Finite Group Databasebeta

← #30639018

Subgroups — #30639018

Order 16256.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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#22549903C212yesmaximal, complemented
3#22549843C211yesmaximal, characteristic
4#22549646C212yesmaximal, complemented
5#410×C127C211yesmaximal, characteristic
6#16861216—24no—
7C2×#14523232—22no—
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2212yes—
10C22×C127⋊C8C2211yescharacteristic
11C4⋊(C127⋊C8)—24no—
12C22⋊(C127⋊C8)—22no—
13C2×#41×C127C2211yescharacteristic
14#117×C127C2212yes—
15#99×C127C414yes—
16#14523232—24no—
17#14523232—22no—
18#14523232—22no—
19#14523232—22no—
20C2×C127⋊C8—24no—
21C2×C127⋊C8—21no—
22C2×C127⋊C8—22no—
23C2×C127⋊C8—21no—
24C2×C127⋊C8D414yes—
25#41×C127—22no—
26#41×C127—22no—
27C2×Q8×C127Q811yescharacteristic
28C2×D4×C127C2×C412yes—
29C2×D4×C127Q811yescharacteristic
30C22×C4×C127D412yes—
31C22×C4×C127C2311yescharacteristic
32C4⋊C4×C127—22no—
33C22⋊C4×C127C2×C414yes—
34C42×C127—22no—
35C127⋊C8—24no—
36C127⋊C8—41no—
37C127⋊C8—41no—
38C23×C127C4⋊C412yes—
39C23×C127C22×C411yescharacteristic
40Q8×C127—22no—
41D4×C127—24no—
42D4×C127—22no—
43C2×C4×C127—22no—
44C2×C4×C127—22no—
45C2×C4×C127—24no—
46C2×C4×C127C4⋊C412yes—
47C2×C4×C127C2×D412yes—
48C2×C4×C127C2×D411yescharacteristic
49C2×C4×C127C2×Q811yescharacteristic
50C2×C4×C127#4112yes—
51C22×C127—22no—
52C22×C127—41no—
53C22×C127—42no—
54C22×C127C2×C4⋊C411yescharacteristic
55C22×C127C4×D412yes—
56C4×C127—22no—
57C4×C127—44no—
58C4×C127#12011yescharacteristic
59C4×C127C22⋊Q812yes—
60C4×C127#12311yescharacteristic
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#410D12711yescharacteristic
70#408—1271no—
71#344—1272no—
72C2×#41C2×D12711yescharacteristic
73#131—2544no—
74C2×M16—1272no—
75C2×M16—1271no—
76C2×M16—2542no—
77C22×C8—1271no—
78#117C2×D12712yes—
79C4⋊C8—2544no—
80#99Dic12714yes—
81C22⋊C8—2542no—
82#41—22no—
83#41—22no—
84C2×Q8Dic25411yescharacteristic
85C2×D4C2×Dic12712yes—
86C2×D4Dic25411yescharacteristic
87C22×C4D50812yes—
88C22×C4C22×D12711yescharacteristic
89M16—2542no—
90M16—2544no—
91M16—2542no—
92M16—2542no—
93C2×C8—1274no—
94C2×C8—2542no—
95C2×C8—2541no—
96C2×C8—2541no—
97C2×C8—2544no—
98C4⋊C4—22no—
99C22⋊C4C2×Dic12714yes—
100C42—22no—
101C23C4⋊Dic12712yes—
102C23C22×Dic12711yescharacteristic
103Q8—22no—
104D4—24no—
105D4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—24no—
109C2×C4C4⋊Dic12712yes—
110C2×C4D4×D12712yes—
111C2×C4C2×D50811yescharacteristic
112C2×C4C2×Dic25411yescharacteristic
113C2×C4#1452324712yes—
114C8—5081no—
115C8—5081no—
116C8—2544no—
117C22—22no—
118C22—41no—
119C22—42no—
120C22C2×C4⋊Dic12711yescharacteristic
121C22D4×Dic12712yes—
122C4—22no—
123C4—44no—
124C4#1686117911yescharacteristic
125C4#1686118612yes—
126C4#1686119311yescharacteristic
127C2—22no—
128C2—21no—
129C2—42no—
130C2#2254975111yescharacteristic
subgroupG/H
centerC4#16861179
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#385
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#410D127
hypocenterC127#1868
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22549903C2semidirect
4. #22549646C2semidirect
66. C127#1868semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225499032
#225498432
#225496462
#410×C1272
#1868127