Finite Group Databasebeta

← #24071

Subgroups — #24071

Order 256.

501 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 131 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×#541C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic
3C2×#483C211yesmaximal, characteristic
4#2317C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2043C212yesmaximal, complemented
7#1873C211yesmaximal, characteristic
8#1873C211yesmaximal, characteristic
9#1865C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1818C211yesmaximal, characteristic
12#541—24no—
13#541—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—21no—
16C2×#126—21no—
17C2×#125—21no—
18C2×#125—21no—
19C2×#125C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#484—24no—
28#483—24no—
29C2×#107C2212yes—
30C2×#106C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#389C2212yes—
38#389C2212yes—
39#385—21no—
40#385—21no—
41#384—21no—
42#382—21no—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8C2311yescharacteristic
48C22×Q8C2311yescharacteristic
49C23×C4—21no—
50C23×C4C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55#126—28no—
56#126—28no—
57#125—28no—
58#125—28no—
59#125—22no—
60#125—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8D414yes—
64C22⋊Q8D414yes—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×Q8—22no—
69#117—22no—
70#117—22no—
71#117—24no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2312yes—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—24no—
102#106—24no—
103#103—24no—
104#103—24no—
105#103—24no—
106#103—24no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109#95—22no—
110#95—22no—
111C24C2×D411yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8C2×D412yes—
119C2×Q8C2×D412yes—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—82no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C4⋊C4#4114yes—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C42—24no—
171C42—24no—
172C42—24no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C23—21no—
178C23—21no—
179C23—22no—
180C23—22no—
181C23#12012yes—
182C23C22×D411yescharacteristic
183Q8—24no—
184Q8—24no—
185Q8—42no—
186Q8—42no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—84no—
216C2×C4#12012yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C2×C4C2×#4112yes—
220C2×C42−1+412yes—
221C8—42no—
222C8—42no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22C2×#12011yescharacteristic
229C22#52511yescharacteristic
230C22#53611yescharacteristic
231C22#57712yes—
232C22#57812yes—
233C4—22no—
234C4—44no—
235C4—42no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—42no—
243C4—82no—
244C4#54712yes—
245C4#54712yes—
246C2—42no—
247C2#240611yescharacteristic
248C2#301011yescharacteristic
249C2#301111yescharacteristic
250C2#312012yes—
251C2#324212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2059C2semidirect
6. #2043C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4842
C2×#4832
#23172
#20592
#20432
#18732
#18652
#18632
#18182