Finite Group Databasebeta

← #2814

Subgroups — #2814

Order 128.

239 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 92 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#398C211yesmaximal, characteristic, complemented
4#398C212yesmaximal, complemented
5#397C211yesmaximal, characteristic, complemented
6#397C212yesmaximal, complemented
7#396C212yesmaximal, complemented
8#395C211yesmaximal, characteristic, complemented
9#394C212yesmaximal
10#393C211yesmaximal, characteristic, complemented
11#380C211yesmaximal, 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×C4⋊C4C2212yes—
24C2×C22⋊C4C2212yes—
25C2×C22⋊C4C2211yescharacteristic
26C2×C22⋊C4C2211yescharacteristic
27C2×C22⋊C4C2211yescharacteristic
28C2×C22⋊C4C2212yes—
29C2×C22⋊C4C2212yes—
30C2×C22⋊C4C2212yes—
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×C4C2311yescharacteristic
46C22×C4C2312yes—
47C22×C4C2312yes—
48C22×C4C2311yescharacteristic
49C22×C4C2312yes—
50C22×C4C2312yes—
51C22×C4C2311yescharacteristic
52C22×C4C2311yescharacteristic
53C22×C4C2311yescharacteristic
54C4⋊C4—22no—
55C4⋊C4—24no—
56C4⋊C4—22no—
57C4⋊C4—22no—
58C4⋊C4—42no—
59C4⋊C4—42no—
60C4⋊C4—42no—
61C22⋊C4—24no—
62C22⋊C4—22no—
63C22⋊C4—22no—
64C22⋊C4—22no—
65C22⋊C4—42no—
66C22⋊C4—42no—
67C22⋊C4—42no—
68C23—21no—
69C23—22no—
70C23—22no—
71C23—21no—
72C23—21no—
73C23—21no—
74C23—22no—
75C23—22no—
76C23—21no—
77C23C2×D412yes—
78C23C2411yescharacteristic
79Q8—44no—
80D4—44no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—21no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—44no—
103C2×C4C2×D412yes—
104C2×C4C2×D412yes—
105C22—22no—
106C22—22no—
107C22—22no—
108C22—42no—
109C22—42no—
110C22—41no—
111C22—41no—
112C22—41no—
113C22—42no—
114C22—42no—
115C22C22×D411yescharacteristic
116C22H(5,𝔽2)12yes—
117C22H(5,𝔽2)11yescharacteristic
118C222−1+412yes—
119C222−1+411yescharacteristic
120C4—22no—
121C4—22no—
122C4—41no—
123C4—42no—
124C4—41no—
125C4—41no—
126C4—42no—
127C4—42no—
128C4—42no—
129C2—42no—
130C2—81no—
131C2#53411yescharacteristic
132C2#53611yescharacteristic
133C2#53711yescharacteristic
134C2#56212yes—
135C2#56312yes—
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. #398C2semidirect
4. #398C2semidirect
5. #397C2semidirect
6. #397C2semidirect
7. #396C2semidirect
8. #395C2semidirect
10. #393C2semidirect
11. #380C2semidirect

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
#3982
#3972
#3962
#3952
#3942
#3932
#3802