Finite Group Databasebeta

← #30647975

Subgroups — #30647975

Order 16256.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861242C211yesmaximal, characteristic, complemented
3C2×#16861237C211yesmaximal, characteristic, complemented
4#22550387C211yesmaximal, characteristic, complemented
5#22550040C214yesmaximal, complemented
6#22550023C214yesmaximal, complemented
7#22549886C211yesmaximal, characteristic, complemented
8#22549882C211yesmaximal, characteristic, complemented
9#407×C127C211yesmaximal, characteristic, complemented
10C2×#14523248—24no—
11C2×#14523248—24no—
12C2×#14523248—22no—
13C2×#14523248C2214yes—
14C2×#14523248C2211yescharacteristic
15C22×Dic254C2211yescharacteristic
16C22×D508C2211yescharacteristic
17C22×C22⋊D127—21no—
18C22×C4×D127—21no—
19#16861242—24no—
20#16861237—24no—
21C2×Dic508C2212yescomplemented
22C2×#14523237C2212yescomplemented
23C2×#14523237C2212yescomplemented
24C2×D1016C2212yescomplemented
25#16861192—21no—
26#16861188C2212yescomplemented
27#16861183C2212yescomplemented
28#16861178—21no—
29C4×C22⋊D127—22no—
30#16861168C2211yescharacteristic
31#16861134C2214yescomplemented
32#16861131C2214yescomplemented
33C23×C4×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic
35C2×M16×C127C2211yescharacteristic
36C22⋊C8×C127C2214yescomplemented
37C23×D127—21no—
38#14523248—216no—
39#14523248—28no—
40#14523248—28no—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254—22no—
44C2×Dic254—21no—
45C2×Dic254D414yescomplemented
46C2×Dic254C2311yescharacteristic
47C2×Dic254C2312yes—
48C2×D508—22no—
49C2×D508—21no—
50C2×D508D414yescomplemented
51C2×D508C2311yescharacteristic
52C2×D508C2312yes—
53C2×C22⋊D127—28no—
54C2×C22⋊D127—24no—
55C2×C22⋊D127—21no—
56C2×C22⋊D127—22no—
57C22×Dic127—21no—
58C2×C4×D127—28no—
59C2×C4×D127—24no—
60C2×C4×D127—21no—
61Dic508—24no—
62#14523237—24no—
63#14523237—24no—
64D1016—24no—
65C4⋊Dic127C2312yes—
66Dic127⋊C4—22no—
67C22⋊Dic127—22no—
68C22⋊Dic127—21no—
69(C2×D127)⋊C4—22no—
70C4×Dic127—21no—
71C24×C127D411yescharacteristic
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127D411yescharacteristic
75C22×C4×C127D412yes—
76C22×C4×C127C2312yes—
77C22×C4×C127C2311yescharacteristic
78M16×C127—22no—
79M16×C127—22no—
80C2×C8×C127C2312yes—
81C2×C8×C127C2312yes—
82C22×D127—24no—
83C22×D127—22no—
84C22×D127—21no—
85C22×D127—44no—
86Dic254—24no—
87Dic254—22no—
88Dic254—21no—
89Dic254—21no—
90Dic254C2×D414yes—
91D508—24no—
92D508—22no—
93D508—21no—
94D508—21no—
95D508C2×D414yes—
96C22⋊D127—216no—
97C22⋊D127—44no—
98C22⋊D127—42no—
99C2×Dic127—24no—
100C2×Dic127—22no—
101C2×Dic127—22no—
102C2×Dic127—21no—
103C4×D127—216no—
104C4×D127—42no—
105C23×C127—22no—
106C23×C127—42no—
107C23×C127C2×D412yes—
108C23×C127C2×D411yescharacteristic
109C2×C4×C127—22no—
110C2×C4×C127—24no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127C2×D412yes—
114C2×C4×C127C2×D412yes—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2411yescharacteristic
118C8×C127—22no—
119C8×C127—22no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—44no—
125C22×C127#12012yes—
126C22×C127C22×D411yescharacteristic
127C2×D127—24no—
128C2×D127—48no—
129C2×D127—44no—
130C2×D127—41no—
131C2×D127—42no—
132C4×C127—22no—
133C4×C127#12012yes—
134C4×C127C22×D411yescharacteristic
135C4×C127C22×D411yescharacteristic
136Dic127—24no—
137Dic127—41no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—42no—
141C2×C127C2×#12011yescharacteristic
142C2×C127#57512yes—
