Finite Group Databasebeta

← #30647559

Subgroups — #30647559

Order 16256.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861233C211yesmaximal, characteristic
2#22550355C211yesmaximal, characteristic, complemented
3#22550348C211yesmaximal, characteristic, complemented
4#22549985C211yesmaximal, characteristic, complemented
5#22549978C211yesmaximal, characteristic, complemented
6#22549973C211yesmaximal, characteristic, complemented
7#22549963C211yesmaximal, characteristic, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9D8×Dic127C211yesmaximal, characteristic, complemented
10C4×#14523233C211yesmaximal, characteristic, complemented
11#22549915C211yesmaximal, characteristic, complemented
12#22549869C211yesmaximal, characteristic, complemented
13#102×D127C211yesmaximal, characteristic, complemented
14#22549833C211yesmaximal, characteristic, complemented
15#442×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18#16861233C418yescomplemented
19C2×#14523237C2211yescharacteristic
20C2×#14523236C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22C2×#14523231C2211yescharacteristic
23C2×C8×D127C2211yescharacteristic
24C4×Dic254C2211yescharacteristic, complemented
25#16861174C2211yescharacteristic, complemented
26C4×D508C2211yescharacteristic
27D4×Dic127C2211yescharacteristic
28D4×Dic127C2211yescharacteristic, complemented
29#16861171C2211yescharacteristic
30C4×C22⋊D127—21no—
31C4×C22⋊D127—21no—
32(C22⋊D127)⋊C4—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167C2211yescharacteristic
35#16861166C2211yescharacteristic
36#16861165—21no—
37C2×Dic127⋊C4—21no—
38C4⋊C4×D127C2211yescharacteristic
39C2×(C2×D127)⋊C4—21no—
40C22⋊C4×D127—21no—
41#16861143C2211yescharacteristic, complemented
42#16861142C2211yescharacteristic, complemented
43#16861140C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861132C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C4×C127⋊C8C2211yescharacteristic, complemented
52C8×Dic127C2211yescharacteristic, complemented
53C2×D8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic, complemented
56#106×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic, complemented
59#97×C127C2211yescharacteristic, complemented
60C23×D127—21no—
61#14523247—22no—
62#14523247C2×C414yescomplemented
63C2×Dic254C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67D4×D127—22no—
68D4×D127C2×C414yescomplemented
69C22×Dic127—21no—
70C22×Dic127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127C2311yescharacteristic
76#14523237C2×C414yescomplemented
77#14523236C2×C414yescomplemented
78#14523233C2×C414yescomplemented
79#14523231C2×C414yescomplemented
80C2×C127⋊C8C2311yescharacteristic
81C2×C127⋊C8C2311yescharacteristic
82C8×D127—22no—
83C4⋊Dic127C2311yescharacteristic
84C4⋊Dic127C2311yescharacteristic
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4D411yescharacteristic
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96(C2×D127)⋊C4D411yescharacteristic
97C4×Dic127—21no—
98C4×Dic127—21no—
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic
101C2×D4×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104C22×C4×C127—21no—
105D8×C127C2×C414yescomplemented
106C2×C8×C127C2311yescharacteristic
107C2×C8×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C22⋊C4×C127—21no—
