Finite Group Databasebeta

← #30647955

Subgroups — #30647955

Order 16256.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D127C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic, complemented
3#22550383C211yesmaximal, characteristic, complemented
4#22550025C214yesmaximal, complemented
5#22550008C214yesmaximal, complemented
6#22549882C211yesmaximal, characteristic, complemented
7#22549881C211yesmaximal, characteristic, complemented
8#22549839C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
10C24×D127—21no—
11C22×D508C2211yescharacteristic
12C22×C22⋊D127—21no—
13C2×D4×D127—24no—
14C2×D4×D127—22no—
15C2×D4×D127—24no—
16C2×D4×D127C2214yes—
17C22×C4×D127C2211yescharacteristic
18#16861234—24no—
19C2×#14523235C2212yescomplemented
20C2×#14523233C2212yescomplemented
21C2×#14523232C2211yescharacteristic, complemented
22#16861190—21no—
23#16861185C2212yescomplemented
24#16861180C2212yescomplemented
25#16861177—21no—
26C4×C22⋊D127—22no—
27#16861168C2211yescharacteristic, complemented
28#16861131C2214yescomplemented
29#16861130C2214yescomplemented
30(C2×D127)⋊C8C2214yescomplemented
31C2×#41×C127C2211yescharacteristic, complemented
32C22×D4×C127C2211yescharacteristic
33AGL(1,ℤ/8)×C127—24no—
34C2×SD16×C127C2212yescomplemented
35C2×D8×C127C2212yescomplemented
36C2×M16×C127C2211yescharacteristic, complemented
37C23×D127—28no—
38C23×D127—24no—
39C23×D127—21no—
40C23×D127—48no—
41C23×D127D411yescharacteristic
42C2×D508—21no—
43C2×D508—22no—
44C2×D508D414yescomplemented
45C2×D508C2312yes—
46C2×C22⋊D127—28no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—24no—
49D4×D127—216no—
50D4×D127—28no—
51D4×D127—28no—
52C22×Dic127D411yescharacteristic
53C2×C4×D127—22no—
54C2×C4×D127—22no—
55C2×C4×D127D412yes—
56C2×C4×D127C2312yes—
57#14523235—24no—
58#14523233—24no—
59#14523232—22no—
60C2×C127⋊C8C2312yescomplemented
61C4⋊Dic127C2312yescomplemented
62Dic127⋊C4—22no—
63C22⋊Dic127—22no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—22no—
66C4×Dic127—21no—
67C24×C127—21no—
68#41×C127—22no—
69#41×C127—22no—
70C2×Q8×C127C2311yescharacteristic, complemented
71C2×D4×C127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—22no—
74C2×D4×C127D414yescomplemented
75C2×D4×C127C2312yes—
76C2×D4×C127C2311yescharacteristic, complemented
77C22×C4×C127—21no—
78C22×C4×C127C2311yescharacteristic
79SD16×C127—24no—
80D8×C127—24no—
81M16×C127—22no—
82C2×C8×C127C2312yescomplemented
83C22×D127—22no—
84C22×D127—24no—
85C22×D127—216no—
86C22×D127—24no—
87C22×D127—22no—
88C22×D127—432no—
89C22×D127—48no—
90C22×D127—44no—
91C22×D127—48no—
92C22×D127C2×D412yes—
93D508—24no—
94D508—22no—
95D508C2×D414yes—
96C22⋊D127—216no—
97C22⋊D127—44no—
98C2×Dic127—22no—
99C2×Dic127—22no—
100C2×Dic127C2×D412yes—
101C4×D127—24no—
102C4×D127—22no—
103C4×D127—22no—
104C127⋊C8—22no—
105C23×C127—24no—
106C23×C127—22no—
107C23×C127—21no—
108C23×C127—44no—
109C23×C127C2×D411yescharacteristic
110Q8×C127—21no—
111Q8×C127—21no—
112D4×C127—24no—
113D4×C127—22no—
114D4×C127—21no—
115D4×C127—21no—
116D4×C127—42no—
117D4×C127C2×D414yes—
118C2×C4×C127—21no—
119C2×C4×C127—42no—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D412yes—
123C2×C4×C127C2411yescharacteristic
124C8×C127—22no—
125C22×C127—22no—
126C22×C127—24no—
127C22×C127—48no—
128C22×C127—44no—
129C22×C127—41no—
130C22×C127—42no—
131C22×C127#12012yes—
132C22×C127C22×D411yescharacteristic
133C2×D127—28no—
134C2×D127—22no—
135C2×D127—22no—
136C2×D127—24no—
137C2×D127—432no—
138C2×D127—48no—
139C2×D127—44no—
140C2×D127AGL(1,ℤ/8)14yescomplemented
141C4×C127—41no—
142C4×C127#12012yes—
143C4×C127C22×D411yescharacteristic
144C4×C127C22×D411yescharacteristic
145Dic127—22no—
146Dic127—42no—
147C2×C127—22no—
148C2×C127—44no—
149C2×C127—81no—
150C2×C127C2×#12011yescharacteristic
151C2×C127C2×AGL(1,ℤ/8)12yes—
152D127—24no—
153D127—44no—
154#3003—1271nomaximal, Sylow
155C127#300311yescharacteristic, complemented, Sylow
156C23×D4—1271no—
157C2×AGL(1,ℤ/8)—1271no—
158C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
159#525—1271no—
160#450—1274no—
