Finite Group Databasebeta

← #30650328

Subgroups — #30650328

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×#16861236C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3#22550528C211yesmaximal, characteristic, complemented
4D4×C22⋊D127C211yesmaximal, characteristic, complemented
5#22550247C211yesmaximal, characteristic, complemented
6#22550124C211yesmaximal, characteristic, complemented
7#22550121C211yesmaximal, characteristic, complemented
8#22550079C211yesmaximal, characteristic, complemented
9#22550070C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22550023C211yesmaximal, characteristic, complemented
12#22550020C211yesmaximal, characteristic, complemented
13#22549956C211yesmaximal, characteristic, complemented
14#22549951C211yesmaximal, characteristic, complemented
15#450×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C2×#14523248C2211yescharacteristic, complemented
18C22×C22⋊D127—21no—
19C2×D4×D127C2211yescharacteristic
20#16861236—24no—
21#16861235—24no—
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523236C2211yescharacteristic
24C2×#14523236C2211yescharacteristic, complemented
25C2×#14523235C2211yescharacteristic, complemented
26C2×D1016C2211yescharacteristic, complemented
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic, complemented
29C2×#14523232C2211yescharacteristic, complemented
30C2×#14523231C2211yescharacteristic, complemented
31#16861207C2211yescharacteristic, complemented
32#16861205C2211yescharacteristic
33#16861194—21no—
34#16861191—21no—
35#16861189C2211yescharacteristic, complemented
36#16861183C2211yescharacteristic
37#16861182—21no—
38#16861181—21no—
39#16861180C2211yescharacteristic
40#16861178—21no—
41#16861177—21no—
42D4×Dic127C2211yescharacteristic
43#16861171C2211yescharacteristic, complemented
44C4×C22⋊D127C2211yescharacteristic
45C4⋊C4×D127—21no—
46#16861142C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861133C2211yescharacteristic, complemented
50#16861131C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C8×Dic127C2211yescharacteristic, complemented
54C22×D4×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56C2×SD16×C127C2211yescharacteristic, complemented
57C22⋊Q8×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic, complemented
60C22⋊C8×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64#14523248—22no—
65#14523248—41no—
66C2×Dic254C2311yescharacteristic
67C2×D508—21no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—44no—
77C2×C22⋊D127D411yescharacteristic
78D4×D127—22no—
79D4×D127—41no—
80C22×Dic127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127—21no—
84C2×C4×D127C2311yescharacteristic
85#14523237—41no—
86#14523236—22no—
87#14523236—22no—
88#14523235—22no—
89D1016—22no—
90#14523233—22no—
91#14523233D414yescomplemented
92#14523232—22no—
93#14523231—22no—
94C2×C127⋊C8C2311yescharacteristic
95C2×C127⋊C8C2311yescharacteristic
96C4⋊Dic127—21no—
97C4⋊Dic127C2311yescharacteristic
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4D411yescharacteristic
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108C4×Dic127C2311yescharacteristic
109C24×C127—21no—
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—22no—
112C2×D4×C127—21no—
113C2×D4×C127—41no—
114C2×D4×C127C2311yescharacteristic
115C2×D4×C127C2311yescharacteristic
116C22×C4×C127C2311yescharacteristic
117SD16×C127—22no—
118SD16×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127—21no—
122C4⋊C4×C127C2311yescharacteristic
123C22⋊C4×C127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—81no—
129C22×D127C2×D411yescharacteristic
130Dic254—21no—
131Dic254—21no—
