Finite Group Databasebeta

← #30637044

Subgroups — #30637044

Order 16256.

264 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 109 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#22549929C212yesmaximal, complemented
2#22549602C212yesmaximal, complemented
3#22549600C212yesmaximal
4C2×#117×C127C211yesmaximal, characteristic
5C4⋊(C127⋊C8)—22no—
6C22⋊(C127⋊C8)C2214yes—
7C4×C127⋊C8—22no—
8C23×C4×C127C411yescharacteristic
9#117×C127—22no—
10#117×C127—22no—
11C2×C4⋊C4×C127C411yescharacteristic
12C2×C4⋊C4×C127C2211yescharacteristic
13C2×C22⋊C4×C127C411yescharacteristic
14C2×C22⋊C4×C127C2211yescharacteristic
15C2×C42×C127C411yescharacteristic
16C2×C42×C127C2211yescharacteristic
17C2×C127⋊C8—24no—
18C24×C127C2×C411yescharacteristic
19C22×C4×C127—21no—
20C22×C4×C127—21no—
21C22×C4×C127—22no—
22C22×C4×C127C814yescomplemented
23C22×C4×C127C2×C411yescharacteristic
24C22×C4×C127C2×C411yescharacteristic
25C22×C4×C127C2×C411yescharacteristic
26C22×C4×C127C2×C411yescharacteristic
27C22×C4×C127C2×C411yescharacteristic
28C22×C4×C127C2311yescharacteristic
29C4⋊C4×C127—22no—
30C4⋊C4×C127—22no—
31C22⋊C4×C127—22no—
32C22⋊C4×C127—22no—
33C42×C127—21no—
34C42×C127—21no—
35C42×C127D412yes—
36C42×C127D412yes—
37C127⋊C8—44no—
38C23×C127—21no—
39C23×C127—21no—
40C23×C127—41no—
41C23×C127—41no—
42C23×C127C2×C812yes—
43C23×C127C22×C411yescharacteristic
44C2×C4×C127—21no—
45C2×C4×C127—21no—
46C2×C4×C127—24no—
47C2×C4×C127—24no—
48C2×C4×C127—21no—
49C2×C4×C127—21no—
50C2×C4×C127—41no—
51C2×C4×C127—41no—
52C2×C4×C127—41no—
53C2×C4×C127—41no—
54C2×C4×C127C22⋊C412yes—
55C2×C4×C127C22⋊C411yescharacteristic
56C2×C4×C127C22⋊C411yescharacteristic
57C2×C4×C127C2×C812yes—
58C2×C4×C127C2×C812yes—
59C2×C4×C127M1612yes—
60C2×C4×C127C2×D412yes—
61C22×C127—21no—
62C22×C127—21no—
63C22×C127—22no—
64C22×C127—42no—
65C22×C127—42no—
66C22×C127—41no—
67C22×C127—41no—
68C22×C127C2×C22⋊C411yescharacteristic
69C22×C127C22×C811yescharacteristic
70C22×C127C2×M1611yescharacteristic
71C4×C127—22no—
72C4×C127—22no—
73C4×C127—22no—
74C4×C127—41no—
75C4×C127—41no—
76C4×C127C22⋊C812yes—
77C4×C127C22⋊C812yes—
78C2×C127—21no—
79C2×C127—21no—
80C2×C127—42no—
81C2×C127C2×C22⋊C811yescharacteristic
82C2×C127#41011yescharacteristic
83C2×C127#41311yescharacteristic
84#1458—1271nomaximal, Sylow
85C127#145811yescharacteristic, complemented, Sylow
86C2×#117D12711yescharacteristic
87#431—1272no—
88#324—1272no—
89#323—1272no—
90C23×C4Dic12711yescharacteristic
91#117—22no—
92#117—22no—
93C2×C4⋊C4Dic12711yescharacteristic
94C2×C4⋊C4C2×D12711yescharacteristic
95C2×C22⋊C4Dic12711yescharacteristic
96C2×C22⋊C4C2×D12711yescharacteristic
97C2×C42Dic12711yescharacteristic
98C2×C42C2×D12711yescharacteristic
99C4⋊C8—2542no—
100C22⋊C8—1274no—
101C4×C8—2542no—
102C24C2×Dic12711yescharacteristic
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4C127⋊C814yescomplemented
107C22×C4C2×Dic12711yescharacteristic
108C22×C4C2×Dic12711yescharacteristic
109C22×C4C2×Dic12711yescharacteristic
110C22×C4C2×Dic12711yescharacteristic
111C22×C4C2×Dic12711yescharacteristic
112C22×C4C22×D12711yescharacteristic
113C2×C8—2544no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C42—21no—
119C42—21no—
120C42C22⋊D12712yes—
121C42C22⋊D12712yes—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23C2×C127⋊C812yes—
127C23C22×Dic12711yescharacteristic
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4C22⋊Dic12712yes—
139C2×C4C22⋊Dic12711yescharacteristic
140C2×C4C22⋊Dic12711yescharacteristic
141C2×C4C2×C127⋊C812yes—
142C2×C4C2×C127⋊C812yes—
143C2×C4#1452323212yes—
144C2×C4C2×C22⋊D12712yes—
145C8—5084no—
146C22—21no—
147C22—21no—
148C22—22no—
149C22—42no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22C2×C22⋊Dic12711yescharacteristic
154C22C22×C127⋊C811yescharacteristic
155C22C2×#1452323211yescharacteristic
156C4—22no—
157C4—22no—
158C4—22no—
159C4—41no—
160C4—41no—
161C4C22⋊(C127⋊C8)12yes—
162C4C22⋊(C127⋊C8)12yes—
163C2—21no—
164C2—21no—
165C2—42no—
166C2C2×C22⋊(C127⋊C8)11yescharacteristic
167C2#2254985011yescharacteristic
168C2#2254986011yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC22×C127C22×C8
FrattiniC22×C4C22×D127
FittingC2×#117×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×C127C2×C4
cosocleC22×C4×C127C23
hypercenterC2×#117D127
hypocenterC127#1458
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #22549929C2semidirect
2. #22549602C2semidirect
22. C22×C4×C127C8semidirect
85. C127#1458semidirect
106. C22×C4C127⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225499292
#225496022
#225496002
C2×#117×C1272
#1458127