Finite Group Databasebeta

← #2664

Subgroups — #2664

Order 128.

217 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 96 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×#126C211yesmaximal, characteristic
2C2×#125C211yesmaximal, characteristic, complemented
3#400C211yesmaximal, characteristic, complemented
4#399C211yesmaximal, characteristic
5#398C211yesmaximal, characteristic, complemented
6#398C212yesmaximal, complemented
7#397C211yesmaximal, characteristic
8#396C212yesmaximal
9#389C212yesmaximal, complemented
10#388C212yesmaximal
11C4×C4⋊C4C211yesmaximal, characteristic, complemented
12#126—24no—
13#125—24no—
14C2×C4⋊C4C2212yes—
15C2×C4⋊C4C2212yes—
16C2×C4⋊C4C2211yescharacteristic
17C2×C4⋊C4C2211yescharacteristic
18C2×C4⋊C4C2211yescharacteristic
19C2×C4⋊C4C2211yescharacteristic
20C2×C4⋊C4C2211yescharacteristic
21C2×C4⋊C4C2212yes—
22C2×C4⋊C4C2212yes—
23C2×C4⋊C4C2212yes—
24C2×C22⋊C4C2211yescharacteristic
25C2×C22⋊C4C2211yescharacteristic
26C2×C22⋊C4C2211yescharacteristic
27C2×C22⋊C4C2212yes—
28C2×C22⋊C4C2212yes—
29C2×C42C2212yes—
30C2×C42C2211yescharacteristic
31#95C2211yescharacteristic
32#95C2211yescharacteristic
33#95C2212yes—
34#95C2212yes—
35#95C2212yes—
36#95C2212yes—
37C24C2311yescharacteristic
38C22×C4C2312yes—
39C22×C4C2311yescharacteristic
40C22×C4C2311yescharacteristic
41C22×C4C2311yescharacteristic
42C22×C4C2312yes—
43C22×C4C2312yes—
44C22×C4C2311yescharacteristic
45C22×C4C2311yescharacteristic
46C22×C4C2311yescharacteristic
47C22×C4C2312yes—
48C4⋊C4—24no—
49C4⋊C4—24no—
50C4⋊C4—22no—
51C4⋊C4—22no—
52C4⋊C4—22no—
53C4⋊C4—22no—
54C4⋊C4—22no—
55C4⋊C4—24no—
56C4⋊C4—24no—
57C4⋊C4—42no—
58C22⋊C4—22no—
59C22⋊C4—22no—
60C22⋊C4—22no—
61C22⋊C4—24no—
62C22⋊C4—42no—
63C42—42no—
64C42D414yes—
65C23—21no—
66C23—22no—
67C23—21no—
68C23—22no—
69C23—21no—
70C23C2411yescharacteristic
71C2×C4—22no—
72C2×C4—21no—
73C2×C4—22no—
74C2×C4—22no—
75C2×C4—21no—
76C2×C4—22no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—22no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—22no—
84C2×C4—22no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—22no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—21no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4C2×D412yes—
94C2×C4C2×D412yes—
95C2×C4C2×D412yes—
96C2×C4#4114yes—
97C22—41no—
98C22—41no—
99C22—41no—
100C22—42no—
101C22—42no—
102C22C22×D411yescharacteristic
103C22C2×#4112yes—
104C22H(5,𝔽2)11yescharacteristic
105C222−1+412yes—
106C222−1+411yescharacteristic
107C4—22no—
108C4—22no—
109C4—22no—
110C4—24no—
111C4—41no—
112C4—42no—
113C4—41no—
114C4—41no—
115C4—42no—
116C4—42no—
117C2—81no—
118C2#53211yescharacteristic
119C2#53611yescharacteristic
120C2#53711yescharacteristic
121C2#53912yes—
122C2#54112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C24
prime radicalC24C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#125C2semidirect
3. #400C2semidirect
5. #398C2semidirect
6. #398C2semidirect
9. #389C2semidirect
11. C4×C4⋊C4C2semidirect

Maximal subgroups

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

subgroupindex
C2×#1262
C2×#1252
#4002
#3992
#3982
#3972
#3962
#3892
#3882
C4×C4⋊C42