Finite Group Databasebeta

← #24069

Subgroups — #24069

Order 256.

565 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 131 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×#540C211yesmaximal, characteristic
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#480C211yesmaximal, characteristic, complemented
4#2317C211yesmaximal, characteristic, complemented
5#2059C212yesmaximal, complemented
6#2041C212yesmaximal, complemented
7#1875C211yesmaximal, characteristic, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1818C211yesmaximal, characteristic, complemented
12#540—24no—
13#540—24no—
14C2×#127C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#123—21no—
17C2×#123—21no—
18C2×#121—21no—
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#481—24no—
28#480—24no—
29C2×#107C2212yes—
30C2×#106C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#389C2212yes—
38#389C2212yes—
39#385—21no—
40#385—21no—
41#384—21no—
42#382—21no—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C22×D4C2311yescharacteristic
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C2311yescharacteristic
57C22×C8C2312yes—
58C22×C8C2312yes—
59#127—22no—
60#127—22no—
61#125—22no—
62#125—22no—
63#123—28no—
64#123—28no—
65#121—28no—
66#121—28no—
67#121—22no—
68#121—22no—
69#121D414yes—
70#121D414yes—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75#117—22no—
76#117—22no—
77#117—24no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2312yes—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—24no—
102#106—24no—
103#102—24no—
104#102—24no—
105#102—24no—
106#102—24no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109#95—22no—
110#95—22no—
111C24—21no—
112C24—21no—
113C24C2×D411yescharacteristic
114C2×D4—24no—
115C2×D4—24no—
116C2×D4—24no—
117C2×D4—24no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—82no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—82no—
128C2×D4C2×D412yes—
129C2×D4C2×D412yes—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D412yes—
148C22×C4C2×D412yes—
149C22×C4C2411yescharacteristic
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C4⋊C4#4114yes—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C42—24no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23#12012yes—
198C23C22×D411yescharacteristic
199D4—24no—
200D4—24no—
201D4—42no—
202D4—42no—
203D4—84no—
204D4—84no—
205D4—84no—
206D4—84no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—84no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4#12012yes—
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C2×#4112yes—
236C2×C42−1+412yes—
237C8—42no—
238C8—42no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22—82no—
252C22—82no—
253C22—82no—
254C22C2×#12011yescharacteristic
255C22#52511yescharacteristic
256C22#53611yescharacteristic
257C22#57612yes—
258C22#57712yes—
259C4—22no—
260C4—44no—
261C4—41no—
262C4—41no—
263C4—42no—
264C4—41no—
265C4—41no—
266C4—42no—
267C4—82no—
268C4#54712yes—
269C4#54712yes—
270C2—42no—
271C2—82no—
272C2—82no—
273C2#240611yescharacteristic
274C2#300811yescharacteristic
275C2#300911yescharacteristic
276C2#311912yes—
277C2#324112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#540C2
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×#481C2semidirect
3. C2×#480C2semidirect
4. #2317C2semidirect
5. #2059C2semidirect
6. #2041C2semidirect
7. #1875C2semidirect
8. #1875C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5402
C2×#4812
C2×#4802
#23172
#20592
#20412
#18752
#18642
#18622
#18182