143D127—44no—
144D127—81no—
145#3005—1271nomaximal, Sylow
146C127#300511yescharacteristic, complemented, Sylow
147C22×#41—1271no—
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#407D12711yescharacteristic, complemented
156C2×#41—1271no—
157C2×#41—1274no—
158C2×#41—2544no—
159C2×#41—2544no—
160C2×#41—2542no—
161C22×Q8—1271no—
162C22×D4—1271no—
163C22×D4—2541no—
164C23×C4—2541no—
165C23×C4C2×D12711yescharacteristic
166#137—2544no—
167AGL(1,ℤ/8)—2544no—
168C2×Q16—1272no—
169C2×SD16—1272no—
170C2×SD16—1272no—
171C2×D8—1272no—
172C2×M16C2×D12711yescharacteristic
173C2×M16C2×D12711yescharacteristic
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⋊C8C2×D12714yescomplemented
183C24—2541no—
184C24D50811yescharacteristic
185#41—2548no—
186#41—2548no—
187#41—25416no—
188#41—2542no—
189#41—2542no—
190C2×Q8—1274no—
191C2×Q8—1272no—
192C2×Q8—1271no—
193C2×Q8—2542no—
194C2×Q8—2541no—
195C2×D4—1274no—
196C2×D4—1272no—
197C2×D4—1271no—
198C2×D4—2542no—
199C2×D4—2541no—
200C2×D4—2548no—
201C2×D4—2544no—
202C2×D4—2542no—
203C2×D4—2541no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—2548no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4—2544no—
210C22×C4D50812yes—
211C22×C4D50811yescharacteristic
212C22×C4C22×D12712yes—
213C22×C4C22×D12711yescharacteristic
214Q16—2544no—
215SD16—2544no—
216SD16—2544no—
217D8—2544no—
218M16—22no—
219M16—22no—
220C2×C8C22×D12712yes—
221C2×C8C22×D12712yes—
222C4⋊C4—1272no—
223C4⋊C4—2542no—
224C22⋊C4—2541no—
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C42—2541no—
228C23—22no—
229C23—42no—
230C23—5084no—
231C23—2541no—
232C23—2542no—
233C23—2544no—
234C23C2×D50812yes—
235C23C2×D50811yescharacteristic
236Q8—1274no—
237Q8—2541no—
238Q8—2541no—
239Q8—2542no—
240Q8—2544no—
241D4—1274no—
242D4—5084no—
243D4—5082no—
244D4—25416no—
245D4—2541no—
246D4—2541no—
247D4—2542no—
248D4—2544no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—5082no—
254C2×C4—25416no—
255C2×C4—2542no—
256C2×C4—2541no—
257C2×C4—2542no—
258C2×C4—2544no—
259C2×C4D4×D12712yes—
260C2×C4D4×D12712yes—
261C2×C4C2×D50812yes—
262C2×C4C2×D50811yescharacteristic
263C2×C4C23×D12711yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—5088no—
272C22—5081no—
273C22—5082no—
274C22—5084no—
275C22—2544no—
276C22#1686117912yes—
277C22C22×D50811yescharacteristic
278C4—22no—
279C4—5081no—
280C4—5082no—
281C4—2544no—
282C4#1686117912yes—
283C4C2×D4×D12711yescharacteristic
284C4C2×D4×D12711yescharacteristic
285C2—22no—
286C2—42no—
287C2—10161no—
288C2—5084no—
289C2C2×#1686117911yescharacteristic
290C2#2255067612yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#407×C127C2
socleC22×C127C22×D4
hypercenter#407D127
hypocenterC127#3005
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×#16861242C2semidirect
3. C2×#16861237C2semidirect
4. #22550387C2semidirect
5. #22550040C2semidirect
6. #22550023C2semidirect
7. #22549886C2semidirect
8. #22549882C2semidirect
9. #407×C127C2semidirect
21. C2×Dic508C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523237C22semidirect
24. C2×D1016C22semidirect
26. #16861188C22semidirect
27. #16861183C22semidirect
31. #16861134C22semidirect
32. #16861131C22semidirect
36. C22⋊C8×C127C22semidirect
45. C2×Dic254D4semidirect
50. C2×D508D4semidirect
146. C127#3005semidirect
155. #407D127semidirect
182. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168612422
C2×#168612372
#225503872
#225500402
#225500232
#225498862
#225498822
#407×C1272
#3005127