Finite Group Databasebeta

← #27153227

Subgroups — #27153227

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×#15690836C211yesmaximal, characteristic, complemented
2C22×#13095499C211yesmaximal, characteristic, complemented
3#20261394C211yesmaximal, characteristic, complemented
4#20261122C211yesmaximal, characteristic, complemented
5#20260979C211yesmaximal, characteristic, complemented
6#20260969C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic, complemented
8#20260932C211yesmaximal, characteristic, complemented
9#20260886C211yesmaximal, characteristic, complemented
10#20260878C211yesmaximal, characteristic, complemented
11#20260873C211yesmaximal, characteristic, complemented
12#20260869C211yesmaximal, characteristic, complemented
13#20260835C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15D42×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic, complemented
17C22×Dic194C2211yescharacteristic, complemented
18#15690836—24no—
19C2×#13095499—24no—
20C2×#13095499C2211yescharacteristic
21C2×#13095499C2211yescharacteristic
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095499C2211yescharacteristic, complemented
24C2×#13095499C2211yescharacteristic, complemented
25C2×#13095499C2211yescharacteristic, complemented
26C2×#13095496C2211yescharacteristic
27C2×#13095496C2211yescharacteristic, complemented
28C2×#13095496C2211yescharacteristic, complemented
29C2×#13095495C2211yescharacteristic, complemented
30C22×#11715895C2211yescharacteristic, complemented
31#15690798C2211yescharacteristic, complemented
32#15690796—21no—
33#15690789C2211yescharacteristic, complemented
34#15690788—21no—
35#15690784C2211yescharacteristic, complemented
36#15690782—21no—
37C4×Dic194C2211yescharacteristic, complemented
38#15690770—21no—
39#15690743C2211yescharacteristic, complemented
40C4⋊#11715895C2211yescharacteristic, complemented
41#15690733C2211yescharacteristic, complemented
42#15690733C2211yescharacteristic, complemented
43#15690731C2211yescharacteristic
44#15690731C2211yescharacteristic, complemented
45#15690731C2211yescharacteristic, complemented
46#15690730C2211yescharacteristic, complemented
47C22⋊#11715895C2211yescharacteristic, complemented
48C22⋊#11715895C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50C22×D4×C97—21no—
51C22×D4×C97C2211yescharacteristic
52C22×D4×C97C2211yescharacteristic
53#127×C97C2211yescharacteristic
54#121×C97—21no—
55#121×C97C2211yescharacteristic
56#121×C97C2211yescharacteristic
57#120×C97—21no—
58#120×C97—21no—
59C4×D4×C97C2211yescharacteristic
60C4×D4×C97C2211yescharacteristic
61#13095511—22no—
62#13095511—41no—
63C2×Dic194—22no—
64C2×Dic194—21no—
65C2×Dic194—41no—
66C2×Dic194C2311yescharacteristic
67C2×Dic194C2311yescharacteristic
68C2×D388C2311yescharacteristic
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C22×Dic97—21no—
72C2×C4×D97—21no—
73#13095499—22no—
74#13095499—22no—
75#13095499—22no—
76#13095499—22no—
77#13095499—41no—
78#13095499D414yescomplemented
79#13095496—22no—
80#13095496—22no—
81#13095496—41no—
82#13095495—22no—
83C2×#11715895—22no—
84C2×#11715895C2311yescharacteristic
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C4⋊Dic97C2311yescharacteristic
89#13095490—21no—
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95C4×Dic97—21no—
96C24×C97—21no—
97C24×C97—21no—
98C2×D4×C97—22no—
99C2×D4×C97—22no—
100C2×D4×C97—21no—
101C2×D4×C97—21no—
102C2×D4×C97—21no—
103C2×D4×C97—21no—
104C2×D4×C97—21no—
105C2×D4×C97—21no—
106C2×D4×C97—41no—
107C2×D4×C97—41no—
