Finite Group Databasebeta

← #27153212

Subgroups — #27153212

Order 12416.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690836C211yesmaximal, characteristic, complemented
2C22×#13095496C211yesmaximal, characteristic, complemented
3#20261381C211yesmaximal, characteristic, complemented
4#20261122C211yesmaximal, characteristic, complemented
5#20260986C211yesmaximal, characteristic, complemented
6#20260962C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic, complemented
8#20260927C211yesmaximal, characteristic, complemented
9#20260886C211yesmaximal, characteristic, complemented
10#20260878C211yesmaximal, characteristic, complemented
11#20260869C211yesmaximal, characteristic, complemented
12#20260868C211yesmaximal, characteristic, complemented
13#20260845C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15D42×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic, complemented
17C22×D388C2211yescharacteristic, complemented
18#15690836—24no—
19C2×#13095499C2211yescharacteristic, complemented
20C2×#13095499C2211yescharacteristic, complemented
21C2×#13095499C2211yescharacteristic, complemented
22C2×#13095496—24no—
23C2×#13095496C2211yescharacteristic
24C2×#13095496C2211yescharacteristic
25C2×#13095496C2211yescharacteristic, complemented
26C2×#13095496C2211yescharacteristic, complemented
27C2×#13095496C2211yescharacteristic, complemented
28C2×#13095496C2211yescharacteristic, complemented
29C2×#13095495C2211yescharacteristic, complemented
30C22×#11715895C2211yescharacteristic, complemented
31#15690798C2211yescharacteristic, complemented
32#15690795—21no—
33#15690789C2211yescharacteristic, complemented
34#15690784C2211yescharacteristic, complemented
35#15690783—21no—
36#15690778—21no—
37C4×D388C2211yescharacteristic, complemented
38C22⋊C4×D97—21no—
39#15690741C2211yescharacteristic, complemented
40C4⋊#11715895C2211yescharacteristic, complemented
41#15690733C2211yescharacteristic, complemented
42#15690731C2211yescharacteristic, complemented
43#15690731C2211yescharacteristic, complemented
44#15690731C2211yescharacteristic, complemented
45#15690730C2211yescharacteristic, complemented
46#15690730C2211yescharacteristic, complemented
47C22⋊#11715895C2211yescharacteristic, complemented
48C22⋊#11715895C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50C22×D4×C97—21no—
51C22×D4×C97C2211yescharacteristic
52C22×D4×C97C2211yescharacteristic, complemented
53#127×C97C2211yescharacteristic, complemented
54#121×C97—21no—
55#121×C97C2211yescharacteristic
56#121×C97C2211yescharacteristic, complemented
57#120×C97—21no—
58#120×C97—21no—
59C4×D4×C97C2211yescharacteristic
60C4×D4×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095511—22no—
63#13095511—41no—
64C2×Dic194C2311yescharacteristic
65C2×D388—22no—
66C2×D388—21no—
67C2×D388—41no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74#13095499—22no—
75#13095499—22no—
76#13095499—41no—
77#13095496—22no—
78#13095496—22no—
79#13095496—22no—
80#13095496—22no—
81#13095496—41no—
82#13095496D414yescomplemented
83#13095495—22no—
84C2×#11715895—22no—
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C2×#11715895C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96C24×C97—21no—
97C24×C97—21no—
98C2×D4×C97—22no—
99C2×D4×C97—22no—
100C2×D4×C97—21no—
101C2×D4×C97—21no—
102C2×D4×C97—21no—
103C2×D4×C97—21no—
104C2×D4×C97—21no—
105C2×D4×C97—21no—
106C2×D4×C97—41no—
107C2×D4×C97—41no—
108C2×D4×C97—44no—
109C2×D4×C97D411yescharacteristic
110C2×D4×C97C2311yescharacteristic
111C2×D4×C97C2311yescharacteristic
112C2×D4×C97C2311yescharacteristic
113C22×C4×C97—21no—
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C4⋊C4×C97D411yescharacteristic
117C4⋊C4×C97C2311yescharacteristic
118C22⋊C4×C97—21no—
119C22⋊C4×C97—21no—
120C22⋊C4×C97—21no—
