Finite Group Databasebeta

← #27153342

Subgroups — #27153342

Order 12416.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic, complemented
2C2×#15690834C211yesmaximal, characteristic, complemented
3#20261380C211yesmaximal, characteristic, complemented
4#20261378C211yesmaximal, characteristic, complemented
5#20261117C211yesmaximal, characteristic, complemented
6#20260984C211yesmaximal, characteristic, complemented
7#20260980C211yesmaximal, characteristic, complemented
8#20260951C211yesmaximal, characteristic, complemented
9#20260900C211yesmaximal, characteristic, complemented
10#20260894C211yesmaximal, characteristic, complemented
11#20260870C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14#20260812C211yesmaximal, characteristic, complemented
15#463×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095510C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic, complemented
19C2×D4×D97C2211yescharacteristic, complemented
20#15690837—24no—
21#15690834—24no—
22C2×#13095501C2211yescharacteristic, complemented
23C2×#13095500C2211yescharacteristic, complemented
24C2×#13095499C2211yescharacteristic, complemented
25C2×#13095498C2211yescharacteristic, complemented
26C2×D776C2211yescharacteristic
27C2×#13095496C2211yescharacteristic
28C2×#13095496C2211yescharacteristic
29C2×#13095494C2211yescharacteristic, complemented
30C2×#13095494C2211yescharacteristic, complemented
31#15690798C2211yescharacteristic, complemented
32#15690794—21no—
33#15690792—21no—
34#15690790C2211yescharacteristic, complemented
35#15690786C2211yescharacteristic, complemented
36#15690785C2211yescharacteristic
37#15690784—21no—
38#15690783—21no—
39#15690781C2211yescharacteristic
40#15690780—21no—
41#15690780—21no—
42C4×D388C2211yescharacteristic
43C4×D388C2211yescharacteristic, complemented
44C2×#13095488—21no—
45C22⋊C4×D97—21no—
46#15690743C2211yescharacteristic, complemented
47#15690735C2211yescharacteristic, complemented
48#15690732C2211yescharacteristic, complemented
49#15690731C2211yescharacteristic, complemented
50#15690730C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53C4×#11715895C2211yescharacteristic, complemented
54C2×SD16×C97C2211yescharacteristic, complemented
55C2×D8×C97C2211yescharacteristic
56#124×C97C2211yescharacteristic, complemented
57C4×D4×C97C2211yescharacteristic, complemented
58C4⋊C8×C97C2211yescharacteristic, complemented
59#103×C97C2211yescharacteristic, complemented
60#102×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63#13095512—22no—
64#13095512—41no—
65#13095510—22no—
66#13095510—41no—
67C2×Dic194C2311yescharacteristic
68C2×D388—21no—
69C2×D388—21no—
70C2×D388D411yescharacteristic
71C2×D388C2311yescharacteristic
72C2×D388C2311yescharacteristic
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76C2×C22⋊D97—21no—
77D4×D97—22no—
78D4×D97—22no—
79D4×D97—41no—
80D4×D97—41no—
81C22×Dic97—21no—
82C2×C4×D97—21no—
83C2×C4×D97—21no—
84C2×C4×D97C2311yescharacteristic
85C2×C4×D97C2311yescharacteristic
86#13095501—41no—
87#13095500—22no—
88#13095499—22no—
89#13095498—22no—
90D776—22no—
91#13095496—41no—
92#13095496D414yescomplemented
93#13095494—22no—
94#13095494—22no—
95C2×#11715895C2311yescharacteristic
96C2×#11715895C2311yescharacteristic
97C4⋊Dic97—21no—
98C4⋊Dic97D411yescharacteristic
99C4⋊Dic97C2311yescharacteristic
100#13095490—21no—
101C22⋊Dic97—21no—
102#13095488—21no—
103#13095488—21no—
104#13095488—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488D411yescharacteristic
109#13095488D411yescharacteristic
110C2×Q8×C97C2311yescharacteristic
111C2×D4×C97C2311yescharacteristic
112C2×D4×C97C2311yescharacteristic
113C22×C4×C97—21no—
114SD16×C97—22no—
115D8×C97—22no—
116C2×C8×C97C2311yescharacteristic
117C2×C8×C97C2311yescharacteristic
