Finite Group Databasebeta

← #27153218

Subgroups — #27153218

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×#15690834C211yesmaximal, characteristic, complemented
2C2×D8×D97C211yesmaximal, characteristic, complemented
3#20261380C211yesmaximal, characteristic, complemented
4D4×D388C211yesmaximal, characteristic, complemented
5#20261124C211yesmaximal, characteristic, complemented
6#20260989C211yesmaximal, characteristic, complemented
7#20260965C211yesmaximal, characteristic, complemented
8#20260950C211yesmaximal, characteristic, complemented
9#20260930C211yesmaximal, characteristic, complemented
10#20260887C211yesmaximal, characteristic, complemented
11#20260881C211yesmaximal, characteristic, complemented
12#20260870C211yesmaximal, characteristic, complemented
13#20260870C211yesmaximal, characteristic, complemented
14#20260808C211yesmaximal, characteristic, complemented
15#442×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic, complemented
17C22×D388—21no—
18C2×D4×D97C2211yescharacteristic
19C2×D4×D97C2211yescharacteristic, complemented
20C2×D4×D97C2211yescharacteristic, complemented
21#15690834—24no—
22C2×#13095500C2211yescharacteristic, complemented
23C2×#13095500C2211yescharacteristic, complemented
24C2×#13095499C2211yescharacteristic, complemented
25C2×D776C2211yescharacteristic, complemented
26C2×D776C2211yescharacteristic, complemented
27C2×#13095496C2211yescharacteristic
28C2×#13095496C2211yescharacteristic, complemented
29C2×#13095496C2211yescharacteristic, complemented
30D8×D97—24no—
31C2×#13095494C2211yescharacteristic, complemented
32C2×C8×D97C2211yescharacteristic, complemented
33#15690807C2211yescharacteristic, complemented
34#15690798C2211yescharacteristic, complemented
35#15690794—21no—
36#15690790C2211yescharacteristic, complemented
37#15690785C2211yescharacteristic
38#15690785C2211yescharacteristic, complemented
39#15690785C2211yescharacteristic, complemented
40#15690784—21no—
41#15690784—21no—
42#15690780—21no—
43#15690780—21no—
44#15690780—21no—
45C4×D388C2211yescharacteristic
46C2×#13095488—21no—
47C4⋊#11715895C2211yescharacteristic, complemented
48#15690733C2211yescharacteristic, complemented
49#15690730C2211yescharacteristic, complemented
50#15690730C2211yescharacteristic, complemented
51#15690730C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53#15690722C2211yescharacteristic, complemented
54C2×D8×C97C2211yescharacteristic
55C4×D4×C97C2211yescharacteristic, complemented
56C4×D4×C97C2211yescharacteristic, complemented
57#106×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60#97×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63C23×D97—21no—
64#13095510—22no—
65#13095510—41no—
66C2×Dic194C2311yescharacteristic
67C2×D388—21no—
68C2×D388—21no—
69C2×D388—21no—
70C2×D388—21no—
71C2×D388—21no—
72C2×D388—44no—
73C2×D388D411yescharacteristic
74C2×D388C2311yescharacteristic
75C2×D388C2311yescharacteristic
76C2×D388C2311yescharacteristic
77C2×C22⋊D97—21no—
78C2×C22⋊D97—21no—
79C2×C22⋊D97—21no—
80C2×C22⋊D97—21no—
81D4×D97—22no—
82D4×D97—22no—
83D4×D97—22no—
84D4×D97—41no—
85D4×D97—41no—
86D4×D97—41no—
87C22×Dic97—21no—
88C2×C4×D97C2311yescharacteristic
89C2×C4×D97C2311yescharacteristic
90#13095500—22no—
91#13095500—41no—
92#13095499—22no—
93D776—22no—
94D776—41no—
95#13095496—22no—
96#13095496—22no—
97#13095496—22no—
98#13095494—22no—
99C2×#11715895C2311yescharacteristic
100C2×#11715895C2311yescharacteristic
101C8×D97—22no—
102C4⋊Dic97—21no—
103C4⋊Dic97D411yescharacteristic
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488—21no—
109#13095488—21no—
110#13095488D411yescharacteristic
111#13095488D411yescharacteristic
112C2×D4×C97C2311yescharacteristic
113C2×D4×C97C2311yescharacteristic
114C22×C4×C97—21no—
115C22×C4×C97—21no—
116D8×C97D414yescomplemented
