Finite Group Databasebeta

← #27153216

Subgroups — #27153216

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
2#20261379C211yesmaximal, characteristic, complemented
3D4×C22⋊D97C211yesmaximal, characteristic, complemented
4#20261123C211yesmaximal, characteristic, complemented
5#20260985C211yesmaximal, characteristic, complemented
6#20260964C211yesmaximal, characteristic, complemented
7#20260948C211yesmaximal, characteristic, complemented
8#20260928C211yesmaximal, characteristic, complemented
9#20260886C211yesmaximal, characteristic, complemented
10#20260877C211yesmaximal, characteristic, complemented
11#20260869C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13#20260847C211yesmaximal, characteristic, complemented
14#20260806C211yesmaximal, characteristic, complemented
15C22×D8×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic, complemented
17C22×C22⋊D97—21no—
18C2×D4×D97C2211yescharacteristic, complemented
19#15690834—24no—
20C2×#13095500C2211yescharacteristic, complemented
21C2×#13095499C2211yescharacteristic, complemented
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095496C2211yescharacteristic, complemented
24C2×#13095496C2211yescharacteristic, complemented
25C2×#13095494C2211yescharacteristic, complemented
26#15690806C2211yescharacteristic, complemented
27#15690797C2211yescharacteristic, complemented
28#15690791—21no—
29#15690789C2211yescharacteristic, complemented
30#15690784C2211yescharacteristic
31#15690782—21no—
32#15690782—21no—
33#15690781C2211yescharacteristic, complemented
34#15690781C2211yescharacteristic, complemented
35#15690779—21no—
36#15690779—21no—
37#15690778—21no—
38D4×Dic97C2211yescharacteristic, complemented
39D4×Dic97C2211yescharacteristic, complemented
40C4×C22⋊D97C2211yescharacteristic
41C2×C22⋊Dic97—21no—
42#15690742C2211yescharacteristic, complemented
43#15690739C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic, complemented
46#15690731C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic, complemented
48#15690731C2211yescharacteristic, complemented
49#15690731C2211yescharacteristic, complemented
50C22⋊#11715895C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic, complemented
52#15690720C2211yescharacteristic, complemented
53C22×D4×C97C2211yescharacteristic
54C22×D4×C97C2211yescharacteristic, complemented
55C2×D8×C97—24no—
56C2×D8×C97C2211yescharacteristic
57C2×D8×C97C2211yescharacteristic
58C2×D8×C97C2211yescharacteristic, complemented
59C2×D8×C97C2211yescharacteristic, complemented
60C22×C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095510—22no—
63#13095510—41no—
64C2×Dic194C2311yescharacteristic
65C2×D388C2311yescharacteristic
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—21no—
73C2×C22⋊D97—44no—
74C2×C22⋊D97D411yescharacteristic
75D4×D97—22no—
76D4×D97—41no—
77C22×Dic97—21no—
78C22×Dic97—21no—
79C22×Dic97—21no—
80C2×C4×D97C2311yescharacteristic
81#13095500—22no—
82#13095499—22no—
83#13095499—41no—
84#13095496—22no—
85#13095496—41no—
86#13095494—22no—
87C2×#11715895C2311yescharacteristic
88C2×#11715895C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90C4⋊Dic97C2311yescharacteristic
91#13095490—21no—
92#13095490D411yescharacteristic
93C22⋊Dic97—21no—
94C22⋊Dic97—21no—
95C22⋊Dic97—21no—
96C22⋊Dic97—21no—
97C22⋊Dic97—21no—
98C22⋊Dic97—21no—
