Finite Group Databasebeta

← #30639663

Subgroups — #30639663

Order 16256.

344 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×C4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
2C2×#16861133C212yesmaximal, complemented
3C2×#16861130C212yesmaximal
4C4×C4⋊Dic127C211yesmaximal, characteristic, complemented
5C2×#124×C127C211yesmaximal, characteristic
6C22×C127⋊C8C2212yes—
7C2×C4⋊Dic127C2212yes—
8C2×C4×Dic127—21no—
9C4⋊(C127⋊C8)—22no—
10#16861133—24no—
11#16861130—24no—
12C127⋊#95—21no—
13C22×Q8×C127C2211yescharacteristic
14C22×D4×C127C2211yescharacteristic
15#124×C127—22no—
16#124×C127C414yescomplemented
17C2×C22⋊C4×C127—21no—
18C2×C22⋊C4×C127—21no—
19C2×C42×C127C2211yescharacteristic
20C22×Dic127—22no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23C2×C127⋊C8—22no—
24C4⋊Dic127—22no—
25C4⋊Dic127D414yes—
26C4×Dic127—42no—
27C24×C127—21no—
28C2×Q8×C127—21no—
29C2×Q8×C127—22no—
30C2×Q8×C127—22no—
31C2×Q8×C127C2×C412yes—
32C2×D4×C127—22no—
33C2×D4×C127—21no—
34C2×D4×C127—22no—
35C2×D4×C127C2×C412yes—
36C22×C4×C127—21no—
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127C2311yescharacteristic
40C22⋊C4×C127—24no—
41C22⋊C4×C127—24no—
42C42×C127C2×C412yes—
43C42×C127D412yes—
44C2×Dic127—24no—
45C2×Dic127—42no—
46C2×Dic127—42no—
47C127⋊C8—42no—
48C23×C127—22no—
49C23×C127—42no—
50C23×C127—42no—
51C23×C127—41no—
52C23×C127—41no—
53C23×C127C2×D411yescharacteristic
54Q8×C127—24no—
55Q8×C127—41no—
56Q8×C127—41no—
57D4×C127—24no—
58D4×C127—41no—
59D4×C127—41no—
60C2×C4×C127—22no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—22no—
64C2×C4×C127—42no—
65C2×C4×C127C22⋊C412yes—
66C2×C4×C127C22⋊C412yes—
67C2×C4×C127C22×C411yescharacteristic
68C2×C4×C127C2×D412yes—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127#4112yes—
71C22×C127—42no—
72C22×C127—42no—
73C22×C127—42no—
74C22×C127—82no—
75C22×C127—82no—
76C22×C127C2×C22⋊C411yescharacteristic
77C22×C127#12112yes—
78C22×C127#13512yes—
79C22×C127AGL(1,ℤ/8)11yescharacteristic
80C22×C127#13711yescharacteristic
81C4×C127—22no—
82C4×C127—22no—
83C4×C127—42no—
84C4×C127C4×D412yes—
85C4×C127#12411yescharacteristic
86C4×C127#12711yescharacteristic
87Dic127—44no—
88C2×C127—82no—
89C2×C127#39011yescharacteristic
90C2×C127#41611yescharacteristic
91C2×C127#41711yescharacteristic
92C2×C127#46312yes—
93C2×C127#46412yes—
94#1983—1271nomaximal, Sylow
95C127#198311yescharacteristic, complemented, Sylow
96C2×#124D12711yescharacteristic
97C2×C4⋊C8—1271no—
98C2×#103—1272no—
99C2×#102—1272no—
100C4×C4⋊C4—1271no—
101C22×Q8C2×D12711yescharacteristic
102C22×D4C2×D12711yescharacteristic
103C22×C8—1272no—
104#124—22no—
105#124Dic12714yescomplemented
106C2×C4⋊C4—1272no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C42—2541no—
110C2×C42C2×D12711yescharacteristic
111C4⋊C8—2542no—
112#103—2544no—
113#102—2544no—
114#95—2541no—
115C24—21no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8C2×Dic12712yes—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4C2×Dic12712yes—
124C22×C4—21no—
125C22×C4—2542no—
126C22×C4C22⋊D12711yescharacteristic
127C22×C4C22⋊D12711yescharacteristic
128C22×C4C22×D12711yescharacteristic
129C2×C8—2542no—
130C2×C8—2542no—
131C2×C8—2542no—
132C4⋊C4—1274no—
133C4⋊C4—2542no—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C42—5082no—
137C42C2×Dic12712yes—
138C42C22⋊D12712yes—
139C23—22no—
140C23—42no—
141C23—41no—
142C23—41no—
143C23—42no—
144C23C2×C22⋊D12711yescharacteristic
145Q8—24no—
146Q8—41no—
147Q8—41no—
148D4—24no—
149D4—41no—
150D4—41no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—5082no—
157C2×C4—5082no—
158C2×C4—2544no—
159C2×C4C22⋊Dic12712yes—
160C2×C4C22⋊Dic12712yes—
161C2×C4C22×Dic12711yescharacteristic
162C2×C4D4×D12712yes—
163C2×C4C2×C22⋊D12711yescharacteristic
164C2×C4#1452324712yes—
165C8—5082no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—82no—
170C22—82no—
171C22C2×C22⋊Dic12711yescharacteristic
172C22#1686118012yes—
173C22#1686122912yes—
174C22#1686123411yescharacteristic
175C22#1686123911yescharacteristic
176C4—22no—
177C4—22no—
178C4—42no—
179C4—5084no—
180C4D4×Dic12712yes—
181C4#1686119611yescharacteristic
182C4#1686120511yescharacteristic
183C2—82no—
184C2#2254977911yescharacteristic
185C2#2254987211yescharacteristic
186C2#2254987711yescharacteristic
187C2#2255008812yes—
188C2#2255009512yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#124×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#124D127
hypocenterC127#1983
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C127⋊C8)C2semidirect
2. C2×#16861133C2semidirect
4. C4×C4⋊Dic127C2semidirect
16. #124×C127C4semidirect
95. C127#1983semidirect
105. #124Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C127⋊C8)2
C2×#168611332
C2×#168611302
C4×C4⋊Dic1272
C2×#124×C1272
#1983127