Finite Group Databasebeta

← #27153235

Subgroups — #27153235

Order 12416.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690834C211yesmaximal, characteristic, complemented
2C22×#13095499C211yesmaximal, characteristic, complemented
3#20261391C211yesmaximal, characteristic, complemented
4D4×C22⋊D97C211yesmaximal, characteristic, complemented
5#20261123C211yesmaximal, characteristic, complemented
6#20260980C211yesmaximal, characteristic, complemented
7#20260969C211yesmaximal, characteristic, complemented
8#20260948C211yesmaximal, characteristic, complemented
9#20260934C211yesmaximal, characteristic, complemented
10#20260887C211yesmaximal, characteristic, complemented
11#20260877C211yesmaximal, characteristic, complemented
12#20260876C211yesmaximal, characteristic, complemented
13#20260832C211yesmaximal, characteristic, complemented
14#20260803C211yesmaximal, characteristic, complemented
15#447×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic, complemented
17C22×Dic194C2211yescharacteristic, complemented
18C22×C22⋊D97—21no—
19C2×D4×D97C2211yescharacteristic
20#15690834—24no—
21C2×#13095500C2211yescharacteristic, complemented
22C2×#13095500C2211yescharacteristic, complemented
23C2×#13095499—24no—
24C2×#13095499C2211yescharacteristic
25C2×#13095499C2211yescharacteristic
26C2×#13095499C2211yescharacteristic, complemented
27C2×#13095499C2211yescharacteristic, complemented
28C2×#13095496C2211yescharacteristic, complemented
29C2×#13095494C2211yescharacteristic, complemented
30C22×#11715895C2211yescharacteristic, complemented
31#15690806C2211yescharacteristic
32#15690797C2211yescharacteristic, complemented
33#15690796—21no—
34#15690790C2211yescharacteristic, complemented
35#15690787—21no—
36#15690784C2211yescharacteristic
37#15690782—21no—
38#15690782—21no—
39#15690781C2211yescharacteristic
40#15690779—21no—
41#15690778—21no—
42#15690775C2211yescharacteristic, complemented
43D4×Dic97—21no—
44D4×Dic97C2211yescharacteristic
45C4×C22⋊D97C2211yescharacteristic
46#15690743C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690735C2211yescharacteristic, complemented
50#15690731C2211yescharacteristic, complemented
51#15690730C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53#15690720C2211yescharacteristic, complemented
54C22×D4×C97C2211yescharacteristic
55C2×D8×C97C2211yescharacteristic
56C2×D8×C97C2211yescharacteristic, complemented
57#121×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic
59#102×C97C2211yescharacteristic, complemented
60C22⋊C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095510—22no—
63#13095510—41no—
64C2×Dic194—22no—
65C2×Dic194—21no—
66C2×Dic194—41no—
67C2×Dic194C2311yescharacteristic
68C2×Dic194C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76C2×C22⋊D97—44no—
77C2×C22⋊D97D411yescharacteristic
78D4×D97—22no—
79D4×D97—41no—
80C22×Dic97—21no—
81C22×Dic97—21no—
82C22×Dic97—21no—
83C2×C4×D97C2311yescharacteristic
84#13095500—22no—
85#13095500—41no—
86#13095499—22no—
87#13095499—22no—
88#13095499—22no—
89#13095499D414yescomplemented
90#13095496—22no—
91#13095494—22no—
92C2×#11715895—22no—
93C2×#11715895C2311yescharacteristic
94C2×#11715895C2311yescharacteristic
95C4⋊Dic97—21no—
96C4⋊Dic97C2311yescharacteristic
97#13095490—21no—
98#13095490D411yescharacteristic
99C22⋊Dic97—21no—
100C22⋊Dic97—21no—
101C22⋊Dic97—21no—
102C22⋊Dic97—21no—
103C22⋊Dic97D411yescharacteristic
104#13095488—21no—
105#13095488—21no—
106#13095488D411yescharacteristic
107C4×Dic97—21no—
108C4×Dic97C2311yescharacteristic
109C24×C97—21no—
110C2×D4×C97—22no—
111C2×D4×C97—21no—
112C2×D4×C97—21no—
113C2×D4×C97—41no—
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2311yescharacteristic
