Finite Group Databasebeta

← #30647972

Subgroups — #30647972

Order 16256.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3#22550387C211yesmaximal, characteristic, complemented
4#22550038C214yesmaximal, complemented
5#22550021C214yesmaximal, complemented
6#22549885C211yesmaximal, characteristic, complemented
7#22549881C211yesmaximal, characteristic, complemented
8#22549840C211yesmaximal, characteristic, complemented
9C22×#41×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11#16861240—24no—
12#16861235—24no—
13C2×#14523238C2212yescomplemented
14C2×#14523236C2212yescomplemented
15C2×#14523235C2212yescomplemented
16C2×#14523233C2212yescomplemented
17C2×#14523232C2211yescharacteristic
18C2×#14523232C2211yescharacteristic
19#16861192—21no—
20#16861188C2212yescomplemented
21#16861183C2212yescomplemented
22#16861178—21no—
23C4×C22⋊D127—22no—
24#16861168C2211yescharacteristic
25#16861133C2214yescomplemented
26#16861130C2214yescomplemented
27C22⋊(C127⋊C8)C2214yescomplemented
28C2×#41×C127—24no—
29C2×#41×C127—24no—
30C2×#41×C127—22no—
31C2×#41×C127C2214yes—
32C22×Q8×C127C2211yescharacteristic
33C22×D4×C127—21no—
34C22×D4×C127C2211yescharacteristic
35C23×C4×C127—21no—
36C23×C4×C127C2211yescharacteristic
37#14523248—22no—
38#14523248—22no—
39C2×Dic254C2311yescharacteristic
40C2×D508C2311yescharacteristic
41C2×C22⋊D127—21no—
42C2×C22⋊D127—22no—
43C2×C4×D127—21no—
44#14523238—24no—
45#14523236—24no—
46#14523235—24no—
47#14523233—24no—
48#14523232—22no—
49#14523232—22no—
50C2×C127⋊C8C2312yes—
51C2×C127⋊C8C2312yes—
52C4⋊Dic127C2312yes—
53Dic127⋊C4—22no—
54C22⋊Dic127—22no—
55C22⋊Dic127—21no—
56(C2×D127)⋊C4—22no—
57C4×Dic127—21no—
58C24×C127—21no—
59C24×C127D411yescharacteristic
60#41×C127—216no—
61#41×C127—28no—
62#41×C127—28no—
63C2×Q8×C127—22no—
64C2×Q8×C127—21no—
65C2×Q8×C127D414yescomplemented
66C2×Q8×C127C2312yes—
67C2×D4×C127—28no—
68C2×D4×C127—24no—
69C2×D4×C127—21no—
70C2×D4×C127—22no—
71C2×D4×C127D414yescomplemented
72C2×D4×C127C2312yes—
73C22×C4×C127—28no—
74C22×C4×C127—21no—
75C22×C4×C127—24no—
76C22×C4×C127—22no—
77C22×C4×C127—22no—
78C22×C4×C127D411yescharacteristic
79C22×C4×C127D412yes—
80C22×C4×C127C2312yes—
81C22×C4×C127C2311yescharacteristic
82C22×D127—21no—
83Dic254—21no—
84Dic254—21no—
85D508—21no—
86D508—21no—
87C22⋊D127—44no—
88C22⋊D127—42no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C4×D127—42no—
92C127⋊C8—22no—
93C127⋊C8—22no—
94C23×C127—22no—
95C23×C127—24no—
96C23×C127—22no—
97C23×C127—42no—
98C23×C127—44no—
99C23×C127C2×D412yes—
100C23×C127C2×D411yescharacteristic
101Q8×C127—24no—
102Q8×C127—22no—
103Q8×C127C2×D414yes—
104D4×C127—216no—
105D4×C127—24no—
106D4×C127—22no—
107D4×C127C2×D414yes—
108C2×C4×C127—216no—
109C2×C4×C127—22no—
110C2×C4×C127—24no—
111C2×C4×C127—22no—
112C2×C4×C127—24no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2411yescharacteristic
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—24no—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—44no—
126C22×C127—48no—
127C22×C127—44no—
128C22×C127#12012yes—
129C22×C127C22×D411yescharacteristic
130C2×D127—41no—
131C2×D127—42no—
132C4×C127—22no—
133C4×C127—24no—
134C4×C127#12012yes—
135C4×C127C22×D411yescharacteristic
136C4×C127C22×D411yescharacteristic
137Dic127—41no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—42no—
141C2×C127—44no—
142C2×C127C2×#12011yescharacteristic
143C2×C127#57512yes—
144D127—81no—
145#3005—1271nomaximal, Sylow
146C127#300511yescharacteristic, complemented, Sylow
