Finite Group Databasebeta

← #30647984

Subgroups — #30647984

Order 16256.

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

normal subgroups only

This group has 806 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×#14523248C211yesmaximal, characteristic, complemented
2C2×#16861237C212yesmaximal, complemented
3C2×#16861183C211yesmaximal, characteristic, complemented
4#22550023C214yesmaximal, complemented
5#22550017C212yesmaximal, complemented
6#22550012C212yesmaximal, complemented
7#22549878C212yesmaximal, complemented
8#407×C127C211yesmaximal, characteristic, complemented
9C2×#14523248—28no—
10C2×#14523248—21no—
11C2×#14523248—21no—
12C2×#14523248C2212yes—
13C2×#14523248C2212yescomplemented
14C22×Dic254C2211yescharacteristic, complemented
15C22×D508C2211yescharacteristic, complemented
16C22×D508C2211yescharacteristic, complemented
17C22×C22⋊D127—21no—
18C22×C22⋊D127—21no—
19C22×C4×D127—21no—
20#16861237—28no—
21C2×#14523237C2214yescomplemented
22C2×D1016C2214yescomplemented
23#16861183—21no—
24#16861183—21no—
25#16861183C2212yes—
26#16861183C2212yescomplemented
27C2×C4⋊Dic127C2211yescharacteristic, complemented
28C2×(C2×D127)⋊C4—21no—
29#16861134C2214yescomplemented
30#16861131C2214yescomplemented
31C23×C4×C127C2211yescharacteristic
32C2×M16×C127C2212yescomplemented
33C22⋊C8×C127C2212yescomplemented
34C22⋊C8×C127C2212yescomplemented
35C23×D127—21no—
36C23×D127—21no—
37#14523248—28no—
38#14523248—28no—
39#14523248—24no—
40#14523248—24no—
41#14523248—24no—
42#14523248D418yescomplemented
43C2×Dic254—24no—
44C2×Dic254—21no—
45C2×Dic254C2312yes—
46C2×D508—24no—
47C2×D508—21no—
48C2×D508—24no—
49C2×D508—21no—
50C2×D508C2312yes—
51C2×D508C2312yes—
52C2×C22⋊D127—28no—
53C2×C22⋊D127—22no—
54C2×C22⋊D127—22no—
55C2×C22⋊D127—22no—
56C2×C22⋊D127—22no—
57C2×C22⋊D127—44no—
58C22×Dic127—21no—
59C22×Dic127—21no—
60C2×C4×D127—28no—
61C2×C4×D127—22no—
62C2×C4×D127—22no—
63#14523237—28no—
64D1016—28no—
65C4⋊Dic127—21no—
66C4⋊Dic127C2312yes—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69C24×C127D411yescharacteristic
70C22×C4×C127—24no—
71C22×C4×C127D411yescharacteristic
72C22×C4×C127D411yescharacteristic
73C22×C4×C127D411yescharacteristic
74C22×C4×C127C2311yescharacteristic
75C22×C4×C127C2311yescharacteristic
76C22×C4×C127C2311yescharacteristic
77M16×C127—24no—
78C2×C8×C127C2314yes—
79C22×D127—24no—
80C22×D127—22no—
81C22×D127—22no—
82C22×D127—42no—
83C22×D127—42no—
84C22×D127—42no—
85C22×D127—42no—
86C22×D127—42no—
87Dic254—24no—
88Dic254—22no—
89Dic254C2×D414yescomplemented
90D508—24no—
91D508—22no—
92D508—24no—
93D508—42no—
94D508C2×D414yescomplemented
95C22⋊D127—28no—
96C22⋊D127—28no—
97C22⋊D127—44no—
98C22⋊D127—44no—
99C2×Dic127—24no—
100C2×Dic127—22no—
101C2×Dic127—22no—
102C2×Dic127—42no—
103C4×D127—28no—
104C4×D127—28no—
105C23×C127—22no—
106C23×C127—22no—
107C23×C127—41no—
108C23×C127C2×D411yescharacteristic
109C23×C127C2×D411yescharacteristic
110C23×C127C2×D411yescharacteristic
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127C2×D414yes—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2411yescharacteristic
121C8×C127—24no—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—21no—
125C22×C127—21no—
126C22×C127—21no—
127C22×C127—21no—
128C22×C127—21no—
129C22×C127—44no—
130C22×C127#12012yes—
131C22×C127AGL(1,ℤ/8)12yescomplemented
132C22×C127C22×D411yescharacteristic
133C2×D127—24no—
134C2×D127—48no—
135C2×D127—42no—
136C2×D127—42no—
137C2×D127—42no—
138C2×D127—42no—
139C2×D127—42no—
140C2×D127—84no—
141C4×C127—21no—
142C4×C127—21no—
143C4×C127#12012yes—
144C4×C127C22×D412yes—
145Dic127—24no—
146Dic127—42no—
147C2×C127—22no—
148C2×C127—22no—
149C2×C127—41no—
150C2×C127C2×#12011yescharacteristic
151C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C127#57511yescharacteristic
153D127—44no—
154D127—82no—
155#3006—1271nomaximal, Sylow
156C127#300611yescharacteristic, complemented, Sylow
