Finite Group Databasebeta

← #24328

Subgroups — #24328

Order 256.

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

normal subgroups only

This group has 535 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
1C2×#557C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic
3C2×C4×SD16C211yesmaximal, characteristic
4#2538C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2025C212yesmaximal
7#1990C211yesmaximal, characteristic, complemented
8#1984C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1863C212yesmaximal, complemented
11#1803C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13#557—24no—
14#557—24no—
15C2×C4⋊Q8—21no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—22no—
18C2×#125C2212yes—
19C2×C22⋊Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#474—24no—
26C4×SD16—24no—
27C2×#107C2212yes—
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#102C2212yes—
33C2×#102C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#396—22no—
36#390C2212yes—
37#389C2212yes—
38#384C2211yescharacteristic
39#380—21no—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8D411yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×SD16Q814yescomplemented
53C22×C8C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C4⋊Q8—28no—
57C4⋊Q8—22no—
58C4⋊Q8D414yes—
59#125—216no—
60#125—24no—
61#125—24no—
62C22⋊Q8—44no—
63C4×Q8—28no—
64C4×Q8—24no—
65C4×Q8—24no—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8D414yes—
70C4×D4—22no—
71C4×D4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D412yes—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C42—22no—
90C2×C42—21no—
91C2×C42C2312yes—
92C2×C42C2311yescharacteristic
93#107—24no—
94#106—22no—
95C4⋊C8—22no—
96#103—22no—
97#103—24no—
98#102—22no—
99#102—24no—
100C4×C8—22no—
101#95—22no—
102C24—21no—
103C2×Q8—22no—
104C2×Q8—24no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8C2×D412yes—
113C2×Q8C2×Q812yes—
114C2×Q8#4112yes—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4C2×Q812yes—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4C2×D412yes—
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2411yescharacteristic
134SD16—28no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—22no—
143C2×C8C2×Q812yes—
144C4⋊C4—28no—
145C4⋊C4—28no—
146C4⋊C4—28no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—22no—
150C4⋊C4—24no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—44no—
158C4⋊C4—42no—
159C4⋊C4—81no—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D414yes—
162C22⋊C4—44no—
163C22⋊C4—42no—
164C22⋊C4—82no—
165C42—28no—
166C42—24no—
167C42—22no—
168C42—22no—
169C42C2×D412yes—
170C42C2×D412yes—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23C22×D411yescharacteristic
179Q8—22no—
180Q8—22no—
181Q8—22no—
182Q8—44no—
183Q8—48no—
184Q8C22⋊Q814yescomplemented
185D4—22no—
186D4—22no—
187D4—41no—
188D4—41no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—84no—
213C2×C4#12012yes—
214C2×C4#12012yes—
215C2×C4#13512yes—
216C2×C4AGL(1,ℤ/8)12yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C2×C4C22×Q811yescharacteristic
220C2×C4C2×#4111yescharacteristic
221C2×C42−1+411yescharacteristic
222C2×C42−1+411yescharacteristic
223C8—22no—
224C8—41no—
225C8—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22C2×#12011yescharacteristic
234C22#54811yescharacteristic
235C22D4×Q811yescharacteristic
236C22C2×#13511yescharacteristic
237C22C2×AGL(1,ℤ/8)11yescharacteristic
238C22#57711yescharacteristic
239C22#57811yescharacteristic
240C4—24no—
241C4—22no—
242C4—42no—
243C4—44no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—81no—
248C4#44914yes—
249C4C2×C22⋊Q811yescharacteristic
250C4#53611yescharacteristic
251C4#54811yescharacteristic
252C4D4×Q811yescharacteristic
253C2—82no—
254C2#240911yescharacteristic
255C2C2×#44911yescharacteristic
256C2#301311yescharacteristic
257C2#336411yescharacteristic
258C2#336811yescharacteristic
259C2#338111yescharacteristic
260C2#338611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#557C2semidirect
5. #2059C2semidirect
7. #1990C2semidirect
8. #1984C2semidirect
10. #1863C2semidirect
11. #1803C2semidirect
52. C2×SD16Q8semidirect
184. Q8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4742
C2×C4×SD162
#25382
#20592
#20252
#19902
#19842
#18802
#18632
#18032