Finite Group Databasebeta

← #23757

Subgroups — #23757

Order 256.

398 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 137 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×C22⋊Q8C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2095C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2076C212yesmaximal
7#2072C211yesmaximal, characteristic, complemented
8#2036C212yesmaximal
9C4×#107C212yesmaximal, complemented
10C4×#103C212yesmaximal, complemented
11C4×C22⋊C8C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×#117—21no—
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#107C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×C22⋊C8C2212yes—
21C2×C4×C8C2212yes—
22#391C2211yescharacteristic
23#389C2211yescharacteristic
24#387—21no—
25#385—21no—
26#384C2211yescharacteristic
27#384C2211yescharacteristic
28#382—21no—
29#380—21no—
30#380C2211yescharacteristic
31#380C2212yes—
32C4×C4⋊C4—21no—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2212yes—
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—21no—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8C2311yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51C4×Q8—24no—
52#117—44no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C42—24no—
69C2×C42—22no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—24no—
77#107—24no—
78#103—24no—
79#103—24no—
80C22⋊C8—24no—
81C4×C8—24no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—21no—
89C24C2×D411yescharacteristic
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8#4112yes—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—41no—
109C22×C4Q1614yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8#4114yes—
122C4⋊C4—24no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4#4112yes—
137C22⋊C4—24no—
138C22⋊C4—42no—
139C22⋊C4—44no—
140C42—22no—
141C42—22no—
142C42—22no—
143C42—42no—
144C42—42no—
145C42—44no—
146C42#4114yes—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23C2×Q1612yes—
154C23C22×D411yescharacteristic
155Q8—24no—
156Q8—42no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—44no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4#12312yes—
189C2×C4#12312yes—
190C2×C4C2×Q1612yes—
191C2×C4C2×Q1612yes—
192C2×C4#13512yes—
193C2×C4AGL(1,ℤ/8)12yes—
194C2×C4C2×#4112yes—
195C2×C4C2×#4111yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C4C2×#4112yes—
198C8—24no—
199C8—42no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22C2×#12311yescharacteristic
208C22#52511yescharacteristic
209C22#53211yescharacteristic
210C22C22×Q1611yescharacteristic
211C22C2×#13511yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C22#57511yescharacteristic
214C4—22no—
215C4—22no—
216C4—22no—
217C4—42no—
218C4—41no—
219C4—42no—
220C4—81no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4#48414yes—
225C4#52912yes—
226C4#52912yes—
227C2—42no—
228C2#239511yescharacteristic
229C2C2×#48411yescharacteristic
230C2#309111yescharacteristic
231C2#310411yescharacteristic
232C2#310611yescharacteristic
233C2#317011yescharacteristic
234C2#317111yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutator, Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2100C2semidirect
4. #2095C2semidirect
5. #2083C2semidirect
7. #2072C2semidirect
9. C4×#107C2semidirect
10. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#22852
#21002
#20952
#20832
#20762
#20722
#20362
C4×#1072
C4×#1032
C4×C22⋊C82