108C2×D4×C97—44no—
109C2×D4×C97D411yescharacteristic
110C2×D4×C97C2311yescharacteristic
111C2×D4×C97C2311yescharacteristic
112C2×D4×C97C2311yescharacteristic
113C22×C4×C97—21no—
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C4⋊C4×C97D411yescharacteristic
117C4⋊C4×C97C2311yescharacteristic
118C22⋊C4×C97—21no—
119C22⋊C4×C97—21no—
120C22⋊C4×C97—21no—
121C22⋊C4×C97D411yescharacteristic
122C22⋊C4×C97D411yescharacteristic
123C42×C97C2311yescharacteristic
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C23×C97—42no—
128C23×C97—42no—
129C23×C97—42no—
130C23×C97—42no—
131C23×C97—81no—
132C23×C97—81no—
133C23×C97C2×D411yescharacteristic
134C23×C97C2×D411yescharacteristic
135C22×D97—21no—
136D4×C97—22no—
137D4×C97—21no—
138D4×C97—21no—
139D4×C97—21no—
140D4×C97—21no—
141D4×C97—21no—
142D4×C97—42no—
143D4×C97—42no—
144D4×C97—42no—
145D4×C97—42no—
146D4×C97—42no—
147D4×C97—81no—
148D4×C97C2×D412yes—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97—21no—
152C2×C4×C97—21no—
153C2×C4×C97—21no—
154C2×C4×C97—42no—
155C2×C4×C97C2×D411yescharacteristic
156C2×C4×C97C2×D411yescharacteristic
157C2×C4×C97C2×D411yescharacteristic
158C2×C4×C97C2×D411yescharacteristic
159C2×C4×C97C2411yescharacteristic
160C22⋊D97—42no—
161C22⋊D97—42no—
162C2×Dic97—21no—
163C2×Dic97—21no—
164C2×Dic97—21no—
165C2×Dic97—42no—
166D388—21no—
167D388—21no—
168C4×D97—42no—
169Dic194—21no—
170Dic194—21no—
171Dic194—21no—
172Dic194—42no—
173Dic194C2×D412yescomplemented
174#11715895—21no—
175#11715895—21no—
176#11715895—21no—
177#11715895C2×D412yescomplemented
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97—22no—
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×C97—41no—
191C22×C97—41no—
192C22×C97—84no—
193C22×C97—84no—
194C22×C97AGL(1,ℤ/8)12yescomplemented
195C22×C97C22×D411yescharacteristic
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
199C4×C97—21no—
200C4×C97—21no—
201C4×C97—21no—
202C4×C97—21no—
203C4×C97—41no—
204C4×C97C22×D411yescharacteristic
205C4×C97H(5,𝔽2)11yescharacteristic
206Dic97—22no—
207Dic97—41no—
208Dic97—41no—
209C2×C97—22no—
210C2×C97—41no—
211C2×C97—42no—
212C2×C97—81no—
213C2×C97—81no—
214C2×C97D4211yescharacteristic
215C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C97#57711yescharacteristic
217D97—81no—
218#3265—971nomaximal, Sylow
219C97#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—971no—
221C22×SD16—971no—
222#548—971no—
223D42D9711yescharacteristic, complemented
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#447—971no—
233#440—971no—
234#435—971no—
235C2×#41—971no—
236C22×Q8—971no—
237C22×D4—21no—
238C22×D4C2×D9711yescharacteristic
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×D8—971no—
250C2×D8—971no—
251C2×M16—971no—
252C22×C8—971no—
253#127C2×D9711yescharacteristic
254#124—971no—
255#124—1941no—
256C22⋊Q8—971no—
257C22⋊Q8—1941no—
258#121—21no—
259#121—971no—
260#121—1941no—
261#121C2×D9711yescharacteristic
262#121C2×D9711yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—971no—
266C4×D4—1941no—
267C4×D4C2×D9711yescharacteristic
268C4×D4C2×D9711yescharacteristic
269#107—971no—
270C4⋊C8—971no—
271#103—971no—
272#103—971no—
273#102—971no—
274#102—971no—
275#102—971no—
276#102—971no—
277C22⋊C8—971no—
278C22⋊C8—971no—
279#97—971no—
280C24—21no—
281C24—21no—
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—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—971no—
301C2×D4—1941no—
302C2×D4—1941no—
303C2×D4C22⋊D9711yescharacteristic
304C2×D4C22×D9711yescharacteristic