118C4⋊C4×C97C2311yescharacteristic
119C22⋊C4×C97—21no—
120C22⋊C4×C97—21no—
121C22⋊C4×C97—21no—
122C42×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97—21no—
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—42no—
128C22×D97—42no—
129C22×D97—42no—
130C22×D97—42no—
131C22×D97—81no—
132C22×D97—81no—
133C22×D97C2×D411yescharacteristic
134C22×D97C2×D411yescharacteristic
135Q8×C97—21no—
136Q8×C97—21no—
137D4×C97—21no—
138D4×C97—21no—
139D4×C97—21no—
140D4×C97C2×D412yes—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—81no—
144C2×C4×C97C2×D411yescharacteristic
145C2×C4×C97C2×D411yescharacteristic
146C2×C4×C97C2411yescharacteristic
147C22⋊D97—42no—
148C22⋊D97—42no—
149C22⋊D97—42no—
150C22⋊D97—81no—
151C2×Dic97—21no—
152C2×Dic97—21no—
153C2×Dic97—42no—
154C2×Dic97C2×D411yescharacteristic
155C2×Dic97C2×D411yescharacteristic
156D388—21no—
157D388—21no—
158D388—21no—
159D388—41no—
160D388—42no—
161D388—81no—
162D388C2×D412yes—
163C4×D97—21no—
164C4×D97—21no—
165C4×D97—21no—
166C4×D97—21no—
167C4×D97—42no—
168C4×D97—81no—
169Dic194—21no—
170Dic194—21no—
171C8×C97—21no—
172C8×C97—21no—
173#11715895—21no—
174#11715895C2×D412yes—
175C22×C97—22no—
176C22×C97—41no—
177C22×C97—41no—
178C22×C97—41no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97C22×D411yescharacteristic
182C2×D97—21no—
183C2×D97—21no—
184C2×D97—22no—
185C2×D97—21no—
186C2×D97—21no—
187C2×D97—21no—
188C2×D97—21no—
189C2×D97—41no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—84no—
195C2×D97—84no—
196C4×C97—21no—
197C4×C97—21no—
198C4×C97—41no—
199C4×C97—41no—
200C4×C97C22×D411yescharacteristic
201C4×C97H(5,𝔽2)11yescharacteristic
202Dic97—21no—
203Dic97—21no—
204Dic97—41no—
205Dic97—41no—
206C2×C97—42no—
207C2×C97—81no—
208C2×C97D4211yescharacteristic
209C2×C97#57511yescharacteristic
210C2×C97#57611yescharacteristic
211D97—41no—
212D97—41no—
213D97—42no—
214D97—81no—
215#3273—971nomaximal, Sylow
216C97#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—971no—
218C2×AGL(1,ℤ/8)—971no—
219#546—971no—
220#546—971no—
221#495—971no—
222#468—971no—
223#468—971no—
224#463—971no—
225#463D9711yescharacteristic, complemented
226#452—971no—
227#452—971no—
228#447—971no—
229#447—971no—
230C4×D8—971no—
231#436—971no—
232C2×#41—971no—
233C2×#41—971no—
234C22×D4—971no—
235C22×D4—971no—
236AGL(1,ℤ/8)—1944no—
237AGL(1,ℤ/8)—1944no—
238C2×Q16—971no—
239C2×SD16—971no—
240C2×SD16—971no—
241C2×SD16—971no—
242C2×SD16C2×D9711yescharacteristic, complemented
243C2×D8—971no—
244C2×D8—971no—
245C2×D8—971no—
246C2×D8C2×D9711yescharacteristic
247C2×M16—971no—
248C2×M16—971no—
249#124—971no—
250#124C2×D9711yescharacteristic, complemented
251#123—1941no—
252#123—1941no—
253C22⋊Q8—971no—
254C22⋊Q8—971no—
255#121—971no—
256#121—971no—
257#121—1941no—
258#121—1941no—
259#120—1941no—
260#120—1941no—
261C4×D4—971no—
262C4×D4—971no—
263C4×D4C2×D9711yescharacteristic, complemented
264C2×C22⋊C4—1941no—
265C2×C22⋊C4—1941no—
266#107—971no—
267C4⋊C8C2×D9711yescharacteristic, complemented
268#103—971no—
269#103C2×D9711yescharacteristic, complemented
270#102—971no—
271#102—971no—
272#102—971no—
273#102C2×D9711yescharacteristic, complemented
274C22⋊C8—971no—
275C22⋊C8—971no—
276C4×C8—971no—
277C24—1941no—
278C24—1941no—
279#41—3881no—
280#41—3881no—
281#41—1942no—
282#41—1942no—
283C2×Q8—971no—
284C2×Q8C22×D9711yescharacteristic
285C2×D4—971no—
286C2×D4—971no—
287C2×D4—971no—
288C2×D4—3881no—
289C2×D4—3881no—
290C2×D4—1941no—
291C2×D4—1941no—
292C2×D4—1941no—
293C2×D4—1941no—
294C2×D4—1942no—
295C2×D4—1941no—
296C2×D4—1941no—
297C2×D4—1942no—
298C2×D4C22×D9711yescharacteristic
299C2×D4C22×D9711yescharacteristic
300C22×C4—21no—
301C22×C4—971no—
302C22×C4—971no—
303C22×C4—1941no—
304C22×C4—1941no—
305C22×C4—1941no—