117C2×C8×C97C2311yescharacteristic
118C2×C8×C97C2311yescharacteristic
119C4⋊C4×C97C2311yescharacteristic
120C4⋊C4×C97C2311yescharacteristic
121C22⋊C4×C97—21no—
122C22⋊C4×C97—21no—
123C42×C97C2311yescharacteristic
124C23×C97—21no—
125C23×C97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—42no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—42no—
135C22×D97—81no—
136C22×D97—81no—
137C22×D97—81no—
138C22×D97C2×D411yescharacteristic
139C22×D97C2×D411yescharacteristic
140D4×C97—21no—
141D4×C97—21no—
142D4×C97C2×D412yes—
143D4×C97C2×D412yescomplemented
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—42no—
147C2×C4×C97—81no—
148C2×C4×C97C2×D411yescharacteristic
149C2×C4×C97C2×D411yescharacteristic
150C2×C4×C97C2411yescharacteristic
151C22⋊D97—42no—
152C22⋊D97—42no—
153C22⋊D97—42no—
154C22⋊D97—42no—
155C2×Dic97—21no—
156C2×Dic97—42no—
157C2×Dic97C2×D411yescharacteristic
158C2×Dic97C2×D411yescharacteristic
159D388—22no—
160D388—21no—
161D388—21no—
162D388—21no—
163D388—21no—
164D388—21no—
165D388—21no—
166D388—42no—
167D388—42no—
168D388—42no—
169D388—81no—
170D388—81no—
171C4×D97—21no—
172C4×D97—21no—
173C4×D97—21no—
174C4×D97—21no—
175Dic194—21no—
176Dic194—21no—
177C8×C97—21no—
178C8×C97C2×D412yescomplemented
179#11715895—21no—
180#11715895—21no—
181C22×C97—22no—
182C22×C97—22no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97C22×D411yescharacteristic
188C2×D97—21no—
189C2×D97—21no—
190C2×D97—21no—
191C2×D97—21no—
192C2×D97—21no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—84no—
203C2×D97—84no—
204C2×D97—84no—
205C2×D97AGL(1,ℤ/8)12yescomplemented
206C4×C97—21no—
207C4×C97—21no—
208C4×C97—41no—
209C4×C97—41no—
210C4×C97C22×D411yescharacteristic
211C4×C97H(5,𝔽2)11yescharacteristic
212Dic97—21no—
213Dic97—21no—
214Dic97—41no—
215C2×C97—42no—
216C2×C97—42no—
217C2×C97D4211yescharacteristic
218C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C97#57611yescharacteristic
220D97—22no—
221D97—41no—
222D97—81no—
223D97—81no—
224D97—81no—
225#3264—971nomaximal, Sylow
226C97#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—971no—
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#442D9711yescharacteristic, complemented
241#435—971no—
242C2×#41—971no—
243C22×D4—971no—
244C22×D4—971no—
245C22×D4—971no—
246C22×D4—1941no—
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—1944no—
257C2×D8C2×D9711yescharacteristic
258C2×M16—971no—
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×D4C2×D9711yescharacteristic, complemented
274C4×D4C2×D9711yescharacteristic, complemented
275C2×C22⋊C4—1941no—
276#106C2×D9711yescharacteristic, complemented
277C4⋊C8—971no—
278#103—971no—
279#102—971no—
280#102—971no—
281#102—971no—
282#102C2×D9711yescharacteristic, complemented
283#102C2×D9711yescharacteristic, complemented
284C22⋊C8—971no—
285C22⋊C8—971no—
286#97C2×D9711yescharacteristic, complemented
287C24—1941no—
288C24—1941no—
289C24—1941no—
290#41—3881no—
291#41—1942no—
292C2×Q8—971no—
293C2×D4—971no—
294C2×D4—971no—
295C2×D4—971no—
296C2×D4—971no—
297C2×D4—3884no—
298C2×D4—3881no—
299C2×D4—3881no—
300C2×D4—3881no—
301C2×D4—1941no—
302C2×D4—1941no—
303C2×D4—1941no—
304C2×D4—1941no—
305C2×D4—1941no—
306C2×D4—1941no—
307C2×D4—1941no—
308C2×D4—1942no—
309C2×D4—1942no—
310C2×D4—1942no—
311C2×D4—1941no—
312C2×D4—1941no—
313C2×D4C22×D9711yescharacteristic
314C2×D4C22×D9711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—971no—
318C22×C4—971no—
319C22×C4—1941no—
320SD16—3881no—
321SD16—1942no—
322SD16—1942no—
323D8—3881no—
324D8—1942no—
325D8—1942no—
326D8—1942no—
327D8—1942no—
328D8D38814yescomplemented
329M16—1942no—
330C2×C8—971no—
331C2×C8—971no—
332C2×C8—1942no—
333C2×C8C22×D9711yescharacteristic
334C2×C8C22×D9711yescharacteristic
335C4⋊C4—971no—
336C4⋊C4—1941no—
