Finite Group Databasebeta

← #391702

Subgroups — #391702

Order 512.

2570 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

normal subgroups only

This group has 2570 conjugacy classes of subgroups — too many to draw. Only its 330 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×#2848C211yesmaximal, characteristic
2C2×#1779C211yesmaximal, characteristic
3#14680C211yesmaximal, characteristic
4#11200C216yesmaximal, complemented
5#10970C216yesmaximal, complemented
6C25×C4C2211yescharacteristic
7C22×C22⋊Q8C2213yes—
8C22×#103C2214yescomplemented
9C22×C22⋊C8C2216yes—
10#2848—24no—
11#2848C418yescomplemented
12C2×#385—23no—
13C2×#380C2213yes—
14C2×#380C2216yes—
15C2×#379—21no—
16C2×#379—21no—
17#1779—24no—
18C2×#340C22112yescomplemented
19C22⋊#95—23no—
20C26D411yescharacteristic
21C23×Q8C2311yescharacteristic
22C24×C4—412no—
23C24×C4C2×C414yes—
24C24×C4D413yes—
25C24×C4D414yes—
26C24×C4C2313yes—
27C23×C8C2314yes—
28C2×C22⋊Q8—26no—
29C2×C22⋊Q8—412no—
30C2×C22⋊Q8C2×C4112yes—
31C22×C4⋊C4—23no—
32C22×C4⋊C4C2314yes—
33C22×C4⋊C4C2313yes—
34C22×C22⋊C4—23no—
35C22×C22⋊C4—23no—
36C22×C22⋊C4—26no—
37C2×#103—412no—
38C2×C22⋊C8—212no—
39C2×C22⋊C8—412no—
40#385—224no—
41#380—26no—
42#380—412no—
43#380C2×C4112yes—
44#379—28no—
45#379—44no—
46C2×#95—23no—
47C2×#95—26no—
48#340—412no—
49C25—46no—
50C25—46no—
51C25C22⋊C414yes—
52C25SD1614yes—
53C25Q1614yes—
54C25C2×D413yes—
55C22×Q8—21no—
56C22×Q8—212no—
57C22×Q8C22×C412yes—
58C23×C4—212no—
59C23×C4—212no—
60C23×C4—212no—
61C23×C4—21no—
62C23×C4—23no—
63C23×C4—24no—
64C23×C4—424no—
65C23×C4—424no—
66C23×C4—46no—
67C23×C4—46no—
68C23×C4—46no—
69C23×C4—448no—
70C23×C4—416no—
71C23×C4C22⋊C4112yes—
72C23×C4C22×C416yes—
73C23×C4C2×D413yes—
74C23×C4C2×D416yes—
75C23×C4C2411yescharacteristic
76C22×C8—24no—
77C22×C8—212no—
78C22×C8—212no—
79C22⋊Q8—424no—
80C22⋊Q8—424no—
81C2×C4⋊C4—212no—
82C2×C4⋊C4—212no—
83C2×C4⋊C4—23no—
84C2×C4⋊C4—212no—
85C2×C4⋊C4—212no—
86C2×C4⋊C4—412no—
87C2×C4⋊C4C22×C416yes—
88C2×C22⋊C4—212no—
89C2×C22⋊C4—212no—
90C2×C22⋊C4—412no—
91C2×C22⋊C4—412no—
92C2×C22⋊C4—412no—
93C2×C22⋊C4—412no—
94C2×C22⋊C4—412no—
95#103—416no—
96C22⋊C8—424no—
97#95—212no—
98#95—412no—
99C24—26no—
100C24—26no—
101C24—26no—
102C24—26no—
103C24—26no—
104C24—26no—
105C24—412no—
106C24—412no—
107C24—424no—
108C24—424no—
109C24—416no—
110C24—448no—
111C24#103116yescomplemented
112C24C2×C22⋊C416yes—
113C24C2×SD1616yes—
114C24C2×Q1616yes—
115C24C22×D411yescharacteristic
116C2×Q8—212no—
117C2×Q8—46no—
118C2×Q8—412no—
119C2×Q8—44no—
120C22×C4—26no—
121C22×C4—26no—
122C22×C4—26no—
123C22×C4—212no—
124C22×C4—212no—
125C22×C4—212no—
126C22×C4—212no—
127C22×C4—212no—
128C22×C4—212no—
129C22×C4—26no—
130C22×C4—22no—
131C22×C4—412no—
132C22×C4—412no—
133C22×C4—412no—
134C22×C4—448no—
135C22×C4—464no—
136C22×C4—464no—
137C22×C4—436no—
138C22×C4—412no—
139C22×C4—46no—
140C22×C4—46no—
141C22×C4—412no—
142C22×C4—412no—
143C22×C4—44no—
144C22×C4—412no—
145C22×C4C2×C22⋊C416yes—
146C22×C4C23×C411yescharacteristic
147C22×C4C22×D411yescharacteristic
148C22×C4H(5,𝔽2)112yes—
149C2×C8—412no—
150C2×C8—412no—
151C2×C8—44no—
152C4⋊C4—212no—
153C4⋊C4—424no—
154C4⋊C4—46no—
155C4⋊C4—412no—
156C4⋊C4—44no—
157C22⋊C4—424no—
158C22⋊C4—424no—
159C22⋊C4—824no—
160C23—26no—
161C23—26no—
162C23—26no—
163C23—26no—
164C23—26no—
165C23—26no—
166C23—212no—
167C23—212no—
168C23—212no—
169C23—212no—
170C23—212no—
171C23—212no—
172C23—448no—
173C23—416no—
174C23—448no—
175C23—464no—
176C23—464no—
177C23—412no—
178C23—412no—
179C23—412no—
180C23—412no—
181C23C2×#103112yes—
182C23C22×C22⋊C411yescharacteristic
183C23#53416yes—
184C23C22×SD1611yescharacteristic
185C23C22×Q1611yescharacteristic
186C23#57516yes—
187Q8—48no—
188Q8—84no—
189C2×C4—26no—
190C2×C4—26no—
191C2×C4—26no—
192C2×C4—448no—
193C2×C4—412no—
194C2×C4—424no—
195C2×C4—424no—
196C2×C4—46no—
197C2×C4—46no—
198C2×C4—412no—
199C2×C4—812no—
200C2×C4—84no—
201C2×C4—812no—
202C2×C4—812no—
203C2×C4—84no—
204C2×C4#51816yes—
205C2×C4#53416yes—
206C2×C4#560112yes—
207C2×C4H(3,𝔽4)14yes—
208C8—84no—
209C22—212no—
210C22—212no—
211C22—212no—
212C22—464no—
213C22—46no—
214C22—46no—
215C22—46no—
216C22—46no—
217C22—424no—
218C22—424no—
219C22#230813yes—
220C22#281214yes—
221C22C22×#10311yescharacteristic
222C22#289613yes—
223C22#318716yes—
224C22#318916yes—
225C22#3197112yes—
226C4—46no—
227C4—86no—
228C4—82no—
229C4—164no—
230C4#237514yes—
231C4#281214yes—
232C2—412no—
233C2#1465711yescharacteristic
234C2#2175913yes—
235C2#2176613yes—
236C2#3270814yes—
237C2#3271014yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11200C2semidirect
5. #10970C2semidirect
8. C22×#103C22semidirect
11. #2848C4semidirect
18. C2×#340C22semidirect
111. C24#103semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#17792
#146802
#112002
#109702