Finite Group Databasebeta

← #24499

Subgroups — #24499

Order 256.

475 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 123 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×#482C211yesmaximal, characteristic
2C2×#419C211yesmaximal, characteristic, complemented
3#2436C211yesmaximal, characteristic
4#2346C211yesmaximal, characteristic, complemented
5#2066C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1983C212yesmaximal, complemented
8#1928C212yesmaximal, complemented
9#1895C211yesmaximal, characteristic
10#1851C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C2×#124C2212yes—
15C2×#121C2211yescharacteristic
16C2×#117C2211yescharacteristic
17#482—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2212yes—
21#419—24no—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2212yes—
29C2×C22⋊C8C2211yescharacteristic
30#393—21no—
31#393—21no—
32#391C2212yes—
33#389C2212yes—
34#388—22no—
35#386—22no—
36#382—21no—
37#380C2211yescharacteristic
38C4×C4⋊C4C2212yes—
39C4×C22⋊C4C2211yescharacteristic
40#376—21no—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—24no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×Q8C2312yes—
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4C2311yescharacteristic
53C2×M16—22no—
54C2×M16—42no—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58#124—24no—
59#124—42no—
60#121—22no—
61#121—22no—
62#117—22no—
63#117D414yes—
64C2×C4⋊C4—22no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4D412yes—
75C2×C42—21no—
76C2×C42—22no—
77C2×C42C2312yes—
78#107—22no—
79#106—22no—
80C4⋊C8—42no—
81#103—24no—
82#103—24no—
83#103—24no—
84#103—42no—
85#102—22no—
86#102—22no—
87#102—42no—
88C22⋊C8—22no—
89#95—21no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95D411yescharacteristic
95#95D411yescharacteristic
96C24—21no—
97C24C2×D411yescharacteristic
98C2×Q8—22no—
99C2×Q8—28no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—44no—
105C2×Q8—48no—
106C2×Q8—44no—
107C2×Q8—42no—
108C2×Q8#4114yes—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—44no—
114C2×D4—42no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—44no—
123C22×C4—42no—
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4#4112yes—
129C22×C4C2411yescharacteristic
130M16—44no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—82no—
145C4⋊C4C2×D412yes—
146C4⋊C4C2×D412yes—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C22⋊C4—44no—
150C22⋊C4—44no—
151C22⋊C4—42no—
152C22⋊C4—81no—
153C42—24no—
154C42—22no—
155C42—22no—
156C42—44no—
157C23—22no—
158C23—22no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—42no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23#12112yes—
167C23C22×D411yescharacteristic
168Q8—24no—
169Q8—24no—
170Q8—48no—
171Q8—42no—
172Q8—42no—
173D4—41no—
174D4—41no—
175D4—42no—
176D4—84no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—48no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4#12112yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C2×#4112yes—
204C2×C4C2×#4111yescharacteristic
205C2×C4H(5,𝔽2)12yes—
206C8—41no—
207C8—41no—
208C8—42no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—82no—
218C22—82no—
219C22C2×#12111yescharacteristic
220C22#54612yes—
221C22#57512yes—
222C22#57711yescharacteristic
223C22#57811yescharacteristic
224C4—22no—
225C4—44no—
226C4—42no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4—82no—
232C4#52511yescharacteristic
233C4#53511yescharacteristic
234C4#55112yes—
235C2—42no—
236C2—161no—
237C2#241111yescharacteristic
238C2#304011yescharacteristic
239C2#304611yescharacteristic
240C2#332112yes—
241C2#332712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#419C2semidirect
4. #2346C2semidirect
5. #2066C2semidirect
6. #2042C2semidirect
7. #1983C2semidirect
8. #1928C2semidirect
10. #1851C2semidirect
11. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4192
#24362
#23462
#20662
#20422
#19832
#19282
#18952
#18512
#17872