Finite Group Databasebeta

← #27153349

Subgroups — #27153349

Order 12416.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic, complemented
2C2×#15690835C211yesmaximal, characteristic, complemented
3#20261404C211yesmaximal, characteristic, complemented
4#20261369C211yesmaximal, characteristic, complemented
5#20261110C211yesmaximal, characteristic, complemented
6#20260987C211yesmaximal, characteristic, complemented
7#20260986C211yesmaximal, characteristic, complemented
8#20260933C211yesmaximal, characteristic, complemented
9#20260929C211yesmaximal, characteristic, complemented
10#20260883C211yesmaximal, characteristic, complemented
11#20260871C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13#20260820C211yesmaximal, characteristic, complemented
14#20260810C211yesmaximal, characteristic, complemented
15#449×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic, complemented
17C22×D388C2211yescharacteristic
18C2×D4×D97—21no—
19C2×D4×D97C2211yescharacteristic
20#15690837—24no—
21#15690835—24no—
22C2×#13095498C2211yescharacteristic
23C2×#13095498C2211yescharacteristic, complemented
24C2×D776C2211yescharacteristic
25C2×D776C2211yescharacteristic, complemented
26C2×#13095496C2211yescharacteristic
27C2×#13095496C2211yescharacteristic, complemented
28C2×#13095496C2211yescharacteristic, complemented
29C2×#13095495C2211yescharacteristic, complemented
30C2×#13095494C2211yescharacteristic, complemented
31#15690806—21no—
32#15690806C2211yescharacteristic
33#15690806C2211yescharacteristic, complemented
34#15690785C2211yescharacteristic
35#15690784—21no—
36#15690784C2211yescharacteristic, complemented
37#15690783—21no—
38#15690781—21no—
39#15690781C2211yescharacteristic, complemented
40#15690780—21no—
41#15690778—21no—
42Q8×Dic97C2211yescharacteristic, complemented
43#15690772C2211yescharacteristic
44C4×C22⋊D97—21no—
45C4×C22⋊D97C2211yescharacteristic
46#15690741C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic, complemented
50#15690731C2211yescharacteristic, complemented
51#15690730C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53C8×Dic97C2211yescharacteristic, complemented
54C2×#41×C97C2211yescharacteristic, complemented
55C2×SD16×C97C2211yescharacteristic, complemented
56C2×D8×C97C2211yescharacteristic, complemented
57#121×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60C22⋊C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63#13095512—22no—
64#13095512—41no—
65C2×D388—22no—
66C2×D388—21no—
67C2×D388—21no—
68C2×D388—21no—
69C2×D388—41no—
70C2×D388C2311yescharacteristic
71C2×D388C2311yescharacteristic
72C2×D388C2311yescharacteristic
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76C2×C22⋊D97—21no—
77C2×C22⋊D97—21no—
78C2×C22⋊D97—21no—
79C2×C22⋊D97—21no—
80C2×C22⋊D97D411yescharacteristic
81D4×D97—22no—
82D4×D97—41no—
83D4×D97—44no—
84C2×C4×D97—21no—
85C2×C4×D97—21no—
86C2×C4×D97C2311yescharacteristic
87#13095498—22no—
88#13095498D414yescomplemented
89D776—22no—
90D776—41no—
91#13095496—22no—
92#13095496—22no—
93#13095496—22no—
94#13095495—22no—
95#13095494—22no—
96C2×#11715895C2311yescharacteristic
97C2×#11715895C2311yescharacteristic
98C4⋊Dic97—21no—
99C4⋊Dic97C2311yescharacteristic
100#13095490D411yescharacteristic
101C22⋊Dic97—21no—
102C22⋊Dic97D411yescharacteristic
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488D411yescharacteristic
107C4×Dic97—21no—
108C4×Dic97C2311yescharacteristic
109#41×C97—22no—
110#41×C97—41no—
111C2×Q8×C97C2311yescharacteristic
112C2×D4×C97—21no—
113C2×D4×C97—21no—
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97—21no—
117C22×C4×C97C2311yescharacteristic
118SD16×C97—22no—
119D8×C97—41no—
120C2×C8×C97C2311yescharacteristic