337C4⋊C4C22×D9711yescharacteristic
338C4⋊C4C22×D9711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—971no—
342C22⋊C4—971no—
343C22⋊C4—1941no—
344C22⋊C4—1941no—
345C22⋊C4—1941no—
346C22⋊C4—1941no—
347C22⋊C4—1941no—
348C22⋊C4—1941no—
349C42C22×D9711yescharacteristic
350C23—21no—
351C23—21no—
352C23—7761no—
353C23—7761no—
354C23—7761no—
355C23—971no—
356C23—971no—
357C23—3882no—
358C23—3882no—
359C23—3882no—
360C23—3882no—
361C23—3882no—
362C23—3882no—
363C23—1941no—
364C23—1941no—
365C23—1941no—
366Q8—1941no—
367Q8—1941no—
368D4—21no—
369D4—21no—
370D4—7761no—
371D4—7761no—
372D4—3882no—
373D4—3882no—
374D4—3882no—
375D4—3882no—
376D4—3882no—
377D4—3882no—
378D4—3882no—
379D4—1942no—
380D4—1941no—
381D4—1941no—
382D4—1941no—
383D4—1941no—
384D4—1941no—
385D4—1941no—
386D4C2×D38812yes—
387D4C2×D38812yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—971no—
393C2×C4—971no—
394C2×C4—3882no—
395C2×C4—1941no—
396C2×C4—1941no—
397C2×C4—1941no—
398C2×C4—1941no—
399C2×C4—1941no—
400C2×C4D4×D9711yescharacteristic
401C2×C4D4×D9711yescharacteristic
402C2×C4C23×D9711yescharacteristic
403C8—21no—
404C8—1941no—
405C8—1941no—
406C8C2×D38812yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—7764no—
414C22—7764no—
415C22—7764no—
416C22—972no—
417C22—3881no—
418C22—3881no—
419C22—3881no—
420C22—3881no—
421C22—3881no—
422C22—3881no—
423C22—3881no—
424C22—3881no—
425C22—3881no—
426C22—1941no—
427C22—1941no—
428C22—1941no—
429C22—1941no—
430C22—1941no—
431C22C2×D4×D9711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—3881no—
437C4—1941no—
438C4—1941no—
439C4C22×D38811yescharacteristic
440C4#1569085611yescharacteristic
441C2—42no—
442C2—42no—
443C2—7761no—
444C2—7761no—
445C2—7761no—
446C2—3881no—
447C2—1942no—
448C2D4×D38811yescharacteristic
449C2AGL(1,ℤ/8)×D9711yescharacteristic
450C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#442×C97C2
socleC22×C97C22×D4
hypercenter#442D97
hypocenterC97#3264
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. C2×D8×D97C2semidirect
3. #20261380C2semidirect
4. D4×D388C2semidirect
5. #20261124C2semidirect
6. #20260989C2semidirect
7. #20260965C2semidirect
8. #20260950C2semidirect
9. #20260930C2semidirect
10. #20260887C2semidirect
11. #20260881C2semidirect
12. #20260870C2semidirect
13. #20260870C2semidirect
14. #20260808C2semidirect
15. #442×C97C2semidirect
16. C2×#13095510C22semidirect
19. C2×D4×D97C22semidirect
20. C2×D4×D97C22semidirect
22. C2×#13095500C22semidirect
23. C2×#13095500C22semidirect
24. C2×#13095499C22semidirect
25. C2×D776C22semidirect
26. C2×D776C22semidirect
28. C2×#13095496C22semidirect
29. C2×#13095496C22semidirect
31. C2×#13095494C22semidirect
32. C2×C8×D97C22semidirect
33. #15690807C22semidirect
34. #15690798C22semidirect
36. #15690790C22semidirect
38. #15690785C22semidirect
39. #15690785C22semidirect
47. C4⋊#11715895C22semidirect
48. #15690733C22semidirect
49. #15690730C22semidirect
50. #15690730C22semidirect
51. #15690730C22semidirect
52. #15690722C22semidirect
53. #15690722C22semidirect
55. C4×D4×C97C22semidirect
56. C4×D4×C97C22semidirect
57. #106×C97C22semidirect
58. #102×C97C22semidirect
59. #102×C97C22semidirect
60. #97×C97C22semidirect
116. D8×C97D4semidirect
143. D4×C97C2×D4semidirect
178. C8×C97C2×D4semidirect
205. C2×D97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
240. #442D97semidirect
273. C4×D4C2×D97semidirect
274. C4×D4C2×D97semidirect
276. #106C2×D97semidirect
282. #102C2×D97semidirect
283. #102C2×D97semidirect
286. #97C2×D97semidirect
328. D8D388semidirect
387. D4C2×D388semidirect
406. C8C2×D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
C2×D8×D972
#202613802
D4×D3882
#202611242
#202609892
#202609652
#202609502
#202609302
#202608872
#202608812
#202608702
#202608082
#442×C972
#326497