Finite Group Databasebeta

← #27153234

Subgroups — #27153234

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×#15690835C211yesmaximal, characteristic, complemented
2C2×SD16×D97C211yesmaximal, characteristic, complemented
3#20261392C211yesmaximal, characteristic, complemented
4#20261369C211yesmaximal, characteristic, complemented
5#20261123C211yesmaximal, characteristic, complemented
6#20260979C211yesmaximal, characteristic, complemented
7#20260970C211yesmaximal, characteristic, complemented
8#20260949C211yesmaximal, characteristic, complemented
9#20260933C211yesmaximal, characteristic, complemented
10#20260888C211yesmaximal, characteristic, complemented
11#20260875C211yesmaximal, characteristic, complemented
12#20260867C211yesmaximal, characteristic, complemented
13#20260838C211yesmaximal, characteristic, complemented
14#20260809C211yesmaximal, characteristic, complemented
15#449×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic, complemented
17C22×D388C2211yescharacteristic
18C2×D4×D97—21no—
19C2×D4×D97C2211yescharacteristic
20#15690835—24no—
21C2×#13095500C2211yescharacteristic
22C2×#13095500C2211yescharacteristic, complemented
23C2×#13095499C2211yescharacteristic, complemented
24C2×#13095498C2211yescharacteristic
25C2×#13095498C2211yescharacteristic, complemented
26SD16×D97—24no—
27C2×#13095496C2211yescharacteristic
28C2×#13095496C2211yescharacteristic, complemented
29C2×#13095495C2211yescharacteristic, complemented
30C2×C8×D97C2211yescharacteristic, complemented
31#15690806C2211yescharacteristic
32#15690797—21no—
33#15690797C2211yescharacteristic, complemented
34#15690789—21no—
35#15690789C2211yescharacteristic, complemented
36#15690785C2211yescharacteristic
37#15690783—21no—
38#15690781—21no—
39#15690781C2211yescharacteristic, complemented
40#15690780—21no—
41#15690778—21no—
42Q8×Dic97C2211yescharacteristic, complemented
43#15690772C2211yescharacteristic
44C4×C22⋊D97—21no—
45C4×C22⋊D97C2211yescharacteristic
46#15690743C2211yescharacteristic, complemented
47#15690739C2211yescharacteristic, complemented
48#15690733C2211yescharacteristic, complemented
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic, complemented
51#15690731C2211yescharacteristic, complemented
52#15690722C2211yescharacteristic, complemented
53#15690720C2211yescharacteristic, complemented
54C2×#41×C97C2211yescharacteristic, complemented
55C2×SD16×C97C2211yescharacteristic, complemented
56C2×D8×C97C2211yescharacteristic, complemented
57#121×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic, complemented
59#102×C97C2211yescharacteristic, complemented
60C22⋊C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C23×D97—21no—
63C2×Dic194—21no—
64C2×Dic194C2311yescharacteristic
65Q8×D97—22no—
66Q8×D97—41no—
67C2×D388—22no—
68C2×D388—21no—
69C2×D388—21no—
70C2×D388—41no—
71C2×D388C2311yescharacteristic
72C2×D388C2311yescharacteristic
73C2×C22⋊D97—21no—
74C2×C22⋊D97—21no—
75C2×C22⋊D97—21no—
76C2×C22⋊D97—21no—
77C2×C22⋊D97D411yescharacteristic
78D4×D97—22no—
79D4×D97—41no—
80D4×D97—44no—
81C2×C4×D97—21no—
82C2×C4×D97—21no—
83C2×C4×D97C2311yescharacteristic
84#13095500—22no—
85#13095500—41no—
86#13095499—22no—
87#13095498—22no—
88#13095498D414yescomplemented
