Finite Group Databasebeta

← C2×#2232 #33639

Subgroups — C2×#2232 #33639

Order 256.

341 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 127 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×#466C211yesmaximal, characteristic
2C22×#107C211yesmaximal, characteristic, complemented
3#2232C218yesmaximal, direct product factor
4C2×#366C212yesmaximal, complemented
5C2×#357C212yesmaximal, complemented
6C2×C22⋊C16C211yesmaximal, characteristic, complemented
7C22×D8C2211yescharacteristic
8C23×C8C2211yescharacteristic
9C2×#121—21no—
10C22×C4⋊C4—21no—
11C22×C16C2212yes—
12#466—22no—
13#466C2214yes—
14C2×#107—22no—
15C2×#107—21no—
16C2×#107C2212yes—
17C2×#107C2211yescharacteristic
18C2×#107C2214yescomplemented
19C2×#102—21no—
20#366C2218yescomplemented
21#357C2218yescomplemented
22C22⋊C16C2214yescomplemented
23C22×D4—21no—
24C22×D4—41no—
25C23×C4—41no—
26C23×C4D411yescharacteristic
27C2×D8—21no—
28C2×D8—24no—
29C2×D8C2312yes—
30C22×C8—24no—
31C22×C8D411yescharacteristic
32C22×C8D412yes—
33C22×C8C2312yes—
34C22×C8C2311yescharacteristic
35#121—24no—
36#121—42no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—24no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—42no—
41C2×C4⋊C4—41no—
42C2×C22⋊C4—41no—
43C2×C16—24no—
44C2×C16C2314yes—
45#107—22no—
46#107—24no—
47#107—21no—
48#107C2314yes—
49#107C2312yes—
50#102—24no—
51C24—41no—
52C24D811yescharacteristic
53C2×D4—22no—
54C2×D4—44no—
55C2×D4—42no—
56C2×D4—41no—
57C2×D4—42no—
58C2×D4—84no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—44no—
62C22×C4—41no—
63C22×C4—42no—
64C22×C4—84no—
65C22×C4D812yes—
66C22×C4D811yescharacteristic
67C22×C4C2×D412yes—
68C22×C4C2×D411yescharacteristic
69D8—24no—
70D8—42no—
71C2×C8—22no—
72C2×C8—24no—
73C2×C8—24no—
74C2×C8C2×D412yes—
75C2×C8C2×D411yescharacteristic
76C2×C8#4114yes—
77C2×C8C2411yescharacteristic
78C4⋊C4—22no—
79C4⋊C4—24no—
80C4⋊C4—44no—
81C4⋊C4—42no—
82C4⋊C4—41no—
83C22⋊C4—44no—
84C16—24no—
85C23—22no—
86C23—22no—
87C23—42no—
88C23—82no—
89C23—82no—
90C23—82no—
91C23D1614yes—
92C23C2×D812yes—
93C23C2×D811yescharacteristic
94D4—42no—
95D4—42no—
96D4—88no—
97D4—82no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—24no—
102C2×C4—42no—
103C2×C4—44no—
104C2×C4—88no—
105C2×C4—84no—
106C2×C4—82no—
107C2×C4#12312yes—
108C2×C4C2×D812yes—
109C2×C4C2×D811yescharacteristic
110C2×C4#13712yes—
111C2×C4C22×D411yescharacteristic
112C8—22no—
113C8#12312yes—
114C8C2×#4112yes—
115C22—22no—
116C22—22no—
117C22—28no—
118C22—82no—
119C22—82no—
120C22—82no—
121C22—164no—
122C22#48012yes—
123C22C2×D1614yes—
124C22C2×D1612yes—
125C22#51012yes—
126C22C22×D811yescharacteristic
127C4—22no—
128C4—84no—
129C4—82no—
130C4#48012yes—
131C4C2×#12311yescharacteristic
132C4C2×#13711yescharacteristic
133C2—24no—
134C2—162no—
135C2#223214yesdirect product factor
136C2C2×#48011yescharacteristic
137C2C22×D1611yescharacteristic
138C2C2×#51011yescharacteristic
subgroupG/H
center, socleC23C2×D8
commutator, Frattini, cosocleC2×C8C24
prime radicalC2×#466C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#107C2semidirect
3. #2232C2direct (N × H)
4. C2×#366C2semidirect
5. C2×#357C2semidirect
6. C2×C22⋊C16C2semidirect
18. C2×#107C22semidirect
20. #366C22semidirect
21. #357C22semidirect
22. C22⋊C16C22semidirect
135. C2#2232direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#4662
C22×#1072
#22322
C2×#3662
C2×#3572
C2×C22⋊C162