121C2×C8×C97C2311yescharacteristic
122C4⋊C4×C97C2311yescharacteristic
123C22⋊C4×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97C2×D411yescharacteristic
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—81no—
135C22×D97—81no—
136C22×D97C2×D411yescharacteristic
137Q8×C97—21no—
138Q8×C97C2×D412yescomplemented
139D4×C97—21no—
140D4×C97—21no—
141D4×C97—21no—
142D4×C97—21no—
143D4×C97—42no—
144D4×C97—42no—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—42no—
150C2×C4×C97C2×D411yescharacteristic
151C2×C4×C97C2411yescharacteristic
152C22⋊D97—22no—
153C22⋊D97—42no—
154C22⋊D97—42no—
155C22⋊D97—42no—
156C22⋊D97—81no—
157C22⋊D97—81no—
158C22⋊D97—81no—
159C22⋊D97—81no—
160C2×Dic97—21no—
161C2×Dic97C2×D411yescharacteristic
162C2×Dic97C2×D411yescharacteristic
163C2×Dic97C2×D411yescharacteristic
164D388—21no—
165D388—21no—
166D388—21no—
167D388—21no—
168D388—21no—
169D388—42no—
170D388—42no—
171D388—42no—
172D388C2×D412yes—
173C4×D97—21no—
174C4×D97—21no—
175C4×D97—42no—
176C4×D97—42no—
177C8×C97—21no—
178C8×C97—21no—
179#11715895—21no—
180#11715895C2×D412yescomplemented
181C22×C97—21no—
182C22×C97—21no—
183C22×C97—21no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97—41no—
190C22×C97C22×D411yescharacteristic
191C2×D97—21no—
192C2×D97—22no—
193C2×D97—21no—
194C2×D97—21no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C2×D97—84no—
204C2×D97—84no—
205C4×C97—22no—
206C4×C97—21no—
207C4×C97—41no—
208C4×C97C22×D411yescharacteristic
209C4×C97H(5,𝔽2)11yescharacteristic
210Dic97—21no—
211Dic97—21no—
212Dic97—41no—
213Dic97AGL(1,ℤ/8)12yescomplemented
214C2×C97—41no—
215C2×C97—81no—
216C2×C97—81no—
217C2×C97D4211yescharacteristic
218C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C97#57611yescharacteristic
220D97—41no—
221D97—42no—
222D97—81no—
223D97—81no—
224#3274—971nomaximal, Sylow
225C97#327411yescharacteristic, complemented, Sylow
226C2×AGL(1,ℤ/8)—971no—
227C2×AGL(1,ℤ/8)—971no—
228#550—971no—
229D42—971no—
230#493—971no—
231#469—971no—
232#469—971no—
233#460—971no—
234#459—971no—
235#449—971no—
236#449D9711yescharacteristic, complemented
237#447—971no—
238#447—971no—
239C4×SD16—971no—
240#436—971no—
241C2×#41—971no—
242C2×#41C2×D9711yescharacteristic, complemented
243C22×D4—971no—
244C22×D4—971no—
245C22×D4—1941no—
246AGL(1,ℤ/8)—1944no—
247AGL(1,ℤ/8)—1944no—
248C2×SD16—971no—
249C2×SD16—971no—
250C2×SD16C2×D9711yescharacteristic, complemented
251C2×D8—971no—
252C2×D8—971no—
253C2×D8—971no—
254C2×D8—971no—
255C2×D8—971no—
256C2×D8C2×D9711yescharacteristic, complemented
257C2×M16—971no—
258C2×M16—971no—
259#127—971no—
260#127—971no—
261#127—1941no—
262#121—971no—
263#121—971no—
264#121—971no—
265#121—1941no—
266#121—1941no—
267#121—1941no—
268#121C2×D9711yescharacteristic
269#120—1941no—
270#120—1941no—
271C4×Q8—971no—
272C4×D4—971no—
273C4×D4—971no—
274C4×D4—1941no—
275#106—971no—
276C4⋊C8—971no—
277#103C2×D9711yescharacteristic, complemented
278#102—971no—
279#102—971no—
280#102—971no—
281#102—971no—
282#102C2×D9711yescharacteristic, complemented
283C22⋊C8—971no—
284C22⋊C8C2×D9711yescharacteristic, complemented
285C4×C8—971no—
286C24—1941no—
287C24—1941no—
288#41—22no—
289#41—41no—
290#41—3881no—
291#41—1942no—
292C2×Q8C22×D9711yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—971no—
296C2×D4—971no—
297C2×D4—971no—
298C2×D4—971no—
299C2×D4—3881no—
300C2×D4—3884no—
301C2×D4—3881no—
302C2×D4—1942no—
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—1941no—
312C2×D4—1941no—
313C2×D4—1941no—
314C2×D4C22×D9711yescharacteristic
315C2×D4C22×D9711yescharacteristic
316C22×C4—21no—
317C22×C4—971no—
318C22×C4—1941no—
319C22×C4—1941no—
320C22×C4C22×D9711yescharacteristic
321SD16—22no—