132D508—21no—
133D508—21no—
134D508—21no—
135D508—42no—
136D508C2×D412yescomplemented
137C22⋊D127—22no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C22⋊D127—42no—
142C22⋊D127—42no—
143C22⋊D127—81no—
144C2×Dic127—21no—
145C2×Dic127—21no—
146C2×Dic127—42no—
147C2×Dic127C2×D411yescharacteristic
148C2×Dic127C2×D411yescharacteristic
149C2×Dic127C2×D411yescharacteristic
150C4×D127—21no—
151C4×D127—21no—
152C4×D127—42no—
153C4×D127—42no—
154C4×D127—42no—
155C127⋊C8—21no—
156C127⋊C8C2×D412yescomplemented
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—42no—
160C23×C127—42no—
161C23×C127—81no—
162C23×C127C2×D411yescharacteristic
163Q8×C127—21no—
164Q8×C127—21no—
165D4×C127—21no—
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—42no—
169D4×C127C2×D412yes—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127C2×D411yescharacteristic
175C2×C4×C127C2411yescharacteristic
176C8×C127—21no—
177C8×C127—21no—
178C22×C127—21no—
179C22×C127—22no—
180C22×C127—21no—
181C22×C127—21no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—84no—
188C22×C127C22×D411yescharacteristic
189C2×D127—21no—
190C2×D127—22no—
191C2×D127—21no—
192C2×D127—21no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—84no—
199C4×C127—21no—
200C4×C127—41no—
201C4×C127—41no—
202C4×C127C22×D411yescharacteristic
203C4×C127H(5,𝔽2)11yescharacteristic
204Dic127—21no—
205Dic127—21no—
206Dic127—41no—
207Dic127—41no—
208Dic127AGL(1,ℤ/8)12yescomplemented
209C2×C127—42no—
210C2×C127—41no—
211C2×C127—81no—
212C2×C127D4211yescharacteristic
213C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C127#57711yescharacteristic
215D127—41no—
216D127—42no—
217D127—81no—
218#3272—1271nomaximal, Sylow
219C127#327211yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1271no—
221C2×AGL(1,ℤ/8)—1271no—
222#547—1271no—
223D42—1271no—
224#492—1271no—
225#468—1271no—
226#468—1271no—
227#461—1271no—
228#459—1271no—
229#450—1271no—
230#450D12711yescharacteristic, complemented
231#449—1271no—
232#449—1271no—
233C4×D8—1271no—
234#436—1271no—
235C2×#41—1271no—
236C2×#41—1271no—
237C22×D4—1271no—
238C22×D4—2541no—
239C22×D4C2×D12711yescharacteristic
240AGL(1,ℤ/8)—2544no—
241AGL(1,ℤ/8)—2544no—
242C2×SD16—1271no—
243C2×SD16—1271no—
244C2×SD16—1271no—
245C2×SD16—1271no—
246C2×SD16C2×D12711yescharacteristic
247C2×SD16C2×D12711yescharacteristic, complemented
248C2×D8—1271no—
249C2×D8—1271no—
250C2×D8—1271no—
251C2×M16—1271no—
252C2×M16—1271no—
253C4⋊Q8—1271no—
254#127—1271no—
255#123—2541no—
256#123—2541no—
257C22⋊Q8—1271no—
258C22⋊Q8C2×D12711yescharacteristic, complemented
259#121—1271no—
260#121—1271no—
261#121—2541no—
262#121—2541no—
263#120—2541no—
264#120—2541no—
265C4×D4—1271no—
266C4×D4—1271no—
267C4×D4—1271no—
268C2×C4⋊C4—2541no—
269#107—1271no—
270C4⋊C8—1271no—
271#103—1271no—
272#103—1271no—
273#102—1271no—
274#102—1271no—
275#102C2×D12711yescharacteristic
276#102C2×D12711yescharacteristic, complemented
277C22⋊C8—1271no—
278C22⋊C8C2×D12711yescharacteristic, complemented
279C4×C8—1271no—
280C24—21no—
281C24—2541no—
282#41—5081no—
283#41—5081no—
284#41—2542no—
285#41—2542no—
286C2×Q8—1271no—
287C2×Q8C22×D12711yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—1271no—
292C2×D4—1271no—
293C2×D4—1271no—
294C2×D4—5084no—
295C2×D4—5081no—
296C2×D4—2541no—
297C2×D4—2541no—
298C2×D4—2541no—
299C2×D4—2541no—
300C2×D4—2541no—
301C2×D4—2541no—
302C2×D4—2541no—
303C2×D4—2542no—
304C2×D4C22×D12711yescharacteristic
305C2×D4C22×D12711yescharacteristic
306C22×C4—1271no—
307C22×C4—2541no—
308C22×C4—2541no—
309C22×C4—2541no—
310C22×C4—2541no—
