Finite Group Databasebeta

← #18321

Subgroups — #18321

Order 256.

491 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 113 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×#539C211yesmaximal, characteristic
2#3195C211yesmaximal, characteristic, complemented
3#3193C211yesmaximal, characteristic
4#2583C211yesmaximal, characteristic, complemented
5#1609C214yesmaximal, complemented
6#1603C214yesmaximal, complemented
7#1508C211yesmaximal, characteristic, complemented
8#1507C211yesmaximal, characteristic
9#1463C211yesmaximal, characteristic, complemented
10#539—22no—
11#539—22no—
12#539—22no—
13#539C2214yes—
14C2×#126—21no—
15C2×#125—21no—
16C2×#123—21no—
17C2×#123—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4—21no—
23C22×C4⋊C4C2211yescharacteristic
24#484—21no—
25#483—21no—
26#483C2212yescomplemented
27#480C2212yes—
28#470—22no—
29#468—22no—
30#451—21no—
31#451C2212yescomplemented
32#450—21no—
33#448C2212yes—
34#407C2211yescharacteristic
35#407C2211yescharacteristic
36#398C2212yescomplemented
37#395C2212yescomplemented
38#389—22no—
39#385—21no—
40#385—21no—
41C4×C22⋊C4C2211yescharacteristic
42#328C2214yescomplemented
43#327C2214yes—
44#324C2214yescomplemented
45C22×Q8D411yescharacteristic
46C22×Q8C2311yescharacteristic
47C22×D4—21no—
48C22×D4D411yescharacteristic
49C23×C4—21no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C2×Q16—22no—
54C2×SD16—22no—
55C2×M16—21no—
56C2×M16—21no—
57#126—24no—
58#126—24no—
59#125—24no—
60#125—24no—
61#123—24no—
62#123—24no—
63#123—24no—
64#123—24no—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C22⋊Q8—21no—
71C22⋊Q8—21no—
72C22⋊Q8—21no—
73C22⋊Q8D412yes—
74C22⋊Q8C2312yes—
75#121—21no—
76#121—21no—
77#121D412yes—
78#121C2312yes—
79C4×D4—24no—
80C4×D4—24no—
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⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4D412yes—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2312yes—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C42—21no—
105C2×C42—21no—
106#107—22no—
107#106—22no—
108#103—22no—
109#103—22no—
110#103—22no—
111#102—22no—
112C22⋊C8—21no—
113C22⋊C8—21no—
114C22⋊C8C2312yes—
115C22⋊C8C2312yes—
116#95—22no—
117#95—22no—
118#95C2312yes—
119C24—21no—
120C24C2×D411yescharacteristic
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—81no—
125C2×Q8—81no—
126C2×Q8C2×D412yes—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—44no—
131C2×D4—81no—
132C2×D4C2×D412yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4—82no—
148C22×C4C2×D412yes—
149C22×C4C2×D412yes—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
156Q16—82no—
157SD16—82no—
158M16—81no—
159M16—81no—
160C2×C8—22no—
161C2×C8—22no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—44no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—82no—
179C4⋊C4C2×D412yes—
180C4⋊C4C2×D412yes—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C42—24no—
192C42—81no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23#12011yescharacteristic
201C23#12011yescharacteristic
202C23C22×D411yescharacteristic
203Q8—42no—
204Q8—42no—
205Q8—42no—
206Q8—81no—
207D4—42no—
208D4—44no—
209D4—44no—
210D4—81no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
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—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—81no—
239C2×C4—84no—
240C2×C4#12011yescharacteristic
241C2×C4#12011yescharacteristic
242C2×C4C22×D411yescharacteristic
243C2×C4C22×D411yescharacteristic
244C8—42no—
245C8—42no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—84no—
258C22#45612yes—
259C22C2×#12011yescharacteristic
260C22#57512yes—
261C4—21no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—44no—
266C4—42no—
267C4—42no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—81no—
272C4—81no—
273C2—22no—
274C2—22no—
275C2—41no—
276C2—82no—
277C2#300511yescharacteristic
278C2C2×#45611yescharacteristic
279C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3195C2semidirect
4. #2583C2semidirect
5. #1609C2semidirect
6. #1603C2semidirect
7. #1508C2semidirect
9. #1463C2semidirect
26. #483C22semidirect
31. #451C22semidirect
36. #398C22semidirect
37. #395C22semidirect
42. #328C22semidirect
44. #324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5392
#31952
#31932
#25832
#16092
#16032
#15082
#15072
#14632