Finite Group Databasebeta

← #29162

Subgroups — #29162

Order 256.

444 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 129 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#2857C211yesmaximal, characteristic, complemented
2C4×#121C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#1919C212yesmaximal, complemented
8#1895C212yesmaximal
9#1862C212yesmaximal, complemented
10#1809C212yesmaximal, complemented
11#1728C211yesmaximal, characteristic, complemented
12C22×M16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×#121C2211yescharacteristic
16C2×C4×D4—21no—
17C2×C4×D4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23#391—21no—
24#390C2211yescharacteristic
25#389—21no—
26#389C2211yescharacteristic
27#389C2212yes—
28#388—21no—
29#386—21no—
30#384C2211yescharacteristic
31#380—21no—
32#380C2211yescharacteristic
33#380C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C2×M16—24no—
51C2×M16—24no—
52C2×M16—42no—
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—42no—
56C22⋊Q8—44no—
57#121—22no—
58#121Q814yescomplemented
59C4×D4—42no—
60C4×D4—44no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C42—21no—
78C2×C42—42no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#107—42no—
84#106—24no—
85#102—24no—
86#102—42no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—22no—
92C24—21no—
93C24C2×D411yescharacteristic
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4—21no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4C2×Q812yes—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×Q812yes—
126C22×C4#4112yes—
127C22×C4C2411yescharacteristic
128M16—48no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8#4114yes—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—44no—
147C4⋊C4C2×Q812yes—
148C22⋊C4—24no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—44no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C42—22no—
155C42—22no—
156C42—41no—
157C42—41no—
158C42—42no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23C22⋊Q812yes—
173C23C22×D411yescharacteristic
174Q8—84no—
175D4—24no—
176D4—48no—
177D4—42no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4C22⋊Q812yes—
210C2×C4AGL(1,ℤ/8)12yes—
211C2×C4AGL(1,ℤ/8)12yes—
212C2×C4#13712yes—
213C2×C4C22×Q811yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C4C2×#4112yes—
216C2×C4H(5,𝔽2)12yes—
217C8—24no—
218C8—42no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22C2×C22⋊Q811yescharacteristic
234C22#53211yescharacteristic
235C22#53411yescharacteristic
236C22C2×AGL(1,ℤ/8)11yescharacteristic
237C22C2×AGL(1,ℤ/8)11yescharacteristic
238C22C2×#13711yescharacteristic
239C22#57511yescharacteristic
240C4—22no—
241C4—22no—
242C4—22no—
243C4—22no—
244C4—42no—
245C4—41no—
246C4—82no—
247C4—81no—
248C4—81no—
249C4—82no—
250C4#55112yes—
251C4#55412yes—
252C2—42no—
253C2—82no—
254C2#256411yescharacteristic
255C2#307711yescharacteristic
256C2#307911yescharacteristic
257C2#317111yescharacteristic
258C2#318211yescharacteristic
259C2#319211yescharacteristic
260C2#319611yescharacteristic
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
1. #2857C2semidirect
3. #2100C2semidirect
4. #2094C2semidirect
5. #2057C2semidirect
6. #2023C2semidirect
7. #1919C2semidirect
9. #1862C2semidirect
10. #1809C2semidirect
11. #1728C2semidirect
58. #121Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
C4×#1212
#21002
#20942
#20572
#20232
#19192
#18952
#18622
#18092
#17282