161#447—1274no—
162#418—1271no—
163#418—1271no—
164#407—1271no—
165C25—2541no—
166C2×#41C2×D12711yescharacteristic, complemented
167C22×D4—1274no—
168C22×D4—1271no—
169C22×D4—2542no—
170C22×D4—2541no—
171C22×D4—2544no—
172C22×D4—2544no—
173C22×D4C2×D12711yescharacteristic
174C23×C4—1271no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2544no—
177C2×SD16—1272no—
178C2×SD16C2×D12712yescomplemented
179C2×D8—1272no—
180C2×D8C2×D12712yescomplemented
181C2×M16—1271no—
182C2×M16C2×D12711yescharacteristic, complemented
183#123—2541no—
184C22⋊Q8—1272no—
185#121—1272no—
186#120—2541no—
187C4×D4—2542no—
188#117—1271no—
189#102—1274no—
190#102—1274no—
191C22⋊C8—1274no—
192C24—21no—
193C24—1271no—
194C24—5088no—
195C24—2544no—
196C24—2548no—
197C24—2541no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D12711yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—1274no—
205C2×D4—1272no—
206C2×D4—2544no—
207C2×D4—2548no—
208C2×D4—25416no—
209C2×D4—2542no—
210C2×D4—2548no—
211C2×D4—2548no—
212C2×D4—2541no—
213C2×D4—2542no—
214C2×D4C22⋊D12714yescomplemented
215C2×D4C22×D12712yes—
216C2×D4C22×D12711yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—1271no—
219C22×C4—1272no—
220C22×C4—1272no—
221C22×C4—2542no—
222C22×C4—2542no—
223C22×C4C22×D12711yescharacteristic
224SD16—24no—
225SD16—2544no—
226D8—24no—
227D8—2544no—
228M16—22no—
229M16—2542no—
230C2×C8—1272no—
231C2×C8C22×D12712yescomplemented
232C4⋊C4—1272no—
233C4⋊C4—2542no—
234C22⋊C4—2542no—
235C22⋊C4—2542no—
236C22⋊C4—2541no—
237C42—2541no—
238C23—22no—
239C23—24no—
240C23—21no—
241C23—44no—
242C23—1272no—
243C23—5084no—
244C23—5088no—
245C23—50832no—
246C23—5088no—
247C23—2544no—
248C23—25416no—
249C23—2542no—
250C23—2544no—
251C23—2542no—
252C23D4×D12711yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—24no—
259D4—42no—
260D4—1274no—
261D4—5084no—
262D4—25416no—
263D4—2542no—
264D4—2544no—
265D4C2×C22⋊D12714yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—1272no—
269C2×C4—2542no—
270C2×C4—2542no—
271C2×C4—2542no—
272C2×C4—2542no—
273C2×C4—2544no—
274C2×C4D4×D12712yes—
275C2×C4D4×D12711yescharacteristic
276C2×C4C2×C22⋊D12712yes—
277C2×C4C23×D12711yescharacteristic
278C8—22no—
279C8—2542no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—1274no—
287C22—5088no—
288C22—50832no—
289C22—5084no—
290C22—2548no—
291C22—2544no—
292C22—2542no—
293C22—2542no—
294C22#1686117712yes—
295C22C2×D4×D12711yescharacteristic
296C4—41no—
297C4—5082no—
298C4—2542no—
299C4#1686117712yes—
300C4C2×D4×D12711yescharacteristic
301C4C22×C22⋊D12711yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—5084no—
306C2—2544no—
307C2C2×#1686117711yescharacteristic
308C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3003
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D127C2semidirect
2. C2×#16861234C2semidirect
3. #22550383C2semidirect
4. #22550025C2semidirect
5. #22550008C2semidirect
6. #22549882C2semidirect
7. #22549881C2semidirect
8. #22549839C2semidirect
9. C2×AGL(1,ℤ/8)×C127C2semidirect
19. C2×#14523235C22semidirect
20. C2×#14523233C22semidirect
21. C2×#14523232C22semidirect
23. #16861185C22semidirect
24. #16861180C22semidirect
27. #16861168C22semidirect
28. #16861131C22semidirect
29. #16861130C22semidirect
30. (C2×D127)⋊C8C22semidirect
31. C2×#41×C127C22semidirect
34. C2×SD16×C127C22semidirect
35. C2×D8×C127C22semidirect
36. C2×M16×C127C22semidirect
44. C2×D508D4semidirect
60. C2×C127⋊C8C23semidirect
61. C4⋊Dic127C23semidirect
70. C2×Q8×C127C23semidirect
74. C2×D4×C127D4semidirect
76. C2×D4×C127C23semidirect
82. C2×C8×C127C23semidirect
140. C2×D127AGL(1,ℤ/8)semidirect
155. C127#3003semidirect
158. C2×AGL(1,ℤ/8)D127semidirect
166. C2×#41C2×D127semidirect
178. C2×SD16C2×D127semidirect
180. C2×D8C2×D127semidirect
182. C2×M16C2×D127semidirect
200. C2×Q8C22×D127semidirect
214. C2×D4C22⋊D127semidirect
216. C2×D4C22×D127semidirect
231. C2×C8C22×D127semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×#168612342
#225503832
#225500252
#225500082
#225498822
#225498812
#225498392
C2×AGL(1,ℤ/8)×C1272
#3003127