311C22×C4C22×D12711yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—5081no—
315SD16—2542no—
316SD16—2542no—
317SD16—2542no—
318D8—1274no—
319D8—2542no—
320D8—2542no—
321M16—2542no—
322M16—2542no—
323C2×C8—1271no—
324C2×C8—1271no—
325C2×C8C22×D12711yescharacteristic
326C2×C8C22×D12711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—1271no—
329C4⋊C4—1271no—
330C4⋊C4—2541no—
331C4⋊C4—2541no—
332C4⋊C4—2541no—
333C4⋊C4C22×D12711yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—1271no—
336C22⋊C4—1271no—
337C22⋊C4—2541no—
338C22⋊C4—2541no—
339C22⋊C4—2541no—
340C22⋊C4—2541no—
341C22⋊C4—2541no—
342C42—1271no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—1271no—
349C23—10161no—
350C23—5082no—
351C23—5082no—
352C23—2541no—
353C23—2541no—
354C23D4×D12711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2541no—
358Q8—2541no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—1272no—
364D4—10161no—
365D4—5082no—
366D4—5082no—
367D4—5082no—
368D4—5082no—
369D4—5082no—
370D4—5082no—
371D4—2542no—
372D4—2541no—
373D4—2541no—
374D4—2541no—
375D4D4×D12712yes—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—1271no—
381C2×C4—1271no—
382C2×C4—1271no—
383C2×C4—5082no—
384C2×C4—5082no—
385C2×C4—5082no—
386C2×C4—5082no—
387C2×C4—2541no—
388C2×C4—2541no—
389C2×C4—2541no—
390C2×C4—2541no—
391C2×C4D4×D12711yescharacteristic
392C2×C4C23×D12711yescharacteristic
393C8—21no—
394C8—21no—
395C8—1272no—
396C8—2541no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—10164no—
408C22—5081no—
409C22—5081no—
410C22—5081no—
411C22—5081no—
412C22—5081no—
413C22—2541no—
414C22—2541no—
415C22—2542no—
416C22—2541no—
417C22C2×D4×D12711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—1272no—
422C4—5081no—
423C4—5081no—
424C4—2541no—
425C4—2541no—
426C4C2×D4×D12711yescharacteristic
427C4#1686125411yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—10161no—
432C2—5081no—
433C2—5082no—
434C2#2255050811yescharacteristic
435C2AGL(1,ℤ/8)×D12711yescharacteristic
436C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
hypercenter#450D127
hypocenterC127#3272
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861235C2semidirect
3. #22550528C2semidirect
4. D4×C22⋊D127C2semidirect
5. #22550247C2semidirect
6. #22550124C2semidirect
7. #22550121C2semidirect
8. #22550079C2semidirect
9. #22550070C2semidirect
10. #22550028C2semidirect
11. #22550023C2semidirect
12. #22550020C2semidirect
13. #22549956C2semidirect
14. #22549951C2semidirect
15. #450×C127C2semidirect
16. C2×#14523249C22semidirect
17. C2×#14523248C22semidirect
22. C2×#14523237C22semidirect
24. C2×#14523236C22semidirect
25. C2×#14523235C22semidirect
26. C2×D1016C22semidirect
28. C2×#14523233C22semidirect
29. C2×#14523232C22semidirect
30. C2×#14523231C22semidirect
31. #16861207C22semidirect
35. #16861189C22semidirect
43. #16861171C22semidirect
46. #16861142C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861133C22semidirect
50. #16861131C22semidirect
51. #16861129C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C8×Dic127C22semidirect
56. C2×SD16×C127C22semidirect
57. C22⋊Q8×C127C22semidirect
59. #102×C127C22semidirect
60. C22⋊C8×C127C22semidirect
91. #14523233D4semidirect
136. D508C2×D4semidirect
156. C127⋊C8C2×D4semidirect
208. Dic127AGL(1,ℤ/8)semidirect
219. C127#3272semidirect
230. #450D127semidirect
247. C2×SD16C2×D127semidirect
258. C22⋊Q8C2×D127semidirect
276. #102C2×D127semidirect
278. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×#168612352
#225505282
D4×C22⋊D1272
#225502472
#225501242
#225501212
#225500792
#225500702
#225500282
#225500232
#225500202
#225499562
#225499512
#450×C1272
#3272127