Finite Group Databasebeta

← #30647982

Subgroups — #30647982

Order 16256.

806 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C2×#16861234C212yesmaximal, complemented
2C2×#16861183C211yesmaximal, characteristic, complemented
3#22550021C214yesmaximal, complemented
4#22550015C212yesmaximal, complemented
5#22550010C212yesmaximal, complemented
6#22549877C212yesmaximal, complemented
7#22549840C211yesmaximal, characteristic, complemented
8C22×#41×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic, complemented
10C22×C22⋊D127—21no—
11#16861234—28no—
12C2×#14523235C2214yescomplemented
13C2×#14523233C2214yescomplemented
14C2×#14523232C2212yescomplemented
15#16861183—21no—
16#16861183—21no—
17#16861183C2212yes—
18#16861183C2212yescomplemented
19C2×C4⋊Dic127C2211yescharacteristic, complemented
20C2×(C2×D127)⋊C4—21no—
21#16861133C2214yescomplemented
22#16861130C2214yescomplemented
23C22⋊(C127⋊C8)C2212yescomplemented
24C22⋊(C127⋊C8)C2212yescomplemented
25C2×#41×C127—28no—
26C2×#41×C127—21no—
27C2×#41×C127—21no—
28C2×#41×C127C2212yes—
29C2×#41×C127C2212yescomplemented
30C22×Q8×C127C2211yescharacteristic, complemented
31C22×D4×C127—21no—
32C22×D4×C127C2211yescharacteristic, complemented
33C23×C4×C127—21no—
34C23×C4×C127C2211yescharacteristic
35C23×D127—21no—
36C2×D508—24no—
37C2×D508—21no—
38C2×D508C2312yes—
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C2×C22⋊D127—44no—
42C22×Dic127—21no—
43#14523235—28no—
44#14523233—28no—
45#14523232—24no—
46C2×C127⋊C8C2314yes—
47C4⋊Dic127—21no—
48C4⋊Dic127C2312yes—
49(C2×D127)⋊C4—22no—
50(C2×D127)⋊C4—22no—
51C24×C127—21no—
52C24×C127D411yescharacteristic
53#41×C127—28no—
54#41×C127—28no—
55#41×C127—24no—
56#41×C127—24no—
57#41×C127—24no—
58#41×C127D418yescomplemented
59C2×Q8×C127—24no—
60C2×Q8×C127—21no—
61C2×Q8×C127C2312yes—
62C2×D4×C127—28no—
63C2×D4×C127—22no—
64C2×D4×C127—22no—
65C2×D4×C127—24no—
66C2×D4×C127—21no—
67C2×D4×C127C2312yes—
68C22×C4×C127—28no—
69C22×C4×C127—21no—
70C22×C4×C127—22no—
71C22×C4×C127—22no—
72C22×C4×C127—24no—
73C22×C4×C127D411yescharacteristic
74C22×C4×C127D411yescharacteristic
75C22×C4×C127D411yescharacteristic
76C22×C4×C127C2311yescharacteristic
77C22×C4×C127C2311yescharacteristic
78C22×C4×C127C2311yescharacteristic
79C22×D127—22no—
80C22×D127—42no—
81C22×D127—42no—
82C22×D127—42no—
83D508—24no—
84D508—42no—
85C22⋊D127—44no—
86C22⋊D127—44no—
87C2×Dic127—22no—
88C2×Dic127—42no—
89C127⋊C8—24no—
90C23×C127—24no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—41no—
95C23×C127—42no—
96C23×C127—42no—
97C23×C127C2×D411yescharacteristic
98C23×C127C2×D411yescharacteristic
99C23×C127C2×D411yescharacteristic
100Q8×C127—24no—
101Q8×C127—22no—
102Q8×C127C2×D414yescomplemented
103D4×C127—28no—
104D4×C127—24no—
105D4×C127—28no—
106D4×C127—22no—
107D4×C127C2×D414yescomplemented
108C2×C4×C127—24no—
109C2×C4×C127—28no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—28no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D414yes—
121C2×C4×C127C2411yescharacteristic
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—21no—
125C22×C127—21no—
126C22×C127—21no—
127C22×C127—24no—
128C22×C127—21no—
129C22×C127—21no—
130C22×C127—44no—
131C22×C127—48no—
132C22×C127—42no—
133C22×C127—42no—
134C22×C127#12012yes—
135C22×C127AGL(1,ℤ/8)12yescomplemented
136C22×C127C22×D411yescharacteristic
137C2×D127—42no—
138C2×D127—42no—
139C2×D127—42no—
140C2×D127—84no—
141C4×C127—21no—
142C4×C127—24no—
143C4×C127—21no—
144C4×C127#12012yes—
145C4×C127C22×D412yes—
146Dic127—42no—
147C2×C127—22no—
148C2×C127—22no—
149C2×C127—41no—
150C2×C127—44no—
151C2×C127C2×#12011yescharacteristic
152C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C127#57511yescharacteristic
154D127—82no—
155#3006—1271nomaximal, Sylow
156C127#300611yescharacteristic, complemented, Sylow
157C22×#41D12711yescharacteristic, complemented
158C2×AGL(1,ℤ/8)—1272no—
159C2×#121—1271no—
160#449—1274no—
