Finite Group Databasebeta

← #27153231

Subgroups — #27153231

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×#15690837C211yesmaximal, characteristic, complemented
2#20261396C211yesmaximal, characteristic, complemented
3D4×C22⋊D97C211yesmaximal, characteristic, complemented
4#20261121C211yesmaximal, characteristic, complemented
5#20260978C211yesmaximal, characteristic, complemented
6#20260971C211yesmaximal, characteristic, complemented
7#20260952C211yesmaximal, characteristic, complemented
8#20260933C211yesmaximal, characteristic, complemented
9#20260884C211yesmaximal, characteristic, complemented
10#20260879C211yesmaximal, characteristic, complemented
11#20260874C211yesmaximal, characteristic, complemented
12#20260869C211yesmaximal, characteristic, complemented
13#20260836C211yesmaximal, characteristic, complemented
14#20260806C211yesmaximal, characteristic, complemented
15C22×SD16×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic, complemented
17C22×C22⋊D97—21no—
18C2×D4×D97C2211yescharacteristic
19#15690837—24no—
20C2×#13095498C2211yescharacteristic, complemented
21C2×#13095498C2211yescharacteristic, complemented
22C2×D776C2211yescharacteristic, complemented
23C2×#13095496C2211yescharacteristic
24C2×#13095496C2211yescharacteristic, complemented
25C2×#13095494C2211yescharacteristic, complemented
26#15690806C2211yescharacteristic
27#15690799—21no—
28#15690799C2211yescharacteristic, complemented
29#15690786—21no—
30#15690786C2211yescharacteristic, complemented
31#15690784—21no—
32#15690784C2211yescharacteristic
33#15690784C2211yescharacteristic, complemented
34#15690782—21no—
35#15690781C2211yescharacteristic
36#15690779—21no—
37#15690778—21no—
38Q8×Dic97C2211yescharacteristic, complemented
39D4×Dic97C2211yescharacteristic
40C4×C22⋊D97—21no—
41C4×C22⋊D97C2211yescharacteristic
42#15690744C2211yescharacteristic, complemented
43#15690739C2211yescharacteristic, complemented
44#15690734C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic, complemented
47#15690732C2211yescharacteristic, complemented
48#15690731C2211yescharacteristic
49#15690731C2211yescharacteristic, complemented
50C22⋊#11715895C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic, complemented
52#15690720C2211yescharacteristic, complemented
53C22×Q8×C97C2211yescharacteristic, complemented
54C22×D4×C97C2211yescharacteristic
55C2×SD16×C97—24no—
56C2×SD16×C97C2211yescharacteristic
57C2×SD16×C97C2211yescharacteristic
58C2×SD16×C97C2211yescharacteristic, complemented
59C2×SD16×C97C2211yescharacteristic, complemented
60C22×C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095512—22no—
63#13095512—41no—
64C2×D388—21no—
65C2×D388C2311yescharacteristic
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C2×C22⋊D97—44no—
74C2×C22⋊D97D411yescharacteristic
75D4×D97—22no—
76D4×D97—41no—
77C22×Dic97—21no—
78C2×C4×D97—21no—
79C2×C4×D97C2311yescharacteristic
80#13095498—22no—
81#13095498—41no—
82D776—22no—
83#13095496—22no—
84#13095496—41no—
85#13095494—22no—
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C4⋊Dic97—21no—
89C4⋊Dic97C2311yescharacteristic
90C4⋊Dic97C2311yescharacteristic
91#13095490—21no—
92#13095490D411yescharacteristic
93C22⋊Dic97—21no—
94C22⋊Dic97—21no—
95C22⋊Dic97D411yescharacteristic
96#13095488—21no—
97#13095488—21no—
98#13095488—21no—
99#13095488—21no—
100#13095488—21no—
101#13095488D411yescharacteristic
102C4×Dic97—21no—
103C4×Dic97C2311yescharacteristic
104C24×C97—21no—
105C2×Q8×C97—21no—
106C2×Q8×C97—22no—
107C2×Q8×C97—41no—
108C2×Q8×C97C2311yescharacteristic
109C2×Q8×C97C2311yescharacteristic
110C2×D4×C97—22no—
111C2×D4×C97—21no—
