Finite Group Databasebeta

← #27153330

Subgroups — #27153330

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
2#20261384C211yesmaximal, characteristic, complemented
3D4×C22⋊D97C211yesmaximal, characteristic, complemented
4#20261106C211yesmaximal, characteristic, complemented
5#20260981C211yesmaximal, characteristic, complemented
6#20260978C211yesmaximal, characteristic, complemented
7#20260938C211yesmaximal, characteristic, complemented
8#20260928C211yesmaximal, characteristic, complemented
9#20260886C211yesmaximal, characteristic, complemented
10#20260884C211yesmaximal, characteristic, complemented
11#20260880C211yesmaximal, characteristic, complemented
12#20260877C211yesmaximal, characteristic, complemented
13D8×Dic97C211yesmaximal, characteristic, complemented
14#20260811C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic, complemented
17C22×C22⋊D97—21no—
18C2×D4×D97C2211yescharacteristic
19#15690834—24no—
20C2×#13095500C2211yescharacteristic, complemented
21C2×#13095499C2211yescharacteristic
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095498C2211yescharacteristic, complemented
24C2×#13095496C2211yescharacteristic, complemented
25C2×#13095494C2211yescharacteristic, complemented
26#15690808C2211yescharacteristic, complemented
27#15690806C2211yescharacteristic
28#15690793—21no—
29#15690791—21no—
30#15690789C2211yescharacteristic, complemented
31#15690786C2211yescharacteristic, complemented
32#15690784C2211yescharacteristic
33#15690782—21no—
34#15690782—21no—
35#15690781C2211yescharacteristic
36#15690779—21no—
37#15690778—21no—
38D4×Dic97C2211yescharacteristic
39D4×Dic97C2211yescharacteristic, complemented
40C4×C22⋊D97C2211yescharacteristic
41C2×#13095490—21no—
42#15690744C2211yescharacteristic, complemented
43#15690739C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic, complemented
49#15690731C2211yescharacteristic, complemented
50C22⋊#11715895C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic, complemented
52C8×Dic97C2211yescharacteristic, complemented
53C2×#41×C97C2211yescharacteristic, complemented
54C22×D4×C97C2211yescharacteristic
55AGL(1,ℤ/8)×C97—24no—
56C2×SD16×C97C2211yescharacteristic
57C2×SD16×C97C2211yescharacteristic, complemented
58C2×D8×C97C2211yescharacteristic
59C2×D8×C97C2211yescharacteristic, complemented
60C2×M16×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095510—22no—
63#13095510—41no—
64C2×Dic194C2311yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—44no—
73C2×C22⋊D97D411yescharacteristic
74D4×D97—22no—
75D4×D97—41no—
76C22×Dic97—21no—
77C22×Dic97—21no—
78C22×Dic97—21no—
79C2×C4×D97C2311yescharacteristic
80#13095500—22no—
81#13095499—22no—
82#13095499—41no—
83#13095498—41no—
84#13095496—22no—
85#13095494—22no—
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C4⋊Dic97C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91#13095490—21no—
92#13095490—21no—
93#13095490—21no—
94#13095490D411yescharacteristic
95C22⋊Dic97—21no—
96C22⋊Dic97—21no—
97C22⋊Dic97—21no—
98C22⋊Dic97—21no—
99C22⋊Dic97D411yescharacteristic
100#13095488—21no—
101#13095488—21no—
102#13095488D411yescharacteristic
103C4×Dic97C2311yescharacteristic
104C24×C97—21no—
105#41×C97—22no—
106#41×C97—41no—
107C2×Q8×C97C2311yescharacteristic
108C2×D4×C97—22no—
109C2×D4×C97—21no—
110C2×D4×C97—21no—
