Finite Group Databasebeta

← #30639015

Subgroups — #30639015

Order 16256.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#22550643C211yesmaximal, characteristic
2#22549902C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic
4#22549648C212yesmaximal, complemented
5#408×C127C211yesmaximal, characteristic
6#16861216—24no—
7C2×#14523232—22no—
8C2×#14523232C2212yes—
9#16861168C2212yes—
10Dic127⋊C8—24no—
11#16861124C414yes—
12C2×#41×C127C2211yescharacteristic
13C2×M16×C127C2211yescharacteristic
14C22×C8×C127C2211yescharacteristic
15C22⋊C8×C127—22no—
16#14523232—24no—
17#14523232—22no—
18#14523232—22no—
19C2×C127⋊C8—24no—
20C2×C127⋊C8D414yes—
21C4⋊Dic127—22no—
22C22⋊Dic127C2×C414yes—
23C4×Dic127—22no—
24#41×C127—22no—
25#41×C127—22no—
26C2×Q8×C127Q811yescharacteristic
27C2×D4×C127C2×C412yes—
28C2×D4×C127Q811yescharacteristic
29C22×C4×C127D412yes—
30C22×C4×C127C2311yescharacteristic
31M16×C127—22no—
32C2×C8×C127—21no—
33C2×C8×C127—21no—
34C2×C8×C127—22no—
35C2×Dic127—24no—
36C127⋊C8—24no—
37C23×C127C4⋊C412yes—
38C23×C127C22×C411yescharacteristic
39Q8×C127—22no—
40D4×C127—24no—
41D4×C127—22no—
42C2×C4×C127—22no—
43C2×C4×C127—22no—
44C2×C4×C127C4⋊C412yes—
45C2×C4×C127C2×D412yes—
46C2×C4×C127C2×D411yescharacteristic
47C2×C4×C127C2×Q811yescharacteristic
48C2×C4×C127#4112yes—
49C8×C127—41no—
50C8×C127—41no—
51C22×C127—22no—
52C22×C127—41no—
53C22×C127—42no—
54C22×C127C2×C4⋊C411yescharacteristic
55C22×C127C4×D412yes—
56C4×C127—22no—
57C4×C127#12011yescharacteristic
58C4×C127C22⋊Q812yes—
59C4×C127#12311yescharacteristic
60Dic127—44no—
61C2×C127—22no—
62C2×C127—21no—
63C2×C127—42no—
64C2×C127#38511yescharacteristic
65#1868—1271nomaximal, Sylow
66C127#186811yescharacteristic, complemented, Sylow
67#568—1271no—
68#424—1272no—
69#410—1271no—
70#408D12711yescharacteristic
71#344—1272no—
72C2×#41C2×D12711yescharacteristic
73#131—2544no—
74C2×M16—1272no—
75C2×M16—2542no—
76C2×M16C2×D12711yescharacteristic
77C22×C8C2×D12711yescharacteristic
78#117—1272no—
79C4⋊C8—2544no—
80#99—1274no—
81C22⋊C8—22no—
82#41—22no—
83#41—22no—
84C2×Q8Dic25411yescharacteristic
85C2×D4C4×D12712yes—
86C2×D4Dic25411yescharacteristic
87C22×C4C22⋊D12712yes—
88C22×C4C22×D12711yescharacteristic
89M16—22no—
90M16—2544no—
91M16—2542no—
92M16—2542no—
93C2×C8—22no—
94C2×C8—21no—
95C2×C8—21no—
96C2×C8—1274no—
97C2×C8—2544no—
98C4⋊C4—2542no—
99C22⋊C4—1274no—
100C42—2542no—
101C23Dic127⋊C412yes—
102C23C2×C4×D12711yescharacteristic
103Q8—22no—
104D4—24no—
105D4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—2544no—
109C2×C4Dic127⋊C412yes—
110C2×C4D4×D12712yes—
111C2×C4C2×C22⋊D12711yescharacteristic
112C2×C4C2×Dic25411yescharacteristic
113C2×C4#1452324712yes—
114C8—41no—
115C8—41no—
116C8—2544no—
117C22—22no—
118C22—41no—
119C22—42no—
120C22C2×Dic127⋊C411yescharacteristic
121C22(C22⋊D127)⋊C412yes—
122C4—22no—
123C4—5084no—
124C4#1686117711yescharacteristic
125C4#1686118612yes—
126C4#1686119011yescharacteristic
127C2—22no—
128C2—21no—
129C2—42no—
130C2#2254974811yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC2×C127#385
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#408D127
hypocenterC127#1868
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #22549902C2semidirect
4. #22549648C2semidirect
66. C127#1868semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225499022
#225498502
#225496482
#408×C1272
#1868127