Finite Group Databasebeta

← #18329

Subgroups — #18329

Order 256.

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

normal subgroups only

This group has 539 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×#538C211yesmaximal, characteristic
2#3195C211yesmaximal, characteristic, complemented
3#3193C211yesmaximal, characteristic, complemented
4#2561C211yesmaximal, characteristic, complemented
5#1612C214yesmaximal, complemented
6#1606C214yesmaximal, complemented
7#1508C211yesmaximal, characteristic, complemented
8#1507C211yesmaximal, characteristic, complemented
9#1462C211yesmaximal, characteristic, complemented
10#538—22no—
11#538—22no—
12#538—22no—
13#538C2214yes—
14C2×#126—21no—
15C2×#124—21no—
16C2×#123—21no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#120—21no—
22C2×C4×D4—21no—
23C22×C22⋊C4C2211yescharacteristic
24#484—21no—
25#483—21no—
26#483C2212yescomplemented
27#480C2212yescomplemented
28#470—22no—
29#468—22no—
30#451—21no—
31#451C2212yescomplemented
32#450—21no—
33#448C2212yescomplemented
34#407C2211yescharacteristic
35#407C2211yescharacteristic
36#397C2212yescomplemented
37#393C2212yescomplemented
38#388—22no—
39#381—21no—
40#379—21no—
41C4×C22⋊C4C2211yescharacteristic
42#328C2214yescomplemented
43#327C2214yescomplemented
44#323C2214yescomplemented
45C25D411yescharacteristic
46C22×Q8D411yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4D411yescharacteristic
52C23×C4—21no—
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C2×Q16—22no—
56C2×SD16—22no—
57C2×M16—21no—
58C2×M16—21no—
59#126—24no—
60#126—24no—
61#124—24no—
62#124—24no—
63#123—24no—
64#123—24no—
65C22⋊Q8—21no—
66C22⋊Q8—21no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C22⋊Q8—21no—
71C22⋊Q8D412yes—
72C22⋊Q8C2312yes—
73#121—24no—
74#121—24no—
75#121—21no—
76#121—21no—
77#121D412yes—
78#121C2312yes—
79#120—24no—
80#120—24no—
81C4×D4—24no—
82C4×D4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4D412yes—
103C2×C22⋊C4C2312yes—
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—
120C24—21no—
121C24—21no—
122C24—42no—
123C24—42no—
124C24—81no—
125C24C2×D411yescharacteristic
126C24C2×D412yes—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—81no—
131C2×Q8—81no—
132C2×Q8C2×D412yes—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—44no—
139C2×D4—81no—
140C2×D4—82no—
141C2×D4—82no—
142C2×D4C2×D412yes—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—42no—
153C22×C4—81no—
154C22×C4—82no—
155C22×C4C2×D412yes—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162Q16—82no—
163SD16—82no—
164M16—81no—
165M16—81no—
166C2×C8—22no—
167C2×C8—22no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4C2×D412yes—
178C4⋊C4C2×D412yes—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—42no—
191C22⋊C4—44no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—82no—
195C42—24no—
196C42—81no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—81no—
214C23—81no—
215C23—88no—
216C23#12011yescharacteristic
217C23#12011yescharacteristic
218C23C22×D411yescharacteristic
219Q8—42no—
220Q8—42no—
221Q8—42no—
222Q8—81no—
223D4—42no—
224D4—44no—
225D4—44no—
226D4—81no—
227D4—84no—
228D4—84no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—81no—
252C2×C4—84no—
253C2×C4#12011yescharacteristic
254C2×C4#12011yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C8—42no—
258C8—42no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—44no—
267C22—44no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—82no—
275C22—84no—
276C22—84no—
277C22—84no—
278C22#45512yes—
279C22C2×#12011yescharacteristic
280C22#57512yes—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4—81no—
292C2—22no—
293C2—22no—
294C2—41no—
295C2—82no—
296C2—82no—
297C2#300511yescharacteristic
298C2C2×#45511yescharacteristic
299C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#538C2
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
3. #3193C2semidirect
4. #2561C2semidirect
5. #1612C2semidirect
6. #1606C2semidirect
7. #1508C2semidirect
8. #1507C2semidirect
9. #1462C2semidirect
26. #483C22semidirect
27. #480C22semidirect
31. #451C22semidirect
33. #448C22semidirect
36. #397C22semidirect
37. #393C22semidirect
42. #328C22semidirect
43. #327C22semidirect
44. #323C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5382
#31952
#31932
#25612
#16122
#16062
#15082
#15072
#14622