Finite Group Databasebeta

← #27678

Subgroups — #27678

Order 256.

485 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C4×#121C211yesmaximal, characteristic
2C2×#384C211yesmaximal, characteristic, complemented
3#2100C212yesmaximal, complemented
4#2094C212yesmaximal, complemented
5#1948C212yesmaximal, complemented
6#1874C212yesmaximal
7#1853C212yesmaximal, complemented
8#1802C212yesmaximal, complemented
9#1797C211yesmaximal, characteristic, complemented
10C2×#121C2211yescharacteristic
11C2×C4×D4—21no—
12C2×C4×D4C2211yescharacteristic
13C22×C4⋊C4—21no—
14C22×C4⋊C4—21no—
15C22×C4⋊C4C2212yes—
16C22×C42C2211yescharacteristic
17C2×#106C2214yes—
18C2×C4⋊C8C2212yes—
19C2×#102C2214yes—
20C2×C22⋊C8C2212yes—
21#390C2211yescharacteristic
22#389C2211yescharacteristic
23#388—21no—
24#386—21no—
25#384—21no—
26#384—21no—
27#384C2212yes—
28#384C2212yes—
29#380C2211yescharacteristic
30C4×C4⋊C4C2211yescharacteristic
31C4×C22⋊C4—21no—
32#340C2214yes—
33#340C2214yes—
34#336C2212yes—
35C22×D4—21no—
36C22×D4C2311yescharacteristic
37C23×C4—22no—
38C23×C4—21no—
39C23×C4D411yescharacteristic
40C23×C4D411yescharacteristic
41C23×C4C2311yescharacteristic
42C22×C8C2314yes—
43#121—22no—
44#121Q814yes—
45C4×D4—24no—
46C4×D4—44no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4—44no—
55C2×C4⋊C4—44no—
56C2×C4⋊C4C2314yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C42—24no—
66C2×C42—21no—
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#106—44no—
72C4⋊C8—24no—
73#102—44no—
74C22⋊C8—24no—
75#95—21no—
76#95—21no—
77#95—21no—
78C24—21no—
79C24C2×D411yescharacteristic
80C2×D4—24no—
81C2×D4—22no—
82C2×D4—21no—
83C2×D4—22no—
84C2×D4—42no—
85C2×D4—42no—
86C2×D4C2×Q812yes—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—24no—
98C22×C4—44no—
99C22×C4—44no—
100C22×C4—44no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4SD1614yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×Q812yes—
111C22×C4#4112yes—
112C22×C4C2411yescharacteristic
113C2×C8—24no—
114C2×C8—24no—
115C2×C8—24no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—24no—
121C4⋊C4—44no—
122C4⋊C4—44no—
123C4⋊C4—44no—
124C4⋊C4—44no—
125C4⋊C4—42no—
126C4⋊C4—44no—
127C4⋊C4C2×Q812yes—
128C22⋊C4—24no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C42—22no—
134C42—22no—
135C42—22no—
136C42—42no—
137C42#4114yes—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23C22⋊Q812yes—
150C23C2×SD1612yes—
151C23AGL(1,ℤ/8)12yes—
152C23C22×D411yescharacteristic
153D4—24no—
154D4—48no—
155D4—42no—
156C2×C4—28no—
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—24no—
166C2×C4—24no—
167C2×C4—28no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—44no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—88no—
181C2×C4—84no—
182C2×C4C22⋊Q812yes—
183C2×C4#12312yes—
184C2×C4#12312yes—
185C2×C4C2×SD1612yes—
186C2×C4C2×SD1612yes—
187C2×C4AGL(1,ℤ/8)12yes—
188C2×C4#13712yes—
189C2×C4C22×Q811yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4112yes—
192C2×C4H(5,𝔽2)12yes—
193C8—44no—
194C22—22no—
195C22—22no—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22#47614yes—
208C22C2×C22⋊Q811yescharacteristic
209C22C2×#12311yescharacteristic
210C22#53511yescharacteristic
211C22C22×SD1611yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C22C2×AGL(1,ℤ/8)11yescharacteristic
214C22C2×#13711yescharacteristic
215C4—22no—
216C4—22no—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—81no—
221C4—81no—
222C4—84no—
223C4#48114yes—
224C4#55312yes—
225C4#55412yes—
226C2—24no—
227C2—82no—
228C2#248711yescharacteristic
229C2C2×#47611yescharacteristic
230C2#307911yescharacteristic
231C2C2×#48111yescharacteristic
232C2#309211yescharacteristic
233C2#320611yescharacteristic
234C2#321711yescharacteristic
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. C2×#384C2semidirect
3. #2100C2semidirect
4. #2094C2semidirect
5. #1948C2semidirect
7. #1853C2semidirect
8. #1802C2semidirect
9. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
C2×#3842
#21002
#20942
#19482
#18742
#18532
#18022
#17972