Finite Group Databasebeta

← #27153347

Subgroups — #27153347

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×#15690836C211yesmaximal, characteristic, complemented
2C2×#15690834C211yesmaximal, characteristic, complemented
3#20261382C211yesmaximal, characteristic, complemented
4#20261379C211yesmaximal, characteristic, complemented
5#20261118C211yesmaximal, characteristic, complemented
6#20260980C211yesmaximal, characteristic, complemented
7#20260979C211yesmaximal, characteristic, complemented
8#20260949C211yesmaximal, characteristic, complemented
9#20260948C211yesmaximal, characteristic, complemented
10#20260899C211yesmaximal, characteristic, complemented
11#20260898C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13#20260815C211yesmaximal, characteristic, complemented
14#20260810C211yesmaximal, characteristic, complemented
15#447×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095510C2211yescharacteristic
18C2×D4×D97C2211yescharacteristic, complemented
19#15690836—24no—
20#15690834—24no—
21C2×Dic388C2211yescharacteristic, complemented
22C2×#13095500C2211yescharacteristic, complemented
23C2×#13095499C2211yescharacteristic, complemented
24C2×#13095499C2211yescharacteristic, complemented
25C2×#13095499C2211yescharacteristic, complemented
26C2×#13095496C2211yescharacteristic
27C2×#13095496C2211yescharacteristic
28C2×#13095495C2211yescharacteristic, complemented
29C2×#13095494C2211yescharacteristic, complemented
30#15690797C2211yescharacteristic, complemented
31#15690797C2211yescharacteristic, complemented
32#15690792—21no—
33#15690791—21no—
34#15690790C2211yescharacteristic, complemented
35#15690789C2211yescharacteristic, complemented
36#15690782—21no—
37#15690781—21no—
38#15690781C2211yescharacteristic
39#15690779—21no—
40#15690778—21no—
41D4×Dic97C2211yescharacteristic, complemented
42#15690772C2211yescharacteristic, complemented
43C4×C22⋊D97C2211yescharacteristic
44C2×C22⋊Dic97—21no—
45C22⋊C4×D97—21no—
46#15690743C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690733C2211yescharacteristic, complemented
50#15690732C2211yescharacteristic, complemented
51#15690731C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53C8×Dic97C2211yescharacteristic, complemented
54C22×D4×C97C2211yescharacteristic, complemented
55C2×D8×C97C2211yescharacteristic
56C2×D8×C97C2211yescharacteristic
57#121×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60C22⋊C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095511—22no—
63#13095511—41no—
64#13095510—22no—
65#13095510—41no—
66C2×Dic194C2311yescharacteristic
67C2×Dic194C2311yescharacteristic
68C2×D388C2311yescharacteristic
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97D411yescharacteristic
74D4×D97—22no—
75D4×D97—41no—
76C22×Dic97—21no—
77C22×Dic97—21no—
78C2×C4×D97—21no—
79C2×C4×D97—21no—
80C2×C4×D97C2311yescharacteristic
81Dic388—41no—
82#13095500—22no—
83#13095499—22no—
84#13095499—22no—
85#13095499—22no—
86#13095496—22no—
87#13095496D414yescomplemented
88#13095495—22no—
89#13095494—22no—
90C2×#11715895C2311yescharacteristic
91C2×#11715895C2311yescharacteristic
92C4⋊Dic97—21no—
93C4⋊Dic97C2311yescharacteristic
94#13095490—21no—
95#13095490D411yescharacteristic
96C22⋊Dic97—21no—
97C22⋊Dic97—21no—
98C22⋊Dic97—21no—
99C22⋊Dic97—21no—
100C22⋊Dic97—21no—
101C22⋊Dic97—21no—
102C22⋊Dic97—21no—
103C22⋊Dic97D411yescharacteristic
104#13095488—21no—
105#13095488—21no—
106#13095488D411yescharacteristic
107C4×Dic97C2311yescharacteristic