157C22×#41—1271no—
158C2×AGL(1,ℤ/8)—1272no—
159C2×#121—1271no—
160#449—1274no—
161#448—1272no—
162#447—1272no—
163#417—1272no—
164#407D12711yescharacteristic, complemented
165C2×#41—1272no—
166C2×#41—1272no—
167C2×#41—2548no—
168C2×#41—2541no—
169C2×#41—2541no—
170C22×Q8—1271no—
171C22×D4—1271no—
172C22×D4—1271no—
173C22×D4—2541no—
174C22×D4—2541no—
175C23×C4—2541no—
176C23×C4C2×D12711yescharacteristic
177AGL(1,ℤ/8)—2548no—
178C2×SD16—1274no—
179C2×D8—1274no—
180C2×M16C2×D12712yescomplemented
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⋊C8C2×D12712yescomplemented
190C22⋊C8C2×D12712yescomplemented
191C24—2541no—
192C24—2541no—
193C24D50811yescharacteristic
194#41—1278no—
195#41—2544no—
196#41—2544no—
197#41—2548no—
198#41—2548no—
199#41—2544no—
200C2×Q8—1272no—
201C2×Q8—2544no—
202C2×Q8—2541no—
203C2×D4—1272no—
204C2×D4—1272no—
205C2×D4—5084no—
206C2×D4—2544no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4—2548no—
210C2×D4—2542no—
211C2×D4—2542no—
212C2×D4—2544no—
213C2×D4—2541no—
214C2×D4—2541no—
215C22×C4—24no—
216C22×C4—2548no—
217C22×C4—2541no—
218C22×C4—2541no—
219C22×C4—2542no—
220C22×C4—2542no—
221C22×C4D50811yescharacteristic
222C22×C4D50811yescharacteristic
223C22×C4D50811yescharacteristic
224C22×C4C22×D12711yescharacteristic
225C22×C4C22×D12711yescharacteristic
226C22×C4C22×D12711yescharacteristic
227SD16—2548no—
228D8—2548no—
229M16—24no—
230C2×C8C22×D12714yes—
231C4⋊C4—1272no—
232C4⋊C4—2541no—
233C22⋊C4—2542no—
234C22⋊C4—2542no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—5082no—
239C23—5082no—
240C23—5082no—
241C23—5082no—
242C23—5082no—
243C23—2544no—
244C23—2542no—
245C23—2542no—
246C23C2×D50811yescharacteristic
247C23C2×D50811yescharacteristic
248C23C2×D50811yescharacteristic
249Q8—1274no—
250Q8—2544no—
251Q8—2542no—
252D4—1274no—
253D4—5084no—
254D4—5084no—
255D4—5082no—
256D4—2544no—
257D4—2548no—
258D4—2548no—
259D4—2544no—
260D4—2542no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—5082no—
267C2×C4—2548no—
268C2×C4—2542no—
269C2×C4—2544no—
270C2×C4—2542no—
271C2×C4—2548no—
272C2×C4D4×D12714yes—
273C2×C4C2×D50811yescharacteristic
274C2×C4C2×D50811yescharacteristic
275C2×C4C2×D50811yescharacteristic
276C2×C4C23×D12711yescharacteristic
277C8—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—10164no—
287C22—5088no—
288C22—5082no—
289C22—5082no—
290C22—5082no—
291C22—5082no—
292C22—5082no—
293C22—2544no—
294C22#1686117912yes—
295C22#1686123712yescomplemented
296C22C22×D50811yescharacteristic
297C4—21no—
298C4—21no—
299C4—5082no—
300C4—2544no—
301C4#1686117912yes—
302C4C2×D4×D12712yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—10162no—
307C2—5084no—
308C2C2×#1686117911yescharacteristic
309C2C2×#1686123711yescharacteristic
310C2#2255067611yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#407×C127C2
socleC22×C127C22×D4
hypercenter#407D127
hypocenterC127#3006
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#14523248C2semidirect
2. C2×#16861237C2semidirect
3. C2×#16861183C2semidirect
4. #22550023C2semidirect
5. #22550017C2semidirect
6. #22550012C2semidirect
7. #22549878C2semidirect
8. #407×C127C2semidirect
13. C2×#14523248C22semidirect
14. C22×Dic254C22semidirect
15. C22×D508C22semidirect
16. C22×D508C22semidirect
21. C2×#14523237C22semidirect
22. C2×D1016C22semidirect
26. #16861183C22semidirect
27. C2×C4⋊Dic127C22semidirect
29. #16861134C22semidirect
30. #16861131C22semidirect
32. C2×M16×C127C22semidirect
33. C22⋊C8×C127C22semidirect
34. C22⋊C8×C127C22semidirect
42. #14523248D4semidirect
89. Dic254C2×D4semidirect
94. D508C2×D4semidirect
131. C22×C127AGL(1,ℤ/8)semidirect
156. C127#3006semidirect
164. #407D127semidirect
180. C2×M16C2×D127semidirect
189. C22⋊C8C2×D127semidirect
190. C22⋊C8C2×D127semidirect
295. C22#16861237semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168612372
C2×#168611832
#225500232
#225500172
#225500122
#225498782
#407×C1272
#3006127