Finite Group Databasebeta

← #32989

Subgroups — #32989

Order 256.

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

normal subgroups only

This group has 650 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×#484C211yesmaximal, characteristic, complemented
3C2×#485C211yesmaximal, characteristic, complemented
4#2616C211yesmaximal, characteristic
5#1873C212yesmaximal, complemented
6#1872C212yesmaximal, complemented
7#1854C212yesmaximal, complemented
8#1852C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic, complemented
10#1823C211yesmaximal, characteristic, complemented
11#1780C211yesmaximal, characteristic
12C22×#41C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2212yes—
15#520—24no—
16#520—24no—
17C2×C4×Q8—21no—
18C2×C4×D4—22no—
19C2×C4×D4—21no—
20C2×#117C2212yes—
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2212yes—
23C22×C4⋊C4C2211yescharacteristic
24#485—24no—
25#484—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2212yes—
28C2×#106C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2212yes—
35C2×C22⋊C8C2212yes—
36#395—21no—
37#394—21no—
38#394—22no—
39#380C2211yescharacteristic
40#380C2212yes—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—24no—
48C2×#41—24no—
49C22×Q8D411yescharacteristic
50C22×Q8C2311yescharacteristic
51C22×D4D411yescharacteristic
52C22×D4D412yes—
53C23×C4—22no—
54C23×C4—21no—
55C23×C4C2311yescharacteristic
56C23×C4C2312yes—
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C22×C8C2312yes—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64C4×Q8—28no—
65C4×D4—216no—
66C4×D4—28no—
67#117—22no—
68#117—24no—
69#117—24no—
70#117D414yes—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4D414yes—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C42—22no—
91C2×C42—21no—
92#107—24no—
93#107—22no—
94#106—22no—
95#103—24no—
96#103—22no—
97#103—22no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100C22⋊C8—24no—
101C22⋊C8—24no—
102#95—21no—
103#95—22no—
104C24C2×D411yescharacteristic
105C24C2×D412yes—
106#41—28no—
107#41—28no—
108#41—216no—
109C2×Q8—22no—
110C2×Q8—24no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×D4—24no—
116C2×D4—24no—
117C2×D4—24no—
118C2×D4—22no—
119C2×D4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—28no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—81no—
135C22×C4—82no—
136C22×C4—81no—
137C22×C4—82no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D412yes—
140C22×C4C2×D412yes—
141C22×C4C2×D412yes—
142C22×C4#4114yes—
143C22×C4#4114yes—
144C22×C4C2411yescharacteristic
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C4⋊C4—28no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D414yes—
166C22⋊C4—28no—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—44no—
170C22⋊C4—81no—
171C22⋊C4—82no—
172C42—28no—
173C42—24no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—21no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23#12012yes—
185C23#12314yes—
186C23C22×D411yescharacteristic
187Q8—28no—
188Q8—42no—
189Q8—41no—
190Q8—41no—
191D4—28no—
192D4—216no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—84no—
220C2×C4—88no—
221C2×C4#12012yes—
222C2×C4#12314yes—
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4112yes—
226C2×C4C2×#4112yes—
227C8—41no—
228C8—42no—
229C8—41no—
230C22—24no—
231C22—24no—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—22no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22C2×#12011yescharacteristic
243C22C2×#12312yes—
244C22#57611yescharacteristic
245C22#57711yescharacteristic
246C22#57812yes—
247C4—22no—
248C4—24no—
249C4—42no—
250C4—44no—
251C4—42no—
252C4—82no—
253C4—81no—
254C4—81no—
255C4#52511yescharacteristic
256C4#52511yescharacteristic
257C4#52512yes—
258C2—44no—
259C2—42no—
260C2#284911yescharacteristic
261C2#301111yescharacteristic
262C2#301211yescharacteristic
263C2#309712yes—
264C2#309812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#484C2semidirect
3. C2×#485C2semidirect
5. #1873C2semidirect
6. #1872C2semidirect
7. #1854C2semidirect
8. #1852C2semidirect
9. #1825C2semidirect
10. #1823C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
C2×#4842
C2×#4852
#26162
#18732
#18722
#18542
#18522
#18252
#18232
#17802