Finite Group Databasebeta

← #27585

Subgroups — #27585

Order 256.

564 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 133 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×#390C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic, complemented
3#2094C212yesmaximal, complemented
4#2071C212yesmaximal, complemented
5#1926C214yesmaximal, complemented
6#1853C212yesmaximal, complemented
7#1817C211yesmaximal, characteristic, complemented
8#1786C212yesmaximal
9C23×D4C2211yescharacteristic
10C2×#117—21no—
11C22×C4⋊C4C2211yescharacteristic
12C22×C22⋊C4—21no—
13C22×C22⋊C4—21no—
14C22×C42C2211yescharacteristic
15C2×#106C2214yescomplemented
16C2×#102C2214yes—
17C2×#102C2214yes—
18C2×C22⋊C8C2212yes—
19#390—21no—
20#390—21no—
21#390C2212yes—
22#390C2212yes—
23#384C2211yescharacteristic, complemented
24#384C2211yescharacteristic, complemented
25#382—21no—
26#380—21no—
27#380C2212yes—
28C4×C4⋊C4—21no—
29C4×C4⋊C4C2212yescomplemented
30#340C2214yescomplemented
31#336C2212yescomplemented
32#336C2212yescomplemented
33C25—21no—
34C22×D4—24no—
35C22×D4—21no—
36C22×D4—24no—
37C22×D4—42no—
38C22×D4C2312yes—
39C23×C4—21no—
40C23×C4D411yescharacteristic
41C23×C4D411yescharacteristic
42C23×C4C2311yescharacteristic
43C22×C8C2314yes—
44#117—44no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—42no—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2312yes—
54C2×C22⋊C4—22no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—44no—
60C2×C22⋊C4—44no—
61C2×C42—24no—
62C2×C42—22no—
63C2×C42D411yescharacteristic
64C2×C42D411yescharacteristic
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67#106—44no—
68#102—28no—
69#102—44no—
70C22⋊C8—24no—
71#95—22no—
72#95—22no—
73C24—24no—
74C24—22no—
75C24—42no—
76C24—42no—
77C24—42no—
78C24—42no—
79C24—42no—
80C24—42no—
81C24C2×D411yescharacteristic
82C2×D4—24no—
83C2×D4—24no—
84C2×D4—22no—
85C2×D4—24no—
86C2×D4—416no—
87C2×D4—42no—
88C2×D4—42no—
89C2×D4#4114yes—
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—41no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4SD1614yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4#4114yes—
110C22×C4C2411yescharacteristic
111C2×C8—24no—
112C2×C8—24no—
113C2×C8—24no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—24no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C22⋊C4—44no—
123C22⋊C4—44no—
124C22⋊C4—44no—
125C22⋊C4—44no—
126C22⋊C4—44no—
127C42—22no—
128C42—22no—
129C42—22no—
130C42—22no—
131C42—44no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—24no—
137C23—48no—
138C23—48no—
139C23—48no—
140C23—42no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—84no—
147C23—84no—
148C23—84no—
149C23—84no—
150C23#12312yes—
151C23C2×SD1612yes—
152C23AGL(1,ℤ/8)12yes—
153C23C22×D411yescharacteristic
154D4—28no—
155D4—48no—
156D4—44no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—28no—
166C2×C4—28no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—84no—
181C2×C4#12312yes—
182C2×C4C2×SD1612yes—
183C2×C4C2×SD1612yes—
184C2×C4AGL(1,ℤ/8)12yes—
185C2×C4C2×#4112yes—
186C2×C4C2×#4112yes—
187C2×C4H(5,𝔽2)12yes—
188C8—44no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—44no—
196C22—44no—
197C22—44no—
198C22—816no—
199C22—82no—
200C22—82no—
201C22—82no—
202C22—82no—
203C22#48114yes—
204C22C2×#12311yescharacteristic
205C22#52511yescharacteristic
206C22#53511yescharacteristic
207C22C22×SD1611yescharacteristic
208C22C2×AGL(1,ℤ/8)11yescharacteristic
209C22C2×AGL(1,ℤ/8)11yescharacteristic
210C22#57511yescharacteristic
211C4—22no—
212C4—22no—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—82no—
217C4—82no—
218C4—82no—
219C4#55114yes—
220C2—24no—
221C2—84no—
222C2#248211yescharacteristic
223C2C2×#48111yescharacteristic
224C2#309411yescharacteristic
225C2#310111yescharacteristic
226C2#310911yescharacteristic
227C2#320811yescharacteristic
228C2#321311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2285C2semidirect
3. #2094C2semidirect
4. #2071C2semidirect
5. #1926C2semidirect
6. #1853C2semidirect
7. #1817C2semidirect
15. C2×#106C22semidirect
23. #384C22semidirect
24. #384C22semidirect
29. C4×C4⋊C4C22semidirect
30. #340C22semidirect
31. #336C22semidirect
32. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#3902
#22852
#20942
#20712
#19262
#18532
#18172
#17862