112C42×C127C2311yescharacteristic
113C22×D127—21no—
114C22×D127—24no—
115C22×D127—81no—
116C22×D127—81no—
117C22×D127C2×D411yescharacteristic
118Dic254—21no—
119Dic254C22×C412yes—
120D508—21no—
121D508C22×C412yes—
122C22⋊D127—24no—
123C22⋊D127—24no—
124C2×Dic127—21no—
125C2×Dic127—24no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—81no—
129C2×Dic127—81no—
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C4×D127—81no—
134C4×D127—81no—
135C4×D127—81no—
136C4×D127C22×C412yes—
137C4×D127#4112yes—
138C127⋊C8—21no—
139C127⋊C8C22×C412yes—
140C23×C127—21no—
141C23×C127—21no—
142D4×C127—21no—
143D4×C127—21no—
144D4×C127C22×C412yes—
145D4×C127C22×C412yes—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—81no—
149C2×C4×C127—81no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127C22×C412yes—
155C22×C127—22no—
156C22×C127—22no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—22no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—84no—
168C2×D127C4×D412yes—
169C4×C127—21no—
170C4×C127—21no—
171C4×C127—41no—
172C4×C127—41no—
173C4×C127C23×C411yescharacteristic
174C4×C127C2×#4111yescharacteristic
175Dic127—22no—
176Dic127—21no—
177Dic127—21no—
178Dic127—41no—
179Dic127—41no—
180Dic127C4×D412yes—
181C2×C127—42no—
182C2×C127—42no—
183C2×C127C2×C4×D411yescharacteristic
184C2×C127#57611yescharacteristic
185C2×C127#57711yescharacteristic
186D127—22no—
187D127—42no—
188#2946—1271nomaximal, Sylow
189C127#294611yescharacteristic, complemented, Sylow
190C2×AGL(1,ℤ/8)—1271no—
191#520—1271no—
192#518—1271no—
193#442—1271no—
194#442D12711yescharacteristic, complemented
195#440—1271no—
196#440—1271no—
197C4×SD16—1271no—
198C4×SD16—1271no—
199C4×D8—1271no—
200C4×D8—1271no—
201#428—1271no—
202#416—1271no—
203C2×#102—1271no—
204#405—1271no—
205C2×#41—1271no—
206C22×D4—1271no—
207AGL(1,ℤ/8)—1278no—
208C2×SD16—1271no—
209C2×SD16—1271no—
210C2×D8—1271no—
211C2×D8C2×D12711yescharacteristic
212C2×M16—1271no—
213C22×C8—1271no—
214C4×Q8—1271no—
215C4×Q8—1271no—
216C4×D4—1271no—
217C4×D4—1271no—
218C4×D4—1271no—
219C4×D4—1271no—
220C4×D4—2541no—
221C4×D4—2541no—
222C4×D4—2541no—
223C4×D4—2541no—
224C4×D4C2×D12711yescharacteristic
225C4×D4C2×D12711yescharacteristic, complemented
226#117—1271no—
227#117—1271no—
228#117—2541no—
229C2×C4⋊C4—1271no—
230C2×C4⋊C4—2541no—
231C2×C22⋊C4—2541no—
232C2×C22⋊C4—2541no—
233#107—1271no—
234#107—1271no—
235#106—1271no—
236#106C2×D12711yescharacteristic, complemented
237#103—1271no—
238#103—1271no—
239#102—1271no—
240#102—1271no—
241#102—1271no—
242#102—1271no—
243#102C2×D12711yescharacteristic
244#102C2×D12711yescharacteristic, complemented
245#97—1271no—
246#97C2×D12711yescharacteristic, complemented
247C4×C8—1271no—
248C4×C8—1271no—
249C24—2541no—
250#41—1274no—
251#41—2542no—
252C2×Q8—1271no—
253C2×D4—1274no—
254C2×D4—1271no—
255C2×D4—2542no—
256C2×D4—2541no—
257C2×D4—2541no—
258C2×D4C22×D12711yescharacteristic
259C2×D4C22×D12711yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—1271no—
263C22×C4—2541no—
264C22×C4—2541no—
265C22×C4—2541no—
266C22×C4—2541no—
267C22×C4—2541no—
268C22×C4—2541no—
269SD16—1274no—
270SD16—1274no—
271D8—1274no—
272D8C4×D12714yescomplemented
273M16—1274no—
274C2×C8—1271no—
275C2×C8—1271no—
276C2×C8—2542no—
277C2×C8C22×D12711yescharacteristic
278C2×C8C22×D12711yescharacteristic