111C2×D4×C97—41no—
112C2×D4×C97C2311yescharacteristic
113C2×D4×C97C2311yescharacteristic
114C2×D4×C97C2311yescharacteristic
115C22×C4×C97—21no—
116C22×C4×C97C2311yescharacteristic
117SD16×C97—22no—
118SD16×C97—22no—
119D8×C97—22no—
120D8×C97D414yescomplemented
121M16×C97—22no—
122C2×C8×C97C2311yescharacteristic
123C2×C8×C97C2311yescharacteristic
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
136Q8×C97—21no—
137Q8×C97—21no—
138D4×C97—21no—
139D4×C97—21no—
140D4×C97—21no—
141D4×C97—21no—
142D4×C97—42no—
143D4×C97—42no—
144D4×C97C2×D412yes—
145D4×C97C2×D412yescomplemented
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—42no—
150C2×C4×C97C2×D411yescharacteristic
151C2×C4×C97C2411yescharacteristic
152C22⋊D97—22no—
153C22⋊D97—42no—
154C22⋊D97—42no—
155C22⋊D97—42no—
156C22⋊D97—42no—
157C22⋊D97—42no—
158C22⋊D97—81no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—42no—
163C2×Dic97—42no—
164C2×Dic97—42no—
165C2×Dic97C2×D411yescharacteristic
166C2×Dic97C2×D411yescharacteristic
167C2×Dic97C2×D411yescharacteristic
168D388—21no—
169D388—21no—
170C4×D97—21no—
171C4×D97—21no—
172Dic194—21no—
173Dic194—21no—
174C8×C97—21no—
175C8×C97C2×D412yescomplemented
176#11715895—21no—
177#11715895—21no—
178C22×C97—22no—
179C22×C97—22no—
180C22×C97—21no—
181C22×C97—21no—
182C22×C97—21no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97—41no—
190C22×C97—84no—
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—21no—
204Dic97—21no—
205Dic97—41no—
206Dic97—41no—
207Dic97—41no—
208Dic97AGL(1,ℤ/8)12yescomplemented
209C2×C97—42no—
210C2×C97—41no—
211C2×C97—42no—
212C2×C97—81no—
213C2×C97D4211yescharacteristic
214C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C97#57711yescharacteristic
216D97—41no—
217D97—81no—
218#3272—971nomaximal, Sylow
219C97#327211yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—971no—
221C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
222#547—971no—
223D42—971no—
224#492—971no—
225#468—971no—
226#468—971no—
227#461—971no—
228#459—971no—
229#450—971no—
230#450—971no—
231#449—971no—
232#449—971no—
233C4×D8—971no—
234#436—971no—
235C2×#41—971no—
236C2×#41C2×D9711yescharacteristic, complemented
237C22×D4—971no—
238C22×D4—1941no—
239C22×D4C2×D9711yescharacteristic
240AGL(1,ℤ/8)—24no—
241AGL(1,ℤ/8)—1944no—
242C2×SD16—971no—
243C2×SD16—971no—
244C2×SD16—971no—
245C2×SD16—971no—
246C2×SD16C2×D9711yescharacteristic
247C2×SD16C2×D9711yescharacteristic, complemented
248C2×D8—971no—
249C2×D8C2×D9711yescharacteristic
250C2×D8C2×D9711yescharacteristic, complemented
251C2×M16—971no—
252C2×M16C2×D9711yescharacteristic, complemented
253C4⋊Q8—971no—
254#127—971no—
255#123—1941no—
256#123—1941no—
257C22⋊Q8—971no—
258C22⋊Q8—971no—
259#121—971no—
260#121—971no—
261#121—1941no—
262#121—1941no—
263#120—1941no—
264#120—1941no—
265C4×D4—971no—
266C4×D4—971no—
267C4×D4—971no—
268C2×C4⋊C4—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—
279C4×C8—971no—
280C24—21no—
281C24—1941no—
282#41—22no—
283#41—41no—
284#41—3881no—
285#41—1942no—
286C2×Q8—971no—
287C2×Q8C22×D9711yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—971no—
293C2×D4—971no—
294C2×D4—3884no—
295C2×D4—3881no—
296C2×D4—1941no—
297C2×D4—1941no—
298C2×D4—1941no—
299C2×D4—1941no—
300C2×D4—1941no—
301C2×D4—1941no—
302C2×D4—1942no—
303C2×D4C22×D9711yescharacteristic