89#13095496—22no—
90#13095496—22no—
91#13095495—22no—
92C2×#11715895C2311yescharacteristic
93C2×#11715895C2311yescharacteristic
94C8×D97—22no—
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
109#41×C97—22no—
110#41×C97—41no—
111C2×Q8×C97C2311yescharacteristic
112C2×D4×C97—21no—
113C2×D4×C97—21no—
114C2×D4×C97C2311yescharacteristic
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97—21no—
117C22×C4×C97C2311yescharacteristic
118SD16×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×C97C2×D411yescharacteristic
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—42no—
130C22×D97—42no—
131C22×D97—42no—
132C22×D97—42no—
133C22×D97—81no—
134C22×D97—81no—
135C22×D97C2×D411yescharacteristic
136Q8×C97—21no—
137Q8×C97C2×D412yescomplemented
138D4×C97—21no—
139D4×C97—21no—
140D4×C97—21no—
141D4×C97—21no—
142D4×C97—42no—
143D4×C97—42no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—42no—
149C2×C4×C97C2×D411yescharacteristic
150C2×C4×C97C2411yescharacteristic
151C22⋊D97—22no—
152C22⋊D97—42no—
153C22⋊D97—42no—
154C22⋊D97—42no—
155C22⋊D97—81no—
156C2×Dic97—21no—
157C2×Dic97—21no—
158C2×Dic97C2×D411yescharacteristic
159C2×Dic97C2×D411yescharacteristic
160C2×Dic97C2×D411yescharacteristic
161D388—21no—
162D388—21no—
163D388—21no—
164D388—42no—
165D388—42no—
166D388C2×D412yes—
167C4×D97—21no—
168C4×D97—21no—
169C4×D97—42no—
170C4×D97—42no—
171Dic194—21no—
172Dic194—21no—
173Dic194—42no—
174C8×C97—21no—
175C8×C97—21no—
176#11715895—21no—
177#11715895C2×D412yescomplemented
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97C22×D411yescharacteristic
188C2×D97—21no—
189C2×D97—21no—
190C2×D97—22no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—41no—
196C2×D97—84no—
197C2×D97—84no—
198C2×D97AGL(1,ℤ/8)12yescomplemented
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—41no—
210C2×C97—81no—
211C2×C97—81no—
212C2×C97D4211yescharacteristic
213C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C97#57711yescharacteristic
215D97—22no—
216D97—42no—
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#449D9711yescharacteristic, complemented
231#448—971no—
232#447—971no—
233#440—971no—
234#435—971no—
235C2×#41C2×D9711yescharacteristic, complemented
236C22×Q8—971no—
237C22×D4—971no—
238C22×D4—971no—
239C22×D4—1941no—
240AGL(1,ℤ/8)—1944no—
241C2×SD16—971no—
242C2×SD16—971no—
243C2×SD16—971no—
244C2×SD16—971no—
245C2×SD16—971no—
246C2×SD16—1944no—
247C2×SD16C2×D9711yescharacteristic, complemented
248C2×D8—971no—
249C2×D8—971no—
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
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#103C2×D9711yescharacteristic, complemented
273#102—971no—
274#102—971no—
275#102—971no—
276#102C2×D9711yescharacteristic, complemented
277C22⋊C8—971no—
278C22⋊C8C2×D9711yescharacteristic, complemented
279#97—971no—
280C24—1941no—
281C24—1941no—
282#41—22no—
283#41—41no—
284C2×Q8—971no—
285C2×Q8—3881no—
286C2×Q8—1941no—
287C2×Q8—1942no—
288C2×Q8C22×D9711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—971no—
292C2×D4—971no—
293C2×D4—971no—
294C2×D4—3881no—
295C2×D4—3884no—
296C2×D4—3881no—
297C2×D4—1942no—
298C2×D4—1941no—
299C2×D4—1941no—
300C2×D4—1941no—
301C2×D4—1941no—
302C2×D4—1942no—
