Finite Group Databasebeta

← #30647557

Subgroups — #30647557

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×#16861236C211yesmaximal, characteristic
2#22550358C211yesmaximal, characteristic, complemented
3#22550348C211yesmaximal, characteristic, complemented
4#22549987C211yesmaximal, characteristic, complemented
5#22549986C211yesmaximal, characteristic, complemented
6#22549977C211yesmaximal, characteristic, complemented
7#22549974C211yesmaximal, characteristic, complemented
8#22549961C211yesmaximal, characteristic, complemented
9C4×D1016C211yesmaximal, characteristic, complemented
10#22549956C211yesmaximal, characteristic, complemented
11#22549915C211yesmaximal, characteristic, complemented
12#22549871C211yesmaximal, characteristic, complemented
13#102×D127C211yesmaximal, characteristic, complemented
14#22549830C211yesmaximal, characteristic, complemented
15C4×SD16×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18#16861236C418yescomplemented
19C2×#14523235C2211yescharacteristic
20C2×D1016C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22C2×#14523231C2211yescharacteristic
23C2×C8×D127C2211yescharacteristic
24Q8×Dic127C2211yescharacteristic, complemented
25C4×D508—21no—
26C4×D508C2211yescharacteristic
27C4×D508C2211yescharacteristic, complemented
28D4×Dic127C2211yescharacteristic
29#16861171—21no—
30#16861171C2211yescharacteristic
31#16861171C2211yescharacteristic, complemented
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167C2211yescharacteristic
35#16861166—21no—
36#16861166C2211yescharacteristic
37C4⋊C4×D127—21no—
38C4⋊C4×D127C2211yescharacteristic
39C2×(C2×D127)⋊C4—21no—
40C22⋊C4×D127—21no—
41#16861143C2211yescharacteristic, complemented
42#16861142C2211yescharacteristic, complemented
43#16861140C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic, complemented
50#16861120C2211yescharacteristic, complemented
51#16861119C2211yescharacteristic, complemented
52C8×Dic127C2211yescharacteristic, complemented
53C2×SD16×C127C2211yescharacteristic
54C4×Q8×C127C2211yescharacteristic, complemented
55C4×D4×C127C2211yescharacteristic
56#106×C127C2211yescharacteristic, complemented
57#103×C127C2211yescharacteristic, complemented
58#102×C127C2211yescharacteristic
59C4×C8×C127C2211yescharacteristic, complemented
60C23×D127—21no—
61#14523249—22no—
62#14523249C2×C414yescomplemented
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67D4×D127—22no—
68D4×D127C2×C414yescomplemented
69C22×Dic127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127C2311yescharacteristic
77#14523235C2×C414yescomplemented
78D1016C2×C414yescomplemented
79#14523233C2×C414yescomplemented
80#14523231C2×C414yescomplemented
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C8×D127—22no—
84C4⋊Dic127—21no—
85C4⋊Dic127C2311yescharacteristic
86C4⋊Dic127C2311yescharacteristic
87Dic127⋊C4—21no—
88Dic127⋊C4D411yescharacteristic
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4D411yescharacteristic
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127—21no—
99C4×Dic127—21no—
100C4×Dic127C2311yescharacteristic
101C2×Q8×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104SD16×C127C2×C414yescomplemented
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127—21no—
108C4⋊C4×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C42×C127—21no—
112C42×C127C2311yescharacteristic
113C22×D127—21no—
114C22×D127—21no—
115C22×D127—24no—
116C22×D127—81no—
117C22×D127—81no—
118C22×D127C2×D411yescharacteristic
119D508—24no—
120D508—21no—
121D508—21no—
122D508C22×C412yes—
123D508C22×C412yes—
124C22⋊D127—24no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—81no—
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C4×D127—24no—
132C4×D127—81no—
133C4×D127—81no—
134C4×D127—81no—
135C4×D127—81no—
136C4×D127—81no—
137C4×D127C22×C412yes—
138C4×D127#4112yes—
139C127⋊C8—21no—
140C127⋊C8C22×C412yes—
141C23×C127—21no—
142Q8×C127—21no—
143Q8×C127C22×C412yes—
144D4×C127—21no—
145D4×C127C22×C412yes—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
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—41no—
157C22×C127—41no—
158C22×C127C22×D411yescharacteristic
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—22no—
162C2×D127—22no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—84no—
168C2×D127C4×D412yes—
169C4×C127—22no—
170C4×C127—21no—
171C4×C127—21no—
172C4×C127—41no—
173C4×C127—41no—
174C4×C127C23×C411yescharacteristic
175C4×C127C2×#4111yescharacteristic
176Dic127—21no—
177Dic127—21no—
178Dic127—41no—
179Dic127—41no—
180Dic127C4×D412yes—
181C2×C127—42no—
182C2×C127C2×C4×D411yescharacteristic
