Finite Group Databasebeta

← #30649183

Subgroups — #30649183

Order 16256.

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

normal subgroups only

This group has 656 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
1C2×#16861234C214yesmaximal, complemented
2C2×#16861206C211yesmaximal, characteristic, complemented
3#22550266C214yesmaximal, complemented
4#22550252C212yesmaximal, complemented
5#22550250C212yesmaximal, complemented
6C4×#14523232C211yesmaximal, characteristic, complemented
7#532×C127C211yesmaximal, characteristic, complemented
8C22×D508—22no—
9C22×D508C2212yes—
10#16861234—216no—
11C2×#14523235C2218yescomplemented
12C2×#14523233C2218yescomplemented
13C2×#14523232C2212yes—
14#16861206—22no—
15#16861206C2212yes—
16#16861206C2212yes—
17#16861120C2212yescomplemented
18C4×C127⋊C8C2212yescomplemented
19C2×#41×C127C2212yes—
20C4⋊Q8×C127C2211yescharacteristic, complemented
21#127×C127C2211yescharacteristic, complemented
22#124×C127C2212yescomplemented
23#121×C127—22no—
24C4×D4×C127—22no—
25C2×C42×C127C2211yescharacteristic
26C23×D127—22no—
27C2×D508—28no—
28C2×D508—22no—
29C2×D508—24no—
30C2×D508—24no—
31C2×D508—48no—
32C2×D508C2314yes—
33#14523235—216no—
34#14523233—216no—
35#14523232D418yescomplemented
36C2×C127⋊C8C2314yes—
37#41×C127—28no—
38C2×Q8×C127C2312yes—
39C2×D4×C127—22no—
40C2×D4×C127—22no—
41C2×D4×C127C2312yes—
42C22×C4×C127—22no—
43C22×C4×C127D412yes—
44C22×C4×C127C2311yescharacteristic
45C4⋊C4×C127—22no—
46C22⋊C4×C127—24no—
47C42×C127D412yes—
48C42×C127C2311yescharacteristic
49C42×C127C2311yescharacteristic
50C22×D127—24no—
51C22×D127—48no—
52C22×D127—44no—
53D508—28no—
54D508—48no—
55D508—44no—
56D508—48no—
57C127⋊C8C2×D418yescomplemented
58C23×C127—22no—
59C23×C127C2×D411yescharacteristic
60Q8×C127—24no—
61D4×C127—24no—
62D4×C127—44no—
63D4×C127—44no—
64C2×C4×C127—24no—
65C2×C4×C127—22no—
66C2×C4×C127—44no—
67C2×C4×C127C2×D414yes—
68C2×C4×C127C2×D412yes—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D412yes—
71C2×C4×C127C2411yescharacteristic
72C22×C127—22no—
73C22×C127—42no—
74C22×C127—44no—
75C22×C127#12712yes—
76C22×C127C22×D411yescharacteristic
77C2×D127—44no—
78C2×D127—48no—
79C2×D127—88no—
80C4×C127—22no—
81C4×C127—42no—
82C4×C127#12712yes—
83C4×C127AGL(1,ℤ/8)14yescomplemented
84C4×C127C22×D412yes—
85C2×C127—22no—
86C2×C127—82no—
87C2×C127C2×#12711yescharacteristic
88C2×C127C2×AGL(1,ℤ/8)12yes—
89D127—84no—
90#3151—1271nomaximal, Sylow
91C127#315111yescharacteristic, complemented, Sylow
92C2×AGL(1,ℤ/8)—1274no—
93#532D12711yescharacteristic, complemented
94C2×#127—1271no—
95#496—1274no—
96#493—1272no—
97#492—1272no—
98C4×M16—1271no—
99C2×#41C2×D12712yes—
100C22×D4—1272no—
101C22×D4—2542no—
102AGL(1,ℤ/8)—25416no—
103C2×SD16—1278no—
104C2×D8—1278no—
105C2×M16—1272no—
106C4⋊Q8C2×D12711yescharacteristic, complemented
107#127—1272no—
108#127—1272no—
109#127—2542no—
110#127C2×D12711yescharacteristic, complemented
111#124C2×D12712yescomplemented
112#121—22no—
113C4×D4—22no—
114C2×C42C2×D12711yescharacteristic
115#97—1272no—
116C4×C8—1272no—
117C24—2542no—
118#41—28no—
119C2×Q8C22×D12712yes—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—1274no—
123C2×D4—5088no—
124C2×D4—2548no—
125C2×D4—2544no—
126C2×D4—2544no—
127C2×D4—2542no—
128C2×D4C22×D12712yes—
129C22×C4—22no—
130C22×C4C22⋊D12712yes—
131C22×C4C22×D12711yescharacteristic
132SD16—25416no—
133D8—25416no—
134M16—1278no—
135C2×C8—1274no—
136C4⋊C4—22no—
137C22⋊C4—24no—
138C42C22⋊D12712yes—
139C42C22×D12711yescharacteristic
140C42C22×D12711yescharacteristic
141C23—22no—
142C23—5088no—
143C23—5084no—
144C23—2544no—
145C23C2×C22⋊D12711yescharacteristic
146Q8—24no—
147D4—24no—
148D4—44no—
149D4—44no—
150D4—5088no—
151D4—5084no—
152D4—5088no—
153D4—2548no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—44no—
157C2×C4D4×D12714yes—
158C2×C4C2×C22⋊D12712yes—
159C2×C4C2×C22⋊D12711yescharacteristic
160C2×C4C2×C22⋊D12712yes—
161C2×C4C23×D12711yescharacteristic
162C8—1278no—
163C22—22no—
164C22—44no—
165C22—42no—
166C22—10168no—
167C22—5088no—
168C22—5084no—
169C22#1686120512yes—
170C22C22×C22⋊D12711yescharacteristic
171C4—22no—
172C4—42no—
173C4#1686120512yes—
174C4#1686123414yescomplemented
175C4C2×D4×D12712yes—
176C2—22no—
177C2—82no—
178C2—10164no—
179C2C2×#1686120511yescharacteristic
180C2C2×#1686123412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#532×C127C2
socleC22×C127C22×D4
hypercenter#532D127
hypocenterC127#3151
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×#16861234C2semidirect
2. C2×#16861206C2semidirect
3. #22550266C2semidirect
4. #22550252C2semidirect
5. #22550250C2semidirect
6. C4×#14523232C2semidirect
7. #532×C127C2semidirect
11. C2×#14523235C22semidirect
12. C2×#14523233C22semidirect
17. #16861120C22semidirect
18. C4×C127⋊C8C22semidirect
20. C4⋊Q8×C127C22semidirect
21. #127×C127C22semidirect
22. #124×C127C22semidirect
35. #14523232D4semidirect
57. C127⋊C8C2×D4semidirect
83. C4×C127AGL(1,ℤ/8)semidirect
91. C127#3151semidirect
93. #532D127semidirect
106. C4⋊Q8C2×D127semidirect
110. #127C2×D127semidirect
111. #124C2×D127semidirect
174. C4#16861234semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168612062
#225502662
#225502522
#225502502
C4×#145232322
#532×C1272
#3151127