Finite Group Databasebeta

← #24745

Subgroups — #24745

Order 256.

504 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 135 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×#463C212yesmaximal
2C2×#461C211yesmaximal, characteristic, complemented
3C2×#442C211yesmaximal, characteristic
4#2547C211yesmaximal, characteristic, complemented
5#2349C211yesmaximal, characteristic, complemented
6#2066C212yesmaximal, complemented
7#2042C212yesmaximal, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1877C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×D4C2211yescharacteristic
17C2×C4×D4C2212yes—
18C2×C4×D4C2211yescharacteristic
19#463—28no—
20#461—24no—
21#442—24no—
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32#397—21no—
33#397—21no—
34#391C2212yes—
35#389C2212yes—
36#385—22no—
37#382C2211yescharacteristic
38#380—22no—
39#380—21no—
40C4×C4⋊C4C2211yescharacteristic
41C2×#95—21no—
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2312yes—
45C22×D4C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—22no—
50C2×SD16—24no—
51C2×SD16—24no—
52C2×SD16—24no—
53C2×D8—22no—
54C2×D8—22no—
55C2×D8D414yes—
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61#124—24no—
62#124—42no—
63C4×D4—24no—
64C4×D4—22no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—42no—
68C4×D4D414yes—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D412yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C42C2312yes—
82C2×C42C2311yescharacteristic
83#106—22no—
84C4⋊C8—24no—
85C4⋊C8—22no—
86#103—24no—
87#103—24no—
88#103—42no—
89#102—22no—
90#102—24no—
91#102—22no—
92#97—22no—
93#95—21no—
94#95—22no—
95#95—22no—
96#95—22no—
97#95—21no—
98#95D411yescharacteristic
99#95D411yescharacteristic
100C24—21no—
101C24—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4C2×D412yes—
117C2×D4C2×D412yes—
118C2×D4C2×D412yes—
119C2×D4#4112yes—
120C22×C4—21no—
121C22×C4—24no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—82no—
130C22×C4—82no—
131C22×C4C2×D412yes—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136SD16—48no—
137D8—28no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8C2×D412yes—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—82no—
156C4⋊C4C2×D412yes—
157C22⋊C4—24no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C22⋊C4—44no—
161C22⋊C4—42no—
162C42—22no—
163C42—22no—
164C42C2×D412yes—
165C42#4112yes—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—42no—
171C23—42no—
172C23—41no—
173C23—41no—
174C23—42no—
175C23—42no—
176C23C22×D411yescharacteristic
177Q8—42no—
178Q8—42no—
179Q8—42no—
180Q8—42no—
181D4—22no—
182D4—22no—
183D4—22no—
184D4—22no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4#12114yes—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—84no—
208C2×C4—84no—
209C2×C4—88no—
210C2×C4#12112yes—
211C2×C4#12112yes—
212C2×C4C22×D411yescharacteristic
213C2×C4C22×D411yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C4C2×#4111yescharacteristic
216C2×C4H(5,𝔽2)11yescharacteristic
217C2×C42−1+411yescharacteristic
218C8—22no—
219C8—42no—
220C8—41no—
221C22—24no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—44no—
228C22—82no—
229C22—81no—
230C22—81no—
231C22C2×#12111yescharacteristic
232C22D4211yescharacteristic
233C22#54811yescharacteristic
234C22#57512yes—
235C22#57712yes—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—82no—
243C4—81no—
244C4C2×#12111yescharacteristic
245C4#53711yescharacteristic
246C4#54711yescharacteristic
247C4#55311yescharacteristic
248C2—44no—
249C2—82no—
250C2#241611yescharacteristic
251C2#304011yescharacteristic
252C2#304211yescharacteristic
253C2#326712yes—
254C2#337612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#461C2semidirect
4. #2547C2semidirect
5. #2349C2semidirect
6. #2066C2semidirect
7. #2042C2semidirect
8. #1960C2semidirect
9. #1940C2semidirect
10. #1877C2semidirect
11. #1804C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4612
C2×#4422
#25472
#23492
#20662
#20422
#19602
#19402
#18772
#18042