×C97C22semidirect
67. C2×D8×C97C22semidirect
68. C2×M16×C97C22semidirect
71. #13095512C23semidirect
87. #13095511C23semidirect
95. #13095510C23semidirect
99. C2×Dic194C23semidirect
101. Q8×D97C23semidirect
105. C2×D388C23semidirect
121. D4×D97C23semidirect
127. Dic388C23semidirect
128. #13095501C23semidirect
129. #13095500C23semidirect
130. #13095500C23semidirect
131. #13095499C23semidirect
132. #13095499C23semidirect
133. #13095498C23semidirect
134. D776C23semidirect
135. #13095496C23semidirect
136. #13095496C23semidirect
138. #13095495C23semidirect
139. #13095494C23semidirect
140. C2×#11715895C23semidirect
141. C8×D97C23semidirect
146. #41×C97C23semidirect
147. C2×Q8×C97C23semidirect
155. C2×D4×C97C23semidirect
158. SD16×C97C23semidirect
159. D8×C97C23semidirect
161. M16×C97C23semidirect
162. C2×C8×C97C23semidirect
269. C97#3585semidirect
281. C2×AGL(1,ℤ/8)D97semidirect
302. C2×#41C2×D97semidirect
306. C22×D4C2×D97semidirect
321. AGL(1,ℤ/8)C2×D97semidirect
322. AGL(1,ℤ/8)C2×D97semidirect
329. C2×SD16C2×D97semidirect
332. C2×D8C2×D97semidirect
337. C2×M16C2×D97semidirect
369. #41C22×D97semidirect
376. C2×Q8C22×D97semidirect
404. C2×D4C22×D97semidirect
419. SD16C22×D97semidirect
423. D8C22×D97semidirect
428. M16C22×D97semidirect
431. C2×C8C22×D97semidirect

Maximal subgroups

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

subgroupindex
#202615712
C2×#156908552
#202615452
#202615412
#202615392
#202615362
#202615352
#202615332
#202615312
C2×#156908362
AGL(1,ℤ/8)×D972
#202615012
C2×AGL(1,ℤ/8)×C972
#358597