Finite Group Databasebeta

← #24269

Subgroups — #24269

Order 256.

790 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 151 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, complemented
2C2×#468C212yesmaximal, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4#2409C211yesmaximal, characteristic, complemented
5#2408C211yesmaximal, characteristic
6#2055C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1973C212yesmaximal, complemented
9#1808C211yesmaximal, characteristic, complemented
10#1787C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×SD16—28no—
15C22×SD16—24no—
16C22×SD16C2212yes—
17C22×SD16C2212yes—
18C22×M16C2211yescharacteristic, complemented
19C23×C8C2211yescharacteristic
20C2×C22⋊Q8C2212yescomplemented
21C2×#121C2212yes—
22C2×#120—21no—
23C2×C4×D4C2211yescharacteristic
24#468—28no—
25#435—24no—
26C2×#107C2212yescomplemented
27C2×C4⋊C8C2211yescharacteristic, complemented
28C2×#103C2212yescomplemented
29C2×#103C2212yescomplemented
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic, complemented
34C2×#97C2211yescharacteristic, complemented
35#395—21no—
36#394—21no—
37#393—21no—
38#392—21no—
39#391C2212yescomplemented
40#390C2212yes—
41#386—21no—
42#385—21no—
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#379—21no—
46#340C2212yescomplemented
47C25—21no—
48C22×Q8—21no—
49C22×Q8—24no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—42no—
53C22×Q8C2312yes—
54C22×D4—24no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—21no—
58C22×D4—22no—
59C22×D4—42no—
60C22×D4D411yescharacteristic
61C22×D4C2312yes—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C2×SD16—232no—
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—24no—
71C2×SD16D418yescomplemented
72C2×M16—22no—
73C2×M16—24no—
74C22×C8—24no—
75C22×C8D414yescomplemented
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22⋊Q8—24no—
81C22⋊Q8—42no—
82#121—24no—
83#121—42no—
84#120—44no—
85C4×D4—22no—
86C4×D4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2312yes—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4D411yescharacteristic
98C2×C22⋊C4D411yescharacteristic
99C2×C42C2311yescharacteristic
100#107—24no—
101C4⋊C8—22no—
102#103—24no—
103#103—42no—
104#102—24no—
105#102—42no—
106C22⋊C8—22no—
107C22⋊C8—22no—
108#97—22no—
109#95—21no—
110#95—21no—
111#95—22no—
112#95—22no—
113C24—22no—
114C24—44no—
115C24—42no—
116C24—42no—
117C24—42no—
118C24—82no—
119C24C2×D411yescharacteristic
120C24C2×D411yescharacteristic
121C2×Q8—22no—
122C2×Q8—28no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—44no—
128C2×Q8—48no—
129C2×Q8—44no—
130C2×Q8—42no—
131C2×Q8C2×D414yes—
132C2×D4—28no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
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—44no—
145C2×D4—44no—
146C2×D4—42no—
147C2×D4—82no—
148C2×D4C2×D414yes—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D412yes—
165C22×C4#4112yes—
166C22×C4C2411yescharacteristic
167SD16—216no—
168SD16—216no—
169M16—44no—
170C2×C8—22no—
171C2×C8—28no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8C2×D412yes—
177C2×C8C2×D412yes—
178C2×C8C2×D414yes—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—44no—
188C22⋊C4—44no—
189C22⋊C4—44no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—82no—
193C42—21no—
194C42—21no—
195C23—22no—
196C23—22no—
197C23—21no—
198C23—22no—
199C23—24no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—48no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—88no—
211C23—88no—
212C23—84no—
213C23—84no—
214C23#12112yes—
215C23AGL(1,ℤ/8)12yes—
216C23#13712yes—
217C23C22×D411yescharacteristic
218Q8—24no—
219Q8—24no—
220Q8—48no—
221Q8—42no—
222Q8—42no—
223D4—24no—
224D4—24no—
225D4—48no—
226D4—44no—
227D4—42no—
228D4—42no—
229D4—88no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4#12112yes—
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D412yes—
257C2×C4C2×#4111yescharacteristic
258C2×C4H(5,𝔽2)12yes—
259C8—22no—
260C8—41no—
261C8—41no—
262C8#12014yescomplemented
263C22—22no—
264C22—24no—
265C22—24no—
266C22—22no—
267C22—41no—
268C22—44no—
269C22—42no—
270C22—44no—
271C22—42no—
272C22—44no—
273C22—41no—
274C22—41no—
275C22—816no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22#46814yescomplemented
281C22C2×#12111yescharacteristic
282C22D4212yes—
283C22C2×AGL(1,ℤ/8)11yescharacteristic
284C22C2×#13711yescharacteristic
285C22#57712yes—
286C4—22no—
287C4—41no—
288C4—44no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4C2×#12011yescharacteristic
294C4#53511yescharacteristic
295C4#54612yes—
296C2—24no—
297C2—81no—
298C2—84no—
299C2#240811yescharacteristic
300C2C2×#46811yescharacteristic
301C2#305711yescharacteristic
302C2#326512yes—
303C2#326612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
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. C23×SD16C2semidirect
2. C2×#468C2semidirect
3. C2×#435C2semidirect
4. #2409C2semidirect
6. #2055C2semidirect
7. #2000C2semidirect
8. #1973C2semidirect
9. #1808C2semidirect
10. #1787C2semidirect
11. #1786C2semidirect
18. C22×M16C22semidirect
20. C2×C22⋊Q8C22semidirect
26. C2×#107C22semidirect
27. C2×C4⋊C8C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
33. C2×C22⋊C8C22semidirect
34. C2×#97C22semidirect
39. #391C22semidirect
46. #340C22semidirect
71. C2×SD16D4semidirect
75. C22×C8D4semidirect
262. C8#120semidirect
280. C22#468semidirect

Maximal subgroups

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

subgroupindex
C23×SD162
C2×#4682
C2×#4352
#24092
#24082
#20552
#20002
#19732
#18082
#17872
#17862