Finite Group Databasebeta

← #27857

Subgroups — #27857

Order 256.

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

normal subgroups only

This group has 425 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
1#2854C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic, complemented
3#2100C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#1926C212yesmaximal, complemented
8#1919C212yesmaximal, complemented
9#1893C212yesmaximal
10#1807C212yesmaximal, complemented
11C2×#336C211yesmaximal, characteristic, complemented
12C23×C8C2212yes—
13C2×#123—21no—
14C2×#121C2211yescharacteristic
15C2×#117—21no—
16C22×C42C2211yescharacteristic
17C2×#106C2212yes—
18C2×#106C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21#390C2211yescharacteristic
22#390C2211yescharacteristic
23#389C2211yescharacteristic
24#389C2211yescharacteristic
25#388—21no—
26#388—21no—
27#386—21no—
28#384C2211yescharacteristic
29#384C2211yescharacteristic
30#382—21no—
31#380—21no—
32#380C2211yescharacteristic
33#380C2212yes—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8—24no—
49C22×C8—24no—
50C22×C8—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53#123—44no—
54#121—24no—
55#117—44no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—24no—
75C2×C42—22no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#106—24no—
81#106—42no—
82#102—24no—
83#102—42no—
84#95—22no—
85#95—22no—
86#95—21no—
87#95—21no—
88#95—21no—
89C24—21no—
90C24C2×D411yescharacteristic
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—42no—
96C2×D4—42no—
97C2×D4—42no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—41no—
111C22×C4SD1614yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4#4114yes—
118C22×C4C2411yescharacteristic
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—48no—
125C2×C8#4114yes—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—44no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42—22no—
151C42—44no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23#12312yes—
164C23C2×SD1612yes—
165C23AGL(1,ℤ/8)12yes—
166C23C22×D411yescharacteristic
167D4—42no—
168D4—42no—
169D4—84no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—28no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—44no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4#12312yes—
201C2×C4C2×SD1612yes—
202C2×C4C2×SD1612yes—
203C2×C4#13512yes—
204C2×C4C2×#4112yes—
205C2×C4C2×#4112yes—
206C2×C4H(5,𝔽2)11yescharacteristic
207C2×C42−1+411yescharacteristic
208C8—24no—
209C8—42no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22#48114yes—
224C22C2×#12311yescharacteristic
225C22#53211yescharacteristic
226C22#53711yescharacteristic
227C22C22×SD1611yescharacteristic
228C22C2×#13511yescharacteristic
229C22C2×AGL(1,ℤ/8)11yescharacteristic
230C22#57511yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—22no—
235C4—41no—
236C4—81no—
237C4—81no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4#55112yes—
242C4#55212yes—
243C2—24no—
244C2—161no—
245C2#249811yescharacteristic
246C2C2×#48111yescharacteristic
247C2#309111yescharacteristic
248C2#316711yescharacteristic
249C2#318011yescharacteristic
250C2#324611yescharacteristic
251C2#324711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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. #2100C2semidirect
4. #2094C2semidirect
5. #2084C2semidirect
6. #2071C2semidirect
7. #1926C2semidirect
8. #1919C2semidirect
10. #1807C2semidirect
11. C2×#336C2semidirect

Maximal subgroups

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

subgroupindex
#28542
#22852
#21002
#20942
#20842
#20712
#19262
#19192
#18932
#18072
C2×#3362