Finite Group Databasebeta

← #24104

Subgroups — #24104

Order 256.

573 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 573 conjugacy classes of subgroups — too many to draw. Only its 133 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×#555C211yesmaximal, characteristic
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#428C211yesmaximal, characteristic, complemented
4#2486C211yesmaximal, characteristic, complemented
5#2058C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#1895C211yesmaximal, characteristic, complemented
8#1870C211yesmaximal, characteristic, complemented
9#1862C212yesmaximal, complemented
10#1854C212yesmaximal, complemented
11#1825C211yesmaximal, characteristic, complemented
12C22×M16C2211yescharacteristic
13#555—24no—
14#555—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124—21no—
17C2×#124—21no—
18C2×#123—21no—
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117—21no—
25C2×#117C2211yescharacteristic
26C2×#117C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#481—24no—
29#428—24no—
30C2×#107C2212yes—
31C2×#106C2212yes—
32C2×#106C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2212yes—
37#398—21no—
38#397—21no—
39#389C2212yes—
40#381—21no—
41#380C2211yescharacteristic
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4D411yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C2×M16—22no—
57C2×M16—24no—
58C22×C8C2312yes—
59C22×C8C2312yes—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62#124—28no—
63#124—28no—
64#123—44no—
65#121—28no—
66#121—22no—
67#121—22no—
68#121—22no—
69#121D414yes—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74#117—28no—
75#117—22no—
76#117—22no—
77#117—22no—
78#117D414yes—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4D411yescharacteristic
98C2×C22⋊C4D411yescharacteristic
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102#107—24no—
103#106—24no—
104#106—24no—
105#102—24no—
106#102—24no—
107C22⋊C8—22no—
108C22⋊C8—24no—
109#95—22no—
110#95—21no—
111#95—22no—
112C24—21no—
113C24C2×D411yescharacteristic
114C24C2×D411yescharacteristic
115C2×Q8—24no—
116C2×Q8—82no—
117C2×D4—22no—
118C2×D4—24no—
119C2×D4—24no—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—41no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4C2×D412yes—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D412yes—
149C22×C4C2×D412yes—
150C22×C4C2×D411yescharacteristic
151C22×C4C2411yescharacteristic
152M16—28no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8#4114yes—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—24no—
179C22⋊C4—22no—
180C22⋊C4—42no—
181C22⋊C4—44no—
182C42—24no—
183C42—24no—
184C42—21no—
185C42—21no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23—22no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23—22no—
196C23—42no—
197C23—41no—
198C23—41no—
199C23—42no—
200C23#12012yes—
201C23C22×D411yescharacteristic
202Q8—84no—
203D4—24no—
204D4—44no—
205D4—41no—
206D4—41no—
207D4—84no—
208D4—84no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—48no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—84no—
237C2×C4#12012yes—
238C2×C4AGL(1,ℤ/8)12yes—
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4C2×#4112yes—
242C2×C42−1+412yes—
243C8—24no—
244C8—42no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—82no—
259C22—82no—
260C22C2×#12011yescharacteristic
261C22#54712yes—
262C22C2×AGL(1,ℤ/8)11yescharacteristic
263C22#57511yescharacteristic
264C22#57712yes—
265C4—22no—
266C4—22no—
267C4—41no—
268C4—42no—
269C4—41no—
270C4—42no—
271C4—42no—
272C4—82no—
273C4—82no—
274C4#52511yescharacteristic
275C4#53611yescharacteristic
276C4#54712yes—
277C2—42no—
278C2—81no—
279C2—82no—
280C2#240611yescharacteristic
281C2#300611yescharacteristic
282C2#300911yescharacteristic
283C2#334812yes—
284C2#335412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#555C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ 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×#428C2semidirect
4. #2486C2semidirect
5. #2058C2semidirect
6. #2034C2semidirect
7. #1895C2semidirect
8. #1870C2semidirect
9. #1862C2semidirect
10. #1854C2semidirect
11. #1825C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4812
C2×#4282
#24862
#20582
#20342
#18952
#18702
#18622
#18542
#18252