Finite Group Databasebeta

← #392079

Subgroups — #392079

Order 512.

694 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 234 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
1#20800C211yesmaximal, characteristic
2C2×#1841C212yesmaximal
3#11214C214yesmaximal
4#11213C214yesmaximal
5#11212C212yesmaximal
6#11210C212yesmaximal
7C2×#428C2214yes—
8#2905C2212yes—
9C2×C4×M16C2211yescharacteristic
10#2316C2212yes—
11#2285C2211yescharacteristic
12C2×#384C2211yescharacteristic
13#1847C414yescomplemented
14#1845C412yescomplemented
15#1844C412yescomplemented
16#1841—28no—
17C2×#344C2214yes—
18C2×#343C2212yes—
19#1740C2212yes—
20#1736C2214yes—
21#1733C2214yes—
22C2×#339C2214yes—
23C2×#338C2212yes—
24C2×H(3,ℤ/4)C2212yes—
25C22×M16C2314yes—
26C22×M16C2312yes—
27C2×#117C2314yes—
28C2×#117C2312yes—
29C22×C4⋊C4—41no—
30C22×C4⋊C4C2312yes—
31C22×C42C2311yescharacteristic
32#428—44no—
33#428—44no—
34C2×#107C2×C418yes—
35C2×#106C2×C418yes—
36#424—44no—
37#423—42no—
38C4×M16D414yes—
39C4×M16Q814yes—
40C2×#97C2×C412yes—
41C2×C4×C8C2×C412yes—
42#389—44no—
43#384—41no—
44#384C2×C411yescharacteristic
45#384C2×C412yes—
46#384C2×C411yescharacteristic
47#384C2×C412yes—
48#382C2×C412yes—
49#380—42no—
50#380—41no—
51C4×C4⋊C4—41no—
52#344—28no—
53#343—42no—
54#342—42no—
55#340—44no—
56#339—28no—
57#339—44no—
58#338—24no—
59H(3,ℤ/4)—24no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4C2411yescharacteristic
63C2×M16—24no—
64C2×M16—24no—
65C2×M16—24no—
66C2×M16—24no—
67C2×M16—24no—
68C2×M16—24no—
69C2×M16—24no—
70C2×M16—24no—
71C2×M16C2×D414yes—
72C2×M16C2×Q814yes—
73C22×C8—24no—
74C22×C8C22×C414yes—
75#117—24no—
76#117—24no—
77#117—24no—
78#117—24no—
79#117—24no—
80#117—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—84no—
91C2×C4⋊C4C22×C414yes—
92C2×C4⋊C4C22×C412yes—
93C2×C4⋊C4C22×C412yes—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—24no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—22no—
100C2×C42—24no—
101C2×C42C22×C411yescharacteristic
102C2×C42C22×C411yescharacteristic
103C2×C42C2×D412yes—
104C2×C42C2×Q812yes—
105#107—48no—
106#106—48no—
107C4⋊C8—48no—
108#97C4⋊C418yescomplemented
109C4×C8C4⋊C418yescomplemented
110#95—42no—
111#95—44no—
112C24H(5,𝔽2)11yescharacteristic
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—44no—
125C22×C4—84no—
126C22×C4C23×C411yescharacteristic
127C22×C4C22×D411yescharacteristic
128C22×C4C22×Q811yescharacteristic
129C22×C4H(5,𝔽2)14yes—
130C22×C4H(5,𝔽2)11yescharacteristic
131C22×C42−1+414yes—
132C22×C42−1+411yescharacteristic
133C22×C42−1+411yescharacteristic
134M16—28no—
135M16—28no—
136M16—44no—
137M16—44no—
138M16—44no—
139M16—44no—
140C2×C8—24no—
141C2×C8—24no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8—44no—
145C2×C8C2×C4⋊C418yes—
146C4⋊C4—24no—
147C4⋊C4—28no—
148C4⋊C4—24no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—44no—
152C4⋊C4—82no—
153C4⋊C4—82no—
154C4⋊C4—84no—
155C4⋊C4—84no—
156C4⋊C4—164no—
157C22⋊C4—44no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C42—21no—
161C42—21no—
162C42—22no—
163C42—22no—
164C42—42no—
165C42—42no—
166C42—44no—
167C42—44no—
168C42C2×C4⋊C412yes—
169C42C2×C4⋊C412yes—
170C23—41no—
171C23—41no—
172C23#51811yescharacteristic
173C23#53711yescharacteristic
174C23#54411yescharacteristic
175C23#56012yes—
176C23#56312yes—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—84no—
189C2×C4—84no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4—84no—
195C2×C4—162no—
196C2×C4C22×C4⋊C411yescharacteristic
197C2×C4#51812yes—
198C2×C4#51912yes—
199C2×C4#51911yescharacteristic
200C2×C4#53711yescharacteristic
201C2×C4#53714yes—
202C2×C4#54414yes—
203C2×C4#54411yescharacteristic
204C2×C4#56012yes—
205C2×C4#56312yes—
206C2×C4#56314yes—
207C2×C4#56314yes—
208C8—28no—
209C8—84no—
210C22—21no—
211C22—22no—
212C22—21no—
213C22—22no—
214C22—41no—
215C22—41no—
216C22—42no—
217C22#231611yescharacteristic
218C22#238011yescharacteristic
219C22#238111yescharacteristic
220C22#282312yes—
221C22#284312yes—
222C4—24no—
223C4—42no—
224C4—82no—
225C4—82no—
226C4—84no—
227C4—164no—
228C4#231612yes—
229C4#238112yes—
230C4#282312yes—
231C4#284312yes—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—21no—
236C2#1467611yescharacteristic
237C2#2336011yescharacteristic
238C2#2336111yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1847C4semidirect
14. #1845C4semidirect
15. #1844C4semidirect
108. #97C4⋊C4semidirect
109. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#208002
C2×#18412
#112142
#112132
#112122
#112102