Finite Group Databasebeta

← #30639580

Subgroups — #30639580

Order 16256.

312 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 97 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#22549900C211yesmaximal, characteristic, complemented
2C2×#16861128C212yesmaximal, complemented
3C2×#16861126C212yesmaximal
4C4×#14523232C211yesmaximal, characteristic, complemented
5C2×#124×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×#14523232C2212yes—
8C4⋊(C127⋊C8)—22no—
9#16861128—24no—
10#16861126—24no—
11#16861120—21no—
12C4×C127⋊C8—21no—
13C22×Q8×C127C2211yescharacteristic
14C22×D4×C127C2211yescharacteristic
15#124×C127—21no—
16#124×C127—21no—
17#124×C127—22no—
18C2×C22⋊C4×C127C414yes—
19C2×C42×C127C2211yescharacteristic
20#14523232—22no—
21#14523232—22no—
22#14523232—22no—
23#14523232D414yes—
24C2×C127⋊C8—22no—
25C2×C127⋊C8—22no—
26C24×C127C2×C412yes—
27C2×Q8×C127—21no—
28C2×Q8×C127—21no—
29C2×Q8×C127—22no—
30C2×Q8×C127—22no—
31C2×D4×C127—21no—
32C2×D4×C127—21no—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C22×C4×C127C2×C412yes—
36C22×C4×C127C2×C412yes—
37C22×C4×C127C2311yescharacteristic
38C22⋊C4×C127—24no—
39C22⋊C4×C127—24no—
40C42×C127D411yescharacteristic
41C42×C127D411yescharacteristic
42C42×C127D412yes—
43C127⋊C8—24no—
44C127⋊C8—42no—
45C23×C127—22no—
46C23×C127—24no—
47C23×C127—44no—
48C23×C127C22×C411yescharacteristic
49Q8×C127—42no—
50Q8×C127—41no—
51Q8×C127—41no—
52D4×C127—42no—
53D4×C127—41no—
54D4×C127—41no—
55C2×C4×C127—22no—
56C2×C4×C127—24no—
57C2×C4×C127—42no—
58C2×C4×C127C22⋊C412yes—
59C2×C4×C127C22⋊C412yes—
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127#4112yes—
64C22×C127—22no—
65C22×C127—42no—
66C22×C127—44no—
67C22×C127—84no—
68C22×C127C2×C22⋊C411yescharacteristic
69C22×C127C4×D412yes—
70C4×C127—22no—
71C4×C127—22no—
72C4×C127—42no—
73C4×C127#12112yes—
74C4×C127#12411yescharacteristic
75C4×C127#12711yescharacteristic
76C2×C127—22no—
77C2×C127—82no—
78C2×C127#39011yescharacteristic
79C2×C127#41312yes—
80#1967—1271nomaximal, Sylow
81C127#196711yescharacteristic, complemented, Sylow
82C2×#124D12711yescharacteristic
83#423—1271no—
84C2×#101—1272no—
85C2×#100—1272no—
86C4×M16—1271no—
87C22×Q8C2×D12711yescharacteristic
88C22×D4C2×D12711yescharacteristic
89C2×M16—1272no—
90C2×M16—1272no—
91#124—21no—
92#124—21no—
93#124—22no—
94C2×C22⋊C4Dic12714yes—
95C2×C42C2×D12711yescharacteristic
96C4⋊C8—2542no—
97#101—2544no—
98#100—2544no—
99#97—2541no—
100C4×C8—2541no—
101C24C2×Dic12712yes—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—22no—
110C22×C4C2×Dic12712yes—
111C22×C4C2×Dic12712yes—
112C22×C4C22×D12711yescharacteristic
113M16—1274no—
114M16—2542no—
115M16—2542no—
116M16—2542no—
117C2×C8—2542no—
118C2×C8—2542no—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C42C22⋊D12711yescharacteristic
122C42C22⋊D12711yescharacteristic
123C42C22⋊D12712yes—
124C23—22no—
125C23—24no—
126C23—44no—
127C23C22×Dic12711yescharacteristic
128Q8—41no—
129Q8—41no—
130Q8—42no—
131D4—41no—
132D4—41no—
133D4—42no—
134C2×C4—22no—
135C2×C4—24no—
136C2×C4—42no—
137C2×C4C22⋊Dic12712yes—
138C2×C4C22⋊Dic12712yes—
139C2×C4D4×D12712yes—
140C2×C4C2×C22⋊D12711yescharacteristic
141C2×C4C2×C22⋊D12711yescharacteristic
142C2×C4#1452324712yes—
143C8—5082no—
144C8—2544no—
145C22—22no—
146C22—42no—
147C22—44no—
148C22—84no—
149C22C2×C22⋊Dic12711yescharacteristic
150C22D4×Dic12712yes—
151C4—22no—
152C4—22no—
153C4—42no—
154C4#1686118012yes—
155C4#1686119611yescharacteristic
156C4#1686120511yescharacteristic
157C2—22no—
158C2—82no—
159C2#2254977911yescharacteristic
160C2#2254986012yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#124×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#124D127
hypocenterC127#1967
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #22549900C2semidirect
2. C2×#16861128C2semidirect
4. C4×#14523232C2semidirect
81. C127#1967semidirect

Maximal subgroups

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

subgroupindex
#225499002
C2×#168611282
C2×#168611262
C4×#145232322
C2×#124×C1272
#1967127