Finite Group Databasebeta

← #2663

Subgroups — #2663

Order 128.

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

normal subgroups only

This group has 249 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×#123C211yesmaximal, characteristic
2C2×C22⋊Q8C211yesmaximal, characteristic, complemented
3#400C211yesmaximal, characteristic, complemented
4#399C211yesmaximal, characteristic, complemented
5#397C212yesmaximal, complemented
6#395C211yesmaximal, characteristic, complemented
7#394C212yesmaximal
8#393C211yesmaximal, characteristic, complemented
9#389C212yesmaximal, complemented
10#388C212yesmaximal, complemented
11C4×C22⋊C4C211yesmaximal, characteristic, complemented
12C22×Q8C2211yescharacteristic
13C22×D4C2211yescharacteristic
14C23×C4C2211yescharacteristic
15#123—24no—
16C22⋊Q8—24no—
17C2×C4⋊C4C2212yes—
18C2×C4⋊C4C2211yescharacteristic
19C2×C4⋊C4C2211yescharacteristic
20C2×C4⋊C4C2211yescharacteristic
21C2×C4⋊C4C2212yes—
22C2×C4⋊C4C2212yes—
23C2×C22⋊C4C2212yes—
24C2×C22⋊C4C2211yescharacteristic
25C2×C22⋊C4C2211yescharacteristic
26C2×C22⋊C4C2211yescharacteristic
27C2×C22⋊C4C2212yes—
28C2×C22⋊C4C2212yes—
29C2×C22⋊C4C2212yes—
30C2×C42C2212yes—
31#95C2211yescharacteristic
32#95C2211yescharacteristic
33#95C2212yes—
34#95C2212yes—
35#95C2212yes—
36#95C2212yes—
37C24C2311yescharacteristic
38C24C2311yescharacteristic
39C2×Q8—22no—
40C2×Q8—42no—
41C2×D4—22no—
42C2×D4—42no—
43C22×C4—42no—
44C22×C4D414yes—
45C22×C4C2312yes—
46C22×C4C2311yescharacteristic
47C22×C4C2311yescharacteristic
48C22×C4C2312yes—
49C22×C4C2312yes—
50C22×C4C2311yescharacteristic
51C22×C4C2311yescharacteristic
52C22×C4C2311yescharacteristic
53C22×C4C2312yes—
54C4⋊C4—22no—
55C4⋊C4—24no—
56C4⋊C4—22no—
57C4⋊C4—22no—
58C4⋊C4—24no—
59C4⋊C4—42no—
60C22⋊C4—24no—
61C22⋊C4—22no—
62C22⋊C4—22no—
63C22⋊C4—22no—
64C22⋊C4—24no—
65C22⋊C4—24no—
66C22⋊C4—42no—
67C42—42no—
68C23—22no—
69C23—22no—
70C23—21no—
71C23—21no—
72C23—21no—
73C23—22no—
74C23—22no—
75C23—21no—
76C23—21no—
77C23C2×D412yes—
78C23C2411yescharacteristic
79Q8—44no—
80D4—44no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—22no—
84C2×C4—22no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—22no—
101C2×C4—44no—
102C2×C4C2×D412yes—
103C2×C4C2×D412yes—
104C2×C4#4114yes—
105C22—22no—
106C22—22no—
107C22—22no—
108C22—42no—
109C22—42no—
110C22—41no—
111C22—41no—
112C22—41no—
113C22—42no—
114C22—42no—
115C22C22×D411yescharacteristic
116C22C2×#4112yes—
117C22H(5,𝔽2)12yes—
118C222−1+411yescharacteristic
119C222−1+411yescharacteristic
120C4—22no—
121C4—22no—
122C4—24no—
123C4—41no—
124C4—42no—
125C4—41no—
126C4—42no—
127C4—41no—
128C4—42no—
129C2—42no—
130C2—81no—
131C2#53211yescharacteristic
132C2#53411yescharacteristic
133C2#53611yescharacteristic
134C2#53912yes—
135C2#54212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C24
prime radicalC22×D4C22
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×C22⋊Q8C2semidirect
3. #400C2semidirect
4. #399C2semidirect
5. #397C2semidirect
6. #395C2semidirect
8. #393C2semidirect
9. #389C2semidirect
10. #388C2semidirect
11. C4×C22⋊C4C2semidirect

Maximal subgroups

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

subgroupindex
C2×#1232
C2×C22⋊Q82
#4002
#3992
#3972
#3952
#3942
#3932
#3892
#3882
C4×C22⋊C42