Finite Group Databasebeta

← #25577

Subgroups — #25577

Order 256.

711 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 711 conjugacy classes of subgroups — too many to draw. Only its 135 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×#547C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4#2540C211yesmaximal, characteristic, complemented
5#2000C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1874C212yesmaximal, complemented
10#1872C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×SD16C2211yescharacteristic
14C22×D8C2211yescharacteristic
15#547—24no—
16#547—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×#123—22no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8C2212yes—
22C2×#121C2212yes—
23C2×#121C2212yes—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2212yes—
27#489—24no—
28#463—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#102C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2212yes—
37C2×#97C2211yescharacteristic
38#399—21no—
39#392—21no—
40#390—22no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—28no—
46C2×#41—48no—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—22no—
50C22×D4—22no—
51C22×D4D411yescharacteristic
52C22×D4C2311yescharacteristic
53C22×D4C2311yescharacteristic
54C23×C4—22no—
55C23×C4—21no—
56C23×C4C2312yes—
57C2×SD16—22no—
58C2×SD16—42no—
59C2×D8—22no—
60C2×D8—42no—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2312yes—
64C4⋊Q8—22no—
65C4⋊Q8D414yescomplemented
66#124—22no—
67#124—22no—
68#123—216no—
69C22⋊Q8—28no—
70C22⋊Q8—24no—
71C22⋊Q8—42no—
72#121—24no—
73#121—24no—
74#121—24no—
75#121—42no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4D414yescomplemented
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2312yes—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4D412yes—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C4⋊C8—22no—
99#103—22no—
100#103—24no—
101#102—22no—
102#102—24no—
103#102—24no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106#97—22no—
107#95—21no—
108C24—21no—
109C24—21no—
110C24C2×D412yes—
111#41—432no—
112C2×Q8—24no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—44no—
118C2×D4—24no—
119C2×D4—28no—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—44no—
132C2×D4—82no—
133C2×D4C2×D412yes—
134C22×C4—28no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4C2×D412yes—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2411yescharacteristic
153SD16—44no—
154D8—44no—
155C2×C8—22no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C4⋊C4—28no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—44no—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D414yes—
171C4⋊C4#4114yescomplemented
172C22⋊C4—28no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C42—81no—
180C42C2×D412yes—
181C42C2×D412yes—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—42no—
190C23—42no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—41no—
197C23—41no—
198C23AGL(1,ℤ/8)14yes—
199C23C22×D411yescharacteristic
200Q8—48no—
201Q8—41no—
202Q8—41no—
203Q8—42no—
204D4—28no—
205D4—22no—
206D4—22no—
207D4—416no—
208D4—41no—
209D4—41no—
210D4—42no—
211D4—42no—
212D4—88no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—48no—
223C2×C4—416no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4#12012yes—
236C2×C4#12012yes—
237C2×C4C22×D411yescharacteristic
238C2×C4C22×D411yescharacteristic
239C2×C4C2×#4112yes—
240C2×C4H(5,𝔽2)12yes—
241C8—41no—
242C8—41no—
243C8—42no—
244C22—24no—
245C22—24no—
246C22—24no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—82no—
255C22—82no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22C2×#12011yescharacteristic
262C22#54612yes—
263C22C2×AGL(1,ℤ/8)12yes—
264C22#57711yescharacteristic
265C22#57811yescharacteristic
266C4—22no—
267C4—22no—
268C4—42no—
269C4—42no—
270C4—44no—
271C4—44no—
272C4—81no—
273C4#53211yescharacteristic
274C4#53511yescharacteristic
275C4#55012yes—
276C2—44no—
277C2—82no—
278C2—161no—
279C2#243111yescharacteristic
280C2#300311yescharacteristic
281C2#301311yescharacteristic
282C2#330612yes—
283C2#330812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#547C2semidirect
2. C2×#489C2semidirect
3. C2×#463C2semidirect
4. #2540C2semidirect
5. #2000C2semidirect
6. #1999C2semidirect
7. #1987C2semidirect
8. #1955C2semidirect
9. #1874C2semidirect
10. #1872C2semidirect
11. #1762C2semidirect
65. C4⋊Q8D4semidirect
79. C4×D4D4semidirect
171. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4892
C2×#4632
#25402
#20002
#19992
#19872
#19552
#18742
#18722
#17622