Finite Group Databasebeta

← #30638923

Subgroups — #30638923

Order 16256.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16861140C212yesmaximal, complemented
2C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
3#22549736C211yesmaximal, characteristic
4#22549638C212yesmaximal, complemented
5C22×C4⋊C4×C127C211yesmaximal, characteristic
6C22×C127⋊C8C2212yes—
7C2×C4⋊Dic127C2211yescharacteristic
8C2×C4⋊Dic127C2211yescharacteristic
9C2×C22⋊Dic127—21no—
10#16861140—24no—
11C22⋊(C127⋊C8)C414yescomplemented
12C127⋊#95—21no—
13C23×C4×C127—21no—
14C23×C4×C127C2211yescharacteristic
15C2×C4⋊C4×C127—21no—
16C2×C4⋊C4×C127—24no—
17C2×C4⋊C4×C127C2212yes—
18C22×Dic127—21no—
19C22×Dic127—21no—
20C2×C127⋊C8—22no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8C2×C414yescomplemented
23C4⋊Dic127—22no—
24C4⋊Dic127—22no—
25C22⋊Dic127—42no—
26C24×C127D411yescharacteristic
27C22×C4×C127—22no—
28C22×C4×C127—22no—
29C22×C4×C127—24no—
30C22×C4×C127—22no—
31C22×C4×C127—42no—
32C22×C4×C127—42no—
33C22×C4×C127C2×C412yes—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127Q812yes—
36C22×C4×C127C2311yescharacteristic
37C4⋊C4×C127—24no—
38C4⋊C4×C127—22no—
39C4⋊C4×C127D414yes—
40C2×Dic127—41no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C127⋊C8—24no—
47C23×C127—21no—
48C23×C127—21no—
49C23×C127—21no—
50C23×C127—21no—
51C23×C127C4⋊C412yes—
52C23×C127SD1612yes—
53C23×C127SD1612yes—
54C23×C127C2×D411yescharacteristic
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—24no—
58C2×C4×C127—24no—
59C2×C4×C127—24no—
60C2×C4×C127—48no—
61C2×C4×C127—42no—
62C2×C4×C127—42no—
63C2×C4×C127C4⋊C412yes—
64C2×C4×C127C22×C411yescharacteristic
65C2×C4×C127C2×D412yes—
66C2×C4×C127C2×Q811yescharacteristic
67C2×C4×C127#4112yes—
68C22×C127—22no—
69C22×C127—22no—
70C22×C127—22no—
71C22×C127—22no—
72C22×C127—22no—
73C22×C127—22no—
74C22×C127#10614yescomplemented
75C22×C127C2×C4⋊C411yescharacteristic
76C22×C127#12011yescharacteristic
77C22×C127#12311yescharacteristic
78C22×C127C2×SD1611yescharacteristic
79C22×C127C2×SD1611yescharacteristic
80C22×C127AGL(1,ℤ/8)11yescharacteristic
81C22×C127#13711yescharacteristic
82C4×C127—22no—
83C4×C127—44no—
84C4×C127C4×D412yes—
85C4×C127C22⋊Q812yes—
86Dic127—81no—
87Dic127—81no—
88C2×C127—24no—
89C2×C127#38511yescharacteristic
90C2×C127C2×#10611yescharacteristic
91C2×C127#42811yescharacteristic
92C2×C127#44811yescharacteristic
93C2×C127#45011yescharacteristic
94C2×C127#48111yescharacteristic
95C2×C127#48311yescharacteristic
96#1853—1271nomaximal, Sylow
97C127#185311yescharacteristic, complemented, Sylow
98C22×C4⋊C4D12711yescharacteristic
99C2×#106—1272no—
100C2×C22⋊C8—1271no—
101#380—1271no—
102#340—1272no—
103C23×C4—21no—
104C23×C4C2×D12711yescharacteristic
105C22×C8—1272no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—1271no—
109C2×C4⋊C4—1271no—
110C2×C4⋊C4C2×D12712yes—
111C2×C22⋊C4—2541no—
112#106—2544no—
113C22⋊C8—1274no—
114#95—2541no—
115C24C22⋊D12711yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—2541no—
123C22×C4—2541no—
124C22×C4C4×D12712yes—
125C22×C4C22⋊D12711yescharacteristic
126C22×C4Dic25412yes—
127C22×C4C22×D12711yescharacteristic
128C2×C8—1274no—
129C2×C8—2542no—
130C2×C8—2542no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—2542no—
134C4⋊C4—2542no—
135C4⋊C4C22⋊D12714yes—
136C22⋊C4—5082no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23Dic127⋊C412yes—
142C23#1452323512yes—
143C23#1452323612yes—
144C23C2×C22⋊D12711yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—48no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—5081no—
154C2×C4—5081no—
155C2×C4—5081no—
156C2×C4—5081no—
157C2×C4—5081no—
158C2×C4—5081no—
159C2×C4Dic127⋊C412yes—
160C2×C4C2×C4×D12711yescharacteristic
161C2×C4C2×C22⋊D12712yes—
162C2×C4C2×Dic25411yescharacteristic
163C2×C4#1452324812yes—
164C8—2544no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#1686114014yescomplemented
172C22C2×Dic127⋊C411yescharacteristic
173C22#1686117811yescharacteristic
174C22#1686119211yescharacteristic
175C22C2×#1452323511yescharacteristic
176C22C2×#1452323611yescharacteristic
177C22#1686123511yescharacteristic
178C22#1686124011yescharacteristic
179C4—22no—
180C4—44no—
181C4—10161no—
182C4—10161no—
183C4C4×C22⋊D12712yes—
184C4#1686118812yes—
185C2—24no—
186C2#2254974911yescharacteristic
187C2C2×#1686114011yescharacteristic
188C2#2254991611yescharacteristic
189C2#2255001511yescharacteristic
190C2#2255002711yescharacteristic
191C2#2255019411yescharacteristic
192C2#2255020611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC22×C4⋊C4×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
cosocleC22×C4×C127C23
hypercenterC22×C4⋊C4D127
hypocenterC127#1853
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16861140C2semidirect
4. #22549638C2semidirect
11. C22⋊(C127⋊C8)C4semidirect
22. C2×C127⋊C8C2×C4semidirect
74. C22×C127#106semidirect
97. C127#1853semidirect
171. C22#16861140semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
C2×C22⋊(C127⋊C8)2
#225497362
#225496382
C22×C4⋊C4×C1272
#1853127