Finite Group Databasebeta

← #30637155

Subgroups — #30637155

Order 16256.

312 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 105 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
1#22549929C211yesmaximal, characteristic, complemented
2#22549900C211yesmaximal, characteristic, complemented
3D4⋊(C127⋊C8)C214yesmaximal, complemented
4C2×C4×D4×C127C211yesmaximal, characteristic
5C2×#14523232—21no—
6C4⋊(C127⋊C8)—21no—
7C4⋊(C127⋊C8)—21no—
8C4⋊(C127⋊C8)C2212yescomplemented
9C22⋊(C127⋊C8)—21no—
10C4×C127⋊C8C2212yescomplemented
11C22×D4×C127C411yescharacteristic
12C23×C4×C127—21no—
13C4×D4×C127—21no—
14C4×D4×C127—21no—
15C4×D4×C127C412yes—
16C4×D4×C127C2212yes—
17C2×C4⋊C4×C127C411yescharacteristic
18C2×C22⋊C4×C127—21no—
19C2×C42×C127C2211yescharacteristic
20#14523232—42no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23C24×C127—21no—
24C2×D4×C127—24no—
25C2×D4×C127—21no—
26C2×D4×C127C2×C412yes—
27C22×C4×C127—21no—
28C22×C4×C127—22no—
29C22×C4×C127—22no—
30C22×C4×C127—42no—
31C22×C4×C127—42no—
32C22×C4×C127C2×C411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C4⋊C4×C127—21no—
36C4⋊C4×C127C2×C412yes—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C42×C127C2×C411yescharacteristic
40C42×C127D411yescharacteristic
41C42×C127D411yescharacteristic
42C42×C127C2311yescharacteristic
43C127⋊C8—42no—
44C127⋊C8—42no—
45C23×C127—24no—
46C23×C127—22no—
47C23×C127—42no—
48C23×C127—42no—
49C23×C127C22⋊C411yescharacteristic
50D4×C127—24no—
51D4×C127—22no—
52D4×C127M1614yescomplemented
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—41no—
56C2×C4×C127—42no—
57C2×C4×C127—44no—
58C2×C4×C127—42no—
59C2×C4×C127—42no—
60C2×C4×C127C22⋊C411yescharacteristic
61C2×C4×C127C22⋊C411yescharacteristic
62C2×C4×C127C22⋊C411yescharacteristic
63C2×C4×C127D812yes—
64C2×C4×C127SD1612yes—
65C2×C4×C127C22×C411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—24no—
71C22×C127—48no—
72C22×C127—42no—
73C22×C127—42no—
74C22×C127#10212yes—
75C22×C127C2×C22⋊C411yescharacteristic
76C4×C127—21no—
77C4×C127—21no—
78C4×C127—21no—
79C4×C127—42no—
80C4×C127#10212yes—
81C4×C127C2×M1612yes—
82C4×C127C2×D811yescharacteristic
83C4×C127C2×SD1611yescharacteristic
84C2×C127—22no—
85C2×C127—44no—
86C2×C127#40711yescharacteristic
87C2×C127C2×#10211yescharacteristic
88C2×C127#42111yescharacteristic
89#1486—1271nomaximal, Sylow
90C127#148611yescharacteristic, complemented, Sylow
91C2×C4×D4D12711yescharacteristic
92#431—1271no—
93#423—1271no—
94D4⋊C8—1274no—
95C22×D4Dic12711yescharacteristic
96C23×C4—21no—
97C2×M16—2541no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4Dic12712yes—
101C4×D4C2×D12712yes—
102C2×C4⋊C4Dic12711yescharacteristic
103C2×C22⋊C4—21no—
104C2×C42C2×D12711yescharacteristic
105C4⋊C8—1272no—
106C4⋊C8—2541no—
107C4⋊C8—2541no—
108C22⋊C8—2541no—
109C4×C8—1272no—
110C24—21no—
111C2×D4—24no—
112C2×D4—21no—
113C2×D4C2×Dic12712yes—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4C2×Dic12711yescharacteristic
120C22×C4C22⋊D12711yescharacteristic
121C22×C4C22⋊D12711yescharacteristic
122M16—5082no—
123C2×C8—2542no—
124C2×C8—2542no—
125C4⋊C4—21no—
126C4⋊C4C2×Dic12712yes—
127C22⋊C4—22no—
128C22⋊C4—22no—
129C42C2×Dic12711yescharacteristic
130C42C22⋊D12711yescharacteristic
131C42C22⋊D12711yescharacteristic
132C42C22×D12711yescharacteristic
133C23—24no—
134C23—22no—
135C23—42no—
136C23—42no—
137C23C22⋊Dic12711yescharacteristic
138D4—24no—
139D4—22no—
140D4#1452323214yescomplemented
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4C22⋊Dic12711yescharacteristic
149C2×C4C22⋊Dic12711yescharacteristic
150C2×C4C22⋊Dic12711yescharacteristic
151C2×C4#1452323312yes—
152C2×C4#1452323612yes—
153C2×C4C22×Dic12711yescharacteristic
154C2×C4C2×C22⋊D12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C8—5082no—
157C8—5082no—
158C22—21no—
159C22—21no—
160C22—24no—
161C22—48no—
162C22—42no—
163C22—42no—
164C22#1686113012yes—
165C22C2×C22⋊Dic12711yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—42no—
170C4#1686113012yes—
171C4C2×#1452323212yes—
172C4C2×#1452323311yescharacteristic
173C4C2×#1452323611yescharacteristic
174C2—22no—
175C2—44no—
176C2#2254984011yescharacteristic
177C2C2×#1686113011yescharacteristic
178C2#2254989311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC2×C4×D4×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C4
cosocleC42×C127C23
hypercenterC2×C4×D4D127
hypocenterC127#1486
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22549929C2semidirect
2. #22549900C2semidirect
3. D4⋊(C127⋊C8)C2semidirect
8. C4⋊(C127⋊C8)C22semidirect
10. C4×C127⋊C8C22semidirect
52. D4×C127M16semidirect
90. C127#1486semidirect
140. D4#14523232semidirect

Maximal subgroups

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

subgroupindex
#225499292
#225499002
D4⋊(C127⋊C8)2
C2×C4×D4×C1272
#1486127