Finite Group Databasebeta

← #24343

Subgroups — #24343

Order 256.

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

normal subgroups only

This group has 532 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
2C2×#475C211yesmaximal, characteristic, complemented
3C2×#440C211yesmaximal, characteristic
4#2581C211yesmaximal, characteristic, complemented
5#2059C212yesmaximal, complemented
6#2026C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal
10#1864C212yesmaximal
11#1804C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125C2212yes—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2212yes—
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4—21no—
24#475—24no—
25#440—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2212yes—
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2212yes—
30C2×#103C2211yescharacteristic
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#400—21no—
35#395—21no—
36#391C2212yes—
37#389C2212yes—
38#387—22no—
39#382C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8D411yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4—22no—
51C2×SD16—22no—
52C2×SD16—22no—
53C2×SD16Q814yescomplemented
54C22×C8C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C4⋊Q8—22no—
58C4⋊Q8D414yes—
59#125—24no—
60#125—24no—
61C22⋊Q8—28no—
62C22⋊Q8—216no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×D4—24no—
68C4×D4—24no—
69C4×D4—22no—
70C4×D4D414yes—
71C2×C4⋊C4—28no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
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×C42—21no—
86C2×C42C2312yes—
87C2×C42C2311yescharacteristic
88#107—22no—
89#106—24no—
90C4⋊C8—22no—
91#103—22no—
92#103—24no—
93#102—22no—
94#102—24no—
95#97—22no—
96#95—21no—
97#95—21no—
98#95—22no—
99#95—22no—
100#95—22no—
101C24—21no—
102C2×Q8—22no—
103C2×Q8—24no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—42no—
110C2×Q8—82no—
111C2×Q8C2×Q812yes—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4C2×D412yes—
116C2×D4C2×Q812yes—
117C2×D4#4112yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—28no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—44no—
126C22×C4—84no—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2411yescharacteristic
132SD16—28no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8C2×Q812yes—
142C4⋊C4—28no—
143C4⋊C4—28no—
144C4⋊C4—22no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4C2×D414yes—
156C4⋊C4C2×D412yes—
157C22⋊C4—28no—
158C22⋊C4—24no—
159C42—22no—
160C42—22no—
161C42—42no—
162C42C2×D412yes—
163C42C2×D412yes—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—42no—
168C23—42no—
169C23C22×D411yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—44no—
173Q8—41no—
174Q8—41no—
175Q8—84no—
176D4—22no—
177D4—22no—
178D4—22no—
179D4C22⋊Q814yes—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—48no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—88no—
201C2×C4#12012yes—
202C2×C4#12012yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C22×D411yescharacteristic
205C2×C4C22×Q811yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C42−1+411yescharacteristic
208C2×C42−1+411yescharacteristic
209C8—22no—
210C8—41no—
211C8—42no—
212C22—24no—
213C22—44no—
214C22—44no—
215C22—42no—
216C22—42no—
217C22C2×#12011yescharacteristic
218C22#54811yescharacteristic
219C22D4×Q811yescharacteristic
220C22#57512yes—
221C22#57712yes—
222C4—22no—
223C4—22no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—42no—
228C4—44no—
229C4—82no—
230C4—81no—
231C4C2×C22⋊Q811yescharacteristic
232C4#53611yescharacteristic
233C4#54811yescharacteristic
234C4D4×Q811yescharacteristic
235C2—44no—
236C2#240911yescharacteristic
237C2#300511yescharacteristic
238C2#300911yescharacteristic
239C2#337612yes—
240C2#338812yes—
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
2. C2×#475C2semidirect
4. #2581C2semidirect
5. #2059C2semidirect
6. #2026C2semidirect
7. #1993C2semidirect
8. #1985C2semidirect
53. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4752
C2×#4402
#25812
#20592
#20262
#19932
#19852
#18772
#18642
#18042