Finite Group Databasebeta

← #30638555

Subgroups — #30638555

Order 16256.

388 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861130C212yesmaximal
2C2×#16861120C211yesmaximal, characteristic, complemented
3C4×C4⋊Dic127C211yesmaximal, characteristic, complemented
4#22549638C212yesmaximal, complemented
5C2×C4×D4×C127C211yesmaximal, characteristic
6C22×C127⋊C8C2212yes—
7C2×C4⋊Dic127C2212yes—
8C2×C4×Dic127—21no—
9#16861130C418yescomplemented
10#16861120—22no—
11C127⋊#95—21no—
12C22×D4×C127C2211yescharacteristic
13C23×C4×C127—21no—
14C4×D4×C127—22no—
15C4×D4×C127C414yescomplemented
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C22⋊C4×C127—21no—
18C2×C42×C127C2211yescharacteristic
19C22×Dic127—22no—
20C2×C127⋊C8—22no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8C2×C414yescomplemented
23C4⋊Dic127—22no—
24C4⋊Dic127C2×C414yescomplemented
25C4×Dic127—42no—
26C24×C127—21no—
27C2×D4×C127—21no—
28C2×D4×C127—22no—
29C2×D4×C127C2×C414yes—
30C2×D4×C127C2×C412yes—
31C22×C4×C127—21no—
32C22×C4×C127—24no—
33C22×C4×C127—44no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C4⋊C4×C127—21no—
38C4⋊C4×C127C2×C412yes—
39C22⋊C4×C127—24no—
40C42×C127C2×C412yes—
41C42×C127D412yes—
42C2×Dic127—24no—
43C2×Dic127—42no—
44C2×Dic127—42no—
45C127⋊C8—24no—
46C23×C127—22no—
47C23×C127—24no—
48C23×C127—42no—
49C23×C127—41no—
50C23×C127—41no—
51C23×C127C2×D411yescharacteristic
52D4×C127—22no—
53D4×C127—24no—
54D4×C127C4214yescomplemented
55C2×C4×C127—22no—
56C2×C4×C127—21no—
57C2×C4×C127—21no—
58C2×C4×C127—22no—
59C2×C4×C127—48no—
60C2×C4×C127—42no—
61C2×C4×C127C22⋊C412yes—
62C2×C4×C127C22⋊C412yes—
63C2×C4×C127C22×C412yes—
64C2×C4×C127C22×C411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127#4112yes—
67C22×C127—24no—
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—44no—
71C22×C127—44no—
72C22×C127C2×C22⋊C411yescharacteristic
73C22×C127C4×D412yes—
74C22×C127AGL(1,ℤ/8)11yescharacteristic
75C22×C127AGL(1,ℤ/8)12yes—
76C22×C127#13711yescharacteristic
77C4×C127—22no—
78C4×C127—22no—
79C4×C127—42no—
80C4×C127C2×C4211yescharacteristic
81C4×C127#11711yescharacteristic
82C4×C127C4×D412yes—
83Dic127—44no—
84C2×C127—44no—
85C2×C127C4×C22⋊C411yescharacteristic
86C2×C127#41711yescharacteristic
87C2×C127#41811yescharacteristic
88C2×C127#44012yes—
89C2×C127#44212yes—
90#1762—1271nomaximal, Sylow
91C127#176211yescharacteristic, complemented, Sylow
92C2×C4×D4D12711yescharacteristic
93C2×#102—1272no—
94C2×#97—1271no—
95C4×C4⋊C4—1271no—
96#340—1272no—
97C22×D4C2×D12711yescharacteristic
98C23×C4—21no—
99C22×C8—1272no—
100C4×D4—22no—
101C4×D4Dic12714yescomplemented
102C2×C4⋊C4—1272no—
103C2×C4⋊C4C2×D12711yescharacteristic
104C2×C22⋊C4—21no—
105C2×C42—2541no—
106C2×C42C2×D12711yescharacteristic
107#102—1278no—
108#97—2542no—
109#95—2541no—
110C24—21no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4C4×D12714yes—
114C2×D4C2×Dic12712yes—
115C22×C4—21no—
116C22×C4—24no—
117C22×C4—44no—
118C22×C4—2542no—
119C22×C4C22⋊D12711yescharacteristic
120C22×C4C22⋊D12711yescharacteristic
121C22×C4C22×D12711yescharacteristic
122C2×C8—1274no—
123C2×C8—2542no—
124C2×C8—2542no—
125C4⋊C4—21no—
126C4⋊C4—1274no—
127C4⋊C4—2542no—
128C4⋊C4C2×Dic12712yes—
129C22⋊C4—24no—
130C42—5082no—
131C42C2×Dic12712yes—
132C42C22⋊D12712yes—
133C23—22no—
134C23—24no—
135C23—41no—
136C23—42no—
137C23—41no—
138C23C2×C22⋊D12711yescharacteristic
139D4—22no—
140D4—24no—
141D4C4×Dic12714yescomplemented
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—48no—
147C2×C4—42no—
148C2×C4—5082no—
149C2×C4—5082no—
150C2×C4—2544no—
151C2×C4C22⋊Dic12712yes—
152C2×C4C22⋊Dic12712yes—
153C2×C4C2×C4×D12712yes—
154C2×C4C22×Dic12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C2×C4#1452324812yes—
157C8—2544no—
158C22—24no—
159C22—42no—
160C22—42no—
161C22—44no—
162C22—44no—
163C22C2×C22⋊Dic12711yescharacteristic
164C22C4×C22⋊D12712yes—
165C22#1686123411yescharacteristic
166C22#1686123512yes—
167C22#1686123911yescharacteristic
168C4—22no—
169C4—22no—
170C4—42no—
171C4—5084no—
172C4C2×C4×Dic12711yescharacteristic
173C4#1686116811yescharacteristic
174C4C4×C22⋊D12712yes—
175C2—44no—
176C2C4×C22⋊Dic12711yescharacteristic
177C2#2254987711yescharacteristic
178C2#2254988111yescharacteristic
179C2#2254997612yes—
180C2#2254998612yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC4×C127C2×C42
FrattiniC22×C4C22×D127
FittingC2×C4×D4×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×C4×D4D127
hypocenterC127#1762
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#16861120C2semidirect
3. C4×C4⋊Dic127C2semidirect
4. #22549638C2semidirect
9. #16861130C4semidirect
15. C4×D4×C127C4semidirect
22. C2×C127⋊C8C2×C4semidirect
24. C4⋊Dic127C2×C4semidirect
54. D4×C127C42semidirect
91. C127#1762semidirect
101. C4×D4Dic127semidirect
141. D4C4×Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611302
C2×#168611202
C4×C4⋊Dic1272
#225496382
C2×C4×D4×C1272
#1762127