Finite Group Databasebeta

← #30647960

Subgroups — #30647960

Order 16256.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D508C211yesmaximal, characteristic, complemented
2C2×#16861237C212yesmaximal, complemented
3#22550387C211yesmaximal, characteristic, complemented
4#22550029C214yesmaximal, complemented
5#22550012C214yesmaximal, complemented
6#22549882C212yesmaximal, complemented
7#407×C127C211yesmaximal, characteristic, complemented
8C24×D127—21no—
9C2×#14523248C2211yescharacteristic, complemented
10C22×D508—28no—
11C22×D508—21no—
12C22×D508—22no—
13C22×D508C2214yes—
14C22×D508C2212yes—
15#16861237—28no—
16C2×#14523237C2214yescomplemented
17C2×D1016C2214yescomplemented
18#16861192—21no—
19#16861188C2212yescomplemented
20#16861183C2212yescomplemented
21#16861178—21no—
22C4×C22⋊D127—22no—
23#16861168C2211yescharacteristic, complemented
24#16861131C2218yescomplemented
25C23×C4×C127C2211yescharacteristic
26C2×M16×C127C2212yescomplemented
27C22⋊C8×C127C2214yescomplemented
28C23×D127—28no—
29C23×D127—22no—
30C23×D127—24no—
31C23×D127—48no—
32#14523248—24no—
33C2×Dic254C2311yescharacteristic, complemented
34C2×D508—28no—
35C2×D508—216no—
36C2×D508—216no—
37C2×D508—24no—
38C2×D508—24no—
39C2×D508—22no—
40C2×D508D418yescomplemented
41C2×D508C2314yes—
42C2×D508C2311yescharacteristic, complemented
43C2×C22⋊D127—21no—
44C2×C22⋊D127—22no—
45C2×C4×D127—21no—
46#14523237—28no—
47D1016—28no—
48C4⋊Dic127C2312yescomplemented
49Dic127⋊C4—22no—
50C22⋊Dic127—22no—
51C22⋊Dic127—21no—
52(C2×D127)⋊C4—22no—
53C4×Dic127—21no—
54C24×C127D411yescharacteristic
55C22×C4×C127—24no—
56C22×C4×C127D411yescharacteristic
57C22×C4×C127D412yes—
58C22×C4×C127C2312yes—
59C22×C4×C127C2311yescharacteristic
60M16×C127—24no—
61C2×C8×C127C2314yescomplemented
62C22×D127—216no—
63C22×D127—28no—
64C22×D127—24no—
65C22×D127—21no—
66C22×D127—432no—
67C22×D127—48no—
68C22×D127—48no—
69C22×D127—48no—
70Dic254—22no—
71D508—216no—
72D508—28no—
73D508—24no—
74D508—22no—
75D508C2×D418yes—
76C22⋊D127—42no—
77C22⋊D127—44no—
78C2×Dic127—22no—
79C2×Dic127—21no—
80C4×D127—42no—
81C23×C127—22no—
82C23×C127—24no—
83C23×C127C2×D411yescharacteristic
84C23×C127C2×D412yes—
85C2×C4×C127—22no—
86C2×C4×C127—24no—
87C2×C4×C127—24no—
88C2×C4×C127C2×D414yes—
89C2×C4×C127C2×D412yes—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2411yescharacteristic
92C8×C127—24no—
93C22×C127—28no—
94C22×C127—22no—
95C22×C127—22no—
96C22×C127—22no—
97C22×C127#12012yes—
98C22×C127AGL(1,ℤ/8)14yescomplemented
99C22×C127C22×D411yescharacteristic
100C2×D127—28no—
101C2×D127—416no—
102C2×D127—432no—
103C2×D127—48no—
104C2×D127—41no—
105C2×D127—42no—
106C4×C127—22no—
107C4×C127#12012yes—
108C4×C127C22×D412yes—
109Dic127—41no—
110Dic127—42no—
111C2×C127—22no—
112C2×C127—24no—
113C2×C127C2×#12011yescharacteristic
114C2×C127C2×AGL(1,ℤ/8)12yes—
115D127—48no—
116D127—81no—
117#3003—1271nomaximal, Sylow
118C127#300311yescharacteristic, complemented, Sylow
119C23×D4—1271no—
120C2×AGL(1,ℤ/8)—1272no—
121#525—1271no—
122#450—1274no—
123#447—1274no—
124#418—1272no—
125#407D12711yescharacteristic, complemented
126C25—2541no—
