Finite Group Databasebeta

← #24097

Subgroups — #24097

Order 256.

569 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 147 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×#554C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3C22×#106C211yesmaximal, characteristic, complemented
4#2486C211yesmaximal, characteristic
5#2058C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1892C211yesmaximal, characteristic, complemented
8#1869C211yesmaximal, characteristic
9#1865C212yesmaximal, complemented
10#1853C212yesmaximal, complemented
11#1824C211yesmaximal, characteristic, complemented
12C23×C8C2211yescharacteristic
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×#123—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#483—24no—
29C2×#106—24no—
30C2×#106C2212yes—
31C2×#106C2212yes—
32C2×#106C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
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×Q8C2311yescharacteristic
49C22×D4D411yescharacteristic
50C23×C4—21no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C22×C8—22no—
56C22×C8—24no—
57C22×C8C2312yes—
58C22×C8C2312yes—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61#125—28no—
62#123—44no—
63C22⋊Q8—28no—
64C22⋊Q8—28no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68C22⋊Q8D414yes—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×D4—28no—
72C4×D4—22no—
73C4×D4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D414yes—
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2312yes—
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102#106—24no—
103#106—24no—
104#106—24no—
105#103—24no—
106#103—24no—
107C22⋊C8—22no—
108C22⋊C8—24no—
109#95—22no—
110#95—21no—
111#95—22no—
112C24C2×D411yescharacteristic
113C24C2×D411yescharacteristic
114C2×Q8—24no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—82no—
119C2×Q8C2×D412yes—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4SD1614yes—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D412yes—
146C22×C4C2×D412yes—
147C22×C4C2×D411yescharacteristic
148C22×C4C2411yescharacteristic
149C2×C8—28no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8#4114yes—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—42no—
180C22⋊C4—44no—
181C42—24no—
182C42—21no—
183C42—21no—
184C23—22no—
185C23—22no—
186C23—21no—
187C23—22no—
188C23—21no—
189C23—22no—
190C23#12012yes—
191C23C2×SD1612yes—
192C23C2×SD1612yes—
193C23#13712yes—
194C23C22×D411yescharacteristic
195Q8—24no—
196Q8—41no—
197Q8—41no—
198Q8—84no—
199D4—28no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4—84no—
230C2×C4#12012yes—
231C2×C4C2×SD1612yes—
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C2×C4C2×#4112yes—
235C2×C42−1+412yes—
236C8—24no—
237C8—42no—
238C22—24no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—24no—
243C22—22no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22#44814yes—
248C22C2×#12011yescharacteristic
249C22#54712yes—
250C22C22×SD1611yescharacteristic
251C22C2×#13711yescharacteristic
252C22#57712yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—42no—
259C4—42no—
260C4—42no—
261C4—82no—
262C4—82no—
263C4#52511yescharacteristic
264C4#53611yescharacteristic
265C4#54712yes—
266C2—24no—
267C2—42no—
268C2#240611yescharacteristic
269C2C2×#44811yescharacteristic
270C2#300911yescharacteristic
271C2#333512yes—
272C2#334412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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×#483C2semidirect
3. C22×#106C2semidirect
5. #2058C2semidirect
6. #2035C2semidirect
7. #1892C2semidirect
9. #1865C2semidirect
10. #1853C2semidirect
11. #1824C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4832
C22×#1062
#24862
#20582
#20352
#18922
#18692
#18652
#18532
#18242