Finite Group Databasebeta

← #30638763

Subgroups — #30638763

Order 16256.

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

normal subgroups only

This group has 316 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×#16861143C211yesmaximal, characteristic, complemented
2C2×#16861141C211yesmaximal, characteristic, complemented
3#22549736C212yesmaximal
4#22549639C212yesmaximal, complemented
5C22×M16×C127C211yesmaximal, characteristic
6C2×C4⋊Dic127C2212yes—
7C2×C4⋊Dic127C2212yes—
8C2×C22⋊Dic127—22no—
9#16861143—22no—
10#16861141—22no—
11C127⋊#95—22no—
12C23×C4×C127C2211yescharacteristic
13C2×M16×C127—22no—
14C2×M16×C127—22no—
15C2×M16×C127C414yescomplemented
16C22×C8×C127C2211yescharacteristic
17C22×C8×C127C2211yescharacteristic
18C22×Dic127—22no—
19C22×Dic127—22no—
20C4⋊Dic127—22no—
21C4⋊Dic127—22no—
22C4⋊Dic127—22no—
23C4⋊Dic127—22no—
24C22⋊Dic127—44no—
25C24×C127D411yescharacteristic
26C22×C4×C127—21no—
27C22×C4×C127—21no—
28C22×C4×C127—21no—
29C22×C4×C127—21no—
30C22×C4×C127C2×C412yes—
31C22×C4×C127D411yescharacteristic
32C22×C4×C127Q812yes—
33C22×C4×C127C2311yescharacteristic
34M16×C127—28no—
35C2×C8×C127—21no—
36C2×C8×C127—21no—
37C2×C8×C127C2×C412yes—
38C2×C8×C127C2×C412yes—
39C2×C8×C127D412yes—
40C2×C8×C127D412yes—
41C2×Dic127—42no—
42C2×Dic127—42no—
43C2×Dic127—42no—
44C2×Dic127—42no—
45C2×Dic127—42no—
46C2×Dic127—42no—
47C23×C127—21no—
48C23×C127—21no—
49C23×C127—22no—
50C23×C127—22no—
51C23×C127C4⋊C412yes—
52C23×C127C2×D411yescharacteristic
53C2×C4×C127—21no—
54C2×C4×C127—21no—
55C2×C4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127C4⋊C412yes—
60C2×C4×C127C22×C411yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127C2×Q811yescharacteristic
64C2×C4×C127#4111yescharacteristic
65C2×C4×C127#4111yescharacteristic
66C8×C127—22no—
67C8×C127C22⋊C414yescomplemented
68C22×C127—22no—
69C22×C127—24no—
70C22×C127—42no—
71C22×C127—41no—
72C22×C127—41no—
73C22×C127C2×C4⋊C411yescharacteristic
74C22×C127#12111yescharacteristic
75C22×C127#12111yescharacteristic
76C22×C127AGL(1,ℤ/8)12yes—
77C22×C127#13712yes—
78C4×C127—22no—
79C4×C127C2×C22⋊C411yescharacteristic
80C4×C127#11711yescharacteristic
81C4×C127C22⋊Q811yescharacteristic
82C4×C127C22⋊Q811yescharacteristic
83Dic127—82no—
84Dic127—82no—
85C2×C127—42no—
86C2×C127#38011yescharacteristic
87C2×C127#42812yes—
88C2×C127#46812yes—
89C2×C127#46911yescharacteristic
90C2×C127#47011yescharacteristic
91#1809—1271nomaximal, Sylow
92C127#180911yescharacteristic, complemented, Sylow
93C22×M16D12711yescharacteristic
94C2×#107—1271no—
95C2×#106—1271no—
96#380—1272no—
97#340—1272no—
98C23×C4C2×D12711yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16Dic12714yescomplemented
102C22×C8C2×D12711yescharacteristic
103C22×C8C2×D12711yescharacteristic
104C2×C4⋊C4—1272no—
105C2×C4⋊C4—1272no—
106C2×C22⋊C4—2542no—
107#107—2542no—
108#106—2542no—
109#95—2542no—
110C24D50811yescharacteristic
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—2542no—
116C22×C4—2542no—
117C22×C4C2×Dic12712yes—
118C22×C4D50811yescharacteristic
119C22×C4Dic25412yes—
120C22×C4C22×D12711yescharacteristic
121M16—28no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8C2×Dic12712yes—
125C2×C8C2×Dic12712yes—
126C2×C8C22⋊D12712yes—
127C2×C8C22⋊D12712yes—
128C4⋊C4—2542no—
129C4⋊C4—2542no—
130C4⋊C4—2542no—
131C4⋊C4—2542no—
132C22⋊C4—5084no—
133C23—22no—
134C23—22no—
135C23—21no—
136C23—21no—
137C23C4⋊Dic12712yes—
138C23C2×D50811yescharacteristic
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—5082no—
146C2×C4—5082no—
147C2×C4—5082no—
148C2×C4—5082no—
149C2×C4—5082no—
150C2×C4—5082no—
151C2×C4C4⋊Dic12712yes—
152C2×C4C22×Dic12711yescharacteristic
153C2×C4C2×C22⋊D12711yescharacteristic
154C2×C4C2×C22⋊D12711yescharacteristic
155C2×C4C2×Dic25411yescharacteristic
156C2×C4#1452324811yescharacteristic
157C2×C4#1452324811yescharacteristic
158C8—22no—
159C8C22⋊Dic12714yescomplemented
160C22—24no—
161C22—22no—
162C22—42no—
163C22—41no—
164C22—41no—
165C22C2×C4⋊Dic12711yescharacteristic
166C22#1686118311yescharacteristic
167C22#1686118311yescharacteristic
168C22#1686123712yes—
169C22#1686124212yes—
170C4—22no—
171C4—10162no—
172C4—10162no—
173C4C2×C22⋊Dic12711yescharacteristic
174C4#1686116811yescharacteristic
175C4#1686118811yescharacteristic
176C4#1686118811yescharacteristic
177C2—42no—
178C2#2254973611yescharacteristic
179C2#2254991812yes—
180C2#2255012212yes—
181C2#2255012911yescharacteristic
182C2#2255013411yescharacteristic
subgroupG/H
centerC23C2×D508
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC22×M16×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
cosocleC22×C4×C127C23
hypercenterC22×M16D127
hypocenterC127#1809
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16861143C2semidirect
2. C2×#16861141C2semidirect
4. #22549639C2semidirect
15. C2×M16×C127C4semidirect
67. C8×C127C22⋊C4semidirect
92. C127#1809semidirect
101. C2×M16Dic127semidirect
159. C8C22⋊Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611432
C2×#168611412
#225497362
#225496392
C22×M16×C1272
#1809127