121C22⋊C4×C97D411yescharacteristic
122C22⋊C4×C97D411yescharacteristic
123C42×C97C2311yescharacteristic
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C23×C97—42no—
128C23×C97—42no—
129C23×C97—42no—
130C23×C97—42no—
131C23×C97—81no—
132C23×C97—81no—
133C23×C97C2×D411yescharacteristic
134C23×C97C2×D411yescharacteristic
135C22×D97—21no—
136C22×D97—21no—
137C22×D97—42no—
138C22×D97—42no—
139C22×D97—81no—
140D4×C97—22no—
141D4×C97—21no—
142D4×C97—21no—
143D4×C97—21no—
144D4×C97—21no—
145D4×C97—21no—
146D4×C97—42no—
147D4×C97—42no—
148D4×C97—42no—
149D4×C97—42no—
150D4×C97—42no—
151D4×C97—81no—
152D4×C97C2×D412yes—
153C2×C4×C97—21no—
154C2×C4×C97—21no—
155C2×C4×C97—21no—
156C2×C4×C97—21no—
157C2×C4×C97—21no—
158C2×C4×C97—42no—
159C2×C4×C97C2×D411yescharacteristic
160C2×C4×C97C2×D411yescharacteristic
161C2×C4×C97C2×D411yescharacteristic
162C2×C4×C97C2×D411yescharacteristic
163C2×C4×C97C2411yescharacteristic
164C22⋊D97—42no—
165C22⋊D97—81no—
166C2×Dic97—21no—
167C2×Dic97—21no—
168D388—21no—
169D388—21no—
170D388—21no—
171D388—42no—
172D388C2×D412yescomplemented
173C4×D97—42no—
174C4×D97—81no—
175Dic194—21no—
176Dic194—21no—
177#11715895—21no—
178#11715895—21no—
179#11715895—21no—
180#11715895C2×D412yescomplemented
181C22×C97—21no—
182C22×C97—21no—
183C22×C97—21no—
184C22×C97—22no—
185C22×C97—21no—
186C22×C97—21no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97—41no—
190C22×C97—41no—
191C22×C97—41no—
192C22×C97—41no—
193C22×C97—41no—
194C22×C97—41no—
195C22×C97—84no—
196C22×C97—84no—
197C22×C97AGL(1,ℤ/8)12yescomplemented
198C22×C97C22×D411yescharacteristic
199C2×D97—22no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C2×D97—41no—
204C2×D97—41no—
205C2×D97—84no—
206C4×C97—21no—
207C4×C97—21no—
208C4×C97—21no—
209C4×C97—21no—
210C4×C97—41no—
211C4×C97C22×D411yescharacteristic
212C4×C97H(5,𝔽2)11yescharacteristic
213Dic97—41no—
214Dic97—41no—
215C2×C97—22no—
216C2×C97—41no—
217C2×C97—42no—
218C2×C97—81no—
219C2×C97—81no—
220C2×C97D4211yescharacteristic
221C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C97#57611yescharacteristic
223D97—42no—
224D97—81no—
225#3264—971nomaximal, Sylow
226C97#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—971no—
228C22×D8—971no—
229#546—971no—
230D42D9711yescharacteristic, complemented
231#496—971no—
232#469—971no—
233#465—971no—
234#463—971no—
235#459—971no—
236#450—971no—
237#449—971no—
238#447—971no—
239#447—971no—
240#442—971no—
241#435—971no—
242C2×#41—971no—
243C22×D4—21no—
244C22×D4—971no—
245C22×D4C2×D9711yescharacteristic
246C22×D4C2×D9711yescharacteristic, complemented
247AGL(1,ℤ/8)—1944no—
248C2×SD16—971no—
249C2×SD16—971no—
250C2×SD16—971no—
251C2×D8—971no—
252C2×D8—971no—
253C2×D8—971no—
254C2×D8—971no—
255C2×D8—971no—
256C2×D8—971no—
257C2×D8—1944no—
258C2×M16—971no—
259C22×C8—971no—
260#127C2×D9711yescharacteristic, complemented
261#124—971no—
262#123—1941no—
263C22⋊Q8—971no—
264#121—21no—
265#121—971no—
266#121—1941no—
267#121C2×D9711yescharacteristic
268#121C2×D9711yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—1941no—
272C4×D4—971no—
273C4×D4C2×D9711yescharacteristic
274C4×D4C2×D9711yescharacteristic, complemented
275C2×C22⋊C4—1941no—
276#106—971no—
277C4⋊C8—971no—
278#103—971no—
279#102—971no—
280#102—971no—
281#102—971no—
282#102—971no—
283#102—971no—
284C22⋊C8—971no—
285C22⋊C8—971no—
286#97—971no—
287C24—21no—
288C24—21no—
289C24—1941no—
290#41—3881no—
291#41—1942no—
292C2×Q8—971no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—971no—
305C2×D4—971no—
306C2×D4—3881no—
307C2×D4—1942no—
308C2×D4—1941no—
309C2×D4—1941no—
310C2×D4—1941no—
311C2×D4C22⋊D9711yescharacteristic
312C2×D4C22×D9711yescharacteristic
313C2×D4C22×D9711yescharacteristic
314C2×D4C22×D9711yescharacteristic
315C22×C4—21no—