99C22⋊Dic97—21no—
100C22⋊Dic97D411yescharacteristic
101#13095488—21no—
102#13095488D411yescharacteristic
103C4×Dic97C2311yescharacteristic
104C24×C97—21no—
105C24×C97—21no—
106C2×D4×C97—22no—
107C2×D4×C97—22no—
108C2×D4×C97—21no—
109C2×D4×C97—21no—
110C2×D4×C97—41no—
111C2×D4×C97—41no—
112C2×D4×C97C2311yescharacteristic
113C2×D4×C97C2311yescharacteristic
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117D8×C97—22no—
118D8×C97—22no—
119D8×C97—22no—
120D8×C97D414yescomplemented
121C2×C8×C97—22no—
122C2×C8×C97C2311yescharacteristic
123C2×C8×C97C2311yescharacteristic
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C23×C97—21no—
128C23×C97—42no—
129C23×C97—42no—
130C23×C97—42no—
131C23×C97—42no—
132C23×C97—81no—
133C23×C97—81no—
134C23×C97C2×D411yescharacteristic
135C22×D97—21no—
136C22×D97—42no—
137C22×D97—42no—
138C22×D97—81no—
139C22×D97C2×D411yescharacteristic
140D4×C97—21no—
141D4×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×C97C2×D411yescharacteristic
153C2×C4×C97C2411yescharacteristic
154C22⋊D97—22no—
155C22⋊D97—42no—
156C22⋊D97—42no—
157C22⋊D97—42no—
158C22⋊D97—42no—
159C22⋊D97—42no—
160C22⋊D97—81no—
161C22⋊D97—81no—
162C2×Dic97—21no—
163C2×Dic97—21no—
164C2×Dic97—21no—
165C2×Dic97—42no—
166C2×Dic97—42no—
167C2×Dic97—81no—
168C2×Dic97C2×D411yescharacteristic
169C2×Dic97C2×D411yescharacteristic
170C2×Dic97C2×D411yescharacteristic
171D388—21no—
172D388—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—21no—
182C22×C97—21no—
183C22×C97—22no—
184C22×C97—22no—
185C22×C97—41no—
186C22×C97—41no—
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—21no—
200C2×D97—21no—
201C2×D97—21no—
202C2×D97—41no—
203C2×D97—41no—
204C2×D97—41no—
205C2×D97—84no—
206C4×C97—21no—
207C4×C97C22×D411yescharacteristic
208C4×C97H(5,𝔽2)11yescharacteristic
209Dic97—21no—
210Dic97—21no—
211Dic97—21no—
212Dic97—41no—
213Dic97—41no—
214Dic97—41no—
215C2×C97—22no—
216C2×C97—42no—
217C2×C97—42no—
218C2×C97—81no—
219C2×C97—81no—
220C2×C97D4211yescharacteristic
221C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C97#57611yescharacteristic
223D97—41no—
224D97—81no—
225#3264—971nomaximal, Sylow
226C97#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—971no—
228C22×D8D9711yescharacteristic, complemented
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×#41—971no—
243C22×D4—971no—
244C22×D4—1941no—
245C22×D4C2×D9711yescharacteristic
246C22×D4C2×D9711yescharacteristic, complemented
247AGL(1,ℤ/8)—1944no—
248C2×SD16—971no—
249C2×SD16—971no—
250C2×SD16—971no—
251C2×D8—24no—
252C2×D8—971no—
253C2×D8—971no—
254C2×D8C2×D9711yescharacteristic
255C2×D8C2×D9711yescharacteristic
256C2×D8C2×D9711yescharacteristic, complemented
257C2×D8C2×D9711yescharacteristic, complemented
258C2×M16—971no—
259C22×C8C2×D9711yescharacteristic, complemented
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—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—41no—
298C2×D4—41no—
299C2×D4—971no—
300C2×D4—971no—
301C2×D4—3884no—
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×D4C22×D9711yescharacteristic
312C2×D4C22×D9711yescharacteristic
313C2×D4C22×D9711yescharacteristic
314C2×D4C22×D9711yescharacteristic
315C22×C4—971no—
316C22×C4—1941no—
317C22×C4—1941no—
318C22×C4—1941no—
319C22×C4C22×D9711yescharacteristic
320SD16—3881no—
321SD16—1942no—
322SD16—1942no—
