Finite Group Databasebeta

← #11694

Subgroups — #11694

Order 256.

234 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#2621C211yesmaximal, characteristic
2#1740C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4#1344C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8—21no—
7C2×#117—21no—
8C2×#117C2211yescharacteristic
9#431—21no—
10#398—21no—
11#398C412yescomplemented
12#397—21no—
13#397C2212yes—
14#391—21no—
15#382C411yescharacteristic, complemented
16#342—21no—
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22C4⋊Q8—41no—
23C4⋊Q8—41no—
24C22⋊Q8—42no—
25C22⋊Q8—42no—
26#117—24no—
27#117—41no—
28#117—41no—
29#117—42no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—41no—
34C2×C4⋊C4C2×C412yes—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4D411yescharacteristic
40C2×C22⋊C4C2311yescharacteristic
41C2×C42—41no—
42C2×C42C2×C411yescharacteristic
43C2×C42D411yescharacteristic
44C4⋊C8—41no—
45C22⋊C8—22no—
46C4×C8—41no—
47#95—21no—
48#95—22no—
49#95—22no—
50#95C2×C412yes—
51C24C2×D411yescharacteristic
52C2×Q8—41no—
53C2×Q8—41no—
54C2×Q8—42no—
55C2×Q8—42no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—42no—
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4D812yes—
67C22×C4SD1612yes—
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C2×C8—42no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—42no—
74C4⋊C4—42no—
75C4⋊C4—44no—
76C4⋊C4—42no—
77C4⋊C4—44no—
78C22⋊C4—24no—
79C22⋊C4—41no—
80C22⋊C4—41no—
81C22⋊C4—42no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C42—21no—
85C42—21no—
86C42—21no—
87C42—21no—
88C42—44no—
89C23—21no—
90C23—21no—
91C23—41no—
92C23—41no—
93C23C2×C22⋊C411yescharacteristic
94C23C2×D811yescharacteristic
95C23C2×SD1611yescharacteristic
96Q8—82no—
97Q8—82no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—44no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—82no—
120C2×C4—82no—
121C2×C4#9911yescharacteristic
122C2×C4#9911yescharacteristic
123C2×C4#10212yes—
124C2×C4#10212yes—
125C2×C4#10412yes—
126C8—82no—
127C22—21no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—41no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22—81no—
136C22C2×#9911yescharacteristic
137C22C2×#10211yescharacteristic
138C22C2×#10411yescharacteristic
139C4—22no—
140C4—41no—
141C4—41no—
142C4—41no—
143C4—41no—
144C4—82no—
145C4—84no—
146C4—82no—
147C4#32712yes—
148C4#32712yes—
149C2—21no—
150C2—21no—
151C2—41no—
152C2—41no—
153C2C2×#32711yescharacteristic
154C2#212311yescharacteristic
155C2#212911yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
11. #398C4semidirect
15. #382C4semidirect

Maximal subgroups

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

subgroupindex
#26212
#17402
#14582
#13442