147C22×#41D12711yescharacteristic, complemented
148C2×#137—1271no—
149C2×AGL(1,ℤ/8)—1271no—
150#525—1271no—
151#452—1274no—
152#449—1274no—
153#419—1271no—
154#418—1271no—
155#407—1271no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1271no—
160C2×#41C2×D12714yes—
161C22×Q8C2×D12711yescharacteristic
162C22×D4—21no—
163C22×D4C2×D12711yescharacteristic
164C23×C4—21no—
165C23×C4C2×D12711yescharacteristic
166#137—2544no—
167AGL(1,ℤ/8)—2544no—
168C2×Q16—1272no—
169C2×SD16—1272no—
170C2×SD16—1272no—
171C2×D8—1272no—
172C2×M16—1271no—
173C2×M16—1271no—
174#123—2541no—
175C22⋊Q8—1272no—
176#121—1272no—
177#120—2541no—
178C4×D4—2542no—
179#117—1271no—
180#103—1274no—
181#102—1274no—
182C22⋊C8—1274no—
183C24—21no—
184C24C22⋊D12711yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—2542no—
189#41—2542no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—1271no—
193C2×Q8C22⋊D12714yescomplemented
194C2×Q8C22×D12712yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—1271no—
200C2×D4—2542no—
201C2×D4—2541no—
202C2×D4C22⋊D12714yescomplemented
203C2×D4C22×D12712yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—2541no—
210C22×C4C22⋊D12712yes—
211C22×C4C22⋊D12711yescharacteristic
212C22×C4C22×D12712yes—
213C22×C4C22×D12711yescharacteristic
214Q16—2544no—
215SD16—2544no—
216SD16—2544no—
217D8—2544no—
218M16—2542no—
219M16—2542no—
220C2×C8—1272no—
221C2×C8—1272no—
222C4⋊C4—1272no—
223C4⋊C4—2542no—
224C22⋊C4—2541no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C42—2541no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—42no—
232C23—44no—
233C23—2541no—
234C23C2×C22⋊D12712yes—
235C23C2×C22⋊D12711yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—2541no—
239Q8—2541no—
240Q8C2×C22⋊D12714yes—
241D4—216no—
242D4—22no—
243D4—24no—
244D4—5084no—
245D4—5082no—
246D4—2541no—
247D4—2541no—
248D4C2×C22⋊D12714yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—5082no—
257C2×C4—2542no—
258C2×C4—2541no—
259C2×C4C2×C22⋊D12712yes—
260C2×C4C2×C22⋊D12712yes—
261C2×C4C2×C22⋊D12712yes—
262C2×C4C2×C22⋊D12711yescharacteristic
263C2×C4C23×D12711yescharacteristic
264C8—2542no—
265C8—2542no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—5081no—
275C22—5082no—
276C22#1686117812yes—
277C22C22×C22⋊D12711yescharacteristic
278C4—22no—
279C4—24no—
280C4—5081no—
281C4—5082no—
282C4#1686117812yes—
283C4C22×C22⋊D12711yescharacteristic
284C4C22×C22⋊D12711yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—10161no—
289C2C2×#1686117811yescharacteristic
290C2#2255067412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×#41×C127C2
socleC22×C127C22×D4
hypercenterC22×#41D127
hypocenterC127#3005
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×#16861240C2semidirect
2. C2×#16861235C2semidirect
3. #22550387C2semidirect
4. #22550038C2semidirect
5. #22550021C2semidirect
6. #22549885C2semidirect
7. #22549881C2semidirect
8. #22549840C2semidirect
9. C22×#41×C127C2semidirect
13. C2×#14523238C22semidirect
14. C2×#14523236C22semidirect
15. C2×#14523235C22semidirect
16. C2×#14523233C22semidirect
20. #16861188C22semidirect
21. #16861183C22semidirect
25. #16861133C22semidirect
26. #16861130C22semidirect
27. C22⋊(C127⋊C8)C22semidirect
65. C2×Q8×C127D4semidirect
71. C2×D4×C127D4semidirect
146. C127#3005semidirect
147. C22×#41D127semidirect
193. C2×Q8C22⋊D127semidirect
202. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612352
#225503872
#225500382
#225500212
#225498852
#225498812
#225498402
C22×#41×C1272
#3005127