Finite Group Databasebeta

← #30649196

Subgroups — #30649196

Order 16256.

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

normal subgroups only

This group has 572 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×#16861242C212yesmaximal, complemented
2C2×#16861237C212yesmaximal, complemented
3#22550426C211yesmaximal, characteristic, complemented
4C2×#16861197C211yesmaximal, characteristic, complemented
5#22550270C212yesmaximal, complemented
6#22550265C212yesmaximal, complemented
7#22550260C214yesmaximal, complemented
8C4×M16×C127C211yesmaximal, characteristic, complemented
9C2×#14523248C2212yes—
10C22×Dic254C2211yescharacteristic
11C22×D508C2211yescharacteristic
12#16861242—28no—
13#16861237—28no—
14C2×Dic508C2214yescomplemented
15C2×#14523237C2214yes—
16C2×#14523237C2214yescomplemented
17C2×D1016C2214yes—
18#16861209C2211yescharacteristic, complemented
19#16861206C2211yescharacteristic, complemented
20#16861197—22no—
21#16861197C2212yes—
22#16861197C2212yes—
23#16861197C2212yescomplemented
24#16861183—22no—
25C4×D508—22no—
26C2×(C2×D127)⋊C4—22no—
27C2×M16×C127C2212yes—
28C2×C42×C127C2211yescharacteristic
29#97×C127C2212yescomplemented
30C4×C8×C127C2212yescomplemented
31C23×D127—21no—
32#14523248—24no—
33#14523248—24no—
34C2×Dic254—21no—
35C2×Dic254—24no—
36C2×Dic254C2312yes—
37C2×Dic254C2312yes—
38C2×D508—24no—
39C2×D508—21no—
40C2×D508—22no—
41C2×D508C2312yes—
42C2×D508C2312yes—
43C2×C22⋊D127—22no—
44C22×Dic127—21no—
45C2×C4×D127—22no—
46Dic508—28no—
47#14523237—28no—
48#14523237—28no—
49D1016—28no—
50C4⋊Dic127—22no—
51(C2×D127)⋊C4—24no—
52(C2×D127)⋊C4—24no—
53(C2×D127)⋊C4—24no—
54C22×C4×C127D412yes—
55C22×C4×C127C2311yescharacteristic
56M16×C127D418yescomplemented
57C2×C8×C127C2314yes—
58C42×C127D412yes—
59C42×C127C2311yescharacteristic
60C42×C127C2311yescharacteristic
61C22×D127—22no—
62C22×D127—22no—
63C22×D127—44no—
64C22×D127—42no—
65Dic254—22no—
66Dic254—24no—
67Dic254—22no—
68Dic254—42no—
69D508—22no—
70D508—24no—
71D508—22no—
72D508—44no—
73D508—42no—
74C22⋊D127—44no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—42no—
78C4×D127—44no—
79C23×C127C2×D411yescharacteristic
80C2×C4×C127—42no—
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×D412yes—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D414yes—
85C2×C4×C127C2411yescharacteristic
86C8×C127C2×D418yes—
87C22×C127—22no—
88C22×C127#12712yes—
89C22×C127C22×D411yescharacteristic
90C2×D127—42no—
91C2×D127—42no—
92C2×D127—44no—
93C2×D127—44no—
94C2×D127—84no—
95C4×C127—22no—
96C4×C127—22no—
97C4×C127#12712yes—
98C4×C127C22×D412yes—
99Dic127—42no—
100Dic127—42no—
101C2×C127—22no—
102C2×C127C2×#12711yescharacteristic
103C2×C127#57512yes—
104D127—82no—
105D127—82no—
106#3153—1271nomaximal, Sylow
107C127#315311yescharacteristic, complemented, Sylow
108C2×#137—1272no—
