Finite Group Databasebeta

← #8360

Subgroups — #8360

Order 256.

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

normal subgroups only

This group has 423 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
1C2×#555C211yesmaximal, characteristic
2#1826C212yesmaximal, complemented
3C4×#99C211yesmaximal, characteristic, complemented
4#1507C212yesmaximal, complemented
5#1465C211yesmaximal, characteristic, complemented
6#555—24no—
7#555—24no—
8C2×C4⋊Q8C411yescharacteristic, complemented
9C2×#124—21no—
10C2×#124—21no—
11C2×#121—21no—
12C2×#121C411yescharacteristic, complemented
13C2×#121C2211yescharacteristic
14C2×C4×D4C411yescharacteristic, complemented
15C2×C4×D4C2211yescharacteristic
16C2×#117—21no—
17C2×#117C411yescharacteristic, complemented
18C2×#117C2211yescharacteristic
19C2×#99—21no—
20#407C2212yes—
21C4×C22⋊C4C2212yes—
22#342—21no—
23#339—24no—
24#327—24no—
25#324—21no—
26#323—21no—
27C22×Q8—21no—
28C22×D4—21no—
29C22×D4—21no—
30C22×D4C2×C411yescharacteristic
31C22×D4D411yescharacteristic
32C23×C4—21no—
33C23×C4C2×C411yescharacteristic
34C23×C4C2311yescharacteristic
35C2×M16—22no—
36C4⋊Q8—22no—
37C4⋊Q8—22no—
38#124—28no—
39#124—28no—
40#121—28no—
41#121—22no—
42#121—22no—
43#121—22no—
44#121D414yes—
45C4×D4—22no—
46C4×D4—22no—
47C4×D4—22no—
48C4×D4—22no—
49#117—28no—
50#117—22no—
51#117—22no—
52#117—22no—
53#117D414yes—
54C2×C4⋊C4—21no—
55C2×C4⋊C4C2×C411yescharacteristic
56C2×C4⋊C4C2×C411yescharacteristic
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4D411yescharacteristic
66C2×C22⋊C4D411yescharacteristic
67C2×C42—21no—
68C2×C42—21no—
69C2×C42—22no—
70C2×C42C2×C411yescharacteristic
71#99—24no—
72C22⋊C8—22no—
73C22⋊C8—22no—
74#95—22no—
75C24—21no—
76C24C22⋊C411yescharacteristic
77C24C2×D411yescharacteristic
78C2×Q8—24no—
79C2×Q8—82no—
80C2×D4—22no—
81C2×D4—24no—
82C2×D4—24no—
83C2×D4—22no—
84C2×D4—21no—
85C2×D4—41no—
86C2×D4—42no—
87C2×D4—82no—
88C2×D4—82no—
89C2×D4C22⋊C412yes—
90C22×C4—24no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4C22⋊C411yescharacteristic
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22⋊C411yescharacteristic
104C22×C4C22×C411yescharacteristic
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108M16—44no—
109C2×C8—42no—
110C4⋊C4—24no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4C22⋊C412yes—
116C4⋊C4C22⋊C412yes—
117C4⋊C4C22⋊C412yes—
118C22⋊C4—24no—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C22⋊C4—22no—
123C22⋊C4—24no—
124C22⋊C4—22no—
125C22⋊C4—24no—
126C22⋊C4—42no—
127C42—24no—
128C42—21no—
129C42—21no—
130C42—24no—
131C42—44no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—22no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23—42no—
141C23—42no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23C2×C22⋊C411yescharacteristic
147C23#12012yes—
148Q8—84no—
149D4—44no—
150D4—42no—
151D4—42no—
152D4—84no—
153D4—84no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—48no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—44no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4#10412yes—
178C2×C4C2×C22⋊C411yescharacteristic
179C2×C4C2×C22⋊C411yescharacteristic
180C2×C4#12012yes—
181C8—82no—
182C22—21no—
183C22—21no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—41no—
188C22—42no—
189C22—41no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—82no—
198C22—82no—
199C22#37911yescharacteristic
200C22C2×#10411yescharacteristic
201C22#42111yescharacteristic
202C4—22no—
203C4—21no—
204C4—22no—
205C4—42no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—42no—
210C4—84no—
211C4#41012yes—
212C2—21no—
213C2—21no—
214C2—42no—
215C2—81no—
216C2—82no—
217C2#178211yescharacteristic
218C2#178911yescharacteristic
219C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#555C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1826C2semidirect
3. C4×#99C2semidirect
4. #1507C2semidirect
5. #1465C2semidirect
8. C2×C4⋊Q8C4semidirect
12. C2×#121C4semidirect
14. C2×C4×D4C4semidirect
17. C2×#117C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
#18262
C4×#992
#15072
#14652