Finite Group Databasebeta

← #24545

Subgroups — #24545

Order 256.

504 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 127 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×#452C211yesmaximal, characteristic
2C2×#417C211yesmaximal, characteristic
3#2503C211yesmaximal, characteristic, complemented
4#2339C211yesmaximal, characteristic
5#2074C212yesmaximal, complemented
6#2056C212yesmaximal, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1877C212yesmaximal, complemented
9#1851C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1804C212yesmaximal, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C22×M16C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×D4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19#452—24no—
20C2×C4⋊C8C2212yes—
21#417—24no—
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28#400—21no—
29#400—21no—
30#391C2212yes—
31#391C2212yes—
32#386—22no—
33#382C2212yes—
34#381—22no—
35#381—21no—
36#380C2211yescharacteristic
37C4×C22⋊C4C2211yescharacteristic
38C2×#95—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C2×#41—22no—
44C2×#41—24no—
45C2×#41—42no—
46C2×#41D414yes—
47C22×Q8C2311yescharacteristic
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4C2311yescharacteristic
55C2×Q16—22no—
56C2×Q16—42no—
57C2×SD16—22no—
58C2×SD16—24no—
59C2×M16—22no—
60C2×M16—42no—
61C22×C8C2312yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C4×D4—24no—
67C4×D4—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4D412yes—
79C2×C42C2312yes—
80C4⋊C8—42no—
81#103—24no—
82#103—22no—
83#103—42no—
84#102—24no—
85#102—22no—
86C22⋊C8—22no—
87#95—21no—
88#95—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95D411yescharacteristic
95#95D411yescharacteristic
96C24—21no—
97C24C2×D411yescharacteristic
98#41—28no—
99#41—44no—
100#41—48no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8C2×D412yes—
109C2×D4—24no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—44no—
113C2×D4C2×D412yes—
114C2×D4#4114yes—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—42no—
128C22×C4—82no—
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D412yes—
133C22×C4#4112yes—
134C22×C4C2411yescharacteristic
135Q16—44no—
136SD16—44no—
137M16—44no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—44no—
155C22⋊C4—42no—
156C22⋊C4—81no—
157C42—22no—
158C42—22no—
159C23—22no—
160C23—24no—
161C23—22no—
162C23—22no—
163C23—21no—
164C23—21no—
165C23—42no—
166C23—41no—
167C23—41no—
168C23#12112yes—
169C23C22×D411yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—42no—
173Q8—41no—
174Q8—41no—
175Q8—42no—
176D4—22no—
177D4—24no—
178D4—48no—
179D4#12314yes—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—48no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—88no—
204C2×C4—84no—
205C2×C4#12112yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C2×#4112yes—
208C2×C4C2×#4111yescharacteristic
209C2×C4H(5,𝔽2)12yes—
210C8—41no—
211C8—42no—
212C8—41no—
213C22—22no—
214C22—24no—
215C22—22no—
216C22—22no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—44no—
221C22—44no—
222C22—42no—
223C22C2×#12111yescharacteristic
224C22#54612yes—
225C22#57512yes—
226C22#57711yescharacteristic
227C22#57811yescharacteristic
228C4—22no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—81no—
233C4—82no—
234C4—81no—
235C4—82no—
236C4C2×#12311yescharacteristic
237C4#53411yescharacteristic
238C4#55112yes—
239C2—42no—
240C2—44no—
241C2#241211yescharacteristic
242C2#304011yescharacteristic
243C2#304611yescharacteristic
244C2#331512yes—
245C2#332312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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
3. #2503C2semidirect
5. #2074C2semidirect
6. #2056C2semidirect
7. #1894C2semidirect
8. #1877C2semidirect
9. #1851C2semidirect
10. #1824C2semidirect
11. #1804C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
C2×#4172
#25032
#23392
#20742
#20562
#18942
#18772
#18512
#18242
#18042