127C2×#41—1271no—
128C22×D4—1272no—
129C22×D4—1274no—
130C22×D4—2548no—
131C22×D4—2542no—
132C22×D4—2541no—
133C23×C4C2×D12711yescharacteristic
134AGL(1,ℤ/8)—2548no—
135C2×SD16—1274no—
136C2×D8—1274no—
137C2×M16C2×D12712yescomplemented
138#123—2541no—
139C22⋊Q8—1272no—
140#121—1272no—
141#120—2541no—
142C4×D4—2542no—
143#117—1271no—
144#102—1278no—
145C22⋊C8C2×D12714yescomplemented
146C24—5088no—
147C24—2548no—
148C24—2542no—
149C24—2544no—
150C24D50811yescharacteristic
151#41—2544no—
152C2×Q8—1271no—
153C2×D4—1278no—
154C2×D4—1274no—
155C2×D4—1271no—
156C2×D4—25416no—
157C2×D4—2548no—
158C2×D4—25416no—
159C2×D4—2542no—
160C2×D4—2541no—
161C2×D4—2544no—
162C2×D4—2544no—
163C2×D4—2542no—
164C22×C4—24no—
165C22×C4—2541no—
166C22×C4D50812yes—
167C22×C4D50811yescharacteristic
168C22×C4C22×D12712yes—
169C22×C4C22×D12711yescharacteristic
170SD16—2548no—
171D8—2548no—
172M16—24no—
173C2×C8C22×D12714yescomplemented
174C4⋊C4—1272no—
175C4⋊C4—2542no—
176C22⋊C4—2541no—
177C22⋊C4—2542no—
178C22⋊C4—2542no—
179C42—2541no—
180C23—24no—
181C23—22no—
182C23—50832no—
183C23—5088no—
184C23—5088no—
185C23—5088no—
186C23—25416no—
187C23—2541no—
188C23—2544no—
189C23—2548no—
190C23C2×D50812yes—
191C23C2×D50811yescharacteristic
192Q8—2542no—
193D4—1278no—
194D4—5084no—
195D4—5082no—
196D4—25416no—
197D4—2542no—
198D4—2544no—
199D4—2548no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—5082no—
204C2×C4—2542no—
205C2×C4—2541no—
206C2×C4D4×D12714yes—
207C2×C4C2×D50812yes—
208C2×C4C2×D50811yescharacteristic
209C2×C4C23×D12711yescharacteristic
210C8—24no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—50832no—
216C22—50816no—
217C22—5081no—
218C22—5082no—
219C22—5088no—
220C22—2548no—
221C22#1686117912yes—
222C22#1686123714yescomplemented
223C22C22×D50811yescharacteristic
224C4—22no—
225C4—5081no—
226C4—5082no—
227C4#1686117912yes—
228C4C2×D4×D12712yes—
229C2—24no—
230C2—22no—
231C2—10161no—
232C2—5088no—
233C2C2×#1686117911yescharacteristic
234C2C2×#1686123712yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#407×C127C2
socleC22×C127C22×D4
hypercenter#407D127
hypocenterC127#3003
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. C23×D508C2semidirect
2. C2×#16861237C2semidirect
3. #22550387C2semidirect
4. #22550029C2semidirect
5. #22550012C2semidirect
6. #22549882C2semidirect
7. #407×C127C2semidirect
9. C2×#14523248C22semidirect
16. C2×#14523237C22semidirect
17. C2×D1016C22semidirect
19. #16861188C22semidirect
20. #16861183C22semidirect
23. #16861168C22semidirect
24. #16861131C22semidirect
26. C2×M16×C127C22semidirect
27. C22⋊C8×C127C22semidirect
33. C2×Dic254C23semidirect
40. C2×D508D4semidirect
42. C2×D508C23semidirect
48. C4⋊Dic127C23semidirect
61. C2×C8×C127C23semidirect
98. C22×C127AGL(1,ℤ/8)semidirect
118. C127#3003semidirect
125. #407D127semidirect
137. C2×M16C2×D127semidirect
145. C22⋊C8C2×D127semidirect
173. C2×C8C22×D127semidirect
222. C22#16861237semidirect

Maximal subgroups

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

subgroupindex
C23×D5082
C2×#168612372
#225503872
#225500292
#225500122
#225498822
#407×C1272
#3003127