108C24×C97—21no—
109C2×D4×C97—22no—
110C2×D4×C97—21no—
111C2×D4×C97—21no—
112C2×D4×C97—41no—
113C2×D4×C97C2311yescharacteristic
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117D8×C97—22no—
118D8×C97—41no—
119C2×C8×C97C2311yescharacteristic
120C2×C8×C97C2311yescharacteristic
121C4⋊C4×C97C2311yescharacteristic
122C22⋊C4×C97—21no—
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—42no—
127C23×C97—42no—
128C23×C97—81no—
129C23×C97C2×D411yescharacteristic
130C22×D97—21no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—81no—
134C22×D97C2×D411yescharacteristic
135D4×C97—21no—
136D4×C97—21no—
137D4×C97—21no—
138D4×C97—21no—
139D4×C97—21no—
140D4×C97—42no—
141D4×C97—81no—
142D4×C97C2×D412yes—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97C2×D411yescharacteristic
147C2×C4×C97C2411yescharacteristic
148C22⋊D97—41no—
149C22⋊D97—42no—
150C22⋊D97—42no—
151C22⋊D97—42no—
152C22⋊D97—81no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—42no—
157C2×Dic97—81no—
158C2×Dic97C2×D411yescharacteristic
159C2×Dic97C2×D411yescharacteristic
160C2×Dic97C2×D411yescharacteristic
161D388—21no—
162D388C2×D412yes—
163C4×D97—21no—
164C4×D97—21no—
165C4×D97—42no—
166C4×D97—81no—
167Dic194—21no—
168Dic194—21no—
169Dic194—21no—
170Dic194—21no—
171C8×C97—21no—
172C8×C97—21no—
173#11715895—21no—
174#11715895C2×D412yes—
175C22×C97—22no—
176C22×C97—21no—
177C22×C97—21no—
178C22×C97—21no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—84no—
188C22×C97C22×D411yescharacteristic
189C2×D97—21no—
190C2×D97—22no—
191C2×D97—21no—
192C2×D97—21no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—84no—
196C4×C97—21no—
197C4×C97—41no—
198C4×C97C22×D411yescharacteristic
199C4×C97H(5,𝔽2)11yescharacteristic
200Dic97—21no—
201Dic97—21no—
202Dic97—21no—
203Dic97—41no—
204Dic97—41no—
205Dic97—41no—
206C2×C97—41no—
207C2×C97—42no—
208C2×C97—81no—
209C2×C97—81no—
210C2×C97D4211yescharacteristic
211C2×C97#57511yescharacteristic
212C2×C97#57611yescharacteristic
213D97—41no—
214D97—42no—
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#463—971no—
226#452—971no—
227#452—971no—
228#447—971no—
229#447D9711yescharacteristic, complemented
230C4×D8—971no—
231#436—971no—
232C2×#41—971no—
233C2×#41—971no—
234C22×D4—971no—
235C22×D4C2×D9711yescharacteristic, complemented
236AGL(1,ℤ/8)—1944no—
237AGL(1,ℤ/8)—1944no—
238C2×Q16—971no—
239C2×SD16—971no—
240C2×SD16—971no—
241C2×SD16—971no—
242C2×SD16—971no—
243C2×D8—971no—
244C2×D8—971no—
245C2×D8C2×D9711yescharacteristic
246C2×D8C2×D9711yescharacteristic
247C2×M16—971no—
248C2×M16—971no—
249#124—971no—
250#124—971no—
251#123—1941no—
252#123—1941no—
253C22⋊Q8—971no—
254C22⋊Q8—971no—
255#121—971no—
256#121—1941no—
257#121—1941no—
258#121C2×D9711yescharacteristic
259#120—1941no—
260#120—1941no—
261C4×D4—971no—
262C4×D4—971no—
263C4×D4—971no—
264C2×C22⋊C4—1941no—
265C2×C22⋊C4—1941no—
266#107—971no—
267C4⋊C8—971no—
268#103—971no—
269#103—971no—
270#102—971no—
271#102—971no—
272#102C2×D9711yescharacteristic, complemented
273#102C2×D9711yescharacteristic, complemented
274C22⋊C8—971no—
275C22⋊C8C2×D9711yescharacteristic, complemented
276C4×C8—971no—
277C24—21no—
278C24—1941no—
279#41—3881no—
280#41—3881no—
281#41—1942no—
282#41—1942no—
283C2×Q8—971no—
284C2×Q8—971no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—971no—
290C2×D4—971no—
291C2×D4—3881no—
292C2×D4—1941no—
293C2×D4—1942no—
294C2×D4—1941no—
295C2×D4—1941no—
296C2×D4—1941no—
297C2×D4C22×D9711yescharacteristic