303C2×D4—1941no—
304C2×D4—1941no—
305C2×D4C22×D9711yescharacteristic
306C2×D4C22×D9711yescharacteristic
307C22×C4—21no—
308C22×C4—971no—
309C22×C4—1941no—
310C22×C4—1941no—
311C22×C4C22×D9711yescharacteristic
312SD16—22no—
313SD16—974no—
314SD16—3881no—
315SD16—1942no—
316SD16—1942no—
317SD16—1942no—
318D8—41no—
319D8—1942no—
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—7761no—
346C23—7761no—
347C23—971no—
348C23—3882no—
349C23—3882no—
350C23—3882no—
351C23—3882no—
352C23—1941no—
353C23—1941no—
354C23D4×D9711yescharacteristic
355Q8—21no—
356Q8—3882no—
357Q8—1941no—
358Q8—1941no—
359Q8D4×D9712yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—7761no—
367D4—972no—
368D4—3882no—
369D4—3882no—
370D4—3882no—
371D4—3882no—
372D4—3882no—
373D4—1942no—
374D4—1941no—
375D4—1941no—
376D4—1941no—
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—21no—
395C8—972no—
396C8—1941no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—7764no—
407C22—7764no—
408C22—972no—
409C22—3881no—
410C22—3881no—
411C22—3881no—
412C22—3881no—
413C22—3881no—
414C22—1942no—
415C22—1941no—
416C22—1941no—
417C22C2×D4×D9711yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—3881no—
422C4—3881no—
423C4—1941no—
424C4—1941no—
425C4—1941no—
426C4C2×D4×D9711yescharacteristic
427C4#1569085511yescharacteristic
428C2—41no—
429C2—81no—
430C2—81no—
431C2—7761no—
432C2—3882no—
433C2—1942no—
434C2#2026136711yescharacteristic
435C2AGL(1,ℤ/8)×D9711yescharacteristic
436C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#449×C97C2
socleC22×C97C22×D4
hypercenter#449D97
hypocenterC97#3265
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×SD16×D97C2semidirect
3. #20261392C2semidirect
4. #20261369C2semidirect
5. #20261123C2semidirect
6. #20260979C2semidirect
7. #20260970C2semidirect
8. #20260949C2semidirect
9. #20260933C2semidirect
10. #20260888C2semidirect
11. #20260875C2semidirect
12. #20260867C2semidirect
13. #20260838C2semidirect
14. #20260809C2semidirect
15. #449×C97C2semidirect
16. C2×Q8×D97C22semidirect
22. C2×#13095500C22semidirect
23. C2×#13095499C22semidirect
25. C2×#13095498C22semidirect
28. C2×#13095496C22semidirect
29. C2×#13095495C22semidirect
30. C2×C8×D97C22semidirect
33. #15690797C22semidirect
35. #15690789C22semidirect
39. #15690781C22semidirect
42. Q8×Dic97C22semidirect
46. #15690743C22semidirect
47. #15690739C22semidirect
48. #15690733C22semidirect
50. #15690732C22semidirect
51. #15690731C22semidirect
52. #15690722C22semidirect
53. #15690720C22semidirect
54. C2×#41×C97C22semidirect
55. C2×SD16×C97C22semidirect
56. C2×D8×C97C22semidirect
58. #103×C97C22semidirect
59. #102×C97C22semidirect
60. C22⋊C8×C97C22semidirect
88. #13095498D4semidirect
137. Q8×C97C2×D4semidirect
177. #11715895C2×D4semidirect
198. C2×D97AGL(1,ℤ/8)semidirect
219. C97#3265semidirect
230. #449D97semidirect
235. C2×#41C2×D97semidirect
247. C2×SD16C2×D97semidirect
250. C2×D8C2×D97semidirect
272. #103C2×D97semidirect
276. #102C2×D97semidirect
278. C22⋊C8C2×D97semidirect
359. Q8D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×SD16×D972
#202613922
#202613692
#202611232
#202609792
#202609702
#202609492
#202609332
#202608882
#202608752
#202608672
#202608382
#202608092
#449×C972
#326597