Finite Group Databasebeta

← #30638661

Subgroups — #30638661

Order 16256.

848 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16861130C214yesmaximal, complemented
2C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
3#22549736C211yesmaximal, characteristic, complemented
4C23×D4×C127C211yesmaximal, characteristic
5C22×C127⋊C8C2212yescomplemented
6C2×C4⋊Dic127C2212yescomplemented
7C2×C22⋊Dic127—21no—
8#16861130—28no—
9C22⋊(C127⋊C8)—22no—
10C127⋊#95—21no—
11C25×C127—21no—
12C22×D4×C127—28no—
13C22×D4×C127—21no—
14C22×D4×C127—22no—
15C22×D4×C127C414yescomplemented
16C22×D4×C127C2212yes—
17C23×C4×C127C2211yescharacteristic
18C22×Dic127—22no—
19C2×C127⋊C8—22no—
20C2×C127⋊C8—24no—
21C4⋊Dic127—24no—
22C22⋊Dic127—42no—
23C24×C127—28no—
24C24×C127—24no—
25C24×C127—22no—
26C24×C127—44no—
27C24×C127—42no—
28C24×C127—42no—
29C24×C127D411yescharacteristic
30C2×D4×C127—28no—
31C2×D4×C127—216no—
32C2×D4×C127—216no—
33C2×D4×C127—24no—
34C2×D4×C127—24no—
35C2×D4×C127—22no—
36C2×D4×C127C2×C414yes—
37C2×D4×C127D418yes—
38C22×C4×C127—24no—
39C22×C4×C127C2×C412yes—
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D412yes—
42C22×C4×C127C2311yescharacteristic
43C2×Dic127—42no—
44C2×Dic127—42no—
45C2×Dic127—42no—
46C127⋊C8—42no—
47C23×C127—22no—
48C23×C127—22no—
49C23×C127—216no—
50C23×C127—24no—
51C23×C127—28no—
52C23×C127—48no—
53C23×C127—416no—
54C23×C127—416no—
55C23×C127—48no—
56C23×C127—42no—
57C23×C127—42no—
58C23×C127—44no—
59C23×C127C22⋊C412yes—
60C23×C127D812yes—
61C23×C127SD1612yes—
62C23×C127C2×D411yescharacteristic
63D4×C127—216no—
64D4×C127—24no—
65D4×C127—28no—
66D4×C127C22⋊C418yescomplemented
67C2×C4×C127—22no—
68C2×C4×C127—24no—
69C2×C4×C127—24no—
70C2×C4×C127C22⋊C412yes—
71C2×C4×C127C22×C411yescharacteristic
72C2×C4×C127C2×D414yes—
73C2×C4×C127C2×D411yescharacteristic
74C22×C127—22no—
75C22×C127—22no—
76C22×C127—24no—
77C22×C127—24no—
78C22×C127—28no—
79C22×C127—432no—
80C22×C127—44no—
81C22×C127—44no—
82C22×C127—48no—
83C22×C127—48no—
84C22×C127#10214yescomplemented
85C22×C127C2×C22⋊C411yescharacteristic
86C22×C127#12012yes—
87C22×C127C2×D811yescharacteristic
88C22×C127C2×SD1611yescharacteristic
89C22×C127AGL(1,ℤ/8)12yes—
90C4×C127—22no—
91C4×C127C2×C22⋊C412yes—
92C4×C127#12012yes—
93Dic127—82no—
94C2×C127—24no—
95C2×C127—48no—
96C2×C127#37911yescharacteristic
97C2×C127C2×#10211yescharacteristic
98C2×C127#41811yescharacteristic
99C2×C127#44712yes—
100C2×C127#45012yes—
101#1786—1271nomaximal, Sylow
102C127#178611yescharacteristic, complemented, Sylow
103C23×D4D12711yescharacteristic
104C2×#102—1274no—
105C2×C22⋊C8—1271no—
106#380—1271no—
107C25—21no—
108C22×D4—28no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4Dic12714yescomplemented
112C22×D4C2×D12712yes—
113C23×C4C2×D12711yescharacteristic
114C22×C8—1272no—
115C2×C4⋊C4—1272no—
116C2×C22⋊C4—2541no—
117#102—2548no—
118C22⋊C8—2542no—
119#95—2541no—
120C24—28no—
121C24—24no—
122C24—22no—
123C24—44no—
124C24—42no—
125C24—42no—
126C24C22⋊D12711yescharacteristic
127C2×D4—28no—
128C2×D4—216no—
129C2×D4—216no—
130C2×D4—24no—
131C2×D4—24no—
132C2×D4—22no—
133C2×D4C2×Dic12714yes—
134C2×D4C22⋊D12718yes—
135C22×C4—24no—
136C22×C4—2542no—
137C22×C4C2×Dic12712yes—
138C22×C4C22⋊D12711yescharacteristic
139C22×C4C22⋊D12712yes—
140C22×C4C22×D12711yescharacteristic
141C2×C8—2542no—
142C2×C8—2544no—
143C4⋊C4—2544no—
144C22⋊C4—5082no—
145C23—22no—
146C23—22no—
147C23—216no—
148C23—24no—
149C23—28no—
150C23—416no—
151C23—48no—
152C23—416no—
153C23—42no—
154C23—42no—
155C23—44no—
156C23—48no—
157C23C22⋊Dic12712yes—
158C23#1452323312yes—
159C23#1452323612yes—
160C23C2×C22⋊D12711yescharacteristic
161D4—216no—
162D4—24no—
163D4—28no—
164D4C22⋊Dic12718yescomplemented
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—5082no—
169C2×C4—5082no—
170C2×C4—5082no—
171C2×C4C22⋊Dic12712yes—
172C2×C4C22×Dic12711yescharacteristic
173C2×C4C2×C22⋊D12714yes—
174C2×C4C2×C22⋊D12711yescharacteristic
175C8—5082no—
176C22—24no—
177C22—22no—
178C22—22no—
179C22—24no—
180C22—28no—
181C22—432no—
182C22—48no—
183C22—44no—
184C22—44no—
185C22—48no—
186C22#1686113014yescomplemented
187C22C2×C22⋊Dic12711yescharacteristic
188C22#1686117812yes—
189C22C2×#1452323311yescharacteristic
190C22C2×#1452323611yescharacteristic
191C22#1686123512yes—
192C4—22no—
193C4—10162no—
194C4C2×C22⋊Dic12712yes—
195C4#1686117812yes—
196C2—24no—
197C2—48no—
198C2#2254973211yescharacteristic
199C2C2×#1686113011yescharacteristic
200C2#2254988111yescharacteristic
201C2#2255001012yes—
202C2#2255002712yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC23×D4×C127C2
socleC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC23×D4D127
hypocenterC127#1786
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16861130C2semidirect
2. C2×C22⋊(C127⋊C8)C2semidirect
3. #22549736C2semidirect
5. C22×C127⋊C8C22semidirect
6. C2×C4⋊Dic127C22semidirect
15. C22×D4×C127C4semidirect
66. D4×C127C22⋊C4semidirect
84. C22×C127#102semidirect
102. C127#1786semidirect
111. C22×D4Dic127semidirect
164. D4C22⋊Dic127semidirect
186. C22#16861130semidirect

Maximal subgroups

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

subgroupindex
C2×#168611302
C2×C22⋊(C127⋊C8)2
#225497362
C23×D4×C1272
#1786127