Finite Group Databasebeta

← #30650230

Subgroups — #30650230

Order 16256.

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×#16861237C211yesmaximal, characteristic, complemented
2C22×#14523237C211yesmaximal, characteristic, complemented
3#22550535C211yesmaximal, characteristic, complemented
4D4×D508C211yesmaximal, characteristic, complemented
5#22550265C211yesmaximal, characteristic, complemented
6#22550122C211yesmaximal, characteristic, complemented
7#22550112C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550077C211yesmaximal, characteristic, complemented
10#22550029C211yesmaximal, characteristic, complemented
11#22550023C211yesmaximal, characteristic, complemented
12#22550017C211yesmaximal, characteristic, complemented
13#22550012C211yesmaximal, characteristic, complemented
14#22549978C211yesmaximal, characteristic, complemented
15#435×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic, complemented
17C22×Dic254C2211yescharacteristic, complemented
18C22×D508C2211yescharacteristic
19C22×D508C2211yescharacteristic
20C2×D4×D127—21no—
21#16861237—24no—
22C2×#14523237—24no—
23C2×#14523237C2211yescharacteristic
24C2×#14523237C2211yescharacteristic
25C2×#14523237C2211yescharacteristic, complemented
26C2×#14523237C2211yescharacteristic, complemented
27C2×#14523237C2211yescharacteristic, complemented
28C2×#14523237C2211yescharacteristic, complemented
29C2×D1016C2211yescharacteristic
30C2×D1016C2211yescharacteristic, complemented
31C2×D1016C2211yescharacteristic, complemented
32#16861206C2211yescharacteristic
33#16861197C2211yescharacteristic, complemented
34#16861195—21no—
35#16861188C2211yescharacteristic, complemented
36#16861187—21no—
37#16861184—21no—
38#16861183C2211yescharacteristic
39#16861183C2211yescharacteristic
40#16861183C2211yescharacteristic, complemented
41#16861181—21no—
42#16861179—21no—
43#16861179—21no—
44C4×Dic254C2211yescharacteristic, complemented
45C4×D508C2211yescharacteristic
46(C22⋊D127)⋊C4—21no—
47#16861143C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861134C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic
51#16861131C2211yescharacteristic, complemented
52#16861131C2211yescharacteristic, complemented
53#16861131C2211yescharacteristic, complemented
54C2×M16×C127C2211yescharacteristic, complemented
55C22×C8×C127C2211yescharacteristic, complemented
56C4×D4×C127C2211yescharacteristic
57C4⋊C8×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic, complemented
59C22⋊C8×C127C2211yescharacteristic, complemented
60#97×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523248—22no—
64#14523248—41no—
65C2×Dic254—22no—
66C2×Dic254—21no—
67C2×Dic254—41no—
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70C2×D508—22no—
71C2×D508—22no—
72C2×D508—21no—
73C2×D508—21no—
74C2×D508—21no—
75C2×D508—21no—
76C2×D508—41no—
77C2×D508—41no—
78C2×D508C2311yescharacteristic
79C2×D508C2311yescharacteristic
80C2×D508C2311yescharacteristic
81C2×D508C2311yescharacteristic
82C2×C22⋊D127—21no—
83C2×C22⋊D127—21no—
84C2×C22⋊D127—21no—
85C2×C22⋊D127—21no—
86D4×D127—44no—
87C22×Dic127—21no—
88C2×C4×D127—21no—
89C2×C4×D127—21no—
90#14523237—22no—
91#14523237—22no—
92#14523237—22no—
93#14523237—22no—
94#14523237—41no—
95#14523237D414yescomplemented
96D1016—22no—
97D1016—22no—
98D1016—41no—
99C4⋊Dic127C2311yescharacteristic
100C4⋊Dic127C2311yescharacteristic
101Dic127⋊C4—21no—
102Dic127⋊C4—21no—
103C22⋊Dic127—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109(C2×D127)⋊C4—21no—
110C4×Dic127—21no—
111C2×D4×C127D411yescharacteristic
112C22×C4×C127C2311yescharacteristic
113C22×C4×C127C2311yescharacteristic
114M16×C127—22no—
115C2×C8×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
122C22⋊C4×C127D411yescharacteristic
123C42×C127C2311yescharacteristic
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—21no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—42no—