112C2×D4×C97—41no—
113C2×D4×C97C2311yescharacteristic
114C2×D4×C97C2311yescharacteristic
115C22×C4×C97—21no—
116C22×C4×C97C2311yescharacteristic
117SD16×C97—22no—
118SD16×C97—22no—
119SD16×C97—22no—
120SD16×C97D414yescomplemented
121C2×C8×C97—22no—
122C2×C8×C97C2311yescharacteristic
123C2×C8×C97C2311yescharacteristic
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—42no—
127C23×C97—42no—
128C23×C97—81no—
129C23×C97C2×D411yescharacteristic
130C22×D97—21no—
131C22×D97—21no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—81no—
135C22×D97C2×D411yescharacteristic
136Q8×C97—21no—
137Q8×C97—21no—
138Q8×C97—21no—
139Q8×C97—42no—
140Q8×C97C2×D412yescomplemented
141D4×C97—21no—
142D4×C97—21no—
143D4×C97—21no—
144D4×C97—42no—
145D4×C97C2×D412yes—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—42no—
151C2×C4×C97C2×D411yescharacteristic
152C2×C4×C97C2411yescharacteristic
153C22⋊D97—22no—
154C22⋊D97—42no—
155C22⋊D97—42no—
156C22⋊D97—42no—
157C22⋊D97—42no—
158C22⋊D97—42no—
159C22⋊D97—81no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—42no—
163C2×Dic97C2×D411yescharacteristic
164C2×Dic97C2×D411yescharacteristic
165C2×Dic97C2×D411yescharacteristic
166D388—21no—
167D388—21no—
168D388—21no—
169D388—21no—
170D388—42no—
171C4×D97—21no—
172C4×D97—21no—
173C4×D97—42no—
174C8×C97—21no—
175C8×C97C2×D412yescomplemented
176#11715895—21no—
177#11715895—21no—
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—84no—
187C22×C97AGL(1,ℤ/8)12yescomplemented
188C22×C97C22×D411yescharacteristic
189C2×D97—21no—
190C2×D97—21no—
191C2×D97—21no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—84no—
199C4×C97—22no—
200C4×C97—21no—
201C4×C97—41no—
202C4×C97C22×D411yescharacteristic
203C4×C97H(5,𝔽2)11yescharacteristic
204Dic97—21no—
205Dic97—21no—
206Dic97—21no—
207Dic97—41no—
208Dic97—41no—
209C2×C97—22no—
210C2×C97—42no—
211C2×C97—81no—
212C2×C97D4211yescharacteristic
213C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C97#57711yescharacteristic
215D97—41no—
216D97—81no—
217D97—81no—
218#3265—971nomaximal, Sylow
219C97#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—971no—
221C22×SD16D9711yescharacteristic, complemented
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#447—971no—
233#440—971no—
234#435—971no—
235C2×#41—971no—
236C22×Q8C2×D9711yescharacteristic, complemented
237C22×D4—971no—
238C22×D4—1941no—
239C22×D4C2×D9711yescharacteristic
240AGL(1,ℤ/8)—1944no—
241C2×SD16—24no—
242C2×SD16—971no—
243C2×SD16—971no—
244C2×SD16C2×D9711yescharacteristic
245C2×SD16C2×D9711yescharacteristic
246C2×SD16C2×D9711yescharacteristic, complemented
247C2×SD16C2×D9711yescharacteristic, complemented
248C2×D8—971no—
249C2×D8—971no—
250C2×D8—971no—
251C2×M16—971no—
252C22×C8C2×D9711yescharacteristic, complemented
253#127—971no—
254#124—971no—
255#124—1941no—
256C22⋊Q8—971no—
257C22⋊Q8—1941no—
258#121—971no—
259#121—971no—
260#121—971no—
261#121—1941no—
262#121—1941no—
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#102—971no—
276#102—971no—
277C22⋊C8—971no—
278C22⋊C8—971no—
279#97—971no—
280C24—21no—
281C24—1941no—
282#41—3881no—
283#41—1942no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D9711yescharacteristic
288C2×Q8C22×D9711yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—971no—
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—1941no—
304C2×D4—1942no—
305C2×D4C22×D9711yescharacteristic
306C2×D4C22×D9711yescharacteristic
307C22×C4—21no—
308C22×C4—971no—
309C22×C4—1941no—
310C22×C4—1941no—
311C22×C4C22×D9711yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—22no—
315SD16—3881no—
316SD16—1942no—
317SD16C22⋊D9714yescomplemented