305C2×D4C22×D9711yescharacteristic
306C2×D4C22×D9711yescharacteristic
307C22×C4—21no—
308C22×C4—1941no—
309C22×C4—1941no—
310C22×C4C22×D9711yescharacteristic
311C22×C4C22×D9711yescharacteristic
312SD16—974no—
313SD16—3881no—
314SD16—1942no—
315SD16—1942no—
316SD16—1942no—
317SD16—1942no—
318D8—3881no—
319D8—1942no—
320D8—1942no—
321M16—1942no—
322C2×C8—971no—
323C2×C8—971no—
324C2×C8—971no—
325C2×C8—971no—
326C2×C8—1942no—
327C4⋊C4—971no—
328C4⋊C4—1941no—
329C4⋊C4—1941no—
330C4⋊C4C22⋊D9711yescharacteristic
331C4⋊C4C22×D9711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—1941no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C22⋊C4—1941no—
339C22⋊C4C22⋊D9711yescharacteristic
340C22⋊C4C22⋊D9711yescharacteristic
341C42—1941no—
342C42C22×D9711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—1941no—
353C23C2×C22⋊D9711yescharacteristic
354C23C2×C22⋊D9711yescharacteristic
355Q8—972no—
356Q8—3882no—
357Q8—1941no—
358Q8—1941no—
359Q8—1941no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—3882no—
373D4—3882no—
374D4—1941no—
375D4—1941no—
376D4D4×D9712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—3882no—
384C2×C4—3882no—
385C2×C4—1941no—
386C2×C4—1941no—
387C2×C4—1941no—
388C2×C4C2×C22⋊D9711yescharacteristic
389C2×C4C2×C22⋊D9711yescharacteristic
390C2×C4C2×C22⋊D9711yescharacteristic
391C2×C4C2×C22⋊D9711yescharacteristic
392C2×C4C23×D9711yescharacteristic
393C8—972no—
394C8—1941no—
395C8—1941no—
396C8—1941no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—3881no—
414C22—3881no—
415C22—3881no—
416C22#1569083612yescomplemented
417C22C22×C22⋊D9711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—3881no—
424C4—3881no—
425C4—1942no—
426C4C2×D4×D9711yescharacteristic
427C4#1569085511yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—7761no—
434C2D4×C22⋊D9711yescharacteristic
435C2C2×#1569083611yescharacteristic
436C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingD42×C97C2
socleC22×C97C22×D4
hypercenterD42D97
hypocenterC97#3265
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. C22×#13095499C2semidirect
3. #20261394C2semidirect
4. #20261122C2semidirect
5. #20260979C2semidirect
6. #20260969C2semidirect
7. #20260950C2semidirect
8. #20260932C2semidirect
9. #20260886C2semidirect
10. #20260878C2semidirect
11. #20260873C2semidirect
12. #20260869C2semidirect
13. #20260835C2semidirect
14. #20260807C2semidirect
15. D42×C97C2semidirect
16. C2×#13095511C22semidirect
17. C22×Dic194C22semidirect
22. C2×#13095499C22semidirect
23. C2×#13095499C22semidirect
24. C2×#13095499C22semidirect
25. C2×#13095499C22semidirect
27. C2×#13095496C22semidirect
28. C2×#13095496C22semidirect
29. C2×#13095495C22semidirect
30. C22×#11715895C22semidirect
31. #15690798C22semidirect
33. #15690789C22semidirect
35. #15690784C22semidirect
37. C4×Dic194C22semidirect
39. #15690743C22semidirect
40. C4⋊#11715895C22semidirect
41. #15690733C22semidirect
42. #15690733C22semidirect
44. #15690731C22semidirect
45. #15690731C22semidirect
46. #15690730C22semidirect
47. C22⋊#11715895C22semidirect
48. C22⋊#11715895C22semidirect
49. #15690721C22semidirect
78. #13095499D4semidirect
173. Dic194C2×D4semidirect
177. #11715895C2×D4semidirect
194. C22×C97AGL(1,ℤ/8)semidirect
219. C97#3265semidirect
223. D42D97semidirect
416. C22#15690836semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
C22×#130954992
#202613942
#202611222
#202609792
#202609692
#202609502
#202609322
#202608862
#202608782
#202608732
#202608692
#202608352
#202608072
D42×C972
#326597