Finite Group Databasebeta

← #11700

Subgroups — #11700

Order 256.

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

normal subgroups only

This group has 226 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#1742C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4#1346C214yesmaximal, complemented
5C22×M16—21no—
6C2×C4⋊Q8C411yescharacteristic
7C2×C22⋊Q8—21no—
8C2×#117C2211yescharacteristic
9#431—21no—
10#398—21no—
11#398C2212yes—
12#397—21no—
13#397C412yes—
14#391—21no—
15#382C411yescharacteristic
16#343—21no—
17#343C2212yes—
18#324C2212yes—
19#323—21no—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22C2×M16—24no—
23C2×M16—22no—
24C2×M16—41no—
25C2×M16—42no—
26C22×C8—41no—
27C4⋊Q8—41no—
28C4⋊Q8—41no—
29C22⋊Q8—42no—
30C22⋊Q8—42no—
31#117—41no—
32#117—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4C2×C412yes—
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—22no—
40C2×C22⋊C4C2×C411yescharacteristic
41C2×C22⋊C4D411yescharacteristic
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—
51C24C22⋊C411yescharacteristic
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×C4C22⋊C411yescharacteristic
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4D812yes—
65C22×C4SD1612yes—
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D411yescharacteristic
69M16—24no—
70M16—44no—
71M16—42no—
72C2×C8—42no—
73C2×C8—44no—
74C2×C8—42no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4—42no—
78C4⋊C4—42no—
79C4⋊C4—42no—
80C4⋊C4—42no—
81C4⋊C4—42no—
82C22⋊C4—41no—
83C22⋊C4—41no—
84C22⋊C4—44no—
85C42—21no—
86C42—21no—
87C42—21no—
88C42—21no—
89C23—21no—
90C23—21no—
91C23—41no—
92C23—41no—
93C23#10212yes—
94C23C2×C22⋊C411yescharacteristic
95Q8—82no—
96Q8—82no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4#9911yescharacteristic
118C2×C4#9911yescharacteristic
119C2×C4#10212yes—
120C2×C4#10412yes—
121C2×C4C2×D811yescharacteristic
122C2×C4C2×SD1611yescharacteristic
123C8—44no—
124C8—82no—
125C22—21no—
126C22—21no—
127C22—22no—
128C22—42no—
129C22—41no—
130C22—41no—
131C22—81no—
132C22C2×#9911yescharacteristic
133C22C2×#10211yescharacteristic
134C22C2×#10411yescharacteristic
135C4—21no—
136C4—21no—
137C4—41no—
138C4—41no—
139C4—41no—
140C4—41no—
141C4—82no—
142C4—82no—
143C4#32712yes—
144C4#32712yes—
145C2—21no—
146C2—21no—
147C2—42no—
148C2C2×#32711yescharacteristic
149C2#212311yescharacteristic
150C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1346C2semidirect

Maximal subgroups

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

subgroupindex
#26212
#17422
#14582
#13462