Finite Group Databasebeta

← #30637082

Subgroups — #30637082

Order 16256.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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#22549933C211yesmaximal, characteristic, complemented
2#22549929C211yesmaximal, characteristic, complemented
3#22549600C214yesmaximal, complemented
4C2×#124×C127C211yesmaximal, characteristic
5C4⋊(C127⋊C8)—21no—
6C4⋊(C127⋊C8)—21no—
7C22⋊(C127⋊C8)C2212yes—
8C22⋊(C127⋊C8)C2212yes—
9#16861120—21no—
10C4×C127⋊C8—21no—
11C22×Q8×C127C411yescharacteristic
12C22×D4×C127C411yescharacteristic
13#124×C127—21no—
14#124×C127—21no—
15#124×C127—21no—
16#124×C127—21no—
17C2×C22⋊C4×C127C412yes—
18C2×C22⋊C4×C127C2212yes—
19C2×C42×C127C2211yescharacteristic
20C2×C127⋊C8—22no—
21C2×C127⋊C8—22no—
22C24×C127C2×C412yes—
23C2×Q8×C127—21no—
24C2×Q8×C127—21no—
25C2×Q8×C127—42no—
26C2×D4×C127—21no—
27C2×D4×C127—21no—
28C2×D4×C127—42no—
29C22×C4×C127C2×C411yescharacteristic
30C22×C4×C127C2×C411yescharacteristic
31C22×C4×C127C2×C412yes—
32C22×C4×C127C2311yescharacteristic
33C22⋊C4×C127—22no—
34C22⋊C4×C127—22no—
35C22⋊C4×C127—22no—
36C22⋊C4×C127—22no—
37C42×C127D411yescharacteristic
38C42×C127D411yescharacteristic
39C42×C127D411yescharacteristic
40C42×C127D411yescharacteristic
41C127⋊C8—42no—
42C127⋊C8—42no—
43C23×C127—22no—
44C23×C127—24no—
45C23×C127—42no—
46C23×C127—42no—
47C23×C127C22×C411yescharacteristic
48Q8×C127—42no—
49Q8×C127—42no—
50D4×C127—42no—
51D4×C127—42no—
52C2×C4×C127—22no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—41no—
56C2×C4×C127—42no—
57C2×C4×C127C22⋊C411yescharacteristic
58C2×C4×C127C22⋊C411yescharacteristic
59C2×C4×C127C22⋊C411yescharacteristic
60C2×C4×C127C22⋊C411yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C22×C127—21no—
64C22×C127—21no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127—42no—
68C22×C127—84no—
69C22×C127C2×C22⋊C411yescharacteristic
70C22×C127#13112yes—
71C4×C127—21no—
72C4×C127—21no—
73C4×C127—21no—
74C4×C127—21no—
75C4×C127—21no—
76C4×C127—42no—
77C4×C127#10012yes—
78C2×C127—22no—
79C2×C127—82no—
80C2×C127#40811yescharacteristic
81C2×C127#41011yescharacteristic
82C2×C127C2×#10011yescharacteristic
83#1469—1271nomaximal, Sylow
84C127#146911yescharacteristic, complemented, Sylow
85C2×#124D12711yescharacteristic
86#432—1271no—
87#431—1271no—
88#323—1274no—
89C22×Q8Dic12711yescharacteristic
90C22×D4Dic12711yescharacteristic
91#124—21no—
92#124—21no—
93#124—21no—
94#124—21no—
95C2×C22⋊C4Dic12712yes—
96C2×C22⋊C4C2×D12712yes—
97C2×C42C2×D12711yescharacteristic
98C4⋊C8—2541no—
99C4⋊C8—2541no—
100C22⋊C8—1272no—
101C22⋊C8—1272no—
102#97—2541no—
103C4×C8—2541no—
104C24C2×Dic12712yes—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—42no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—42no—
111C22×C4C2×Dic12711yescharacteristic
112C22×C4C2×Dic12711yescharacteristic
113C22×C4C2×Dic12712yes—
114C22×C4C22×D12711yescharacteristic
115C2×C8—2542no—
116C2×C8—2542no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C42C22⋊D12711yescharacteristic
122C42C22⋊D12711yescharacteristic
123C42C22⋊D12711yescharacteristic
124C42C22⋊D12711yescharacteristic
125C23—22no—
126C23—24no—
127C23—42no—
128C23—42no—
129C23C22×Dic12711yescharacteristic
130Q8—42no—
131Q8—42no—
132D4—42no—
133D4—42no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4C22⋊Dic12711yescharacteristic
140C2×C4C22⋊Dic12711yescharacteristic
141C2×C4C22⋊Dic12711yescharacteristic
142C2×C4C22⋊Dic12711yescharacteristic
143C2×C4C2×C22⋊D12711yescharacteristic
144C2×C4C2×C22⋊D12711yescharacteristic
145C8—5082no—
146C8—5082no—
147C22—21no—
148C22—21no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22—84no—
153C22C2×C22⋊Dic12711yescharacteristic
154C22#1686121612yes—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—21no—
160C4—42no—
161C4#1686112612yes—
162C2—22no—
163C2—82no—
164C2#2254984311yescharacteristic
165C2#2254985011yescharacteristic
166C2C2×#1686112611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#124×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C4
cosocleC22×C4×C127C23
hypercenterC2×#124D127
hypocenterC127#1469
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. #22549933C2semidirect
2. #22549929C2semidirect
3. #22549600C2semidirect
84. C127#1469semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225499292
#225496002
C2×#124×C1272
#1469127