Finite Group Databasebeta

← #27890

Subgroups — #27890

Order 256.

469 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#2856C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic, complemented
3#2034C214yesmaximal, complemented
4#1957C211yesmaximal, characteristic, complemented
5#1957C211yesmaximal, characteristic, complemented
6#1926C214yesmaximal, complemented
7#1844C211yesmaximal, characteristic, complemented
8#1834C211yesmaximal, characteristic, complemented
9C4×C22⋊C8C211yesmaximal, characteristic, complemented
10C2×#127C2211yescharacteristic
11C2×#121—21no—
12C2×#121—21no—
13C2×#121C2212yes—
14C2×#117—21no—
15C22×C42C2211yescharacteristic
16C2×#106C2214yes—
17C2×#102C2214yes—
18C2×#102C2214yes—
19C2×C22⋊C8C2212yes—
20C2×C4×C8C2211yescharacteristic
21C2×C4×C8C2211yescharacteristic
22#390—21no—
23#390—21no—
24#390C2212yes—
25#384C2211yescharacteristic
26#384C2211yescharacteristic
27#384C2211yescharacteristic
28#382—21no—
29#380—21no—
30#380C2212yes—
31C4×C4⋊C4—21no—
32C4×C4⋊C4C2212yes—
33#340C2214yes—
34#336C2212yes—
35C22×D4—22no—
36C22×D4—21no—
37C22×D4—21no—
38C22×D4C2312yes—
39C23×C4D411yescharacteristic
40C23×C4D411yescharacteristic
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44#127—22no—
45#127—22no—
46#121—44no—
47#121—44no—
48#121—44no—
49#117—44no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C42—22no—
64C2×C42—22no—
65C2×C42—22no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#106—44no—
71#102—28no—
72#102—44no—
73C22⋊C8—24no—
74C4×C8—22no—
75C4×C8—22no—
76#95—22no—
77#95—22no—
78C24—22no—
79C24C2×D411yescharacteristic
80C2×D4—24no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—24no—
84C2×D4—44no—
85C2×D4—44no—
86C2×D4—44no—
87C2×D4—42no—
88C2×D4—42no—
89C2×D4—42no—
90C2×D4—42no—
91C2×D4—42no—
92C2×D4—42no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4SD1614yes—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8#4114yes—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C22⋊C4—44no—
128C22⋊C4—44no—
129C22⋊C4—44no—
130C22⋊C4—44no—
131C42—22no—
132C42—22no—
133C42—21no—
134C42—21no—
135C42—44no—
136C42#4112yes—
137C42#4112yes—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23C2×SD1612yes—
152C23C22×D411yescharacteristic
153D4—44no—
154D4—44no—
155D4—88no—
156D4—84no—
157D4—84no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—28no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—42no—
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#12312yes—
180C2×C4#12312yes—
181C2×C4C2×SD1612yes—
182C2×C4C2×SD1612yes—
183C2×C4#13512yes—
184C2×C4AGL(1,ℤ/8)12yes—
185C2×C4AGL(1,ℤ/8)12yes—
186C2×C4C2×#4111yescharacteristic
187C2×C4C2×#4112yes—
188C2×C4C2×#4111yescharacteristic
189C2×C4H(5,𝔽2)12yes—
190C8—24no—
191C8—42no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—82no—
200C22—82no—
201C22—82no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22—82no—
206C22C2×#12311yescharacteristic
207C22#53211yescharacteristic
208C22#53511yescharacteristic
209C22C22×SD1611yescharacteristic
210C22C2×#13511yescharacteristic
211C22C2×AGL(1,ℤ/8)11yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C4—22no—
214C4—41no—
215C4—41no—
216C4—82no—
217C4—82no—
218C4—82no—
219C4#48114yes—
220C4#48214yes—
221C4#55312yes—
222C4#55512yes—
223C2—42no—
224C2—162no—
225C2#250011yescharacteristic
226C2C2×#48211yescharacteristic
227C2C2×#48111yescharacteristic
228C2#316711yescharacteristic
229C2#318411yescharacteristic
230C2#320711yescharacteristic
231C2#320811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2856C2
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. #2034C2semidirect
4. #1957C2semidirect
5. #1957C2semidirect
6. #1926C2semidirect
7. #1844C2semidirect
8. #1834C2semidirect
9. C4×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#28562
#22852
#20342
#19572
#19262
#18442
#18342
C4×C22⋊C82