Finite Group Databasebeta

← #24341

Subgroups — #24341

Order 256.

534 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 147 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×Q82C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic
3C2×#440C211yesmaximal, characteristic
4#2580C211yesmaximal, characteristic
5#2057C212yesmaximal, complemented
6#2023C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1863C212yesmaximal, complemented
11#1805C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13Q82—24no—
14Q82—24no—
15C2×C4⋊Q8—22no—
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2212yes—
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4×Q8—21no—
20C2×C4×Q8C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24#474—24no—
25#440—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2212yes—
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2212yes—
30C2×#103C2211yescharacteristic
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—22no—
46C22×Q8—21no—
47C22×Q8D411yescharacteristic
48C22×Q8C2311yescharacteristic
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C2×SD16—22no—
52C2×SD16—22no—
53C2×SD16Q814yescomplemented
54C22×C8C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C4⋊Q8—216no—
58C4⋊Q8—28no—
59C4⋊Q8—24no—
60C4⋊Q8—24no—
61C4⋊Q8—22no—
62C4⋊Q8D414yes—
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⋊C4D412yes—
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2312yes—
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C42—22no—
87C2×C42—21no—
88C2×C42C2312yes—
89C2×C42C2311yescharacteristic
90#107—22no—
91#107—24no—
92C4⋊C8—22no—
93#103—22no—
94#103—24no—
95#102—22no—
96#102—24no—
97#97—22no—
98#95—21no—
99#95—21no—
100#95—22no—
101C24—21no—
102C2×Q8—28no—
103C2×Q8—22no—
104C2×Q8—24no—
105C2×Q8—24no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—82no—
110C2×Q8—82no—
111C2×Q8—82no—
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—22no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—42no—
127C22×C4—82no—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D412yes—
132C22×C4C2411yescharacteristic
133SD16—28no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8C2×Q812yes—
143C4⋊C4—28no—
144C4⋊C4—28no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—82no—
156C4⋊C4C2×D414yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—42no—
159C22⋊C4—44no—
160C22⋊C4—82no—
161C42—28no—
162C42—24no—
163C42—22no—
164C42—22no—
165C42C2×D412yes—
166C42C2×D412yes—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—42no—
172C23C22×D411yescharacteristic
173Q8—28no—
174Q8—22no—
175Q8—22no—
176Q8—22no—
177Q8—84no—
178Q8—88no—
179Q8C22⋊Q814yes—
180D4—22no—
181D4—22no—
182D4—41no—
183D4—41no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—84no—
205C2×C4#12012yes—
206C2×C4#12012yes—
207C2×C4#13714yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C22×Q811yescharacteristic
211C2×C4C2×#4111yescharacteristic
212C2×C42−1+411yescharacteristic
213C2×C42−1+411yescharacteristic
214C8—22no—
215C8—41no—
216C8—42no—
217C22—44no—
218C22—42no—
219C22—82no—
220C22—81no—
221C22—81no—
222C22C2×#12011yescharacteristic
223C22#54811yescharacteristic
224C22D4×Q811yescharacteristic
225C22C2×#13712yes—
226C22#57612yes—
227C4—24no—
228C4—22no—
229C4—24no—
230C4—22no—
231C4—42no—
232C4—44no—
233C4—42no—
234C4—82no—
235C4—81no—
236C4C2×C22⋊Q811yescharacteristic
237C4#53611yescharacteristic
238C4#54811yescharacteristic
239C4D4×Q811yescharacteristic
240C2—82no—
241C2#240911yescharacteristic
242C2#300411yescharacteristic
243C2#300811yescharacteristic
244C2#337512yes—
245C2#338512yes—
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×Q82C2semidirect
5. #2057C2semidirect
7. #1993C2semidirect
8. #1985C2semidirect
10. #1863C2semidirect
11. #1805C2semidirect
53. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q822
C2×#4742
C2×#4402
#25802
#20572
#20232
#19932
#19852
#18802
#18632
#18052