Finite Group Databasebeta

← #25944

Subgroups — #25944

Order 256.

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

normal subgroups only

This group has 550 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×#557C211yesmaximal, characteristic, complemented
2C2×#490C211yesmaximal, characteristic
3C2×#460C211yesmaximal, characteristic
4#2548C211yesmaximal, characteristic
5#2025C212yesmaximal, complemented
6#2023C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal, complemented
10#1880C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13#557—24no—
14#557—24no—
15C2×C4⋊Q8—21no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#125—21no—
19C2×#125—21no—
20C2×#125C2211yescharacteristic
21C2×#125C2211yescharacteristic
22C2×#124—21no—
23C2×C4×Q8—21no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28#490—24no—
29#460—24no—
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2212yes—
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×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2312yes—
53C2×SD16—24no—
54C2×SD16—24no—
55C2×SD16—24no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—28no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61#127—22no—
62#127—22no—
63#125—28no—
64#125—28no—
65#125—22no—
66#125—22no—
67#125—22no—
68#125D414yes—
69#124—44no—
70C4×Q8—28no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8D414yes—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
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—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C4⋊C8—22no—
106#103—24no—
107#102—24no—
108#102—24no—
109#102—24no—
110#97—22no—
111#95—21no—
112C24—22no—
113C2×Q8—24no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—41no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8C2×D412yes—
122C2×Q8#4114yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—82no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2411yescharacteristic
146SD16—48no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D412yes—
172C22⋊C4—44no—
173C22⋊C4—44no—
174C22⋊C4—82no—
175C22⋊C4—82no—
176C42—24no—
177C42—24no—
178C42—24no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—42no—
184C42C2×D412yes—
185C42C2×D412yes—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23C22×D411yescharacteristic
194Q8—22no—
195Q8—24no—
196Q8—44no—
197Q8—48no—
198Q8#12414yescomplemented
199D4—42no—
200D4—42no—
201D4—42no—
202D4—42no—
203D4—84no—
204D4—84no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4#12012yes—
231C2×C4#12012yes—
232C2×C4AGL(1,ℤ/8)12yes—
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C2×#4112yes—
236C2×C42−1+412yes—
237C8—42no—
238C8—42no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22—82no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22C2×#12011yescharacteristic
247C22#54812yes—
248C22C2×AGL(1,ℤ/8)11yescharacteristic
249C22#57511yescharacteristic
250C22#57712yes—
251C4—24no—
252C4—22no—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—41no—
259C4—41no—
260C4—42no—
261C4—82no—
262C4C2×#12411yescharacteristic
263C4#53611yescharacteristic
264C4#54812yes—
265C2—162no—
266C2#243611yescharacteristic
267C2#300611yescharacteristic
268C2#301011yescharacteristic
269C2#336812yes—
270C2#337112yes—
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
1. C2×#557C2semidirect
5. #2025C2semidirect
6. #2023C2semidirect
7. #1989C2semidirect
8. #1985C2semidirect
9. #1880C2semidirect
10. #1880C2semidirect
11. #1763C2semidirect
198. Q8#124semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4902
C2×#4602
#25482
#20252
#20232
#19892
#19852
#18802
#17632