Finite Group Databasebeta

← #25662

Subgroups — #25662

Order 256.

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

normal subgroups only

This group has 514 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
1C2×#556C211yesmaximal, characteristic
2C2×#489C211yesmaximal, characteristic
3C2×#475C211yesmaximal, characteristic, complemented
4#2567C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2042C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1983C211yesmaximal, characteristic
9#1865C212yesmaximal, complemented
10#1864C212yesmaximal
11#1763C211yesmaximal, characteristic, complemented
12#556—24no—
13#556—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#126—21no—
17C2×#126—21no—
18C2×#124C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117—21no—
26#489—24no—
27#475—24no—
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#97C2211yescharacteristic
37#395—21no—
38#395—21no—
39#391—21no—
40#389C2212yes—
41#389C2212yes—
42#384—21no—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8D411yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×Q8C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59#126—28no—
60#126—28no—
61#124—22no—
62#124—22no—
63#124—22no—
64#124D414yes—
65C22⋊Q8—28no—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8D414yes—
72C4×D4—22no—
73C4×D4—22no—
74#117—28no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#106—24no—
100C4⋊C8—22no—
101#103—24no—
102#103—24no—
103#103—24no—
104#103—24no—
105#102—24no—
106#102—24no—
107#97—22no—
108#95—21no—
109#95—22no—
110#95—22no—
111C24—21no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—41no—
121C2×Q8—42no—
122C2×Q8—82no—
123C2×Q8C2×D412yes—
124C2×Q8C2×D412yes—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4C2×D412yes—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2411yescharacteristic
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C4⋊C4C2×D412yes—
165C4⋊C4#4114yes—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—24no—
171C42—24no—
172C42—24no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—42no—
178C42C2×D412yes—
179C42C2×D412yes—
180C23—22no—
181C23—42no—
182C23—41no—
183C23—41no—
184C23—42no—
185C23C22×D411yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—24no—
189Q8—44no—
190Q8—41no—
191Q8—41no—
192Q8—42no—
193Q8—84no—
194D4—24no—
195D4—41no—
196D4—41no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4#12012yes—
224C2×C4#12012yes—
225C2×C4#13712yes—
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C2×C4C2×#4112yes—
229C2×C42−1+412yes—
230C8—42no—
231C8—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—82no—
236C22—82no—
237C22C2×#12011yescharacteristic
238C22#54812yes—
239C22C2×#13711yescharacteristic
240C22#57511yescharacteristic
241C22#57712yes—
242C4—22no—
243C4—22no—
244C4—22no—
245C4—44no—
246C4—42no—
247C4—42no—
248C4—41no—
249C4—42no—
250C4—41no—
251C4—41no—
252C4—82no—
253C4#53211yescharacteristic
254C4#53611yescharacteristic
255C4#54712yes—
256C2—82no—
257C2#243211yescharacteristic
258C2#300711yescharacteristic
259C2#301011yescharacteristic
260C2#336912yes—
261C2#337612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#475C2semidirect
4. #2567C2semidirect
5. #2043C2semidirect
7. #1985C2semidirect
9. #1865C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4892
C2×#4752
#25672
#20432
#20422
#19852
#19832
#18652
#18642
#17632