132C22×D127—81no—
133C22×D127—81no—
134Dic254—21no—
135Dic254—21no—
136Dic254—21no—
137Dic254—42no—
138Dic254C2×D412yescomplemented
139D508—21no—
140D508—21no—
141D508—21no—
142D508—21no—
143D508—21no—
144D508—21no—
145D508—21no—
146D508—42no—
147D508—42no—
148D508—42no—
149D508—81no—
150D508C2×D412yes—
151C22⋊D127—42no—
152C22⋊D127—42no—
153C22⋊D127—42no—
154C22⋊D127—42no—
155C2×Dic127—21no—
156C2×Dic127—21no—
157C2×Dic127—21no—
158C2×Dic127—21no—
159C2×Dic127—42no—
160C4×D127—42no—
161C4×D127—42no—
162C23×C127C2×D411yescharacteristic
163C23×C127C2×D411yescharacteristic
164D4×C127—22no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127C2×D411yescharacteristic
170C2×C4×C127C2×D411yescharacteristic
171C2×C4×C127C2×D411yescharacteristic
172C2×C4×C127C2×D411yescharacteristic
173C2×C4×C127C2411yescharacteristic
174C8×C127—21no—
175C8×C127—21no—
176C8×C127—21no—
177C8×C127C2×D412yescomplemented
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—21no—
181C22×C127—21no—
182C22×C127—21no—
183C22×C127AGL(1,ℤ/8)12yescomplemented
184C22×C127C22×D411yescharacteristic
185C2×D127—22no—
186C2×D127—41no—
187C2×D127—41no—
188C2×D127—41no—
189C2×D127—41no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—84no—
198C2×D127—84no—
199C4×C127—21no—
200C4×C127—21no—
201C4×C127—21no—
202C4×C127—21no—
203C4×C127C22×D411yescharacteristic
204C4×C127H(5,𝔽2)11yescharacteristic
205Dic127—22no—
206Dic127—41no—
207Dic127—41no—
208Dic127—41no—
209C2×C127—22no—
210C2×C127—41no—
211C2×C127D4211yescharacteristic
212C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C127#57711yescharacteristic
214D127—42no—
215D127—81no—
216D127—81no—
217D127—81no—
218#3265—1271nomaximal, Sylow
219C127#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1271no—
221C22×SD16—1271no—
222#548—1271no—
223D42—1271no—
224#496—1271no—
225#468—1271no—
226#466—1271no—
227#463—1271no—
228#460—1271no—
229#450—1271no—
230#449—1271no—
231#448—1271no—
232#447—1271no—
233#440—1271no—
234#435D12711yescharacteristic, complemented
235C2×#41—1271no—
236C22×Q8—1271no—
237C22×D4—1271no—
238C22×D4—1271no—
239C22×D4—2541no—
240AGL(1,ℤ/8)—2544no—
241C2×SD16—1271no—
242C2×SD16—1271no—
243C2×SD16—1271no—
244C2×SD16—1271no—
245C2×SD16—1271no—
246C2×SD16—1271no—
247C2×SD16—2544no—
248C2×D8—1271no—
249C2×D8—1271no—
250C2×D8—1271no—
251C2×M16C2×D12711yescharacteristic, complemented
252C22×C8C2×D12711yescharacteristic, complemented
253#127—1271no—
254#124—1271no—
255#124—2541no—
256C22⋊Q8—1271no—
257C22⋊Q8—2541no—
258#121—1271no—
259#121—1271no—
260#121—1271no—
261#121—2541no—
262#121—2541no—
263#120—2541no—
264#120—2541no—
265C4×Q8—1271no—
266C4×D4—1271no—
267C4×D4—2541no—
268C4×D4C2×D12711yescharacteristic
269#107—1271no—
270C4⋊C8C2×D12711yescharacteristic, complemented
271#103—1271no—
272#103—1271no—
273#102—1271no—
274#102—1271no—
275#102—1271no—
276#102—1271no—
277C22⋊C8C2×D12711yescharacteristic, complemented
278C22⋊C8C2×D12711yescharacteristic, complemented
279#97C2×D12711yescharacteristic, complemented
280C24—2541no—
281C24—2541no—
282#41—5081no—
283#41—2542no—
284C2×Q8—1271no—
285C2×Q8—1271no—
286C2×Q8—5081no—
287C2×Q8—2542no—
288C2×Q8—2541no—
289C2×D4—1271no—
290C2×D4—1271no—
291C2×D4—1271no—
292C2×D4—1271no—
293C2×D4—5081no—
294C2×D4—5081no—
295C2×D4—5084no—
296C2×D4—2542no—
297C2×D4—2542no—
298C2×D4—2541no—
299C2×D4—2541no—
300C2×D4—2541no—
301C2×D4—2541no—
302C2×D4—2541no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4D50811yescharacteristic
307C22×C4—2541no—
308C22×C4—2541no—
309C22×C4—2541no—
310C22×C4C22×D12711yescharacteristic
311C22×C4C22×D12711yescharacteristic
312SD16—1274no—
313SD16—5081no—
314SD16—2542no—
315SD16—2542no—
316SD16—2542no—
317SD16—2542no—
318D8—5081no—
319D8—2542no—
320D8—2542no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D12711yescharacteristic
