Finite Group Databasebeta

← #30639016

Subgroups — #30639016

Order 16256.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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#22549904C212yesmaximal, complemented
2#22549850C211yesmaximal, characteristic
3#22549843C211yesmaximal, characteristic
4#22549647C212yesmaximal, complemented
5#568×C127C211yesmaximal, characteristic
6C2×#14523232C2211yescharacteristic
7C22×C127⋊C8C2211yescharacteristic
8#16861168C2212yes—
9Dic127⋊C8—24no—
10#16861124C414yes—
11C22⋊(C127⋊C8)—22no—
12C2×#41×C127C2211yescharacteristic
13#131×C127—24no—
14C2×M16×C127—22no—
15C2×M16×C127C2212yes—
16#14523232—22no—
17C2×C127⋊C8—21no—
18C2×C127⋊C8—22no—
19C2×C127⋊C8—21no—
20C4⋊Dic127—22no—
21C22⋊Dic127C2×C414yes—
22C4×Dic127—22no—
23#41×C127—22no—
24#41×C127—22no—
25C2×Q8×C127Q811yescharacteristic
26C2×D4×C127C2×C412yes—
27C2×D4×C127Q811yescharacteristic
28C22×C4×C127D412yes—
29C22×C4×C127C2311yescharacteristic
30M16×C127—24no—
31M16×C127—22no—
32M16×C127—22no—
33C2×C8×C127—24no—
34C2×C8×C127D414yes—
35C2×Dic127—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×C127C4⋊C412yes—
46C2×C4×C127C2×D412yes—
47C2×C4×C127C2×D411yescharacteristic
48C2×C4×C127C2×Q811yescharacteristic
49C2×C4×C127#4112yes—
50C8×C127—24no—
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#568D12711yescharacteristic
68#424—1272no—
69#410—1271no—
70#408—1271no—
71#344—1272no—
72C2×#41C2×D12711yescharacteristic
73#131—24no—
74C2×M16—22no—
75C2×M16—1271no—
76C2×M16C2×D12712yes—
77C22×C8—1271no—
78#117—1272no—
79C4⋊C8—2544no—
80#99—1274no—
81C22⋊C8—2542no—
82#41—22no—
83#41—22no—
84C2×Q8Dic25411yescharacteristic
85C2×D4C4×D12712yes—
86C2×D4Dic25411yescharacteristic
87C22×C4C22⋊D12712yes—
88C22×C4C22×D12711yescharacteristic
89M16—24no—
90M16—22no—
91M16—22no—
92M16—2542no—
93C2×C8—24no—
94C2×C8—2541no—
95C2×C8—2541no—
96C2×C8—2542no—
97C2×C8C22⋊D12714yes—
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×C4C2×C22⋊D12712yes—
111C2×C4C2×C22⋊D12711yescharacteristic
112C2×C4C2×Dic25411yescharacteristic
113C2×C4#1452324812yes—
114C8—24no—
115C8—5081no—
116C8—5081no—
117C22—22no—
118C22—41no—
119C22—42no—
120C22C2×Dic127⋊C411yescharacteristic
121C22C4×C22⋊D12712yes—
122C4—22no—
123C4—5084no—
124C4#1686117811yescharacteristic
125C4#1686118812yes—
126C4#1686119211yescharacteristic
127C2—22no—
128C2—21no—
129C2—42no—
130C2#2254974911yescharacteristic
subgroupG/H
centerC4#16861178
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#385
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1868
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #22549904C2semidirect
4. #22549647C2semidirect
66. C127#1868semidirect

Maximal subgroups

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

subgroupindex
#225499042
#225498502
#225498432
#225496472
#568×C1272
#1868127