Finite Group Databasebeta

← #25946

Subgroups — #25946

Order 256.

588 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 137 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, complemented
2C2×#490C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic
4#2548C211yesmaximal, characteristic, complemented
5#2025C212yesmaximal, complemented
6#2023C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal, complemented
10#1879C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×D8C2212yes—
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#124—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8—21no—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4—21no—
28#490—24no—
29#459—24no—
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#97C2211yescharacteristic
36#394—21no—
37#394—21no—
38#390—21no—
39#390C2212yes—
40#390C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8D411yescharacteristic
49C22×D4—21no—
50C22×D4—21no—
51C22×D4C2312yes—
52C22×D4C2311yescharacteristic
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C2×D8—24no—
57C2×D8—24no—
58C2×D8—24no—
59C22×C8C2312yes—
60C22×C8C2312yes—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63#127—22no—
64#127—22no—
65#125—22no—
66#125—22no—
67#125—22no—
68#125D414yes—
69#124—44no—
70C22⋊Q8—28no—
71C22⋊Q8—28no—
72C22⋊Q8—28no—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4D414yes—
81C2×C4⋊C4—28no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
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⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C4⋊C8—22no—
106#102—24no—
107#102—24no—
108#102—24no—
109#102—24no—
110#97—22no—
111#95—21no—
112C24—21no—
113C24—22no—
114C2×Q8—24no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—41no—
118C2×Q8—82no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—82no—
132C2×D4C2×D412yes—
133C2×D4#4114yes—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
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
152D8—48no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—82no—
181C22⋊C4—82no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—42no—
187C42C2×D412yes—
188C42C2×D412yes—
189C23—22no—
190C23—24no—
191C23—41no—
192C23—41no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23C22×D411yescharacteristic
202Q8—44no—
203Q8—84no—
204D4—22no—
205D4—24no—
206D4—42no—
207D4—42no—
208D4—42no—
209D4—42no—
210D4—84no—
211D4—84no—
212D4#12414yescomplemented
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—48no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4#12012yes—
239C2×C4#12012yes—
240C2×C4AGL(1,ℤ/8)12yes—
241C2×C4C22×D411yescharacteristic
242C2×C4C22×D411yescharacteristic
243C2×C4C2×#4112yes—
244C2×C42−1+412yes—
245C8—42no—
246C8—42no—
247C22—24no—
248C22—44no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—82no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—82no—
259C22C2×#12011yescharacteristic
260C22#54812yes—
261C22C2×AGL(1,ℤ/8)11yescharacteristic
262C22#57511yescharacteristic
263C22#57612yes—
264C4—22no—
265C4—22no—
266C4—22no—
267C4—41no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4C2×#12411yescharacteristic
275C4#53611yescharacteristic
276C4#54812yes—
277C2—44no—
278C2—162no—
279C2#243611yescharacteristic
280C2#300611yescharacteristic
281C2#300811yescharacteristic
282C2#336612yes—
283C2#337012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
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
1. C2×#554C2semidirect
2. C2×#490C2semidirect
4. #2548C2semidirect
5. #2025C2semidirect
6. #2023C2semidirect
7. #1989C2semidirect
8. #1985C2semidirect
9. #1879C2semidirect
10. #1879C2semidirect
11. #1762C2semidirect
212. D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4902
C2×#4592
#25482
#20252
#20232
#19892
#19852
#18792
#17622