161#448—1272no—
162#447—1272no—
163#417—1272no—
164#407—1271no—
165C2×#41—28no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41C2×D12712yes—
169C2×#41C2×D12712yescomplemented
170C22×Q8C2×D12711yescharacteristic, complemented
171C22×D4—21no—
172C22×D4—1271no—
173C22×D4—2541no—
174C22×D4C2×D12711yescharacteristic, complemented
175C23×C4—21no—
176C23×C4C2×D12711yescharacteristic
177AGL(1,ℤ/8)—2548no—
178C2×SD16—1274no—
179C2×D8—1274no—
180C2×M16—1272no—
181#121—1272no—
182#121—1272no—
183#121—2541no—
184#121—2541no—
185C2×C4⋊C4—1271no—
186C2×C22⋊C4—2541no—
187#103—1274no—
188#102—1274no—
189C22⋊C8—1272no—
190C22⋊C8—1272no—
191C24—21no—
192C24—2541no—
193C24C22⋊D12711yescharacteristic
194#41—28no—
195#41—28no—
196#41—24no—
197#41—24no—
198#41—24no—
199#41C22⋊D12718yescomplemented
200C2×Q8—24no—
201C2×Q8—21no—
202C2×Q8C22×D12712yes—
203C2×D4—28no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—1272no—
209C2×D4—5084no—
210C2×D4—2542no—
211C2×D4—2542no—
212C2×D4—2541no—
213C2×D4—2544no—
214C2×D4C22×D12712yes—
215C22×C4—28no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—2541no—
221C22×C4C22⋊D12711yescharacteristic
222C22×C4C22⋊D12711yescharacteristic
223C22×C4C22⋊D12711yescharacteristic
224C22×C4C22×D12711yescharacteristic
225C22×C4C22×D12711yescharacteristic
226C22×C4C22×D12711yescharacteristic
227SD16—2548no—
228D8—2548no—
229M16—2544no—
230C2×C8—1274no—
231C4⋊C4—1272no—
232C4⋊C4—2541no—
233C22⋊C4—2542no—
234C22⋊C4—2542no—
235C23—24no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—5082no—
243C23—5082no—
244C23—5082no—
245C23—2542no—
246C23C2×C22⋊D12711yescharacteristic
247C23C2×C22⋊D12711yescharacteristic
248C23C2×C22⋊D12711yescharacteristic
249Q8—24no—
250Q8—22no—
251Q8C2×C22⋊D12714yescomplemented
252D4—28no—
253D4—24no—
254D4—28no—
255D4—22no—
256D4—5084no—
257D4—5084no—
258D4—5082no—
259D4—2544no—
260D4C2×C22⋊D12714yescomplemented
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—28no—
269C2×C4—22no—
270C2×C4—5082no—
271C2×C4—2542no—
272C2×C4C2×C22⋊D12711yescharacteristic
273C2×C4C2×C22⋊D12711yescharacteristic
274C2×C4C2×C22⋊D12714yes—
275C2×C4C2×C22⋊D12711yescharacteristic
276C2×C4C23×D12711yescharacteristic
277C8—2544no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—48no—
288C22—42no—
289C22—42no—
290C22—10164no—
291C22—5082no—
292C22—5082no—
293C22—5082no—
294C22#1686117812yes—
295C22#1686123412yescomplemented
296C22C22×C22⋊D12711yescharacteristic
297C4—21no—
298C4—24no—
299C4—21no—
300C4—5082no—
301C4#1686117812yes—
302C4C22×C22⋊D12712yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—44no—
307C2—10162no—
308C2C2×#1686117811yescharacteristic
309C2C2×#1686123411yescharacteristic
310C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×#41×C127C2
socleC22×C127C22×D4
hypercenterC22×#41D127
hypocenterC127#3006
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861183C2semidirect
3. #22550021C2semidirect
4. #22550015C2semidirect
5. #22550010C2semidirect
6. #22549877C2semidirect
7. #22549840C2semidirect
8. C22×#41×C127C2semidirect
9. C22×D508C22semidirect
12. C2×#14523235C22semidirect
13. C2×#14523233C22semidirect
14. C2×#14523232C22semidirect
18. #16861183C22semidirect
19. C2×C4⋊Dic127C22semidirect
21. #16861133C22semidirect
22. #16861130C22semidirect
23. C22⋊(C127⋊C8)C22semidirect
24. C22⋊(C127⋊C8)C22semidirect
29. C2×#41×C127C22semidirect
30. C22×Q8×C127C22semidirect
32. C22×D4×C127C22semidirect
58. #41×C127D4semidirect
102. Q8×C127C2×D4semidirect
107. D4×C127C2×D4semidirect
135. C22×C127AGL(1,ℤ/8)semidirect
156. C127#3006semidirect
157. C22×#41D127semidirect
169. C2×#41C2×D127semidirect
170. C22×Q8C2×D127semidirect
174. C22×D4C2×D127semidirect
199. #41C22⋊D127semidirect
251. Q8C2×C22⋊D127semidirect
260. D4C2×C22⋊D127semidirect
295. C22#16861234semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168611832
#225500212
#225500152
#225500102
#225498772
#225498402
C22×#41×C1272
#3006127