298C2×D4C22×D9711yescharacteristic
299C2×D4C22×D9711yescharacteristic
300C22×C4—971no—
301C22×C4—1941no—
302C22×C4—1941no—
303C22×C4—1941no—
304C22×C4—1941no—
305C22×C4C22×D9711yescharacteristic
306Q16—3881no—
307SD16—1942no—
308SD16—1942no—
309SD16—1942no—
310SD16—1942no—
311D8—22no—
312D8—41no—
313D8—974no—
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—971no—
328C22⋊C4—971no—
329C22⋊C4—1941no—
330C22⋊C4—1941no—
331C22⋊C4—1941no—
332C22⋊C4—1941no—
333C22⋊C4—1941no—
334C22⋊C4—1941no—
335C22⋊C4—1941no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C42—971no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—7761no—
346C23—971no—
347C23—3882no—
348C23—3882no—
349C23—1941no—
350C23D4×D9711yescharacteristic
351Q8—1941no—
352Q8—1941no—
353Q8—1941no—
354Q8—1941no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—81no—
362D4—7761no—
363D4—972no—
364D4—3882no—
365D4—3882no—
366D4—3881no—
367D4—3882no—
368D4—1941no—
369D4D4×D9712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—7761no—
374C2×C4—7761no—
375C2×C4—971no—
376C2×C4—971no—
377C2×C4—971no—
378C2×C4—3882no—
379C2×C4—3882no—
380C2×C4—1941no—
381C2×C4—1941no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4—1941no—
385C2×C4D4×D9711yescharacteristic
386C2×C4C23×D9711yescharacteristic
387C8—21no—
388C8—21no—
389C8—972no—
390C8—1941no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—7764no—
405C22—3881no—
406C22—3881no—
407C22—1941no—
408C22—1941no—
409C22—1942no—
410C22—1941no—
411C22C2×D4×D9711yescharacteristic
412C4—21no—
413C4—41no—
414C4—3881no—
415C4—3881no—
416C4—3881no—
417C4—1941no—
418C4—1941no—
419C4—1941no—
420C4C2×D4×D9711yescharacteristic
421C4#1569085511yescharacteristic
422C2—41no—
423C2—42no—
424C2—81no—
425C2—81no—
426C2—3881no—
427C2—3882no—
428C2#2026136711yescharacteristic
429C2#2026153111yescharacteristic
430C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#447×C97C2
socleC22×C97C22×D4
hypercenter#447D97
hypocenterC97#3273
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. C2×#15690834C2semidirect
3. #20261382C2semidirect
4. #20261379C2semidirect
5. #20261118C2semidirect
6. #20260980C2semidirect
7. #20260979C2semidirect
8. #20260949C2semidirect
9. #20260948C2semidirect
10. #20260899C2semidirect
11. #20260898C2semidirect
12. #20260867C2semidirect
13. #20260815C2semidirect
14. #20260810C2semidirect
15. #447×C97C2semidirect
18. C2×D4×D97C22semidirect
21. C2×Dic388C22semidirect
22. C2×#13095500C22semidirect
23. C2×#13095499C22semidirect
24. C2×#13095499C22semidirect
25. C2×#13095499C22semidirect
28. C2×#13095495C22semidirect
29. C2×#13095494C22semidirect
30. #15690797C22semidirect
31. #15690797C22semidirect
34. #15690790C22semidirect
35. #15690789C22semidirect
41. D4×Dic97C22semidirect
42. #15690772C22semidirect
46. #15690743C22semidirect
47. #15690739C22semidirect
48. #15690735C22semidirect
49. #15690733C22semidirect
50. #15690732C22semidirect
51. #15690731C22semidirect
52. #15690722C22semidirect
53. C8×Dic97C22semidirect
54. C22×D4×C97C22semidirect
58. #102×C97C22semidirect
59. #102×C97C22semidirect
60. C22⋊C8×C97C22semidirect
87. #13095496D4semidirect
216. C97#3273semidirect
229. #447D97semidirect
235. C22×D4C2×D97semidirect
272. #102C2×D97semidirect
273. #102C2×D97semidirect
275. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
C2×#156908342
#202613822
#202613792
#202611182
#202609802
#202609792
#202609492
#202609482
#202608992
#202608982
#202608672
#202608152
#202608102
#447×C972
#327397