Finite Group Databasebeta

← #27153210

Subgroups — #27153210

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×D8×D97C211yesmaximal, characteristic, complemented
2#20261373C211yesmaximal, characteristic, complemented
3D4×C22⋊D97C211yesmaximal, characteristic, complemented
4#20261121C211yesmaximal, characteristic, complemented
5#20260988C211yesmaximal, characteristic, complemented
6#20260961C211yesmaximal, characteristic, complemented
7#20260952C211yesmaximal, characteristic, complemented
8#20260928C211yesmaximal, characteristic, complemented
9#20260884C211yesmaximal, characteristic, complemented
10#20260880C211yesmaximal, characteristic, complemented
11#20260869C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13#20260847C211yesmaximal, characteristic, complemented
14#20260804C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97—21no—
17C2×D4×D97C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic, complemented
19C2×#13095498C2211yescharacteristic, complemented
20C2×D776C2211yescharacteristic, complemented
21C2×#13095496C2211yescharacteristic
22C2×#13095496C2211yescharacteristic, complemented
23C2×#13095496C2211yescharacteristic, complemented
24D8×D97—24no—
25C2×C8×D97C2211yescharacteristic, complemented
26#15690806C2211yescharacteristic, complemented
27#15690799C2211yescharacteristic, complemented
28#15690793—21no—
29#15690786C2211yescharacteristic, complemented
30#15690784C2211yescharacteristic, complemented
31#15690784C2211yescharacteristic, complemented
32#15690782—21no—
33#15690782—21no—
34#15690781C2211yescharacteristic
35#15690779—21no—
36#15690778—21no—
37#15690778—21no—
38D4×Dic97C2211yescharacteristic, complemented
39D4×Dic97C2211yescharacteristic, complemented
40C4×C22⋊D97C2211yescharacteristic
41C2×#13095488—21no—
42#15690742C2211yescharacteristic, complemented
43#15690739C2211yescharacteristic, complemented
44#15690734C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic, complemented
48#15690731C2211yescharacteristic, complemented
49#15690731C2211yescharacteristic, complemented
50C22⋊#11715895C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic, complemented
52#15690720C2211yescharacteristic, complemented
53C2×#41×C97C2211yescharacteristic, complemented
54C22×D4×C97C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C97—24no—
56C2×SD16×C97C2211yescharacteristic, complemented
57C2×SD16×C97C2211yescharacteristic, complemented
58C2×D8×C97C2211yescharacteristic
59C2×D8×C97C2211yescharacteristic, complemented
60C2×M16×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63C2×D388C2311yescharacteristic
64C2×D388C2311yescharacteristic
65C2×C22⋊D97—21no—
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—44no—
73C2×C22⋊D97D411yescharacteristic
74D4×D97—22no—
75D4×D97—22no—
76D4×D97—41no—
77D4×D97—41no—
78C22×Dic97—21no—
79C22×Dic97—21no—
80C2×C4×D97C2311yescharacteristic
81#13095498—41no—
82D776—22no—
83#13095496—22no—
84#13095496—22no—
85#13095496—41no—
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C8×D97—22no—
89C4⋊Dic97C2311yescharacteristic
90C4⋊Dic97C2311yescharacteristic
91#13095490—21no—
92#13095490D411yescharacteristic
93C22⋊Dic97—21no—
94C22⋊Dic97—21no—
95C22⋊Dic97—21no—
96C22⋊Dic97D411yescharacteristic
97#13095488—21no—
98#13095488—21no—
99#13095488—21no—
100#13095488—21no—
101#13095488—21no—
102#13095488D411yescharacteristic
103C4×Dic97C2311yescharacteristic
104C24×C97—21no—
105#41×C97—22no—
106#41×C97—41no—
107C2×Q8×C97C2311yescharacteristic
108C2×D4×C97—22no—
109C2×D4×C97—21no—
110C2×D4×C97—21no—
111C2×D4×C97—41no—
112C2×D4×C97C2311yescharacteristic
113C2×D4×C97C2311yescharacteristic
114C2×D4×C97C2311yescharacteristic
115C22×C4×C97—21no—
116C22×C4×C97C2311yescharacteristic
117SD16×C97—22no—
118SD16×C97—22no—
119D8×C97—22no—
