Finite Group Databasebeta

← #24339

Subgroups — #24339

Order 256.

556 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 143 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×#554C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic, complemented
3C2×#442C211yesmaximal, characteristic
4#2580C211yesmaximal, characteristic, complemented
5#2059C212yesmaximal, complemented
6#2025C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal
10#1862C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic, complemented
12C22×D8C2211yescharacteristic
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125C2212yes—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2212yes—
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4—21no—
24#474—24no—
25#442—24no—
26C2×#107C2212yes—
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#102C2212yes—
30C2×#102C2211yescharacteristic
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#399—21no—
35#393—21no—
36#390C2212yes—
37#389C2212yes—
38#385—22no—
39#382C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8D411yescharacteristic
47C22×D4C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4—22no—
51C23×C4—21no—
52C2×D8—22no—
53C2×D8—22no—
54C2×D8Q814yescomplemented
55C22×C8C2311yescharacteristic
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C4⋊Q8—22no—
59C4⋊Q8D414yes—
60#125—24no—
61#125—24no—
62C22⋊Q8—28no—
63C22⋊Q8—216no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×D4—24no—
67C4×D4—24no—
68C4×D4—22no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4D414yes—
72C2×C4⋊C4—28no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D412yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C42C2312yes—
89C2×C42C2311yescharacteristic
90#107—24no—
91#106—22no—
92C4⋊C8—22no—
93#102—22no—
94#102—24no—
95#102—22no—
96#102—24no—
97#97—22no—
98#95—21no—
99#95—21no—
100#95—22no—
101C24—21no—
102C24—21no—
103C2×Q8—22no—
104C2×Q8—24no—
105C2×Q8—42no—
106C2×Q8—82no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4C2×D412yes—
116C2×D4C2×Q812yes—
117C2×D4C2×Q812yes—
118C2×D4#4112yes—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—28no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—44no—
126C22×C4—42no—
127C22×C4—82no—
128C22×C4—82no—
129C22×C4—82no—
130C22×C4C2×D412yes—
131C22×C4C2×D412yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2411yescharacteristic
135D8—28no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8C2×Q812yes—
145C4⋊C4—28no—
146C4⋊C4—28no—
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—82no—
157C4⋊C4C2×D412yes—
158C4⋊C4C2×D414yes—
159C22⋊C4—28no—
160C22⋊C4—24no—
161C22⋊C4—44no—
162C22⋊C4—42no—
163C22⋊C4—82no—
164C42—22no—
165C42—22no—
166C42C2×D412yes—
167C42C2×D412yes—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—42no—
173C23—42no—
174C23—41no—
175C23—41no—
176C23—42no—
177C23—42no—
178C23C22×D411yescharacteristic
179Q8—44no—
180Q8—84no—
181D4—22no—
182D4—22no—
183D4—22no—
184D4—22no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4C22⋊Q814yes—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—48no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4—88no—
210C2×C4#12012yes—
211C2×C4#12012yes—
212C2×C4C22×D411yescharacteristic
213C2×C4C22×D411yescharacteristic
214C2×C4C22×Q811yescharacteristic
215C2×C4C2×#4111yescharacteristic
216C2×C42−1+411yescharacteristic
217C2×C42−1+411yescharacteristic
218C8—22no—
219C8—41no—
220C8—42no—
221C22—24no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—42no—
226C22—44no—
227C22—42no—
228C22—82no—
229C22—81no—
230C22—81no—
231C22C2×#12011yescharacteristic
232C22#54811yescharacteristic
233C22D4×Q811yescharacteristic
234C22#57512yes—
235C22#57612yes—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—44no—
242C4—82no—
243C4—81no—
244C4C2×C22⋊Q811yescharacteristic
245C4#53611yescharacteristic
246C4#54811yescharacteristic
247C4D4×Q811yescharacteristic
248C2—44no—
249C2—82no—
250C2#240911yescharacteristic
251C2#300511yescharacteristic
252C2#300811yescharacteristic
253C2#337012yes—
254C2#338912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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×#554C2semidirect
2. C2×#474C2semidirect
4. #2580C2semidirect
5. #2059C2semidirect
6. #2025C2semidirect
7. #1993C2semidirect
8. #1985C2semidirect
10. #1862C2semidirect
11. #1804C2semidirect
54. C2×D8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4742
C2×#4422
#25802
#20592
#20252
#19932
#19852
#18792
#18622
#18042