324C2×C8C22×D12711yescharacteristic
325C2×C8C22×D12711yescharacteristic
326C2×C8C22×D12711yescharacteristic
327C4⋊C4—1271no—
328C4⋊C4—1271no—
329C4⋊C4—2541no—
330C4⋊C4—2541no—
331C4⋊C4D50811yescharacteristic
332C22⋊C4—2541no—
333C22⋊C4—2541no—
334C22⋊C4—2541no—
335C22⋊C4—2541no—
336C22⋊C4—2541no—
337C22⋊C4—2541no—
338C22⋊C4—2541no—
339C22⋊C4D50811yescharacteristic
340C22⋊C4D50811yescharacteristic
341C42—2541no—
342C42C22×D12711yescharacteristic
343C23—10161no—
344C23—10161no—
345C23—5082no—
346C23—5082no—
347C23—5082no—
348C23—5082no—
349C23—2541no—
350C23—2541no—
351C23—2541no—
352C23—2541no—
353C23C2×D50811yescharacteristic
354C23C2×D50811yescharacteristic
355Q8—1272no—
356Q8—5082no—
357Q8—2541no—
358Q8—2541no—
359Q8—2541no—
360D4—22no—
361D4—1272no—
362D4—10161no—
363D4—5082no—
364D4—5082no—
365D4—5082no—
366D4—5082no—
367D4—5082no—
368D4—5082no—
369D4—5082no—
370D4—2541no—
371D4—2541no—
372D4—2541no—
373D4—2541no—
374D4—2541no—
375D4—2541no—
376D4—2541no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—5082no—
382C2×C4—5082no—
383C2×C4—5082no—
384C2×C4—2541no—
385C2×C4—2541no—
386C2×C4—2541no—
387C2×C4—2541no—
388C2×C4C2×D50811yescharacteristic
389C2×C4C2×D50811yescharacteristic
390C2×C4C2×D50811yescharacteristic
391C2×C4C2×D50811yescharacteristic
392C2×C4C23×D12711yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D12712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—10164no—
403C22—10164no—
404C22—5081no—
405C22—5081no—
406C22—5081no—
407C22—5081no—
408C22—5081no—
409C22—5081no—
410C22—5081no—
411C22—5081no—
412C22—5081no—
413C22—5081no—
414C22—5081no—
415C22—2542no—
416C22#1686123712yescomplemented
417C22C22×D50811yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—5081no—
423C4—5081no—
424C4—5081no—
425C4—2542no—
426C4C2×D4×D12711yescharacteristic
427C4#1686125511yescharacteristic
428C2—22no—
429C2—41no—
430C2—10161no—
431C2—10161no—
432C2—10161no—
433C2—5082no—
434C2D4×D50811yescharacteristic
435C2C2×#1686123711yescharacteristic
436C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#435×C127C2
socleC22×C127C22×D4
hypercenter#435D127
hypocenterC127#3265
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C22×#14523237C2semidirect
3. #22550535C2semidirect
4. D4×D508C2semidirect
5. #22550265C2semidirect
6. #22550122C2semidirect
7. #22550112C2semidirect
8. #22550092C2semidirect
9. #22550077C2semidirect
10. #22550029C2semidirect
11. #22550023C2semidirect
12. #22550017C2semidirect
13. #22550012C2semidirect
14. #22549978C2semidirect
15. #435×C127C2semidirect
16. C2×#14523248C22semidirect
17. C22×Dic254C22semidirect
25. C2×#14523237C22semidirect
26. C2×#14523237C22semidirect
27. C2×#14523237C22semidirect
28. C2×#14523237C22semidirect
30. C2×D1016C22semidirect
31. C2×D1016C22semidirect
33. #16861197C22semidirect
35. #16861188C22semidirect
40. #16861183C22semidirect
44. C4×Dic254C22semidirect
47. #16861143C22semidirect
48. #16861134C22semidirect
49. #16861134C22semidirect
51. #16861131C22semidirect
52. #16861131C22semidirect
53. #16861131C22semidirect
54. C2×M16×C127C22semidirect
55. C22×C8×C127C22semidirect
57. C4⋊C8×C127C22semidirect
58. C22⋊C8×C127C22semidirect
59. C22⋊C8×C127C22semidirect
60. #97×C127C22semidirect
95. #14523237D4semidirect
138. Dic254C2×D4semidirect
177. C8×C127C2×D4semidirect
183. C22×C127AGL(1,ℤ/8)semidirect
219. C127#3265semidirect
234. #435D127semidirect
251. C2×M16C2×D127semidirect
252. C22×C8C2×D127semidirect
270. C4⋊C8C2×D127semidirect
277. C22⋊C8C2×D127semidirect
278. C22⋊C8C2×D127semidirect
279. #97C2×D127semidirect
396. C8D4×D127semidirect
416. C22#16861237semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C22×#145232372
#225505352
D4×D5082
#225502652
#225501222
#225501122
#225500922
#225500772
#225500292
#225500232
#225500172
#225500122
#225499782
#435×C1272
#3265127