183C2×C127#57611yescharacteristic
184C2×C127#57711yescharacteristic
185D127—22no—
186D127—42no—
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#442—1271no—
195#440—1271no—
196#440—1271no—
197C4×SD16—1271no—
198C4×SD16D12711yescharacteristic, complemented
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×SD16C2×D12711yescharacteristic
210C2×D8—1271no—
211C2×D8—1271no—
212C2×M16—1271no—
213C22×C8—1271no—
214C4×Q8—1271no—
215C4×Q8C2×D12711yescharacteristic, complemented
216C4×D4—1271no—
217C4×D4—1271no—
218C4×D4—1271no—
219C4×D4—1271no—
220C4×D4—1271no—
221C4×D4—2541no—
222C4×D4—2541no—
223C4×D4—2541no—
224C4×D4—2541no—
225C4×D4C2×D12711yescharacteristic
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#103C2×D12711yescharacteristic, complemented
239#102—1271no—
240#102—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102C2×D12711yescharacteristic
245#97—1271no—
246#97—1271no—
247C4×C8—1271no—
248C4×C8C2×D12711yescharacteristic, complemented
249C24—2541no—
250#41—1274no—
251#41—2542no—
252C2×Q8C22×D12711yescharacteristic
253C2×D4—1274no—
254C2×D4—1271no—
255C2×D4—1271no—
256C2×D4—2541no—
257C2×D4—2542no—
258C2×D4—2541no—
259C2×D4C22×D12711yescharacteristic
260C22×C4—21no—
261C22×C4—1271no—
262C22×C4—2541no—
263C22×C4—2541no—
264C22×C4—2541no—
265C22×C4—2541no—
266C22×C4—2541no—
267C22×C4—2541no—
268C22×C4—2541no—
269SD16—1274no—
270SD16C4×D12714yescomplemented
271D8—1274no—
272D8—1274no—
273M16—1274no—
274C2×C8—1271no—
275C2×C8—1271no—
276C2×C8—2542no—
277C2×C8C22×D12711yescharacteristic
278C2×C8C22×D12711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—1271no—
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4C22×D12711yescharacteristic
287C4⋊C4C22×D12711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—1271no—
290C22⋊C4—1271no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C42—21no—
298C42—1271no—
299C42—2541no—
300C42—2541no—
301C42C22×D12711yescharacteristic
302C23—21no—
303C23—1271no—
304C23—10161no—
305C23—10161no—
306C23—2544no—
307C23—2541no—
308C23—2541no—
309Q8—21no—
310Q8C2×C4×D12712yes—
311D4—21no—
312D4—1272no—
313D4—1272no—
314D4—2544no—
315D4—2544no—
316D4—2541no—
317D4—2541no—
318D4C2×C4×D12712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
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—10161no—
334C2×C4—2544no—
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—41no—
346C22—41no—
347C22—1272no—
348C22—10164no—
349C22—5081no—
350C22—5081no—
351C22—5081no—
352C22—5081no—
353C22—2542no—
354C22—2542no—
355C22—2541no—
356C22—2541no—
357C22C2×D4×D12711yescharacteristic
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—1272no—
364C4—5081no—
365C4—5081no—
366C4—2541no—
367C4—2541no—
368C4C22×C4×D12711yescharacteristic
369C4#41×D12711yescharacteristic
370C2—42no—
371C2—5082no—
372C2—5082no—
373C2—2542no—
374C2C4×D4×D12711yescharacteristic
375C2#2255067811yescharacteristic
376C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
FittingC4×SD16×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861236C2
cosocleC2×C4×C127C24
hypercenterC4×SD16D127
hypocenterC127#2946
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #22550358C2semidirect
3. #22550348C2semidirect
4. #22549987C2semidirect
5. #22549986C2semidirect
6. #22549977C2semidirect
7. #22549974C2semidirect
8. #22549961C2semidirect
9. C4×D1016C2semidirect
10. #22549956C2semidirect
11. #22549915C2semidirect
12. #22549871C2semidirect
13. #102×D127C2semidirect
14. #22549830C2semidirect
15. C4×SD16×C127C2semidirect
18. #16861236C4semidirect
24. Q8×Dic127C22semidirect
27. C4×D508C22semidirect
31. #16861171C22semidirect
41. #16861143C22semidirect
42. #16861142C22semidirect
43. #16861140C22semidirect
44. #16861133C22semidirect
46. #16861131C22semidirect
49. #16861129C22semidirect
50. #16861120C22semidirect
51. #16861119C22semidirect
52. C8×Dic127C22semidirect
54. C4×Q8×C127C22semidirect
56. #106×C127C22semidirect
57. #103×C127C22semidirect
59. C4×C8×C127C22semidirect
62. #14523249C2×C4semidirect
68. D4×D127C2×C4semidirect
77. #14523235C2×C4semidirect
78. D1016C2×C4semidirect
79. #14523233C2×C4semidirect
80. #14523231C2×C4semidirect
104. SD16×C127C2×C4semidirect
189. C127#2946semidirect
198. C4×SD16D127semidirect
215. C4×Q8C2×D127semidirect
236. #106C2×D127semidirect
238. #103C2×D127semidirect
248. C4×C8C2×D127semidirect
270. SD16C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
#225503582
#225503482
#225499872
#225499862
#225499772
#225499742
#225499612
C4×D10162
#225499562
#225499152
#225498712
#102×D1272
#225498302
C4×SD16×C1272
#2946127