109C2×AGL(1,ℤ/8)—1272no—
110#532—1271no—
111C2×#124—1271no—
112#497—1272no—
113#496—1272no—
114#495—1274no—
115C4×M16D12711yescharacteristic, complemented
116C2×#41—1272no—
117C22×Q8—1271no—
118C22×D4—1271no—
119#137—2548no—
120AGL(1,ℤ/8)—2548no—
121C2×Q16—1274no—
122C2×SD16—1274no—
123C2×SD16—1274no—
124C2×D8—1274no—
125C2×M16C2×D12712yes—
126C4⋊Q8—1271no—
127#127—1271no—
128#124—1272no—
129#124—1272no—
130#124—1272no—
131#124—2542no—
132#121—2542no—
133C4×D4—2542no—
134C2×C22⋊C4—2542no—
135C2×C42C2×D12711yescharacteristic
136#97C2×D12712yescomplemented
137C4×C8C2×D12712yescomplemented
138C24—2541no—
139#41—2544no—
140#41—2544no—
141C2×Q8—1272no—
142C2×Q8—1272no—
143C2×Q8—2544no—
144C2×Q8—2541no—
145C2×D4—1272no—
146C2×D4—1272no—
147C2×D4—2542no—
148C2×D4—2542no—
149C2×D4—2541no—
150C2×D4—2544no—
151C22×C4—2541no—
152C22×C4—2542no—
153C22×C4D50812yes—
154C22×C4C22×D12711yescharacteristic
155Q16—2548no—
156SD16—2548no—
157SD16—2548no—
158D8—2548no—
159M16D50818yescomplemented
160C2×C8C22×D12714yes—
161C4⋊C4—2542no—
162C22⋊C4—2544no—
163C22⋊C4—2544no—
164C22⋊C4—2544no—
165C42D50812yes—
166C42C22×D12711yescharacteristic
167C42C22×D12711yescharacteristic
168C23—5084no—
169C23—5082no—
170C23—2542no—
171C23—2542no—
172C23C2×D50811yescharacteristic
173Q8—5082no—
174Q8—2542no—
175Q8—2544no—
176Q8—2542no—
177D4—5084no—
178D4—5084no—
179D4—5082no—
180D4—2542no—
181D4—2544no—
182D4—2542no—
183C2×C4—42no—
184C2×C4—5082no—
185C2×C4—5084no—
186C2×C4—2542no—
187C2×C4—2542no—
188C2×C4C2×D50812yes—
189C2×C4C2×D50812yes—
190C2×C4C2×D50811yescharacteristic
191C2×C4C2×D50814yes—
192C2×C4C23×D12711yescharacteristic
193C8C2×D50818yes—
194C22—22no—
195C22—10164no—
196C22—5082no—
197C22—5082no—
198C22—5084no—
199C22—5084no—
200C22#1686120612yes—
201C22C22×D50811yescharacteristic
202C4—22no—
203C4—22no—
204C4—5082no—
205C4—5082no—
206C4#1686120612yes—
207C4C22×D50812yes—
208C2—22no—
209C2—10162no—
210C2—10162no—
211C2C2×#1686120611yescharacteristic
212C2#2255067612yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×M16×C127C2
socleC22×C127C22×D4
hypercenterC4×M16D127
hypocenterC127#3153
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C2×#16861237C2semidirect
3. #22550426C2semidirect
4. C2×#16861197C2semidirect
5. #22550270C2semidirect
6. #22550265C2semidirect
7. #22550260C2semidirect
8. C4×M16×C127C2semidirect
14. C2×Dic508C22semidirect
16. C2×#14523237C22semidirect
18. #16861209C22semidirect
19. #16861206C22semidirect
23. #16861197C22semidirect
29. #97×C127C22semidirect
30. C4×C8×C127C22semidirect
56. M16×C127D4semidirect
107. C127#3153semidirect
115. C4×M16D127semidirect
136. #97C2×D127semidirect
137. C4×C8C2×D127semidirect
159. M16D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612372
#225504262
C2×#168611972
#225502702
#225502652
#225502602
C4×M16×C1272
#3153127