Finite Group Databasebeta

← #30638950

Subgroups — #30638950

Order 16256.

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

normal subgroups only

This group has 252 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#22549924C212yesmaximal, complemented
2#22549860C211yesmaximal, characteristic
3#22549843C211yesmaximal, characteristic
4#22549640C212yesmaximal, complemented
5#568×C127C211yesmaximal, characteristic
6C2×#14523232C2211yescharacteristic
7C2×#14523232C2212yes—
8C22×C127⋊C8C2211yescharacteristic
9#16861144—24no—
10#16861128C412yes—
11#16861126C412yes—
12C22⋊(C127⋊C8)—22no—
13C2×#41×C127C2211yescharacteristic
14#131×C127—24no—
15C2×M16×C127—22no—
16C2×M16×C127C2212yes—
17#14523232—22no—
18#14523232—22no—
19#14523232C2×C414yes—
20C2×C127⋊C8—22no—
21C2×C127⋊C8—21no—
22C2×C127⋊C8—21no—
23C2×C127⋊C8—22no—
24#41×C127—22no—
25#41×C127—22no—
26C2×Q8×C127C2×C411yescharacteristic
27C2×D4×C127C2×C411yescharacteristic
28C2×D4×C127Q812yes—
29C22×C4×C127D412yes—
30C22×C4×C127C2311yescharacteristic
31M16×C127—24no—
32M16×C127—22no—
33M16×C127D414yes—
34C2×C8×C127—24no—
35C2×C8×C127—22no—
36C127⋊C8—24no—
37C127⋊C8—41no—
38C127⋊C8—41no—
39C23×C127C4⋊C412yes—
40C23×C127C2×Q811yescharacteristic
41Q8×C127—22no—
42D4×C127—24no—
43D4×C127—22no—
44C2×C4×C127—22no—
45C2×C4×C127—22no—
46C2×C4×C127C4⋊C412yes—
47C2×C4×C127C22×C411yescharacteristic
48C2×C4×C127C2×D412yes—
49C2×C4×C127C2×D411yescharacteristic
50C2×C4×C127#4112yes—
51C8×C127—24no—
52C22×C127—22no—
53C22×C127—41no—
54C22×C127—42no—
55C22×C127C2×C4⋊C411yescharacteristic
56C22×C127C22⋊Q812yes—
57C4×C127—22no—
58C4×C127C4×D412yes—
59C4×C127#12011yescharacteristic
60C4×C127#12311yescharacteristic
61C2×C127—21no—
62C2×C127—22no—
63C2×C127—42no—
64C2×C127#38511yescharacteristic
65#1858—1271nomaximal, Sylow
66C127#185811yescharacteristic, complemented, Sylow
67#568D12711yescharacteristic
68#430—1272no—
69#413—1271no—
70#408—1271no—
71#341—1272no—
72C2×#41C2×D12711yescharacteristic
73#131—24no—
74C2×M16—22no—
75C2×M16—1272no—
76C2×M16—1271no—
77C2×M16C2×D12712yes—
78C22×C8—1271no—
79#108—2544no—
80#101—1272no—
81#100—1272no—
82C22⋊C8—2542no—
83#41—22no—
84#41—22no—
85C2×Q8C4×D12711yescharacteristic
86C2×D4C4×D12711yescharacteristic
87C2×D4Dic25412yes—
88C22×C4C22⋊D12712yes—
89C22×C4C22×D12711yescharacteristic
90M16—24no—
91M16—22no—
92M16—1274no—
93M16—2542no—
94M16—2542no—
95M16C22⋊D12714yes—
96C2×C8—24no—
97C2×C8—22no—
98C2×C8—2541no—
99C2×C8—2541no—
100C2×C8—2542no—
101C2×C8—2542no—
102C23Dic127⋊C412yes—
103C23C2×Dic25411yescharacteristic
104Q8—22no—
105D4—24no—
106D4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4Dic127⋊C412yes—
110C2×C4C2×C4×D12711yescharacteristic
111C2×C4C2×C22⋊D12711yescharacteristic
112C2×C4C2×C22⋊D12712yes—
113C2×C4#1452324812yes—
114C8—24no—
115C8—5081no—
116C8—5081no—
117C8—2544no—
118C22—22no—
119C22—41no—
120C22—42no—
121C22C2×Dic127⋊C411yescharacteristic
122C22#1686118812yes—
123C4—22no—
124C4C4×C22⋊D12712yes—
125C4#1686117811yescharacteristic
126C4#1686119211yescharacteristic
127C2—21no—
128C2—22no—
129C2—42no—
130C2#2254974911yescharacteristic
subgroupG/H
centerC4#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#385
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1858
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #22549924C2semidirect
4. #22549640C2semidirect
66. C127#1858semidirect

Maximal subgroups

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

subgroupindex
#225499242
#225498602
#225498432
#225496402
#568×C1272
#1858127