Finite Group Databasebeta

← #25660

Subgroups — #25660

Order 256.

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

normal subgroups only

This group has 546 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×#559C211yesmaximal, characteristic
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#2567C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2041C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1983C211yesmaximal, characteristic, complemented
9#1863C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12#559—24no—
13#559—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#124—21no—
17C2×#124C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×#123—21no—
20C2×#123—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#489—24no—
27#474—24no—
28C2×#107C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#97C2211yescharacteristic
37#395—21no—
38#395—21no—
39#391—21no—
40#389C2212yes—
41#389C2212yes—
42#384—21no—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8D411yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2311yescharacteristic
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60#127—22no—
61#127—22no—
62#124—28no—
63#124—22no—
64#124—22no—
65#124—22no—
66#124D414yes—
67#123—28no—
68#123—28no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×D4—28no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4D414yes—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—24no—
100C4⋊C8—22no—
101#103—24no—
102#103—24no—
103#102—24no—
104#102—24no—
105#102—24no—
106#102—24no—
107#97—22no—
108#95—21no—
109#95—22no—
110#95—22no—
111C24—21no—
112C24—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—41no—
119C2×Q8—42no—
120C2×Q8—82no—
121C2×Q8C2×D412yes—
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—82no—
133C2×D4C2×D412yes—
134C2×D4C2×D412yes—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
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—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4C2×D412yes—
168C4⋊C4#4114yes—
169C22⋊C4—24no—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C42—24no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—42no—
181C42C2×D412yes—
182C42C2×D412yes—
183C23—22no—
184C23—22no—
185C23—42no—
186C23—42no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—42no—
192C23—42no—
193C23C22×D411yescharacteristic
194Q8—24no—
195Q8—44no—
196Q8—41no—
197Q8—41no—
198Q8—84no—
199D4—22no—
200D4—22no—
201D4—24no—
202D4—48no—
203D4—41no—
204D4—41no—
205D4—42no—
206D4—84no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—21no—
219C2×C4—21no—
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#12012yes—
232C2×C4#12012yes—
233C2×C4#13712yes—
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C2×#4112yes—
237C2×C42−1+412yes—
238C8—42no—
239C8—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22—82no—
250C22C2×#12011yescharacteristic
251C22#54812yes—
252C22C2×#13711yescharacteristic
253C22#57511yescharacteristic
254C22#57612yes—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—44no—
259C4—42no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—42no—
264C4—82no—
265C4#53211yescharacteristic
266C4#53611yescharacteristic
267C4#54712yes—
268C2—82no—
269C2—82no—
270C2#243211yescharacteristic
271C2#300711yescharacteristic
272C2#300811yescharacteristic
273C2#337012yes—
274C2#337512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#489C2semidirect
3. C2×#474C2semidirect
4. #2567C2semidirect
5. #2042C2semidirect
7. #1985C2semidirect
8. #1983C2semidirect
9. #1863C2semidirect
10. #1862C2semidirect
11. #1762C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4892
C2×#4742
#25672
#20422
#20412
#19852
#19832
#18632
#18622
#17622