120D8×C97D414yescomplemented
121M16×C97—22no—
122C2×C8×C97C2311yescharacteristic
123C2×C8×C97C2311yescharacteristic
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C23×C97—42no—
128C23×C97—42no—
129C23×C97—81no—
130C23×C97C2×D411yescharacteristic
131C22×D97—21no—
132C22×D97—21no—
133C22×D97—42no—
134C22×D97—42no—
135C22×D97—42no—
136C22×D97—42no—
137C22×D97—81no—
138C22×D97—81no—
139C22×D97C2×D411yescharacteristic
140Q8×C97—21no—
141Q8×C97—21no—
142D4×C97—21no—
143D4×C97—21no—
144D4×C97—21no—
145D4×C97—21no—
146D4×C97—42no—
147D4×C97—42no—
148D4×C97C2×D412yes—
149D4×C97C2×D412yescomplemented
150C2×C4×C97—21no—
151C2×C4×C97—21no—
152C2×C4×C97—21no—
153C2×C4×C97—42no—
154C2×C4×C97C2×D411yescharacteristic
155C2×C4×C97C2411yescharacteristic
156C22⋊D97—22no—
157C22⋊D97—42no—
158C22⋊D97—42no—
159C22⋊D97—42no—
160C22⋊D97—42no—
161C22⋊D97—42no—
162C22⋊D97—81no—
163C22⋊D97—81no—
164C2×Dic97—21no—
165C2×Dic97—21no—
166C2×Dic97—42no—
167C2×Dic97—81no—
168C2×Dic97C2×D411yescharacteristic
169C2×Dic97C2×D411yescharacteristic
170C2×Dic97C2×D411yescharacteristic
171D388—21no—
172D388—21no—
173D388—21no—
174D388—21no—
175C4×D97—21no—
176C4×D97—21no—
177C8×C97—21no—
178C8×C97C2×D412yescomplemented
179#11715895—21no—
180#11715895—21no—
181C22×C97—22no—
182C22×C97—22no—
183C22×C97—21no—
184C22×C97—21no—
185C22×C97—21no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97—41no—
190C22×C97—41no—
191C22×C97—41no—
192C22×C97—41no—
193C22×C97—84no—
194C22×C97C22×D411yescharacteristic
195C2×D97—21no—
196C2×D97—21no—
197C2×D97—41no—
198C2×D97—41no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C2×D97—84no—
204C2×D97—84no—
205C2×D97AGL(1,ℤ/8)12yescomplemented
206C4×C97—21no—
207C4×C97—41no—
208C4×C97C22×D411yescharacteristic
209C4×C97H(5,𝔽2)11yescharacteristic
210Dic97—21no—
211Dic97—21no—
212Dic97—21no—
213Dic97—41no—
214Dic97—41no—
215C2×C97—42no—
216C2×C97—42no—
217C2×C97—41no—
218C2×C97—81no—
219C2×C97D4211yescharacteristic
220C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C97#57611yescharacteristic
222D97—22no—
223D97—81no—
224D97—81no—
225#3264—971nomaximal, Sylow
226C97#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
228C22×D8—971no—
229#546—971no—
230D42—971no—
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×#41C2×D9711yescharacteristic, complemented
243C22×D4—971no—
244C22×D4—971no—
245C22×D4—1941no—
246C22×D4C2×D9711yescharacteristic, complemented
247AGL(1,ℤ/8)—24no—
248C2×SD16—971no—
249C2×SD16C2×D9711yescharacteristic, complemented
250C2×SD16C2×D9711yescharacteristic, complemented
251C2×D8—971no—
252C2×D8—971no—
253C2×D8—971no—
254C2×D8—971no—
255C2×D8—1944no—
256C2×D8C2×D9711yescharacteristic
257C2×D8C2×D9711yescharacteristic, complemented
258C2×M16C2×D9711yescharacteristic, complemented
259C22×C8—971no—
260#127—971no—
261#124—971no—
262#123—1941no—
263C22⋊Q8—971no—
264#121—971no—
265#121—971no—
266#121—971no—
267#121—1941no—
268#121—1941no—
269#120—1941no—
270#120—1941no—
271#120—1941no—
272C4×D4—971no—
273C4×D4—971no—
274C4×D4—971no—
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—1941no—
289C24—1941no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D9711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—971no—
298C2×D4—971no—
299C2×D4—971no—
300C2×D4—3884no—
301C2×D4—3881no—
302C2×D4—3881no—
303C2×D4—1941no—
304C2×D4—1941no—
305C2×D4—1941no—
306C2×D4—1941no—
307C2×D4—1941no—
308C2×D4—1941no—
309C2×D4—1941no—
310C2×D4—1942no—
311C2×D4—1942no—
312C2×D4C22×D9711yescharacteristic
313C2×D4C22×D9711yescharacteristic
314C2×D4C22×D9711yescharacteristic
315C22×C4—21no—
316C22×C4—971no—
317C22×C4—1941no—
318C22×C4—1941no—
319C22×C4C22×D9711yescharacteristic
320SD16—22no—
321SD16—22no—
322SD16—3881no—
323D8—22no—
324D8—3881no—
325D8—1942no—
326D8—1942no—
327D8—1942no—