279C4⋊C4—1271no—
280C4⋊C4—1271no—
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—2541no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4C22×D12711yescharacteristic
287C4⋊C4C22×D12711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—1271no—
291C22⋊C4—1271no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C42—1271no—
298C42—2541no—
299C42—2541no—
300C42—2541no—
301C42C22×D12711yescharacteristic
302C23—21no—
303C23—21no—
304C23—1271no—
305C23—10161no—
306C23—10161no—
307C23—2544no—
308C23—2541no—
309Q8—1272no—
310Q8—2541no—
311D4—21no—
312D4—21no—
313D4—1272no—
314D4—2544no—
315D4—2544no—
316D4—2541no—
317D4C2×C4×D12712yes—
318D4C2×C4×D12712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—1271no—
324C2×C4—1272no—
325C2×C4—1272no—
326C2×C4—1271no—
327C2×C4—1271no—
328C2×C4—10161no—
329C2×C4—10161no—
330C2×C4—10161no—
331C2×C4—10161no—
332C2×C4—10161no—
333C2×C4—2544no—
334C2×C4—2541no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4D4×D12711yescharacteristic
338C2×C4D4×D12711yescharacteristic
339C2×C4C23×D12711yescharacteristic
340C8—21no—
341C8—1272no—
342C8—2541no—
343C8C2×C4×D12712yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—1272no—
351C22—10164no—
352C22—5081no—
353C22—5081no—
354C22—2542no—
355C22—2541no—
356C22—2541no—
357C22C2×D4×D12711yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—1272no—
363C4—5081no—
364C4—5081no—
365C4—2542no—
366C4—2541no—
367C4—2541no—
368C4C22×C4×D12711yescharacteristic
369C4#41×D12711yescharacteristic
370C2—42no—
371C2—42no—
372C2—5082no—
373C2—2542no—
374C2C4×D4×D12711yescharacteristic
375C2#2255068011yescharacteristic
376C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
Fitting#442×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861233C2
cosocleC2×C4×C127C24
hypercenter#442D127
hypocenterC127#2946
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #22550355C2semidirect
3. #22550348C2semidirect
4. #22549985C2semidirect
5. #22549978C2semidirect
6. #22549973C2semidirect
7. #22549963C2semidirect
8. C4×#14523236C2semidirect
9. D8×Dic127C2semidirect
10. C4×#14523233C2semidirect
11. #22549915C2semidirect
12. #22549869C2semidirect
13. #102×D127C2semidirect
14. #22549833C2semidirect
15. #442×C127C2semidirect
18. #16861233C4semidirect
24. C4×Dic254C22semidirect
25. #16861174C22semidirect
28. D4×Dic127C22semidirect
41. #16861143C22semidirect
42. #16861142C22semidirect
43. #16861140C22semidirect
44. #16861134C22semidirect
45. #16861132C22semidirect
48. #16861130C22semidirect
50. #16861119C22semidirect
51. C4×C127⋊C8C22semidirect
52. C8×Dic127C22semidirect
55. C4×D4×C127C22semidirect
56. #106×C127C22semidirect
58. #102×C127C22semidirect
59. #97×C127C22semidirect
62. #14523247C2×C4semidirect
68. D4×D127C2×C4semidirect
76. #14523237C2×C4semidirect
77. #14523236C2×C4semidirect
78. #14523233C2×C4semidirect
79. #14523231C2×C4semidirect
105. D8×C127C2×C4semidirect
189. C127#2946semidirect
194. #442D127semidirect
225. C4×D4C2×D127semidirect
236. #106C2×D127semidirect
244. #102C2×D127semidirect
246. #97C2×D127semidirect
272. D8C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
#225503552
#225503482
#225499852
#225499782
#225499732
#225499632
C4×#145232362
D8×Dic1272
C4×#145232332
#225499152
#225498692
#102×D1272
#225498332
#442×C1272
#2946127