Finite Group Databasebeta

← #27601

Subgroups — #27601

Order 256.

562 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 141 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×#380C211yesmaximal, characteristic
2C2×#380C211yesmaximal, characteristic
3#2077C214yesmaximal, complemented
4#1853C212yesmaximal, complemented
5#1853C212yesmaximal, complemented
6#1824C212yesmaximal, complemented
7#1823C212yesmaximal, complemented
8#1779C211yesmaximal, characteristic
9C24×C4C2211yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4—21no—
15C22×C22⋊C4—21no—
16C2×#106C2214yescomplemented
17C2×C22⋊C8C2212yes—
18C2×C22⋊C8C2212yes—
19C2×C22⋊C8C2212yes—
20#380—21no—
21#380—21no—
22#380—21no—
23#380—21no—
24#380C2212yes—
25#380C2212yes—
26#380C2212yes—
27#380C2212yes—
28C2×#95—21no—
29C2×#95—21no—
30#340C2214yescomplemented
31#340C2214yescomplemented
32#340C2214yescomplemented
33C25D411yescharacteristic
34C23×C4—24no—
35C23×C4—24no—
36C23×C4—21no—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4—21no—
40C23×C4—42no—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2314yes—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—44no—
65C2×C22⋊C4—44no—
66#106—44no—
67C22⋊C8—24no—
68C22⋊C8—24no—
69C22⋊C8—42no—
70#95—22no—
71#95—22no—
72#95—22no—
73#95—22no—
74C24—22no—
75C24—22no—
76C24—22no—
77C24—22no—
78C24—41no—
79C24—41no—
80C24SD1614yes—
81C24C2×D411yescharacteristic
82C24C2×D411yescharacteristic
83C24C2×D411yescharacteristic
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—416no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4#4114yes—
121C22×C4#4114yes—
122C22×C4C2411yescharacteristic
123C2×C8—24no—
124C2×C8—24no—
125C2×C8—24no—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C22⋊C4—44no—
135C22⋊C4—44no—
136C22⋊C4—44no—
137C22⋊C4—44no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—28no—
153C23—28no—
154C23—44no—
155C23—44no—
156C23—44no—
157C23—44no—
158C23—44no—
159C23#12312yes—
160C23#12312yes—
161C23C2×SD1612yes—
162C23C2×SD1612yes—
163C23C2×SD1612yes—
164C23AGL(1,ℤ/8)12yes—
165C23#13712yes—
166C23C22×D411yescharacteristic
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—28no—
172C2×C4—28no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4—84no—
195C2×C4—84no—
196C2×C4#12312yes—
197C2×C4#12312yes—
198C2×C4C2×#4112yes—
199C2×C4C2×#4112yes—
200C2×C4H(5,𝔽2)12yes—
201C8—44no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—416no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22#48114yes—
223C22#48314yes—
224C22C2×#12311yescharacteristic
225C22C2×#12311yescharacteristic
226C22#53411yescharacteristic
227C22C22×SD1611yescharacteristic
228C22C2×AGL(1,ℤ/8)11yescharacteristic
229C22C2×#13711yescharacteristic
230C22#57511yescharacteristic
231C4—22no—
232C4—22no—
233C4—41no—
234C4—82no—
235C4—82no—
236C4—82no—
237C4—82no—
238C4#55114yes—
239C2—24no—
240C2—24no—
241C2—42no—
242C2#248311yescharacteristic
243C2C2×#48311yescharacteristic
244C2C2×#48111yescharacteristic
245C2#309411yescharacteristic
246C2#309511yescharacteristic
247C2#318711yescharacteristic
248C2#319311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2077C2semidirect
4. #1853C2semidirect
5. #1853C2semidirect
6. #1824C2semidirect
7. #1823C2semidirect
16. C2×#106C22semidirect
30. #340C22semidirect
31. #340C22semidirect
32. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#3802
#20772
#18532
#18242
#18232
#17792