316C22×C4—1941no—
317C22×C4—1941no—
318C22×C4C22×D9711yescharacteristic
319C22×C4C22×D9711yescharacteristic
320SD16—3881no—
321SD16—1942no—
322SD16—1942no—
323D8—974no—
324D8—3881no—
325D8—1942no—
326D8—1942no—
327D8—1942no—
328D8—1942no—
329M16—1942no—
330C2×C8—971no—
331C2×C8—971no—
332C2×C8—971no—
333C2×C8—971no—
334C2×C8—1942no—
335C4⋊C4—971no—
336C4⋊C4—1941no—
337C4⋊C4C22⋊D9711yescharacteristic
338C4⋊C4C22×D9711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—1941no—
343C22⋊C4—1941no—
344C22⋊C4—1941no—
345C22⋊C4—1941no—
346C22⋊C4—1941no—
347C22⋊C4C22⋊D9711yescharacteristic
348C22⋊C4C22⋊D9711yescharacteristic
349C42C22×D9711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—7761no—
360C23—3882no—
361C23—3882no—
362C23—1941no—
363C23—1941no—
364C23C2×C22⋊D9711yescharacteristic
365C23C2×C22⋊D9711yescharacteristic
366Q8—1941no—
367Q8—1941no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—7761no—
381D4—972no—
382D4—3882no—
383D4—3882no—
384D4—1941no—
385D4—1941no—
386D4—1941no—
387D4D4×D9712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—7761no—
395C2×C4—3882no—
396C2×C4—1941no—
397C2×C4—1941no—
398C2×C4C2×C22⋊D9711yescharacteristic
399C2×C4C2×C22⋊D9711yescharacteristic
400C2×C4C2×C22⋊D9711yescharacteristic
401C2×C4C2×C22⋊D9711yescharacteristic
402C2×C4C23×D9711yescharacteristic
403C8—972no—
404C8—1941no—
405C8—1941no—
406C8—1941no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—7764no—
424C22—3881no—
425C22—3881no—
426C22—3881no—
427C22—3881no—
428C22—3881no—
429C22—1942no—
430C22#1569083612yescomplemented
431C22C22×C22⋊D9711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—3881no—
438C4—3881no—
439C4C2×D4×D9711yescharacteristic
440C4#1569085511yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—7761no—
447C2—3882no—
448C2D4×C22⋊D9711yescharacteristic
449C2C2×#1569083611yescharacteristic
450C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingD42×C97C2
socleC22×C97C22×D4
hypercenterD42D97
hypocenterC97#3264
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. C22×#13095496C2semidirect
3. #20261381C2semidirect
4. #20261122C2semidirect
5. #20260986C2semidirect
6. #20260962C2semidirect
7. #20260950C2semidirect
8. #20260927C2semidirect
9. #20260886C2semidirect
10. #20260878C2semidirect
11. #20260869C2semidirect
12. #20260868C2semidirect
13. #20260845C2semidirect
14. #20260807C2semidirect
15. D42×C97C2semidirect
16. C2×#13095511C22semidirect
17. C22×D388C22semidirect
19. C2×#13095499C22semidirect
20. C2×#13095499C22semidirect
21. C2×#13095499C22semidirect
25. C2×#13095496C22semidirect
26. C2×#13095496C22semidirect
27. C2×#13095496C22semidirect
28. C2×#13095496C22semidirect
29. C2×#13095495C22semidirect
30. C22×#11715895C22semidirect
31. #15690798C22semidirect
33. #15690789C22semidirect
34. #15690784C22semidirect
37. C4×D388C22semidirect
39. #15690741C22semidirect
40. C4⋊#11715895C22semidirect
41. #15690733C22semidirect
42. #15690731C22semidirect
43. #15690731C22semidirect
44. #15690731C22semidirect
45. #15690730C22semidirect
46. #15690730C22semidirect
47. C22⋊#11715895C22semidirect
48. C22⋊#11715895C22semidirect
49. #15690721C22semidirect
52. C22×D4×C97C22semidirect
53. #127×C97C22semidirect
56. #121×C97C22semidirect
60. C4×D4×C97C22semidirect
82. #13095496D4semidirect
172. D388C2×D4semidirect
180. #11715895C2×D4semidirect
197. C22×C97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
230. D42D97semidirect
246. C22×D4C2×D97semidirect
260. #127C2×D97semidirect
268. #121C2×D97semidirect
274. C4×D4C2×D97semidirect
430. C22#15690836semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
C22×#130954962
#202613812
#202611222
#202609862
#202609622
#202609502
#202609272
#202608862
#202608782
#202608692
#202608682
#202608452
#202608072
D42×C972
#326497