Finite Group Databasebeta

← C2×#22549675 #30640411

Subgroups — C2×#22549675 #30640411

Order 16256.

240 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 77 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×#16861128C212yesmaximal
2#22549675C214yesmaximal, direct product factor
3C2×C4⋊Q8×C127C211yesmaximal, characteristic
4C2×#14523232—22no—
5#16861128—22no—
6#16861128C2214yes—
7C22×Q8×C127C411yescharacteristic
8C22×Q8×C127C2211yescharacteristic
9C4⋊Q8×C127—22no—
10C4⋊Q8×C127C412yes—
11C4⋊Q8×C127C2212yes—
12C2×C4⋊C4×C127—22no—
13C2×C42×C127C411yescharacteristic
14#14523232—24no—
15#14523232—42no—
16C2×C127⋊C8—42no—
17C2×Q8×C127—21no—
18C2×Q8×C127—41no—
19C2×Q8×C127—41no—
20C2×Q8×C127—41no—
21C2×Q8×C127—41no—
22C2×Q8×C127C2×C412yes—
23C2×Q8×C127C2×C411yescharacteristic
24C2×Q8×C127D412yes—
25C2×Q8×C127C2311yescharacteristic
26C22×C4×C127—21no—
27C22×C4×C127—21no—
28C22×C4×C127C2×C411yescharacteristic
29C22×C4×C127D412yes—
30C4⋊C4×C127—24no—
31C4⋊C4×C127—24no—
32C42×C127—21no—
33C42×C127C2×C412yes—
34C127⋊C8—44no—
35C23×C127C22⋊C411yescharacteristic
36Q8×C127—41no—
37Q8×C127—42no—
38Q8×C127—41no—
39Q8×C127—42no—
40Q8×C127—42no—
41C2×C4×C127—24no—
42C2×C4×C127—21no—
43C2×C4×C127—21no—
44C2×C4×C127—22no—
45C2×C4×C127—22no—
46C2×C4×C127—22no—
47C2×C4×C127—22no—
48C2×C4×C127—41no—
49C2×C4×C127—41no—
50C2×C4×C127C22⋊C411yescharacteristic
51C2×C4×C127C22⋊C412yes—
52C2×C4×C127C22×C411yescharacteristic
53C2×C4×C127C2×D412yes—
54C22×C127—22no—
55C22×C127#9911yescharacteristic
56C22×C127#9911yescharacteristic
57C22×C127C2×C22⋊C411yescharacteristic
58C4×C127—24no—
59C4×C127—22no—
60C4×C127—21no—
61C4×C127—21no—
62C4×C127—22no—
63C4×C127—42no—
64C2×C127—21no—
65C2×C127—21no—
66C2×C127#35612yescomplemented
67C2×C127C2×#9911yescharacteristic
68C2×#356—1271nomaximal, Sylow
69C127C2×#35611yescharacteristic, complemented, Sylow
70C2×C4⋊Q8D12711yescharacteristic
71C2×#101—1272no—
72#356—1274no—
73C22×Q8Dic12711yescharacteristic
74C22×Q8C2×D12711yescharacteristic
75C2×M16—2542no—
76C4⋊Q8—22no—
77C4⋊Q8Dic12712yes—
78C4⋊Q8C2×D12712yes—
79C2×C4⋊C4—22no—
80C2×C42Dic12711yescharacteristic
81#101—1274no—
82#101—2542no—
83C2×Q8—21no—
84C2×Q8—41no—
85C2×Q8—41no—
86C2×Q8—41no—
87C2×Q8—41no—
88C2×Q8C2×Dic12712yes—
89C2×Q8C2×Dic12711yescharacteristic
90C2×Q8C22⋊D12712yes—
91C2×Q8C22×D12711yescharacteristic
92C22×C4—21no—
93C22×C4—21no—
94C22×C4C2×Dic12711yescharacteristic
95C22×C4C22⋊D12712yes—
96M16—5082no—
97M16—2544no—
98C2×C8—5082no—
99C4⋊C4—24no—
100C4⋊C4—24no—
101C42—21no—
102C42C2×Dic12712yes—
103C23C22⋊Dic12711yescharacteristic
104Q8—41no—
105Q8—42no—
106Q8—41no—
107Q8—42no—
108Q8—42no—
109C2×C4—24no—
110C2×C4—22no—
111C2×C4—21no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4C22⋊Dic12711yescharacteristic
119C2×C4C22⋊Dic12712yes—
120C2×C4C22×Dic12711yescharacteristic
121C2×C4C2×C22⋊D12712yes—
122C8—5084no—
123C22—22no—
124C22#1686112411yescharacteristic
125C22#1686112411yescharacteristic
126C22C2×C22⋊Dic12711yescharacteristic
127C4—24no—
128C4—22no—
129C4—21no—
130C4—22no—
131C4—21no—
132C4—42no—
133C2—21no—
134C2—21no—
135C2#2254967512yesdirect product factor
136C2C2×#1686112411yescharacteristic
subgroupG/H
centerC22#16861124
commutatorC2×C4×C127C22×C4
FrattiniC2×Q8C22×D127
FittingC2×C4⋊Q8×C127C2
socleC22×C127#99
prime radicalC23×C127C22⋊C4
cosocleC2×Q8×C127C23
hypercenterC2×C4⋊Q8D127
hypocenterC127C2×#356
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22549675C2direct (N × H)
66. C2×C127#356semidirect
69. C127C2×#356semidirect
135. C2#22549675direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611282
#225496752
C2×C4⋊Q8×C1272
C2×#356127