Finite Group Databasebeta

← #24480

Subgroups — #24480

Order 256.

493 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 139 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×#483C211yesmaximal, characteristic
2C22×#103C211yesmaximal, characteristic
3#2434C211yesmaximal, characteristic
4#2344C211yesmaximal, characteristic
5#2066C212yesmaximal, complemented
6#2043C212yesmaximal, complemented
7#1981C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1892C211yesmaximal, characteristic
10#1850C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C4⋊Q8C2212yes—
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117—21no—
17C22×C4⋊C4C2211yescharacteristic
18#483—24no—
19C2×#106C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×#103—24no—
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×C22⋊C8C2211yescharacteristic
31#395—22no—
32#391C2212yes—
33#389C2212yes—
34#387—22no—
35#385—21no—
36#385—21no—
37#381—21no—
38#380C2211yescharacteristic
39C4×C4⋊C4C2212yes—
40C4×C22⋊C4C2211yescharacteristic
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—24no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×Q8C2311yescharacteristic
49C22×Q8C2312yes—
50C23×C4—21no—
51C23×C4—21no—
52C23×C4C2311yescharacteristic
53C22×C8—22no—
54C22×C8—42no—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C4⋊Q8—24no—
59C4⋊Q8—42no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62#117—44no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4D414yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4D411yescharacteristic
77C2×C22⋊C4D411yescharacteristic
78C2×C42—22no—
79C2×C42C2312yes—
80#106—22no—
81#106—22no—
82C4⋊C8—42no—
83#103—24no—
84#103—24no—
85#103—24no—
86#103—22no—
87#103—22no—
88#103—42no—
89#103—42no—
90C22⋊C8—22no—
91#95—22no—
92#95—21no—
93#95—22no—
94#95—22no—
95#95—22no—
96#95D412yes—
97C24C2×D411yescharacteristic
98C2×Q8—22no—
99C2×Q8—28no—
100C2×Q8—22no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—44no—
109C2×Q8—48no—
110C2×Q8—44no—
111C2×Q8—42no—
112C2×Q8#4114yes—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—44no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D412yes—
129C22×C4C2×D412yes—
130C22×C4#4112yes—
131C22×C4C2411yescharacteristic
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—44no—
140C4⋊C4—24no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—42no—
150C4⋊C4—44no—
151C4⋊C4C2×D412yes—
152C4⋊C4C2×D412yes—
153C22⋊C4—22no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4SD1614yes—
157C42—22no—
158C42—22no—
159C42—44no—
160C23—22no—
161C23—22no—
162C23#12112yes—
163C23C2×SD1612yes—
164C23#13712yes—
165C23C22×D411yescharacteristic
166Q8—24no—
167Q8—24no—
168Q8—48no—
169Q8—42no—
170Q8—42no—
171Q8—41no—
172Q8—41no—
173Q8—42no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—48no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4#12112yes—
202C2×C4C2×SD1614yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C2×#4112yes—
205C2×C4C2×#4111yescharacteristic
206C2×C4H(5,𝔽2)12yes—
207C8—41no—
208C8—41no—
209C8—42no—
210C22—22no—
211C22—24no—
212C22—22no—
213C22—24no—
214C22#46114yes—
215C22C2×#12111yescharacteristic
216C22#54612yes—
217C22C22×SD1611yescharacteristic
218C22C2×#13711yescharacteristic
219C22#57812yes—
220C4—22no—
221C4—24no—
222C4—44no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4#52511yescharacteristic
230C4#53511yescharacteristic
231C4#55112yes—
232C2—24no—
233C2#241111yescharacteristic
234C2C2×#46111yescharacteristic
235C2#304411yescharacteristic
236C2#329812yes—
237C2#331212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2066C2semidirect
6. #2043C2semidirect
7. #1981C2semidirect
8. #1927C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C22×#1032
#24342
#23442
#20662
#20432
#19812
#19272
#18922
#18502
#17872