322SD16—974no—
323SD16—1942no—
324D8—41no—
325D8—3881no—
326D8—1942no—
327D8—1942no—
328D8—1942no—
329D8—1942no—
330M16—1942no—
331M16—1942no—
332C2×C8—971no—
333C2×C8—971no—
334C2×C8C22×D9711yescharacteristic
335C2×C8C22×D9711yescharacteristic
336C4⋊C4—971no—
337C4⋊C4—971no—
338C4⋊C4—1941no—
339C4⋊C4C22×D9711yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—971no—
342C22⋊C4—971no—
343C22⋊C4—1941no—
344C22⋊C4—1941no—
345C22⋊C4—1941no—
346C22⋊C4—1941no—
347C42—971no—
348C42—1941no—
349C23—21no—
350C23—21no—
351C23—7761no—
352C23—7761no—
353C23—971no—
354C23—3882no—
355C23—3882no—
356C23—3882no—
357C23—3882no—
358C23—1941no—
359C23—1941no—
360C23—1941no—
361C23D4×D9711yescharacteristic
362Q8—21no—
363Q8D4×D9712yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—42no—
370D4—7761no—
371D4—7761no—
372D4—7761no—
373D4—7761no—
374D4—972no—
375D4—3882no—
376D4—3882no—
377D4—3882no—
378D4—3882no—
379D4—3882no—
380D4—3882no—
381D4—1942no—
382D4—1941no—
383D4—1941no—
384D4—1941no—
385D4—1941no—
386D4—1941no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—971no—
393C2×C4—971no—
394C2×C4—971no—
395C2×C4—3882no—
396C2×C4—3882no—
397C2×C4—1941no—
398C2×C4—1941no—
399C2×C4—1941no—
400C2×C4D4×D9711yescharacteristic
401C2×C4C23×D9711yescharacteristic
402C8—21no—
403C8—21no—
404C8—972no—
405C8—1941no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—7764no—
416C22—7764no—
417C22—3881no—
418C22—3881no—
419C22—3881no—
420C22—3881no—
421C22—3881no—
422C22—3881no—
423C22—3881no—
424C22—3881no—
425C22—1941no—
426C22—1941no—
427C22—1941no—
428C22—1942no—
429C22C2×D4×D9711yescharacteristic
430C4—21no—
431C4—22no—
432C4—41no—
433C4—972no—
434C4—3881no—
435C4—1941no—
436C4—1941no—
437C4C2×D4×D9711yescharacteristic
438C4#1569085511yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—7761no—
443C2—7761no—
444C2—3882no—
445C2—3881no—
446C2#2026136711yescharacteristic
447C2AGL(1,ℤ/8)×D9711yescharacteristic
448C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#449×C97C2
socleC22×C97C22×D4
hypercenter#449D97
hypocenterC97#3274
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×#15690835C2semidirect
3. #20261404C2semidirect
4. #20261369C2semidirect
5. #20261110C2semidirect
6. #20260987C2semidirect
7. #20260986C2semidirect
8. #20260933C2semidirect
9. #20260929C2semidirect
10. #20260883C2semidirect
11. #20260871C2semidirect
12. #20260867C2semidirect
13. #20260820C2semidirect
14. #20260810C2semidirect
15. #449×C97C2semidirect
16. C2×#13095512C22semidirect
23. C2×#13095498C22semidirect
25. C2×D776C22semidirect
27. C2×#13095496C22semidirect
28. C2×#13095496C22semidirect
29. C2×#13095495C22semidirect
30. C2×#13095494C22semidirect
33. #15690806C22semidirect
36. #15690784C22semidirect
39. #15690781C22semidirect
42. Q8×Dic97C22semidirect
46. #15690741C22semidirect
47. #15690739C22semidirect
49. #15690732C22semidirect
50. #15690731C22semidirect
51. #15690730C22semidirect
52. #15690722C22semidirect
53. C8×Dic97C22semidirect
54. C2×#41×C97C22semidirect
55. C2×SD16×C97C22semidirect
56. C2×D8×C97C22semidirect
58. #103×C97C22semidirect
59. #102×C97C22semidirect
60. C22⋊C8×C97C22semidirect
88. #13095498D4semidirect
138. Q8×C97C2×D4semidirect
180. #11715895C2×D4semidirect
213. Dic97AGL(1,ℤ/8)semidirect
225. C97#3274semidirect
236. #449D97semidirect
242. C2×#41C2×D97semidirect
250. C2×SD16C2×D97semidirect
256. C2×D8C2×D97semidirect
277. #103C2×D97semidirect
282. #102C2×D97semidirect
284. C22⋊C8C2×D97semidirect
363. Q8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×#156908352
#202614042
#202613692
#202611102
#202609872
#202609862
#202609332
#202609292
#202608832
#202608712
#202608672
#202608202
#202608102
#449×C972
#327497