Finite Group Databasebeta

← #9035

Subgroups — #9035

Order 256.

306 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 75 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#2905C211yesmaximal, characteristic, complemented
2#2392C211yesmaximal, characteristic
3#1486C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic, complemented
5#1303C212yesmaximal, complemented
6C22×M16—21no—
7C2×C4×D4—21no—
8C2×C4×D4C2211yescharacteristic
9C2×#117C2211yescharacteristic
10#431C2212yes—
11#424—24no—
12#423—21no—
13#423C2212yes—
14#398C411yescharacteristic
15#397C411yescharacteristic
16#396—21no—
17#394—21no—
18#394C411yescharacteristic
19#392C411yescharacteristic
20C4×C4⋊C4C2211yescharacteristic
21C4×C22⋊C4—21no—
22D4⋊C8—24no—
23#324—21no—
24#323—21no—
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C2×M16—22no—
30C2×M16—22no—
31C2×M16—42no—
32C2×M16—42no—
33C22×C8—22no—
34C4×D4—22no—
35C4×D4—22no—
36C4×D4—42no—
37C4×D4—42no—
38#117—22no—
39#117—22no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C4⋊C4C2×C411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4C2×C411yescharacteristic
51C2×C22⋊C4D411yescharacteristic
52C2×C42—21no—
53C2×C42D411yescharacteristic
54C2×C42C2311yescharacteristic
55C4⋊C8—24no—
56C4⋊C8—22no—
57C4⋊C8—22no—
58C4⋊C8D414yescomplemented
59C22⋊C8—22no—
60C4×C8—22no—
61#95—21no—
62#95—21no—
63#95C2×C411yescharacteristic
64#95C2×C411yescharacteristic
65C24—21no—
66C24C22⋊C411yescharacteristic
67C2×D4—22no—
68C2×D4—42no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—42no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4D812yes—
89C22×C4SD1612yes—
90C22×C4C22×C411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93M16—44no—
94M16—42no—
95M16—42no—
96C2×C8—24no—
97C2×C8—44no—
98C2×C8—42no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—42no—
103C4⋊C4—42no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C42—21no—
112C42—21no—
113C42—41no—
114C42—41no—
115C42C2×D412yes—
116C42#4112yes—
117C23—21no—
118C23—21no—
119C23—22no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23#10212yes—
129C23C2×C22⋊C411yescharacteristic
130D4—24no—
131D4—42no—
132D4—48no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4#10212yes—
159C2×C4C4×D412yes—
160C2×C4#12011yescharacteristic
161C2×C4#12112yes—
162C2×C4#12311yescharacteristic
163C2×C4C2×D811yescharacteristic
164C2×C4C2×SD1611yescharacteristic
165C2×C4AGL(1,ℤ/8)11yescharacteristic
166C2×C4#13711yescharacteristic
167C8—44no—
168C8—82no—
169C22—21no—
170C22—21no—
171C22—22no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22#38611yescharacteristic
184C22C2×#10211yescharacteristic
185C22#41711yescharacteristic
186C4—22no—
187C4—22no—
188C4—22no—
189C4—42no—
190C4—41no—
191C4—41no—
192C4—81no—
193C4—81no—
194C4#44711yescharacteristic
195C4#44811yescharacteristic
196C4#48011yescharacteristic
197C4#48111yescharacteristic
198C2—21no—
199C2—21no—
200C2—42no—
201C2—82no—
202C2#187411yescharacteristic
203C2#188411yescharacteristic
204C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#392C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2905C2semidirect
3. #1486C2semidirect
4. #1458C2semidirect
5. #1303C2semidirect
58. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#29052
#23922
#14862
#14582
#13032