Finite Group Databasebeta

← #24971

Subgroups — #24971

Order 256.

421 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 131 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#2538C211yesmaximal, characteristic
2D4×C4⋊C4C211yesmaximal, characteristic
3#2076C212yesmaximal, complemented
4#2036C212yesmaximal, complemented
5#1954C211yesmaximal, characteristic
6#1947C211yesmaximal, characteristic, complemented
7#1872C212yesmaximal, complemented
8#1834C211yesmaximal, characteristic, complemented
9#1824C212yesmaximal, complemented
10#1803C211yesmaximal, characteristic, complemented
11C4×#103C211yesmaximal, characteristic, complemented
12C2×C22⋊Q8C2212yes—
13C2×C4×Q8C2211yescharacteristic
14C2×C4×D4—21no—
15C2×C4×D4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C2×#107C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×#103C2211yescharacteristic
20C2×#103C2211yescharacteristic
21C2×#103C2212yes—
22C2×C22⋊C8C2212yes—
23C2×C22⋊C8C2212yes—
24C2×C4×C8C2211yescharacteristic
25#396—22no—
26#390—21no—
27#389—21no—
28#389—21no—
29#385—22no—
30#384C2211yescharacteristic
31#384C2211yescharacteristic
32#380C2212yes—
33#380C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41C22×Q8C2311yescharacteristic
42C22×D4D411yescharacteristic
43C23×C4—22no—
44C23×C4C2312yes—
45C22×C8C2311yescharacteristic
46C22×C8C2311yescharacteristic
47C22×C8C2312yes—
48C22⋊Q8—24no—
49C22⋊Q8—42no—
50C4×Q8—22no—
51C4×Q8—22no—
52C4×D4—22no—
53C4×D4—22no—
54C4×D4—44no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4D412yes—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C2311yescharacteristic
77#107—42no—
78C4⋊C8—22no—
79#103—22no—
80#103—22no—
81#103—42no—
82C22⋊C8—42no—
83C22⋊C8—42no—
84C4×C8—22no—
85#95—21no—
86#95—22no—
87#95—22no—
88C24C2×D412yes—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8#4112yes—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4Q1614yes—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—82no—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—81no—
142C4⋊C4—81no—
143C4⋊C4#4112yes—
144C22⋊C4—24no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C22⋊C4—82no—
148C42—42no—
149C42—42no—
150C42#4112yes—
151C42#4112yes—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23C2×Q1614yes—
159C23C22×D411yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164D4—28no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—88no—
193C2×C4#12312yes—
194C2×C4#12312yes—
195C2×C4C2×Q1612yes—
196C2×C4AGL(1,ℤ/8)12yes—
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4H(5,𝔽2)11yescharacteristic
202C2×C4H(5,𝔽2)11yescharacteristic
203C8—42no—
204C8—41no—
205C8—41no—
206C22—24no—
207C22—24no—
208C22—24no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22C2×#12311yescharacteristic
214C22#54611yescharacteristic
215C22#54611yescharacteristic
216C22C22×Q1611yescharacteristic
217C22C2×AGL(1,ℤ/8)11yescharacteristic
218C22#57711yescharacteristic
219C22#57711yescharacteristic
220C4—22no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—82no—
225C4—81no—
226C4—81no—
227C4—82no—
228C4#48414yes—
229C4#52911yescharacteristic
230C4#54011yescharacteristic
231C4#55311yescharacteristic
232C4#55511yescharacteristic
233C2—44no—
234C2#242011yescharacteristic
235C2C2×#48411yescharacteristic
236C2#309911yescharacteristic
237C2#329911yescharacteristic
238C2#329911yescharacteristic
239C2#332411yescharacteristic
240C2#333211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2076C2semidirect
4. #2036C2semidirect
6. #1947C2semidirect
7. #1872C2semidirect
8. #1834C2semidirect
9. #1824C2semidirect
10. #1803C2semidirect
11. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
#25382
D4×C4⋊C42
#20762
#20362
#19542
#19472
#18722
#18342
#18242
#18032
C4×#1032