328D8C22⋊D9714yescomplemented
329M16—22no—
330C2×C8—971no—
331C2×C8—971no—
332C2×C8—1942no—
333C2×C8C22×D9711yescharacteristic
334C2×C8C22×D9711yescharacteristic
335C4⋊C4—971no—
336C4⋊C4—971no—
337C4⋊C4—971no—
338C4⋊C4—1941no—
339C22⋊C4—971no—
340C22⋊C4—971no—
341C22⋊C4—1941no—
342C22⋊C4—1941no—
343C22⋊C4—1941no—
344C22⋊C4—1941no—
345C22⋊C4—1941no—
346C22⋊C4—1941no—
347C22⋊C4—1941no—
348C22⋊C4—1941no—
349C42—971no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—7761no—
357C23—7761no—
358C23—971no—
359C23—3882no—
360C23—3882no—
361C23—3882no—
362C23—3882no—
363C23—1941no—
364C23—1941no—
365C23D4×D9711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—7761no—
375D4—7761no—
376D4—3882no—
377D4—3882no—
378D4—3882no—
379D4—3882no—
380D4—3882no—
381D4—1942no—
382D4—1941no—
383D4—1941no—
384D4—1941no—
385D4—1941no—
386D4C2×C22⋊D9712yes—
387D4C2×C22⋊D9712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—7761no—
393C2×C4—971no—
394C2×C4—971no—
395C2×C4—971no—
396C2×C4—3882no—
397C2×C4—1941no—
398C2×C4—1941no—
399C2×C4—1941no—
400C2×C4—1941no—
401C2×C4D4×D9711yescharacteristic
402C2×C4C23×D9711yescharacteristic
403C8—21no—
404C8—1941no—
405C8—1941no—
406C8C2×C22⋊D9712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—84no—
420C22—7764no—
421C22—7764no—
422C22—972no—
423C22—3881no—
424C22—3881no—
425C22—3881no—
426C22—3881no—
427C22—3881no—
428C22—3881no—
429C22—1941no—
430C22—1941no—
431C22C2×D4×D9711yescharacteristic
432C4—21no—
433C4—41no—
434C4—3881no—
435C4—3881no—
436C4—1941no—
437C4—1941no—
438C4—1941no—
439C4C22×C22⋊D9711yescharacteristic
440C4#1569085511yescharacteristic
441C2—41no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—7761no—
446C2—7761no—
447C2—1942no—
448C2D4×C22⋊D9711yescharacteristic
449C2AGL(1,ℤ/8)×D9711yescharacteristic
450C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3264
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×D8×D97C2semidirect
2. #20261373C2semidirect
3. D4×C22⋊D97C2semidirect
4. #20261121C2semidirect
5. #20260988C2semidirect
6. #20260961C2semidirect
7. #20260952C2semidirect
8. #20260928C2semidirect
9. #20260884C2semidirect
10. #20260880C2semidirect
11. #20260869C2semidirect
12. #20260867C2semidirect
13. #20260847C2semidirect
14. #20260804C2semidirect
15. C2×AGL(1,ℤ/8)×C97C2semidirect
18. C2×D4×D97C22semidirect
19. C2×#13095498C22semidirect
20. C2×D776C22semidirect
22. C2×#13095496C22semidirect
23. C2×#13095496C22semidirect
25. C2×C8×D97C22semidirect
26. #15690806C22semidirect
27. #15690799C22semidirect
29. #15690786C22semidirect
30. #15690784C22semidirect
31. #15690784C22semidirect
38. D4×Dic97C22semidirect
39. D4×Dic97C22semidirect
42. #15690742C22semidirect
43. #15690739C22semidirect
44. #15690734C22semidirect
45. #15690732C22semidirect
46. #15690732C22semidirect
47. #15690731C22semidirect
48. #15690731C22semidirect
49. #15690731C22semidirect
50. C22⋊#11715895C22semidirect
51. #15690722C22semidirect
52. #15690720C22semidirect
53. C2×#41×C97C22semidirect
54. C22×D4×C97C22semidirect
56. C2×SD16×C97C22semidirect
57. C2×SD16×C97C22semidirect
59. C2×D8×C97C22semidirect
60. C2×M16×C97C22semidirect
120. D8×C97D4semidirect
149. D4×C97C2×D4semidirect
178. C8×C97C2×D4semidirect
205. C2×D97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
227. C2×AGL(1,ℤ/8)D97semidirect
242. C2×#41C2×D97semidirect
246. C22×D4C2×D97semidirect
249. C2×SD16C2×D97semidirect
250. C2×SD16C2×D97semidirect
257. C2×D8C2×D97semidirect
258. C2×M16C2×D97semidirect
328. D8C22⋊D97semidirect
387. D4C2×C22⋊D97semidirect
406. C8C2×C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D972
#202613732
D4×C22⋊D972
#202611212
#202609882
#202609612
#202609522
#202609282
#202608842
#202608802
#202608692
#202608672
#202608472
#202608042
C2×AGL(1,ℤ/8)×C972
#326497