Finite Group Databasebeta

← #30639983

Subgroups — #30639983

Order 16256.

296 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 71 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#22549902C211yesmaximal, characteristic, complemented
2#22549881C212yesmaximal, complemented
3C2×#16861126C212yesmaximal, complemented
4#22549641C211yesmaximal, characteristic, complemented
5C22×D8×C127C211yesmaximal, characteristic
6C2×#14523232C2211yescharacteristic
7C2×#14523232C2212yes—
8#16861168C2211yescharacteristic
9Dic127⋊C8—22no—
10#16861130—24no—
11#16861126—24no—
12C22×D4×C127C2212yes—
13C2×D8×C127—24no—
14C2×D8×C127—22no—
15C22×C8×C127C2211yescharacteristic
16#14523232—22no—
17#14523232—22no—
18#14523232—22no—
19C2×C127⋊C8—22no—
20C2×C127⋊C8—21no—
21C4⋊Dic127D412yes—
22C22⋊Dic127—21no—
23C4×Dic127—21no—
24C24×C127—22no—
25C2×D4×C127—22no—
26C2×D4×C127—22no—
27C2×D4×C127—22no—
28C2×D4×C127—22no—
29C2×D4×C127—22no—
30C2×D4×C127D414yes—
31C22×C4×C127C2311yescharacteristic
32D8×C127—28no—
33C2×C8×C127—22no—
34C2×C8×C127D412yes—
35C2×Dic127—22no—
36C127⋊C8—41no—
37C127⋊C8—42no—
38C23×C127—24no—
39C23×C127—42no—
40C23×C127—42no—
41C23×C127—84no—
42C23×C127#4111yescharacteristic
43D4×C127—24no—
44D4×C127—24no—
45D4×C127—42no—
46D4×C127—42no—
47C2×C4×C127C2×D411yescharacteristic
48C2×C4×C127C2×D412yes—
49C2×C4×C127C2×D411yescharacteristic
50C2×C4×C127#4112yes—
51C8×C127—22no—
52C22×C127—22no—
53C22×C127—44no—
54C22×C127—84no—
55C22×C127—84no—
56C22×C127—84no—
57C22×C127#12111yescharacteristic
58C22×C127#12312yes—
59C4×C127#12011yescharacteristic
60C4×C127#12112yes—
61C4×C127#12411yescharacteristic
62Dic127—42no—
63C2×C127—22no—
64C2×C127—84no—
65C2×C127#39411yescharacteristic
66C2×C127#47212yes—
67#2045—1271nomaximal, Sylow
68C127#204511yescharacteristic, complemented, Sylow
69C22×D8D12711yescharacteristic
70#424—1271no—
71#418—1272no—
72C2×#100—1272no—
73#341—1271no—
74C22×D4C2×D12712yes—
75C2×D8—24no—
76C2×D8—22no—
77C2×M16—1272no—
78C2×M16—1271no—
79C22×C8C2×D12711yescharacteristic
80#117—1271no—
81C4⋊C8—2542no—
82#102—2544no—
83#100—2544no—
84C24—22no—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4C22⋊D12714yes—
91C22×C4C22×D12711yescharacteristic
92D8—28no—
93M16—2542no—
94M16—2542no—
95M16—2542no—
96C2×C8—22no—
97C2×C8—2542no—
98C2×C8—2541no—
99C2×C8C22⋊D12712yes—
100C4⋊C4—1272no—
101C22⋊C4—2541no—
102C42—2541no—
103C23—24no—
104C23—42no—
105C23—42no—
106C23—84no—
107C23#1452324711yescharacteristic
108D4—24no—
109D4—24no—
110D4—42no—
111D4—42no—
112C2×C4—2542no—
113C2×C4D4×D12711yescharacteristic
114C2×C4C2×C22⋊D12711yescharacteristic
115C2×C4C2×C22⋊D12712yes—
116C2×C4#1452324712yes—
117C8—22no—
118C8—5081no—
119C8—5082no—
120C22—22no—
121C22—44no—
122C22—84no—
123C22—84no—
124C22—84no—
125C22#1686118011yescharacteristic
126C22#1686119012yes—
127C4—5082no—
128C4#1686117811yescharacteristic
129C4#1686118112yes—
130C4#1686119611yescharacteristic
131C2—22no—
132C2—84no—
133C2#2254979011yescharacteristic
134C2#2255014312yes—
subgroupG/H
centerC22#16861180
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radicalC22×D8×C127C2
socleC22×C127#121
hypercenterC22×D8D127
hypocenterC127#2045
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. #22549902C2semidirect
2. #22549881C2semidirect
3. C2×#16861126C2semidirect
4. #22549641C2semidirect
68. C127#2045semidirect

Maximal subgroups

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

subgroupindex
#225499022
#225498812
C2×#168611262
#225496412
C22×D8×C1272
#2045127