Finite Group Databasebeta

← #30649198

Subgroups — #30649198

Order 16256.

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

normal subgroups only

This group has 572 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
1C2×#16861239C212yesmaximal, complemented
2C2×#16861234C212yesmaximal, complemented
3C2×#16861197C211yesmaximal, characteristic, complemented
4#22550268C212yesmaximal, complemented
5#22550263C212yesmaximal, complemented
6#22550258C214yesmaximal, complemented
7C4×#14523232C211yesmaximal, characteristic, complemented
8#532×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2211yescharacteristic
10C22×D508C2211yescharacteristic
11#16861239—28no—
12#16861234—28no—
13C2×#14523238C2214yescomplemented
14C2×#14523236C2214yescomplemented
15C2×#14523235C2214yes—
16C2×#14523233C2214yes—
17C2×#14523232C2212yes—
18#16861197—22no—
19#16861197C2212yes—
20#16861197C2212yes—
21C2×(C2×D127)⋊C4—22no—
22#16861120C2212yescomplemented
23C4×C127⋊C8C2212yescomplemented
24C2×#41×C127C2212yes—
25C4⋊Q8×C127C2211yescharacteristic, complemented
26#127×C127C2211yescharacteristic, complemented
27#124×C127C2212yescomplemented
28#121×C127—22no—
29C4×D4×C127—22no—
30C2×C42×C127C2211yescharacteristic
31C23×D127—21no—
32C2×Dic254—21no—
33C2×Dic254—24no—
34C2×Dic254C2312yes—
35C2×D508—24no—
36C2×D508—21no—
37C2×D508C2312yes—
38C22×Dic127—21no—
39#14523238—28no—
40#14523236—28no—
41#14523235—28no—
42#14523233—28no—
43#14523232D418yescomplemented
44C2×C127⋊C8C2314yes—
45(C2×D127)⋊C4—24no—
46(C2×D127)⋊C4—24no—
47#41×C127—24no—
48#41×C127—24no—
49C2×Q8×C127C2312yes—
50C2×D4×C127—22no—
51C2×D4×C127—22no—
52C2×D4×C127C2312yes—
53C22×C4×C127—22no—
54C22×C4×C127D412yes—
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127—22no—
57C22⋊C4×C127—24no—
58C42×C127D412yes—
59C42×C127C2311yescharacteristic
60C42×C127C2311yescharacteristic
61C22×D127—22no—
62C22×D127—44no—
63C22×D127—42no—
64Dic254—24no—
65Dic254—42no—
66D508—24no—
67D508—42no—
68C2×Dic127—22no—
69C2×Dic127—42no—
70C127⋊C8C2×D418yes—
71C23×C127—22no—
72C23×C127C2×D411yescharacteristic
73Q8×C127—22no—
74Q8×C127—22no—
75D4×C127—22no—
76D4×C127—22no—
77D4×C127—44no—
78D4×C127—44no—
79C2×C4×C127—22no—
80C2×C4×C127—42no—
81C2×C4×C127—44no—
82C2×C4×C127C2×D414yes—
83C2×C4×C127C2×D412yes—
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2411yescharacteristic
87C22×C127—22no—
88C22×C127—42no—
89C22×C127—44no—
90C22×C127#12712yes—
91C22×C127C22×D411yescharacteristic
92C2×D127—42no—
93C2×D127—44no—
94C2×D127—84no—
95C4×C127—22no—
96C4×C127—22no—
97C4×C127—42no—
98C4×C127#12712yes—
99C4×C127C22×D412yes—
100Dic127—42no—
101C2×C127—22no—
102C2×C127—82no—
103C2×C127C2×#12711yescharacteristic
104C2×C127#57512yes—
105D127—82no—
106#3153—1271nomaximal, Sylow
107C127#315311yescharacteristic, complemented, Sylow
108C2×#137—1272no—
109C2×AGL(1,ℤ/8)—1272no—
110#532D12711yescharacteristic, complemented
111C2×#124—1271no—
112#497—1272no—
113#496—1272no—
114#495—1274no—
115C4×M16—1271no—
116C2×#41C2×D12712yes—
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×M16—1272no—
126C4⋊Q8C2×D12711yescharacteristic, complemented
127#127C2×D12711yescharacteristic, complemented
128#124—1272no—
129#124—1272no—
130#124—2542no—
131#124C2×D12712yescomplemented
132#121—22no—
133C4×D4—22no—
134C2×C22⋊C4—2542no—
135C2×C42C2×D12711yescharacteristic
136#97—1272no—
137C4×C8—1272no—
138C24—2541no—
139#41—24no—
140#41—24no—
141C2×Q8—1272no—
142C2×Q8—2544no—
143C2×Q8—2541no—
144C2×Q8C22×D12712yes—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—1272no—
148C2×D4—2544no—
149C2×D4—2541no—
150C2×D4C22×D12712yes—
151C22×C4—22no—
152C22×C4—2541no—
153C22×C4C22⋊D12712yes—
154C22×C4C22×D12711yescharacteristic
155Q16—2548no—
156SD16—2548no—
157SD16—2548no—
158D8—2548no—
159M16—1278no—
160C2×C8—1274no—
161C4⋊C4—22no—
162C22⋊C4—24no—
163C22⋊C4—2544no—
164C22⋊C4—2544no—
165C42C22⋊D12712yes—
166C42C22×D12711yescharacteristic
167C42C22×D12711yescharacteristic
168C23—22no—
169C23—5084no—
170C23—5082no—
171C23—2542no—
172C23C2×C22⋊D12711yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—5082no—
176Q8—2544no—
177D4—22no—
178D4—22no—
179D4—44no—
180D4—44no—
181D4—5082no—
182D4—2544no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—5082no—
187C2×C4—2542no—
188C2×C4D4×D12714yes—
189C2×C4C2×C22⋊D12712yes—
190C2×C4C2×C22⋊D12712yes—
191C2×C4C2×C22⋊D12711yescharacteristic
192C2×C4C23×D12711yescharacteristic
193C8—1278no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—10164no—
198C22—5082no—
199C22—5084no—
200C22#1686120512yes—
201C22C22×C22⋊D12711yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—5082no—
206C4#1686120512yes—
207C4C2×D4×D12712yes—
208C2—22no—
209C2—82no—
210C2—10162no—
211C2C2×#1686120511yescharacteristic
212C2#2255067412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#532×C127C2
socleC22×C127C22×D4
hypercenter#532D127
hypocenterC127#3153
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861234C2semidirect
3. C2×#16861197C2semidirect
4. #22550268C2semidirect
5. #22550263C2semidirect
6. #22550258C2semidirect
7. C4×#14523232C2semidirect
8. #532×C127C2semidirect
13. C2×#14523238C22semidirect
14. C2×#14523236C22semidirect
22. #16861120C22semidirect
23. C4×C127⋊C8C22semidirect
25. C4⋊Q8×C127C22semidirect
26. #127×C127C22semidirect
27. #124×C127C22semidirect
43. #14523232D4semidirect
107. C127#3153semidirect
110. #532D127semidirect
126. C4⋊Q8C2×D127semidirect
127. #127C2×D127semidirect
131. #124C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612342
C2×#168611972
#225502682
#225502632
#225502582
C4×#145232322
#532×C1272
#3153127