Finite Group Databasebeta

← #32918

Subgroups — #32918

Order 256.

686 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 155 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×#519C211yesmaximal, characteristic, complemented
2C2×#464C212yesmaximal, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic
5#2620C211yesmaximal, characteristic
6#1880C212yesmaximal, complemented
7#1878C212yesmaximal, complemented
8#1875C211yesmaximal, characteristic
9#1873C211yesmaximal, characteristic, complemented
10#1803C212yesmaximal, complemented
11#1798C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18#519—28no—
19C2×C4×Q8—22no—
20C2×C4×Q8C2212yes—
21C2×C4×Q8C2212yes—
22C2×#117—22no—
23C2×#117C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#464—28no—
26#451—24no—
27#448—24no—
28C2×C4⋊C8C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#396—22no—
36#394—22no—
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#384C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—24no—
44C22×Q8—24no—
45C22×Q8—21no—
46C22×Q8—22no—
47C22×Q8D414yes—
48C22×Q8C2311yescharacteristic
49C22×Q8C2312yes—
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C2×Q16—24no—
55C2×Q16—24no—
56C2×Q16—24no—
57C2×SD16—24no—
58C2×SD16—24no—
59C2×SD16—24no—
60C22×C8C2312yes—
61C22×C8C2312yes—
62#124—24no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65#121—22no—
66#121—22no—
67C4×Q8—216no—
68C4×Q8—24no—
69C4×Q8—24no—
70C4×Q8—24no—
71C4×Q8D418yescomplemented
72#117—216no—
73#117—24no—
74#117—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4D412yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4D412yes—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—21no—
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94C4⋊C8—24no—
95#103—24no—
96#103—24no—
97#103—24no—
98#102—24no—
99C22⋊C8—22no—
100C22⋊C8—22no—
101#95—22no—
102C24—21no—
103C24C2×D411yescharacteristic
104C2×Q8—24no—
105C2×Q8—216no—
106C2×Q8—24no—
107C2×Q8—216no—
108C2×Q8—24no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8C2×D414yes—
117C2×Q8C2×D414yes—
118C2×Q8#4114yes—
119C2×D4—22no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—42no—
124C2×D4—82no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4#4112yes—
140C22×C4C2411yescharacteristic
141Q16—48no—
142SD16—48no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C4⋊C4—28no—
150C4⋊C4—28no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—42no—
158C4⋊C4—82no—
159C4⋊C4C2×D414yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—42no—
163C22⋊C4—44no—
164C22⋊C4—42no—
165C22⋊C4—82no—
166C42—28no—
167C42—24no—
168C42—28no—
169C42—22no—
170C42C2×D414yes—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—42no—
179C23—42no—
180C23#12112yes—
181C23C22×D411yescharacteristic
182Q8—216no—
183Q8—24no—
184Q8—24no—
185Q8—24no—
186Q8—42no—
187Q8—42no—
188Q8#12118yes—
189D4—42no—
190D4—42no—
191D4—84no—
192C2×C4—216no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4#12014yes—
216C2×C4#12112yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D412yes—
219C2×C4C2×#4112yes—
220C2×C4C2×#4111yescharacteristic
221C8—42no—
222C8—42no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—81no—
229C22—82no—
230C22—82no—
231C22—81no—
232C22—81no—
233C22C2×#12011yescharacteristic
234C22C2×#12111yescharacteristic
235C22#52511yescharacteristic
236C22#57712yes—
237C22#57812yes—
238C4—28no—
239C4—22no—
240C4—24no—
241C4—42no—
242C4—44no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4C2×#12112yes—
247C4#52512yes—
248C2—42no—
249C2—161no—
250C2#284711yescharacteristic
251C2#301312yes—
252C2#304311yescharacteristic
253C2#304411yescharacteristic
254C2#312112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
1. C2×#519C2semidirect
2. C2×#464C2semidirect
3. C2×#451C2semidirect
6. #1880C2semidirect
7. #1878C2semidirect
9. #1873C2semidirect
10. #1803C2semidirect
11. #1798C2semidirect
71. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5192
C2×#4642
C2×#4512
C2×#4482
#26202
#18802
#18782
#18752
#18732
#18032
#17982