306Q16—3881no—
307SD16—22no—
308SD16—1942no—
309SD16—1942no—
310SD16—1942no—
311D8—22no—
312D8—974no—
313D8—3881no—
314D8—1942no—
315M16—1942no—
316M16—1942no—
317C2×C8—971no—
318C2×C8—971no—
319C2×C8C22×D9711yescharacteristic
320C2×C8C22×D9711yescharacteristic
321C4⋊C4—971no—
322C4⋊C4—971no—
323C4⋊C4—1941no—
324C4⋊C4—1941no—
325C4⋊C4C22×D9711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—971no—
330C22⋊C4—971no—
331C22⋊C4—1941no—
332C22⋊C4—1941no—
333C22⋊C4—1941no—
334C22⋊C4—1941no—
335C22⋊C4—1941no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C42C22×D9711yescharacteristic
339C23—21no—
340C23—21no—
341C23—7761no—
342C23—7761no—
343C23—971no—
344C23—971no—
345C23—3882no—
346C23—3882no—
347C23—3882no—
348C23—3882no—
349C23—1941no—
350C23—1941no—
351Q8—21no—
352Q8—21no—
353Q8—1941no—
354Q8—1941no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—7761no—
359D4—7761no—
360D4—972no—
361D4—3881no—
362D4—3882no—
363D4—3882no—
364D4—3882no—
365D4—3882no—
366D4—1941no—
367D4—1941no—
368D4—1941no—
369D4D4×D9712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—7761no—
374C2×C4—971no—
375C2×C4—971no—
376C2×C4—3882no—
377C2×C4—3882no—
378C2×C4—1941no—
379C2×C4—1941no—
380C2×C4—1941no—
381C2×C4—1941no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4D4×D9711yescharacteristic
385C2×C4D4×D9711yescharacteristic
386C2×C4C23×D9711yescharacteristic
387C8—21no—
388C8—21no—
389C8—972no—
390C8—1941no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—7764no—
398C22—7764no—
399C22—3881no—
400C22—3881no—
401C22—3881no—
402C22—3881no—
403C22—3881no—
404C22—1941no—
405C22—1941no—
406C22—1941no—
407C22—1941no—
408C22—1942no—
409C22—1941no—
410C22—1941no—
411C22C2×D4×D9711yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—3881no—
417C4—3881no—
418C4—1941no—
419C4—1941no—
420C4C2×D4×D9711yescharacteristic
421C4#1569085611yescharacteristic
422C2—42no—
423C2—81no—
424C2—7761no—
425C2—3881no—
426C2—3881no—
427C2—3882no—
428C2#2026136911yescharacteristic
429C2#2026153211yescharacteristic
430C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#463×C97C2
socleC22×C97C22×D4
hypercenter#463D97
hypocenterC97#3273
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. C2×#15690834C2semidirect
3. #20261380C2semidirect
4. #20261378C2semidirect
5. #20261117C2semidirect
6. #20260984C2semidirect
7. #20260980C2semidirect
8. #20260951C2semidirect
9. #20260900C2semidirect
10. #20260894C2semidirect
11. #20260870C2semidirect
12. #20260867C2semidirect
13. C4×#13095496C2semidirect
14. #20260812C2semidirect
15. #463×C97C2semidirect
18. C2×D4×D97C22semidirect
19. C2×D4×D97C22semidirect
22. C2×#13095501C22semidirect
23. C2×#13095500C22semidirect
24. C2×#13095499C22semidirect
25. C2×#13095498C22semidirect
29. C2×#13095494C22semidirect
30. C2×#13095494C22semidirect
31. #15690798C22semidirect
34. #15690790C22semidirect
35. #15690786C22semidirect
43. C4×D388C22semidirect
46. #15690743C22semidirect
47. #15690735C22semidirect
48. #15690732C22semidirect
49. #15690731C22semidirect
50. #15690730C22semidirect
51. #15690722C22semidirect
52. #15690722C22semidirect
53. C4×#11715895C22semidirect
54. C2×SD16×C97C22semidirect
56. #124×C97C22semidirect
57. C4×D4×C97C22semidirect
58. C4⋊C8×C97C22semidirect
59. #103×C97C22semidirect
60. #102×C97C22semidirect
92. #13095496D4semidirect
216. C97#3273semidirect
225. #463D97semidirect
242. C2×SD16C2×D97semidirect
250. #124C2×D97semidirect
263. C4×D4C2×D97semidirect
267. C4⋊C8C2×D97semidirect
269. #103C2×D97semidirect
273. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
C2×#156908342
#202613802
#202613782
#202611172
#202609842
#202609802
#202609512
#202609002
#202608942
#202608702
#202608672
C4×#130954962
#202608122
#463×C972
#327397