116C2×D4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118D8×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×C97—21no—
127C23×C97—42no—
128C23×C97—42no—
129C23×C97—81no—
130C23×C97C2×D411yescharacteristic
131C22×D97—21no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—81no—
135C22×D97C2×D411yescharacteristic
136D4×C97—21no—
137D4×C97—21no—
138D4×C97—21no—
139D4×C97—21no—
140D4×C97—21no—
141D4×C97—42no—
142D4×C97—42no—
143D4×C97C2×D412yes—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97C2×D411yescharacteristic
148C2×C4×C97C2411yescharacteristic
149C22⋊D97—22no—
150C22⋊D97—42no—
151C22⋊D97—42no—
152C22⋊D97—42no—
153C22⋊D97—42no—
154C22⋊D97—42no—
155C22⋊D97—81no—
156C2×Dic97—21no—
157C2×Dic97—21no—
158C2×Dic97—21no—
159C2×Dic97—42no—
160C2×Dic97—42no—
161C2×Dic97—42no—
162C2×Dic97C2×D411yescharacteristic
163C2×Dic97C2×D411yescharacteristic
164C2×Dic97C2×D411yescharacteristic
165D388—21no—
166D388—21no—
167C4×D97—21no—
168C4×D97—21no—
169Dic194—21no—
170Dic194—21no—
171Dic194—21no—
172Dic194—42no—
173Dic194C2×D412yescomplemented
174C8×C97—21no—
175C8×C97—21no—
176#11715895—21no—
177#11715895C2×D412yescomplemented
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—22no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97—84no—
190C22×C97AGL(1,ℤ/8)12yescomplemented
191C22×C97C22×D411yescharacteristic
192C2×D97—21no—
193C2×D97—21no—
194C2×D97—21no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—84no—
199C4×C97—21no—
200C4×C97—41no—
201C4×C97C22×D411yescharacteristic
202C4×C97H(5,𝔽2)11yescharacteristic
203Dic97—22no—
204Dic97—21no—
205Dic97—21no—
206Dic97—21no—
207Dic97—41no—
208Dic97—41no—
209C2×C97—22no—
210C2×C97—42no—
211C2×C97—81no—
212C2×C97—81no—
213C2×C97D4211yescharacteristic
214C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C97#57711yescharacteristic
216D97—41no—
217D97—81no—
218#3265—971nomaximal, Sylow
219C97#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—971no—
221C22×SD16—971no—
222#548—971no—
223D42—971no—
224#496—971no—
225#468—971no—
226#466—971no—
227#463—971no—
228#460—971no—
229#450—971no—
230#449—971no—
231#448—971no—
232#447D9711yescharacteristic, complemented
233#440—971no—
234#435—971no—
235C2×#41—971no—
236C22×Q8—971no—
237C22×D4—971no—
238C22×D4—1941no—
239C22×D4C2×D9711yescharacteristic
240AGL(1,ℤ/8)—1944no—
241C2×SD16—971no—
242C2×SD16—971no—
243C2×SD16—971no—
244C2×SD16—971no—
245C2×SD16—971no—
246C2×SD16—971no—
247C2×SD16—1944no—
248C2×D8—971no—
249C2×D8C2×D9711yescharacteristic
250C2×D8C2×D9711yescharacteristic, complemented
251C2×M16—971no—
252C22×C8—971no—
253#127—971no—
254#124—971no—
255#124—1941no—
256C22⋊Q8—971no—
257C22⋊Q8—1941no—
258#121—971no—
259#121—971no—
260#121—1941no—
261#121—1941no—
262#121C2×D9711yescharacteristic, complemented
263#120—1941no—
264#120—1941no—
265C4×Q8—971no—
266C4×D4—971no—
267C4×D4—971no—
268C4×D4—1941no—
269#107—971no—
270C4⋊C8—971no—
271#103—971no—
272#103—971no—
273#102—971no—
274#102—971no—
275#102C2×D9711yescharacteristic
276#102C2×D9711yescharacteristic, complemented
277C22⋊C8—971no—
278C22⋊C8C2×D9711yescharacteristic, complemented
279#97—971no—
280C24—21no—
281C24—1941no—
282#41—3881no—
283#41—1942no—
284C2×Q8—971no—
285C2×Q8—971no—
286C2×Q8—3881no—
287C2×Q8—1942no—
288C2×Q8—1941no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—971no—
294C2×D4—971no—
295C2×D4—3884no—
296C2×D4—3881no—
297C2×D4—1941no—
298C2×D4—1941no—
299C2×D4—1941no—
300C2×D4—1941no—
301C2×D4—1941no—
302C2×D4—1941no—
303C2×D4—1942no—
304C2×D4C22×D9711yescharacteristic
305C2×D4C22×D9711yescharacteristic
306C2×D4C22×D9711yescharacteristic
307C22×C4—971no—
308C22×C4—1941no—
309C22×C4—1941no—
310C22×C4—1941no—