304C2×D4C22×D9711yescharacteristic
305C2×D4C22×D9711yescharacteristic
306C22×C4—21no—
307C22×C4—971no—
308C22×C4—1941no—
309C22×C4—1941no—
310C22×C4—1941no—
311C22×C4C22×D9711yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—3881no—
315SD16—3881no—
316SD16—1942no—
317SD16—1942no—
318D8—22no—
319D8—1942no—
320D8C22⋊D9714yescomplemented
321M16—22no—
322M16—1942no—
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—
332C4⋊C4—1941no—
333C4⋊C4—1941no—
334C22⋊C4—971no—
335C22⋊C4—971no—
336C22⋊C4—1941no—
337C22⋊C4—1941no—
338C22⋊C4—1941no—
339C22⋊C4—1941no—
340C22⋊C4—1941no—
341C22⋊C4—1941no—
342C42—971no—
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—21no—
356Q8—21no—
357Q8—1941no—
358Q8—1941no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—7761no—
366D4—3882no—
367D4—3882no—
368D4—3882no—
369D4—3882no—
370D4—3882no—
371D4—1942no—
372D4—1941no—
373D4—1941no—
374D4C2×C22⋊D9712yes—
375D4C2×C22⋊D9712yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
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—1941no—
395C8—1941no—
396C8C2×C22⋊D9712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—7764no—
411C22—3881no—
412C22—3881no—
413C22—3881no—
414C22—1941no—
415C22—1941no—
416C22—1941no—
417C22C2×D4×D9711yescharacteristic
418C4—21no—
419C4—41no—
420C4—972no—
421C4—3881no—
422C4—3881no—
423C4—3881no—
424C4—1941no—
425C4—1941no—
426C4C22×C22⋊D9711yescharacteristic
427C4#1569085511yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—7761no—
433C2—3881no—
434C2D4×C22⋊D9711yescharacteristic
435C2AGL(1,ℤ/8)×D9711yescharacteristic
436C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3272
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. #20261384C2semidirect
3. D4×C22⋊D97C2semidirect
4. #20261106C2semidirect
5. #20260981C2semidirect
6. #20260978C2semidirect
7. #20260938C2semidirect
8. #20260928C2semidirect
9. #20260886C2semidirect
10. #20260884C2semidirect
11. #20260880C2semidirect
12. #20260877C2semidirect
13. D8×Dic97C2semidirect
14. #20260811C2semidirect
15. C2×AGL(1,ℤ/8)×C97C2semidirect
16. C2×#13095510C22semidirect
20. C2×#13095500C22semidirect
22. C2×#13095499C22semidirect
23. C2×#13095498C22semidirect
24. C2×#13095496C22semidirect
25. C2×#13095494C22semidirect
26. #15690808C22semidirect
30. #15690789C22semidirect
31. #15690786C22semidirect
39. D4×Dic97C22semidirect
42. #15690744C22semidirect
43. #15690739C22semidirect
44. #15690735C22semidirect
45. #15690734C22semidirect
46. #15690732C22semidirect
48. #15690731C22semidirect
49. #15690731C22semidirect
50. C22⋊#11715895C22semidirect
51. #15690722C22semidirect
52. C8×Dic97C22semidirect
53. C2×#41×C97C22semidirect
57. C2×SD16×C97C22semidirect
59. C2×D8×C97C22semidirect
60. C2×M16×C97C22semidirect
120. D8×C97D4semidirect
145. D4×C97C2×D4semidirect
175. C8×C97C2×D4semidirect
208. Dic97AGL(1,ℤ/8)semidirect
219. C97#3272semidirect
221. C2×AGL(1,ℤ/8)D97semidirect
236. C2×#41C2×D97semidirect
247. C2×SD16C2×D97semidirect
250. C2×D8C2×D97semidirect
252. C2×M16C2×D97semidirect
320. D8C22⋊D97semidirect
375. D4C2×C22⋊D97semidirect
396. C8C2×C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
#202613842
D4×C22⋊D972
#202611062
#202609812
#202609782
#202609382
#202609282
#202608862
#202608842
#202608802
#202608772
D8×Dic972
#202608112
C2×AGL(1,ℤ/8)×C972
#327297