Finite Group Databasebeta

← #8310

Subgroups — #8310

Order 256.

760 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C22×C4×D4C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic, complemented
3C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×D4⋊C8C214yesmaximal, complemented
5C23×D4C411yescharacteristic
6C24×C4—21no—
7C2×C4×D4—28no—
8C2×C4×D4—21no—
9C2×C4×D4—21no—
10C2×C4×D4C412yes—
11C2×C4×D4C2212yes—
12C22×C4⋊C4C411yescharacteristic
13C22×C22⋊C4—21no—
14C22×C42C2211yescharacteristic
15C2×C4⋊C8C2212yescomplemented
16C2×C22⋊C8—21no—
17C2×C22⋊C8—21no—
18C2×C4×C8C2212yescomplemented
19#336—21no—
20#336—21no—
21D4⋊C8—28no—
22C25—21no—
23C22×D4—24no—
24C22×D4—24no—
25C22×D4—21no—
26C22×D4—22no—
27C22×D4C814yescomplemented
28C22×D4C2×C412yes—
29C23×C4—28no—
30C23×C4—21no—
31C23×C4—22no—
32C23×C4—22no—
33C23×C4—44no—
34C23×C4—42no—
35C23×C4—42no—
36C23×C4C2×C411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4D411yescharacteristic
39C22×C8—22no—
40C22×C8—22no—
41C4×D4—28no—
42C4×D4—28no—
43C4×D4—24no—
44C4×D4—24no—
45C4×D4—24no—
46C4×D4D418yes—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4C2×C412yes—
50C2×C22⋊C4—28no—
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C42—24no—
54C2×C42C2×C411yescharacteristic
55C2×C42D411yescharacteristic
56C2×C42D411yescharacteristic
57C2×C42C2311yescharacteristic
58C4⋊C8—24no—
59C22⋊C8—24no—
60C22⋊C8—42no—
61C4×C8—24no—
62C24—24no—
63C24—24no—
64C24—24no—
65C24—22no—
66C24—42no—
67C24—42no—
68C24—42no—
69C24—42no—
70C24C22⋊C411yescharacteristic
71C2×D4—24no—
72C2×D4—24no—
73C2×D4—216no—
74C2×D4—216no—
75C2×D4—24no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—22no—
79C2×D4C22⋊C414yes—
80C2×D4C2×C814yes—
81C2×D4M1614yes—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—24no—
87C22×C4—28no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—28no—
92C22×C4—41no—
93C22×C4—44no—
94C22×C4—416no—
95C22×C4—416no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—44no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—44no—
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22⋊C411yescharacteristic
104C22×C4C22⋊C411yescharacteristic
105C22×C4C2×C812yes—
106C22×C4M1612yes—
107C22×C4D812yes—
108C22×C4SD1612yes—
109C22×C4C22×C411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C2×C8—24no—
113C2×C8—42no—
114C2×C8—42no—
115C2×C8—42no—
116C2×C8—42no—
117C2×C8—42no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4C22⋊C414yes—
121C22⋊C4—28no—
122C22⋊C4—28no—
123C42—22no—
124C42—22no—
125C42—22no—
126C42C2×D414yes—
127C23—21no—
128C23—21no—
129C23—21no—
130C23—21no—
131C23—216no—
132C23—24no—
133C23—24no—
134C23—24no—
135C23—48no—
136C23—48no—
137C23—48no—
138C23—48no—
139C23—48no—
140C23—48no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23C22⋊C812yes—
146C23#10212yes—
147C23#10412yes—
148C23C2×C22⋊C411yescharacteristic
149D4—216no—
150D4—24no—
151D4—24no—
152D4—24no—
153D4C22⋊C818yescomplemented
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—44no—
169C2×C4—416no—
170C2×C4—48no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—48no—
174C2×C4—48no—
175C2×C4C22⋊C812yes—
176C2×C4#10212yes—
177C2×C4C2×C22⋊C412yes—
178C2×C4#12012yes—
179C2×C4#12012yes—
180C2×C4C22×C811yescharacteristic
181C2×C4C2×M1612yes—
182C2×C4C2×M1611yescharacteristic
183C2×C4C2×D811yescharacteristic
184C2×C4C2×SD1611yescharacteristic
185C2×C4AGL(1,ℤ/8)12yes—
186C8—44no—
187C8—82no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—28no—
195C22—432no—
196C22—44no—
197C22—44no—
198C22—44no—
199C22—44no—
200C22—44no—
201C22—44no—
202C22D4⋊C814yescomplemented
203C22#37911yescharacteristic
204C22C2×C22⋊C811yescharacteristic
205C22#40711yescharacteristic
206C22C2×#10211yescharacteristic
207C22#41811yescharacteristic
208C22C2×#10411yescharacteristic
209C22#42111yescharacteristic
210C4—22no—
211C4—22no—
212C4—22no—
213C4—44no—
214C4—41no—
215C4C2×C22⋊C812yes—
216C4#40712yes—
217C4#44712yes—
218C4#45012yes—
219C2—24no—
220C2—48no—
221C2C2×D4⋊C811yescharacteristic
222C2#147811yescharacteristic
223C2#148611yescharacteristic
224C2#149011yescharacteristic
225C2#177911yescharacteristic
226C2#178611yescharacteristic
227C2#178911yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC23×D4C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1797C2semidirect
3. C4×C22⋊C8C2semidirect
4. C2×D4⋊C8C2semidirect
15. C2×C4⋊C8C22semidirect
18. C2×C4×C8C22semidirect
27. C22×D4C8semidirect
153. D4C22⋊C8semidirect
202. C22D4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
#17972
C4×C22⋊C82
C2×D4⋊C82