Finite Group Databasebeta

← #33016

Subgroups — #33016

Order 256.

646 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 143 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×#520C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic
4#2625C211yesmaximal, characteristic, complemented
5#1878C212yesmaximal, complemented
6#1877C212yesmaximal, complemented
7#1865C212yesmaximal, complemented
8#1864C212yesmaximal, complemented
9#1824C211yesmaximal, characteristic, complemented
10#1818C211yesmaximal, characteristic, complemented
11#1788C211yesmaximal, characteristic
12C22×#41C2211yescharacteristic
13C22×Q16C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16#520—24no—
17#520—24no—
18C2×C4×Q8C2212yes—
19C2×C4×D4—22no—
20C2×C4×D4C2212yes—
21C2×#117—21no—
22C2×#117C2211yescharacteristic
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2211yescharacteristic
25#483—24no—
26#452—24no—
27C2×#106C2212yes—
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×#102C2211yescharacteristic
32C2×#102C2212yes—
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#400—21no—
36#398—21no—
37#397—22no—
38#391C2212yes—
39#389C2212yes—
40#380C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C2×#41—24no—
46C2×#41—24no—
47C2×#41—22no—
48C2×#41D414yes—
49C22×Q8C2311yescharacteristic
50C22×Q8C2311yescharacteristic
51C22×D4—21no—
52C22×D4C2311yescharacteristic
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—22no—
56C23×C4C2311yescharacteristic
57C2×Q16—22no—
58C2×Q16—24no—
59C2×SD16—22no—
60C2×SD16—24no—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2312yes—
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×D4—216no—
69C4×D4—24no—
70C4×D4—24no—
71#117—28no—
72#117—22no—
73#117—22no—
74C2×C4⋊C4—28no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D414yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4D412yes—
90C2×C42—22no—
91C2×C42C2312yes—
92#106—24no—
93#103—22no—
94#103—24no—
95#103—24no—
96#102—22no—
97#102—24no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100#95—21no—
101#95—22no—
102#95—22no—
103#95—22no—
104C24—21no—
105C24C2×D411yescharacteristic
106#41—216no—
107#41—28no—
108#41—28no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8C2×D412yes—
116C2×Q8#4114yes—
117C2×D4—28no—
118C2×D4—24no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4C2×D412yes—
122C2×D4#4114yes—
123C22×C4—28no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—28no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—84no—
135C22×C4—82no—
136C22×C4C2×D412yes—
137C22×C4C2×D412yes—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D412yes—
140C22×C4C2×D412yes—
141C22×C4C2411yescharacteristic
142Q16—44no—
143SD16—44no—
144C2×C8—21no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C4⋊C4—28no—
152C4⋊C4—22no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4C2×D414yes—
163C22⋊C4—28no—
164C22⋊C4—24no—
165C22⋊C4—42no—
166C22⋊C4—81no—
167C42—28no—
168C42—22no—
169C42—22no—
170C23—22no—
171C23—22no—
172C23—24no—
173C23—21no—
174C23—22no—
175C23—21no—
176C23—42no—
177C23—41no—
178C23—41no—
179C23#12012yes—
180C23C22×D411yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—41no—
184Q8—41no—
185Q8—42no—
186Q8#12314yes—
187D4—216no—
188D4—22no—
189D4—24no—
190D4#12314yes—
191C2×C4—216no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—44no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—84no—
214C2×C4—88no—
215C2×C4#12012yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4112yes—
219C2×C4C2×#4112yes—
220C8—41no—
221C8—41no—
222C8—42no—
223C22—22no—
224C22—24no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—44no—
231C22—42no—
232C22—42no—
233C22C2×#12011yescharacteristic
234C22#52512yes—
235C22#57712yes—
236C22#57711yescharacteristic
237C22#57811yescharacteristic
238C4—22no—
239C4—24no—
240C4—42no—
241C4—44no—
242C4—42no—
243C4—81no—
244C4—82no—
245C4—81no—
246C4C2×#12311yescharacteristic
247C4C2×#12311yescharacteristic
248C4#52512yes—
249C2—42no—
250C2—44no—
251C2#284911yescharacteristic
252C2#300911yescharacteristic
253C2#301311yescharacteristic
254C2#312012yes—
255C2#312512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#483C2semidirect
4. #2625C2semidirect
5. #1878C2semidirect
6. #1877C2semidirect
7. #1865C2semidirect
8. #1864C2semidirect
9. #1824C2semidirect
10. #1818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
C2×#4832
C2×#4522
#26252
#18782
#18772
#18652
#18642
#18242
#18182
#17882