Finite Group Databasebeta

← #25659

Subgroups — #25659

Order 256.

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

normal subgroups only

This group has 556 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×#476C211yesmaximal, characteristic, complemented
4#2546C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2041C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1983C211yesmaximal, characteristic, complemented
9#1865C212yesmaximal, complemented
10#1864C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12#559—24no—
13#559—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17C2×#124—21no—
18C2×#124C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#123—21no—
21C2×#123—21no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27#489—24no—
28#476—24no—
29C2×#106C2212yes—
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×#97C2211yescharacteristic
38#398—21no—
39#398—21no—
40#391—21no—
41#389C2212yes—
42#389C2212yes—
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—
60C4⋊Q8—44no—
61#127—22no—
62#127—22no—
63#124—28no—
64#124—22no—
65#124—22no—
66#124—22no—
67#124D414yes—
68#123—28no—
69#123—28no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×D4—28no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4D414yes—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2312yes—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#106—24no—
101C4⋊C8—22no—
102#103—24no—
103#103—24no—
104#102—24no—
105#102—24no—
106#102—24no—
107#102—24no—
108#97—22no—
109#95—21no—
110#95—22no—
111#95—22no—
112C24—21no—
113C24—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—41no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8C2×D412yes—
124C2×D4—24no—
125C2×D4—24no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—82no—
135C2×D4C2×D412yes—
136C2×D4C2×D412yes—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—82no—
144C22×C4—82no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2411yescharacteristic
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—44no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4C2×D412yes—
170C4⋊C4#4114yes—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C42—24no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—42no—
183C42C2×D412yes—
184C42C2×D412yes—
185C23—22no—
186C23—22no—
187C23—42no—
188C23—42no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—42no—
194C23—42no—
195C23C22×D411yescharacteristic
196Q8—24no—
197Q8—44no—
198Q8—41no—
199Q8—41no—
200Q8—84no—
201D4—22no—
202D4—22no—
203D4—24no—
204D4—48no—
205D4—41no—
206D4—41no—
207D4—42no—
208D4—84no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4#12012yes—
234C2×C4#12012yes—
235C2×C4AGL(1,ℤ/8)12yes—
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C2×C4C2×#4112yes—
239C2×C42−1+412yes—
240C8—42no—
241C8—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—82no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22C2×#12011yescharacteristic
253C22#54812yes—
254C22C2×AGL(1,ℤ/8)11yescharacteristic
255C22#57511yescharacteristic
256C22#57712yes—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—44no—
261C4—42no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—42no—
266C4—82no—
267C4#53211yescharacteristic
268C4#53611yescharacteristic
269C4#54712yes—
270C2—82no—
271C2—82no—
272C2#243211yescharacteristic
273C2#300611yescharacteristic
274C2#300911yescharacteristic
275C2#337112yes—
276C2#337312yes—
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×#476C2semidirect
4. #2546C2semidirect
5. #2042C2semidirect
7. #1985C2semidirect
8. #1983C2semidirect
9. #1865C2semidirect
10. #1864C2semidirect
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×#4762
#25462
#20422
#20412
#19852
#19832
#18652
#18642
#17622