Finite Group Databasebeta

← #8378

Subgroups — #8378

Order 256.

422 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 81 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×#553C211yesmaximal, characteristic
2#1824C212yesmaximal, complemented
3#1799C211yesmaximal, characteristic, complemented
4#1519C212yesmaximal, complemented
5#1459C211yesmaximal, characteristic, complemented
6#553—24no—
7#553—24no—
8C2×#126—21no—
9C2×#125C411yescharacteristic, complemented
10C2×#123—21no—
11C2×#121—21no—
12C2×#121C411yescharacteristic, complemented
13C2×#121C2211yescharacteristic
14C2×C4×D4—21no—
15C2×C4×D4C411yescharacteristic, complemented
16C2×C4×D4C2211yescharacteristic
17C2×#117C411yescharacteristic, complemented
18C22×C4⋊C4C2211yescharacteristic
19C2×#99—21no—
20C2×C22⋊C8C2212yes—
21#380C2212yes—
22#342—21no—
23#340—24no—
24#327—24no—
25#324—21no—
26#323—21no—
27C22×D4—21no—
28C22×D4—21no—
29C22×D4C2×C411yescharacteristic
30C22×D4D411yescharacteristic
31C23×C4—21no—
32C23×C4—21no—
33C23×C4C2×C411yescharacteristic
34C23×C4C2311yescharacteristic
35C22×C8—22no—
36#126—28no—
37#125—22no—
38#125—22no—
39#123—28no—
40#121—28no—
41#121—22no—
42#121—22no—
43#121—22no—
44#121D414yes—
45C4×D4—28no—
46C4×D4—22no—
47C4×D4—22no—
48C4×D4—22no—
49C4×D4—22no—
50#117—22no—
51#117—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C4⋊C4C2×C411yescharacteristic
59C2×C4⋊C4C2×C411yescharacteristic
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D414yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4D411yescharacteristic
68C2×C22⋊C4D411yescharacteristic
69C2×C42—21no—
70C2×C42C2×C411yescharacteristic
71#99—42no—
72C22⋊C8—22no—
73C22⋊C8—22no—
74#95—22no—
75C24—21no—
76C24C22⋊C411yescharacteristic
77C24C2×D411yescharacteristic
78C2×D4—22no—
79C2×D4—24no—
80C2×D4—22no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—24no—
84C2×D4—21no—
85C2×D4—42no—
86C2×D4—42no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—82no—
90C2×D4C22⋊C412yes—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—24no—
95C22×C4—24no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—82no—
102C22×C4—82no—
103C22×C4C22⋊C411yescharacteristic
104C22×C4C22⋊C411yescharacteristic
105C22×C4C22⋊C411yescharacteristic
106C22×C4C22×C411yescharacteristic
107C22×C4C2×D412yes—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C2×C8—44no—
111C2×C8—42no—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—44no—
120C4⋊C4C22⋊C412yes—
121C4⋊C4C22⋊C412yes—
122C4⋊C4C22⋊C412yes—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C22⋊C4—22no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C42—24no—
132C42—21no—
133C42—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—
148C23#13512yes—
149C23AGL(1,ℤ/8)11yescharacteristic
150C23#13711yescharacteristic
151D4—42no—
152D4—42no—
153D4—44no—
154D4—42no—
155D4—44no—
156D4—42no—
157D4—84no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—84no—
179C2×C4—84no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4C2×C22⋊C411yescharacteristic
184C2×C4C2×C22⋊C411yescharacteristic
185C2×C4#12012yes—
186C8—82no—
187C22—21no—
188C22—22no—
189C22—21no—
190C22—22no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22—41no—
197C22—42no—
198C22—41no—
199C22—42no—
200C22—81no—
201C22—81no—
202C22—82no—
203C22—82no—
204C22#37911yescharacteristic
205C22#41611yescharacteristic
206C22#41711yescharacteristic
207C22#44912yes—
208C22#45212yes—
209C4—22no—
210C4—21no—
211C4—42no—
212C4—42no—
213C4—41no—
214C4—41no—
215C4—41no—
216C4—42no—
217C4—162no—
218C4#41012yes—
219C2—22no—
220C2—21no—
221C2—21no—
222C2—22no—
223C2—81no—
224C2—82no—
225C2#178211yescharacteristic
226C2#178811yescharacteristic
227C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#553C2
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. #1824C2semidirect
3. #1799C2semidirect
4. #1519C2semidirect
5. #1459C2semidirect
9. C2×#125C4semidirect
12. C2×#121C4semidirect
15. 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×#5532
#18242
#17992
#15192
#14592