318D8—3881no—
319D8—1942no—
320D8—1942no—
321M16—1942no—
322C2×C8—22no—
323C2×C8—971no—
324C2×C8—971no—
325C2×C8C22×D9711yescharacteristic
326C2×C8C22×D9711yescharacteristic
327C4⋊C4—971no—
328C4⋊C4—971no—
329C4⋊C4—971no—
330C4⋊C4—1941no—
331C4⋊C4—1941no—
332C22⋊C4—971no—
333C22⋊C4—971no—
334C22⋊C4—1941no—
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—42no—
346C23—42no—
347C23—81no—
348C23—7761no—
349C23—971no—
350C23—3882no—
351C23—3882no—
352C23—1941no—
353C23—1941no—
354C23D4×D9711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8C2×C22⋊D9712yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—7761no—
365D4—3882no—
366D4—3882no—
367D4—3882no—
368D4—3882no—
369D4—3882no—
370D4—3882no—
371D4—1942no—
372D4—1941no—
373D4—1941no—
374D4—1941no—
375D4—1941no—
376D4C2×C22⋊D9712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—971no—
383C2×C4—971no—
384C2×C4—971no—
385C2×C4—3882no—
386C2×C4—3882no—
387C2×C4—1941no—
388C2×C4—1941no—
389C2×C4—1941no—
390C2×C4—1941no—
391C2×C4D4×D9711yescharacteristic
392C2×C4C23×D9711yescharacteristic
393C8—21no—
394C8—1941no—
395C8—1941no—
396C8C2×C22⋊D9712yescomplemented
397C22—21no—
398C22—21no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—84no—
406C22—7764no—
407C22—3881no—
408C22—3881no—
409C22—3881no—
410C22—3881no—
411C22—3881no—
412C22—3881no—
413C22—1941no—
414C22—1941no—
415C22—1941no—
416C22#1569083712yescomplemented
417C22C2×D4×D9711yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—3881no—
422C4—3881no—
423C4—1941no—
424C4—1941no—
425C4—1941no—
426C4C22×C22⋊D9711yescharacteristic
427C4#1569085511yescharacteristic
428C2—22no—
429C2—42no—
430C2—81no—
431C2—7761no—
432C2—7761no—
433C2—3881no—
434C2D4×C22⋊D9711yescharacteristic
435C2C2×#1569083711yescharacteristic
436C2#2026154111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC22×C97C22×D4
hypercenterC22×SD16D97
hypocenterC97#3265
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15690837C2semidirect
2. #20261396C2semidirect
3. D4×C22⋊D97C2semidirect
4. #20261121C2semidirect
5. #20260978C2semidirect
6. #20260971C2semidirect
7. #20260952C2semidirect
8. #20260933C2semidirect
9. #20260884C2semidirect
10. #20260879C2semidirect
11. #20260874C2semidirect
12. #20260869C2semidirect
13. #20260836C2semidirect
14. #20260806C2semidirect
15. C22×SD16×C97C2semidirect
16. C2×#13095512C22semidirect
20. C2×#13095498C22semidirect
21. C2×#13095498C22semidirect
22. C2×D776C22semidirect
24. C2×#13095496C22semidirect
25. C2×#13095494C22semidirect
28. #15690799C22semidirect
30. #15690786C22semidirect
33. #15690784C22semidirect
38. Q8×Dic97C22semidirect
42. #15690744C22semidirect
43. #15690739C22semidirect
44. #15690734C22semidirect
45. #15690734C22semidirect
46. #15690732C22semidirect
47. #15690732C22semidirect
49. #15690731C22semidirect
50. C22⋊#11715895C22semidirect
51. #15690722C22semidirect
52. #15690720C22semidirect
53. C22×Q8×C97C22semidirect
58. C2×SD16×C97C22semidirect
59. C2×SD16×C97C22semidirect
60. C22×C8×C97C22semidirect
120. SD16×C97D4semidirect
140. Q8×C97C2×D4semidirect
175. C8×C97C2×D4semidirect
187. C22×C97AGL(1,ℤ/8)semidirect
219. C97#3265semidirect
221. C22×SD16D97semidirect
236. C22×Q8C2×D97semidirect
246. C2×SD16C2×D97semidirect
247. C2×SD16C2×D97semidirect
252. C22×C8C2×D97semidirect
317. SD16C22⋊D97semidirect
359. Q8C2×C22⋊D97semidirect
396. C8C2×C22⋊D97semidirect
416. C22#15690837semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
#202613962
D4×C22⋊D972
#202611212
#202609782
#202609712
#202609522
#202609332
#202608842
#202608792
#202608742
#202608692
#202608362
#202608062
C22×SD16×C972
#326597