323D8—22no—
324D8—22no—
325D8—22no—
326D8—3881no—
327D8—1942no—
328D8C22⋊D9714yescomplemented
329M16—1942no—
330C2×C8—22no—
331C2×C8—971no—
332C2×C8—971no—
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—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—7761no—
361C23—971no—
362C23—3882no—
363C23—3882no—
364C23—1941no—
365C23D4×D9711yescharacteristic
366Q8—1941no—
367Q8—1941no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—7761no—
377D4—7761no—
378D4—3882no—
379D4—3882no—
380D4—3882no—
381D4—3882no—
382D4—3882no—
383D4—1942no—
384D4—1941no—
385D4—1941no—
386D4C2×C22⋊D9712yes—
387D4C2×C22⋊D9712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—7761no—
391C2×C4—971no—
392C2×C4—971no—
393C2×C4—971no—
394C2×C4—3882no—
395C2×C4—3882no—
396C2×C4—1941no—
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—22no—
410C22—22no—
411C22—41no—
412C22—41no—
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—1941no—
428C22—1941no—
429C22—1941no—
430C22#1569083412yescomplemented
431C22C2×D4×D9711yescharacteristic
432C4—21no—
433C4—3881no—
434C4—3881no—
435C4—3881no—
436C4—1941no—
437C4—1941no—
438C4—1941no—
439C4C22×C22⋊D9711yescharacteristic
440C4#1569085511yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—7761no—
447C2—3881no—
448C2D4×C22⋊D9711yescharacteristic
449C2C2×#1569083411yescharacteristic
450C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×D8×C97C2
socleC22×C97C22×D4
hypercenterC22×D8D97
hypocenterC97#3264
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. #20261379C2semidirect
3. D4×C22⋊D97C2semidirect
4. #20261123C2semidirect
5. #20260985C2semidirect
6. #20260964C2semidirect
7. #20260948C2semidirect
8. #20260928C2semidirect
9. #20260886C2semidirect
10. #20260877C2semidirect
11. #20260869C2semidirect
12. #20260867C2semidirect
13. #20260847C2semidirect
14. #20260806C2semidirect
15. C22×D8×C97C2semidirect
16. C2×#13095510C22semidirect
18. C2×D4×D97C22semidirect
20. C2×#13095500C22semidirect
21. C2×#13095499C22semidirect
22. C2×#13095499C22semidirect
23. C2×#13095496C22semidirect
24. C2×#13095496C22semidirect
25. C2×#13095494C22semidirect
26. #15690806C22semidirect
27. #15690797C22semidirect
29. #15690789C22semidirect
33. #15690781C22semidirect
34. #15690781C22semidirect
38. D4×Dic97C22semidirect
39. D4×Dic97C22semidirect
42. #15690742C22semidirect
43. #15690739C22semidirect
44. #15690735C22semidirect
45. #15690732C22semidirect
46. #15690731C22semidirect
47. #15690731C22semidirect
48. #15690731C22semidirect
49. #15690731C22semidirect
50. C22⋊#11715895C22semidirect
51. #15690722C22semidirect
52. #15690720C22semidirect
54. C22×D4×C97C22semidirect
58. C2×D8×C97C22semidirect
59. C2×D8×C97C22semidirect
60. C22×C8×C97C22semidirect
120. D8×C97D4semidirect
149. D4×C97C2×D4semidirect
178. C8×C97C2×D4semidirect
197. C22×C97AGL(1,ℤ/8)semidirect
226. C97#3264semidirect
228. C22×D8D97semidirect
246. C22×D4C2×D97semidirect
256. C2×D8C2×D97semidirect
257. C2×D8C2×D97semidirect
259. C22×C8C2×D97semidirect
328. D8C22⋊D97semidirect
387. D4C2×C22⋊D97semidirect
406. C8C2×C22⋊D97semidirect
430. C22#15690834semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
#202613792
D4×C22⋊D972
#202611232
#202609852
#202609642
#202609482
#202609282
#202608862
#202608772
#202608692
#202608672
#202608472
#202608062
C22×D8×C972
#326497