Finite Group Databasebeta

← #24501

Subgroups — #24501

Order 256.

604 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 141 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×#450C211yesmaximal, characteristic
2C2×#431C211yesmaximal, characteristic, complemented
3#2855C211yesmaximal, characteristic, complemented
4C2×#390C211yesmaximal, characteristic
5#2065C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#1968C212yesmaximal, complemented
8#1934C212yesmaximal, complemented
9#1872C211yesmaximal, characteristic, complemented
10#1853C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C2×C22⋊Q8C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4—21no—
17C22×C22⋊C4—21no—
18C22×C42C2211yescharacteristic
19#450—24no—
20#431—24no—
21C2×#106C2212yescomplemented
22C2×C4⋊C8C2212yescomplemented
23C2×#103C2212yescomplemented
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C4×C8C2212yescomplemented
30#391C2211yescharacteristic, complemented
31#391C2211yescharacteristic, complemented
32#390—21no—
33#390—21no—
34#390C2212yes—
35#390C2212yes—
36#389C2211yescharacteristic, complemented
37#389C2211yescharacteristic, complemented
38#387—21no—
39#387—21no—
40#385—21no—
41#381—21no—
42#380C2211yescharacteristic
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25—21no—
46C22×Q8C2311yescharacteristic
47C22×D4—24no—
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—42no—
52C22×D4C2312yes—
53C23×C4—21no—
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C2×SD16—24no—
58C2×SD16—24no—
59C2×SD16—42no—
60C22×C8C2312yes—
61C22×C8C2312yes—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—44no—
80C2×C22⋊C4—44no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42D411yescharacteristic
84C2×C42D411yescharacteristic
85C2×C42C2311yescharacteristic
86C2×C42C2311yescharacteristic
87#106—42no—
88C4⋊C8—24no—
89#103—42no—
90#102—24no—
91#102—24no—
92#102—42no—
93C22⋊C8—22no—
94C22⋊C8D414yes—
95C4×C8—24no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103C24—24no—
104C24—22no—
105C24—42no—
106C24—42no—
107C24—42no—
108C24—42no—
109C24—42no—
110C24—42no—
111C24C2×D411yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×D4—24no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—48no—
123C2×D4—48no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4#4114yes—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4SD1614yescomplemented
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D412yes—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4#4112yes—
150C22×C4C2411yescharacteristic
151SD16—48no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8C2×D414yes—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C22⋊C4—44no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C22⋊C4—44no—
173C22⋊C4—44no—
174C42—22no—
175C42—22no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—24no—
185C23—48no—
186C23—48no—
187C23—48no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—84no—
195C23—84no—
196C23—84no—
197C23—84no—
198C23#12112yes—
199C23C2×SD1612yes—
200C23#13512yes—
201C23C22×D411yescharacteristic
202Q8—42no—
203Q8—42no—
204D4—24no—
205D4—24no—
206D4—48no—
207D4—42no—
208D4—42no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—28no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—84no—
240C2×C4#12112yes—
241C2×C4C2×SD1612yes—
242C2×C4C2×SD1612yes—
243C2×C4AGL(1,ℤ/8)12yes—
244C2×C4C22×D411yescharacteristic
245C2×C4C2×#4112yes—
246C2×C4C2×#4111yescharacteristic
247C2×C4H(5,𝔽2)12yes—
248C8—24no—
249C8—42no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—44no—
257C22—44no—
258C22—44no—
259C22—816no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22#46514yes—
265C22C2×#12111yescharacteristic
266C22#52511yescharacteristic
267C22#53511yescharacteristic
268C22C22×SD1611yescharacteristic
269C22C2×#13511yescharacteristic
270C22C2×AGL(1,ℤ/8)11yescharacteristic
271C22#57511yescharacteristic
272C4—22no—
273C4—22no—
274C4—22no—
275C4—22no—
276C4—41no—
277C4—81no—
278C4—81no—
279C4—82no—
280C4—81no—
281C4—81no—
282C4#54612yes—
283C4#55112yes—
284C2—24no—
285C2—84no—
286C2#241111yescharacteristic
287C2C2×#46511yescharacteristic
288C2#305511yescharacteristic
289C2#310111yescharacteristic
290C2#310511yescharacteristic
291C2#320811yescharacteristic
292C2#321111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#431C2semidirect
3. #2855C2semidirect
5. #2065C2semidirect
6. #2055C2semidirect
7. #1968C2semidirect
8. #1934C2semidirect
9. #1872C2semidirect
10. #1853C2semidirect
21. C2×#106C22semidirect
22. C2×C4⋊C8C22semidirect
23. C2×#103C22semidirect
29. C2×C4×C8C22semidirect
30. #391C22semidirect
31. #391C22semidirect
36. #389C22semidirect
37. #389C22semidirect
43. #340C22semidirect
44. #340C22semidirect
142. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C2×#4312
#28552
C2×#3902
#20652
#20552
#19682
#19342
#18722
#18532
#17862