311C22×C4C22×D9711yescharacteristic
312SD16—974no—
313SD16—3881no—
314SD16—1942no—
315SD16—1942no—
316SD16—1942no—
317SD16—1942no—
318D8—22no—
319D8—41no—
320D8—1942no—
321M16—1942no—
322C2×C8—971no—
323C2×C8—971no—
324C2×C8—1942no—
325C2×C8C22×D9711yescharacteristic
326C2×C8C22×D9711yescharacteristic
327C4⋊C4—971no—
328C4⋊C4—971no—
329C4⋊C4—1941no—
330C4⋊C4—1941no—
331C4⋊C4C22×D9711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—971no—
334C22⋊C4—971no—
335C22⋊C4—1941no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C22⋊C4—1941no—
339C22⋊C4—1941no—
340C22⋊C4—1941no—
341C42—971no—
342C42—1941no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—7761no—
350C23—971no—
351C23—3882no—
352C23—3882no—
353C23—1941no—
354C23D4×D9711yescharacteristic
355Q8—972no—
356Q8—3882no—
357Q8—1941no—
358Q8—1941no—
359Q8—1941no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—7761no—
368D4—3882no—
369D4—3882no—
370D4—3882no—
371D4—3882no—
372D4—3882no—
373D4—1942no—
374D4—1941no—
375D4—1941no—
376D4D4×D9712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—971no—
381C2×C4—971no—
382C2×C4—971no—
383C2×C4—3882no—
384C2×C4—3882no—
385C2×C4—3882no—
386C2×C4—1941no—
387C2×C4—1941no—
388C2×C4—1941no—
389C2×C4—1941no—
390C2×C4—1941no—
391C2×C4D4×D9711yescharacteristic
392C2×C4C23×D9711yescharacteristic
393C8—21no—
394C8—21no—
395C8—972no—
396C8—1941no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—84no—
409C22—7764no—
410C22—3881no—
411C22—3881no—
412C22—3881no—
413C22—1941no—
414C22—1941no—
415C22—1941no—
416C22#1569083412yescomplemented
417C22C2×D4×D9711yescharacteristic
418C4—21no—
419C4—41no—
420C4—3881no—
421C4—3881no—
422C4—1941no—
423C4—1941no—
424C4—1941no—
425C4—1942no—
426C4C2×D4×D9711yescharacteristic
427C4#1569085511yescharacteristic
428C2—22no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—7761no—
433C2—3881no—
434C2#2026136711yescharacteristic
435C2C2×#1569083411yescharacteristic
436C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#447×C97C2
socleC22×C97C22×D4
hypercenter#447D97
hypocenterC97#3265
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×#15690834C2semidirect
2. C22×#13095499C2semidirect
3. #20261391C2semidirect
4. D4×C22⋊D97C2semidirect
5. #20261123C2semidirect
6. #20260980C2semidirect
7. #20260969C2semidirect
8. #20260948C2semidirect
9. #20260934C2semidirect
10. #20260887C2semidirect
11. #20260877C2semidirect
12. #20260876C2semidirect
13. #20260832C2semidirect
14. #20260803C2semidirect
15. #447×C97C2semidirect
16. C2×#13095510C22semidirect
17. C22×Dic194C22semidirect
21. C2×#13095500C22semidirect
22. C2×#13095500C22semidirect
26. C2×#13095499C22semidirect
27. C2×#13095499C22semidirect
28. C2×#13095496C22semidirect
29. C2×#13095494C22semidirect
30. C22×#11715895C22semidirect
32. #15690797C22semidirect
34. #15690790C22semidirect
42. #15690775C22semidirect
46. #15690743C22semidirect
47. #15690739C22semidirect
48. #15690735C22semidirect
49. #15690735C22semidirect
50. #15690731C22semidirect
51. #15690730C22semidirect
52. #15690722C22semidirect
53. #15690720C22semidirect
56. C2×D8×C97C22semidirect
57. #121×C97C22semidirect
59. #102×C97C22semidirect
60. C22⋊C8×C97C22semidirect
89. #13095499D4semidirect
173. Dic194C2×D4semidirect
177. #11715895C2×D4semidirect
190. C22×C97AGL(1,ℤ/8)semidirect
219. C97#3265semidirect
232. #447D97semidirect
250. C2×D8C2×D97semidirect
262. #121C2×D97semidirect
276. #102C2×D97semidirect
278. C22⋊C8C2×D97semidirect
416. C22#15690834semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
C22×#130954992
#202613912
D4×C22⋊D972
#202611232
#202609802
#202609692
#202609482
#202609342
#202608872
#202608772
#202608762
#202608322
#202608032
#447×C972
#326597