Finite Group Databasebeta

← #9021

Subgroups — #9021

Order 256.

406 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 83 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×#399C211yesmaximal, characteristic
2#2315C211yesmaximal, characteristic
3#1824C212yesmaximal, complemented
4C2×#323C211yesmaximal, characteristic
5#1294C212yesmaximal, complemented
6C22×C4⋊C4—21no—
7C22×C4⋊C4C2211yescharacteristic
8C22×C22⋊C4—21no—
9C22×C22⋊C4—21no—
10C22×C22⋊C4C2211yescharacteristic
11C2×C22⋊C8C2212yes—
12#399—22no—
13#399C414yescomplemented
14#385—24no—
15#380—21no—
16#380C2212yes—
17C2×#95C2211yescharacteristic
18#340—24no—
19#323—22no—
20C25D411yescharacteristic
21C23×C4—21no—
22C23×C4—21no—
23C23×C4—22no—
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—24no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—44no—
33C2×C4⋊C4C2×C412yescomplemented
34C2×C4⋊C4D414yes—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—24no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—44no—
41C2×C22⋊C4—42no—
42C2×C22⋊C4—42no—
43C2×C22⋊C4C2×C412yes—
44C2×C22⋊C4D412yes—
45C22⋊C8—22no—
46C22⋊C8—22no—
47#95—22no—
48#95—21no—
49#95—22no—
50#95C2×C412yescomplemented
51C24—21no—
52C24—21no—
53C24—21no—
54C24—21no—
55C24—42no—
56C24—42no—
57C24C22⋊C412yes—
58C24SD1612yes—
59C24Q1612yes—
60C24C2×D411yescharacteristic
61C22×C4—24no—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—22no—
66C22×C4—24no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—44no—
74C22×C4—42no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—82no—
78C22×C4—82no—
79C22×C4C22⋊C412yes—
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4C2×D411yescharacteristic
83C22×C4#4112yes—
84C2×C8—44no—
85C2×C8—42no—
86C4⋊C4—24no—
87C4⋊C4—44no—
88C4⋊C4—48no—
89C4⋊C4—42no—
90C4⋊C4—82no—
91C22⋊C4—24no—
92C22⋊C4—24no—
93C22⋊C4—48no—
94C22⋊C4—44no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C23—22no—
98C23—22no—
99C23—22no—
100C23—22no—
101C23—22no—
102C23—22no—
103C23—22no—
104C23—22no—
105C23—42no—
106C23—42no—
107C23—44no—
108C23—44no—
109C23—42no—
110C23—42no—
111C23—82no—
112C23—82no—
113C23—81no—
114C23—81no—
115C23#10314yes—
116C23C2×C22⋊C411yescharacteristic
117C23#12011yescharacteristic
118C23#12311yescharacteristic
119C23C2×SD1611yescharacteristic
120C23C2×Q1611yescharacteristic
121C23AGL(1,ℤ/8)11yescharacteristic
122C23#13711yescharacteristic
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—44no—
130C2×C4—44no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—44no—
139C2×C4—44no—
140C2×C4—44no—
141C2×C4—81no—
142C2×C4—82no—
143C2×C4—88no—
144C2×C4—82no—
145C2×C4C4×D412yes—
146C2×C4#12112yes—
147C8—82no—
148C22—22no—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—24no—
157C22—44no—
158C22—44no—
159C22—44no—
160C22—42no—
161C22—42no—
162C22—82no—
163C22—82no—
164C22—82no—
165C22#38611yescharacteristic
166C22C2×#10311yescharacteristic
167C22#41711yescharacteristic
168C22#45011yescharacteristic
169C22#45111yescharacteristic
170C22#45312yes—
171C22#45812yes—
172C22#48311yescharacteristic
173C22#48411yescharacteristic
174C4—22no—
175C4—22no—
176C4—44no—
177C4—42no—
178C4—84no—
179C4—82no—
180C2—21no—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—44no—
185C2#161312yes—
186C2#161412yes—
187C2#187211yescharacteristic
188C2#188711yescharacteristic
189C2#189011yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1824C2semidirect
5. #1294C2semidirect
13. #399C4semidirect
33. C2×C4⋊C4C2×C4semidirect
50. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
#23152
#18242
C2×#3232
#12942