Finite Group Databasebeta

← #23570

Subgroups — #23570

Order 256.

718 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 153 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×#532C211yesmaximal, characteristic
2C2×C4×SD16C212yesmaximal, complemented
3C4×C22⋊Q8C211yesmaximal, characteristic, complemented
4#2058C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2034C212yesmaximal, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2014C211yesmaximal, characteristic, complemented
9#2000C212yesmaximal, complemented
10C4×#102C212yesmaximal, complemented
11C4×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×SD16C2212yes—
14#532—28no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17C2×#124—21no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2212yes—
24C22×C42C2211yescharacteristic
25C4×SD16—28no—
26C2×#106C2212yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×C22⋊C8C2212yes—
32C2×C4×C8C2212yes—
33#391C2211yescharacteristic
34#389C2211yescharacteristic
35#387—21no—
36#385—21no—
37#380C2211yescharacteristic
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2211yescharacteristic
40C4×C22⋊C4—21no—
41#340C2212yes—
42#340C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C2×#41—28no—
46C2×#41—44no—
47C2×#41—44no—
48C22×Q8—21no—
49C22×Q8C2311yescharacteristic
50C22×D4—22no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4C2312yes—
54C23×C4—22no—
55C23×C4D411yescharacteristic
56C23×C4D411yescharacteristic
57C23×C4C2311yescharacteristic
58C2×SD16—24no—
59C2×SD16—24no—
60C2×SD16—24no—
61C22×C8C2312yes—
62C22×C8C2312yes—
63C4⋊Q8—24no—
64#127—24no—
65#124—28no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68#121—28no—
69#121—24no—
70#121—24no—
71C4×Q8—24no—
72C4×D4—28no—
73C4×D4—24no—
74C4×D4D418yes—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C42—24no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42D411yescharacteristic
91C2×C42D411yescharacteristic
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94#106—24no—
95#103—24no—
96#102—24no—
97#102—24no—
98#102—24no—
99C22⋊C8—24no—
100C4×C8—24no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106C24—22no—
107C24C2×D411yescharacteristic
108#41—432no—
109C2×Q8—24no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8#4112yescomplemented
116C2×D4—24no—
117C2×D4—28no—
118C2×D4—24no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4C2×D414yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—28no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4SD1614yescomplemented
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
155SD16—216no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8#4114yescomplemented
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4C2×D414yes—
171C4⋊C4#4112yescomplemented
172C22⋊C4—28no—
173C22⋊C4—24no—
174C22⋊C4—28no—
175C22⋊C4—42no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—42no—
180C42—42no—
181C42C2×D414yes—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—24no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23C2×SD1612yes—
196C23C22×D411yescharacteristic
197Q8—24no—
198Q8—48no—
199Q8—42no—
200D4—24no—
201D4—24no—
202D4—48no—
203D4—48no—
204D4—416no—
205D4—42no—
206D4—42no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—28no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—44no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—416no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4#12012yes—
237C2×C4#12012yes—
238C2×C4C2×SD1612yes—
239C2×C4C2×SD1612yes—
240C2×C4#13512yes—
241C2×C4AGL(1,ℤ/8)12yes—
242C2×C4C22×D412yes—
243C2×C4C2×#4111yescharacteristic
244C2×C4C2×#4112yes—
245C2×C4C2×#4111yescharacteristic
246C8—24no—
247C8—42no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—44no—
256C22—44no—
257C22—44no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22C2×#12011yescharacteristic
263C22#52511yescharacteristic
264C22#53211yescharacteristic
265C22C22×SD1611yescharacteristic
266C22C2×#13511yescharacteristic
267C22C2×AGL(1,ℤ/8)11yescharacteristic
268C22#57511yescharacteristic
269C4—22no—
270C4—22no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—42no—
275C4—44no—
276C4—81no—
277C4—81no—
278C4#45014yes—
279C4#52512yes—
280C4#53212yes—
281C2—42no—
282C2—84no—
283C2#238811yescharacteristic
284C2C2×#45011yescharacteristic
285C2#300211yescharacteristic
286C2#310311yescharacteristic
287C2#310611yescharacteristic
288C2#316711yescharacteristic
289C2#317111yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutator, Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC2×#532C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×SD16C2semidirect
3. C4×C22⋊Q8C2semidirect
4. #2058C2semidirect
5. #2041C2semidirect
6. #2034C2semidirect
7. #2024C2semidirect
8. #2014C2semidirect
9. #2000C2semidirect
10. C4×#102C2semidirect
11. C4×C22⋊C8C2semidirect
115. C2×Q8#41semidirect
148. C22×C4SD16semidirect
161. C2×C8#41semidirect
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×#5322
C2×C4×SD162
C4×C22⋊Q82
#20582
#20412
#20342
#20242
#20142
#20002
C4×#1022
C4×C22⋊C82