Finite Group Databasebeta

← #24306

Subgroups — #24306

Order 256.

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

normal subgroups only

This group has 533 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×#533C211yesmaximal, characteristic
2C2×#479C212yesmaximal, complemented
3C4×C22⋊Q8C211yesmaximal, characteristic
4#2058C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2024C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#1989C212yesmaximal, complemented
9#1873C212yesmaximal
10#1854C212yesmaximal
11#1797C211yesmaximal, characteristic
12#533—28no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2212yes—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×#117C2212yes—
23C22×C42C2211yescharacteristic
24#479—28no—
25C2×#107C2212yes—
26C2×#106C2212yes—
27C2×C4⋊C8C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×C22⋊C8C2212yes—
31#391C2211yescharacteristic
32#389C2211yescharacteristic
33#387—21no—
34#385—21no—
35#380C2211yescharacteristic
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C22×Q8—21no—
46C22×Q8—21no—
47C22×Q8C2311yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—24no—
54C4⋊Q8—24no—
55#125—24no—
56#125D418yes—
57C22⋊Q8—28no—
58C22⋊Q8—28no—
59C22⋊Q8—22no—
60C22⋊Q8Q814yescomplemented
61C4×Q8—28no—
62C4×Q8—28no—
63C4×Q8—24no—
64#117—24no—
65#117—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C42—22no—
82C2×C42—24no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42D411yescharacteristic
86C2×C42D411yescharacteristic
87C2×C42C2311yescharacteristic
88C2×C42C2311yescharacteristic
89#107—24no—
90#106—24no—
91C4⋊C8—24no—
92#103—24no—
93#103—24no—
94C22⋊C8—24no—
95#95—21no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100C24C2×D411yescharacteristic
101C2×Q8—24no—
102C2×Q8—24no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×Q8—42no—
110C2×Q8C2×Q812yes—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×Q812yes—
131C22×C4#4112yes—
132C22×C4C2411yescharacteristic
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—28no—
142C4⋊C4—28no—
143C4⋊C4—24no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4C2×D414yes—
152C4⋊C4C2×D414yes—
153C4⋊C4C2×Q812yes—
154C22⋊C4—28no—
155C22⋊C4—24no—
156C22⋊C4—42no—
157C42—28no—
158C42—22no—
159C42—22no—
160C42—22no—
161C42—42no—
162C42—42no—
163C42C2×D414yes—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23C22⋊Q812yes—
171C23C22×D411yescharacteristic
172Q8—24no—
173Q8—48no—
174Q8—48no—
175Q8—42no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4#12012yes—
209C2×C4#12012yes—
210C2×C4C22⋊Q812yes—
211C2×C4#13512yes—
212C2×C4#13712yes—
213C2×C4C22×D412yes—
214C2×C4C22×Q811yescharacteristic
215C2×C4C2×#4111yescharacteristic
216C2×C42−1+412yes—
217C8—42no—
218C8—42no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22C2×#12011yescharacteristic
227C22C2×C22⋊Q811yescharacteristic
228C22#53611yescharacteristic
229C22C2×#13511yescharacteristic
230C22C2×#13711yescharacteristic
231C22#57511yescharacteristic
232C22#57511yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—44no—
241C4—44no—
242C4—81no—
243C4—81no—
244C4#54812yes—
245C4D4×Q812yes—
246C2—42no—
247C2#240911yescharacteristic
248C2#300211yescharacteristic
249C2#300711yescharacteristic
250C2#307611yescharacteristic
251C2#307811yescharacteristic
252C2#323111yescharacteristic
253C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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×#479C2semidirect
4. #2058C2semidirect
5. #2057C2semidirect
6. #2024C2semidirect
7. #2024C2semidirect
8. #1989C2semidirect
60. C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4792
C4×C22⋊Q82
#20582
#20572
#20242
#19892
#18732
#18542
#17972