Finite Group Databasebeta

← #9683

Subgroups — #9683

Order 256.

315 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 79 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#2857C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic
3C2×#328C214yesmaximal
4#1434C211yesmaximal, characteristic
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8C2212yes—
7C22×C42C2211yescharacteristic
8C2×C22⋊C8C2212yes—
9C2×#97—21no—
10#391C412yescomplemented
11#389—21no—
12#389—21no—
13#384C411yescharacteristic, complemented
14#380—21no—
15#380—21no—
16C2×#95—21no—
17C2×#95—21no—
18C2×#95C2212yes—
19#336—21no—
20#328—28no—
21C22×Q8C2×C412yes—
22C23×C4—22no—
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C4⋊Q8—42no—
28C22⋊Q8—24no—
29C22⋊Q8—24no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4C2×C412yes—
36C2×C22⋊C4—22no—
37C2×C42—42no—
38C2×C42C2×C411yescharacteristic
39C2×C42C2×C411yescharacteristic
40C2×C42D411yescharacteristic
41C2×C42D411yescharacteristic
42C22⋊C8—22no—
43C22⋊C8D414yes—
44#97—42no—
45#95—24no—
46#95—22no—
47#95—22no—
48#95—24no—
49#95—42no—
50C24C2×D411yescharacteristic
51C2×Q8—24no—
52C2×Q8—42no—
53C2×Q8—42no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—24no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D412yes—
78C22×C4#4112yes—
79C2×C8—24no—
80C2×C8—42no—
81C2×C8—42no—
82C4⋊C4—24no—
83C4⋊C4—42no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—44no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C42—22no—
91C42—22no—
92C42—41no—
93C42—41no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23C2×C22⋊C411yescharacteristic
99C23#12411yescharacteristic
100C23#12711yescharacteristic
101C23#13712yes—
102C23#13712yes—
103Q8—84no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—44no—
121C2×C4—44no—
122C2×C4—44no—
123C2×C4—44no—
124C2×C4—44no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—82no—
132C2×C4—82no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4#9912yes—
136C2×C4#10412yes—
137C2×C4C4×D412yes—
138C2×C4#12112yes—
139C8—44no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—22no—
148C22#39011yescharacteristic
149C22C2×#9911yescharacteristic
150C22#41011yescharacteristic
151C22#41911yescharacteristic
152C22#41911yescharacteristic
153C22C2×#10411yescharacteristic
154C22#42111yescharacteristic
155C22#49112yes—
156C22#49712yes—
157C4—22no—
158C4—22no—
159C4—22no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—84no—
164C4—82no—
165C4—82no—
166C2—22no—
167C2—22no—
168C2#150611yescharacteristic
169C2#150811yescharacteristic
170C2#151111yescharacteristic
171C2#151511yescharacteristic
172C2#196011yescharacteristic
173C2#196411yescharacteristic
174C2#198011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
10. #391C4semidirect
13. #384C4semidirect

Maximal subgroups

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

subgroupindex
#28572
#17492
C2×#3282
#14342