Finite Group Databasebeta

← #25784

Subgroups — #25784

Order 256.

809 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 809 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C23×SD16C211yesmaximal, characteristic
2C2×#497C212yesmaximal, complemented
3C2×#467C211yesmaximal, characteristic, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5C2×#450C211yesmaximal, characteristic
6#2435C211yesmaximal, characteristic
7#2434C211yesmaximal, characteristic, complemented
8#2015C212yesmaximal, complemented
9#2013C212yesmaximal, complemented
10#1973C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×Q16C2211yescharacteristic, complemented
15C22×Q16C2212yescomplemented
16C22×SD16—28no—
17C22×SD16—24no—
18C22×SD16C2212yes—
19C22×SD16C2212yes—
20C22×SD16C2212yes—
21C23×C8C2211yescharacteristic
22C2×C4⋊Q8C2212yescomplemented
23C2×#124C2212yes—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26#497—28no—
27#467—24no—
28#451—24no—
29#450—24no—
30C2×#107C2211yescharacteristic, complemented
31C2×#103C2212yescomplemented
32C2×#103C2211yescharacteristic, complemented
33C2×#103C2211yescharacteristic, complemented
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2212yescomplemented
37#395—22no—
38#394—22no—
39#391C2212yescomplemented
40#390C2212yes—
41#385—22no—
42#379—22no—
43C4×C22⋊C4C2211yescharacteristic
44#336C2212yescomplemented
45C25—21no—
46C22×Q8—21no—
47C22×Q8—24no—
48C22×Q8—22no—
49C22×Q8—22no—
50C22×Q8—42no—
51C22×Q8C2311yescharacteristic
52C22×Q8C2311yescharacteristic
53C22×Q8C2312yes—
54C22×D4—24no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—21no—
58C22×D4—42no—
59C22×D4C2312yes—
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C2×Q16—24no—
63C2×Q16—24no—
64C2×Q16—22no—
65C2×Q16—24no—
66C2×Q16—42no—
67C2×SD16—232no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—24no—
71C2×SD16—24no—
72C2×SD16—24no—
73C2×SD16—24no—
74C2×SD16—24no—
75C2×SD16—42no—
76C2×SD16D418yes—
77C22×C8—24no—
78C22×C8D414yes—
79C22×C8C2311yescharacteristic
80C22×C8C2312yes—
81C22×C8C2311yescharacteristic
82C4⋊Q8—24no—
83C4⋊Q8—42no—
84#124—24no—
85#124—42no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4D412yes—
100C2×C42C2312yes—
101#107—22no—
102#103—24no—
103#103—22no—
104#103—22no—
105#102—24no—
106C22⋊C8D414yes—
107#97—24no—
108#95—24no—
109#95D412yes—
110C24—22no—
111C24—44no—
112C24—42no—
113C24—44no—
114C24—82no—
115C24C2×D411yescharacteristic
116C2×Q8—22no—
117C2×Q8—28no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—48no—
132C2×Q8—44no—
133C2×Q8—42no—
134C2×Q8C2×D414yes—
135C2×D4—28no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—48no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—42no—
145C2×D4C2×D414yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4C2×D414yes—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2×D412yes—
159C22×C4C2411yescharacteristic
160Q16—48no—
161Q16—44no—
162SD16—216no—
163SD16—216no—
164SD16—48no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—28no—
169C2×C8C2×D414yes—
170C2×C8C2×D412yes—
171C2×C8C2×D414yes—
172C2×C8C2×D412yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—48no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C22⋊C4—24no—
181C22⋊C4—48no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—84no—
185C42—22no—
186C42—22no—
187C23—22no—
188C23—22no—
189C23—24no—
190C23—48no—
191C23—42no—
192C23—42no—
193C23—44no—
194C23—44no—
195C23—816no—
196C23—84no—
197C23—84no—
198C23#12712yes—
199C23#13714yes—
200C23C22×D411yescharacteristic
201Q8—24no—
202Q8—24no—
203Q8—48no—
204Q8—42no—
205Q8—42no—
206Q8—41no—
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—42no—
211Q8—42no—
212D4—24no—
213D4—24no—
214D4—48no—
215D4—42no—
216D4—42no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—48no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4#12712yes—
237C2×C4C22×D411yescharacteristic
238C2×C4C22×D412yes—
239C2×C4C22×D411yescharacteristic
240C2×C4H(5,𝔽2)12yes—
241C8—22no—
242C8—24no—
243C8#12014yes—
244C22—22no—
245C22—22no—
246C22—28no—
247C22—44no—
248C22—48no—
249C22—816no—
250C22—82no—
251C22—82no—
252C22—84no—
253C22#49714yescomplemented
254C22C2×#12711yescharacteristic
255C22D4212yes—
256C22C2×#13712yes—
257C22#57712yes—
258C4—22no—
259C4—44no—
260C4—44no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4C2×#12011yescharacteristic
266C4#53511yescharacteristic
267C4D4212yes—
268C2—24no—
269C2—84no—
270C2#243311yescharacteristic
271C2C2×#49711yescharacteristic
272C2#315411yescharacteristic
273C2#326614yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#497C2semidirect
3. C2×#467C2semidirect
4. C2×#451C2semidirect
7. #2434C2semidirect
8. #2015C2semidirect
9. #2013C2semidirect
10. #1973C2semidirect
11. #1752C2semidirect
14. C22×Q16C22semidirect
15. C22×Q16C22semidirect
22. C2×C4⋊Q8C22semidirect
30. C2×#107C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
33. C2×#103C22semidirect
36. C2×#97C22semidirect
39. #391C22semidirect
44. #336C22semidirect
253. C22#497semidirect

Maximal subgroups

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

subgroupindex
C23×SD162
C2×#4972
C2×#4672
C2×#4512
C2×#4502
#24352
#24342
#20152
#20132
#19732
#17522