Finite Group Databasebeta

← #243204

Subgroups — #243204

Order 512.

634 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 202 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#21987C212yesmaximal
2#21351C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic
4#10658C214yesmaximal, complemented
5#10657C214yesmaximal, complemented
6#8040C212yesmaximal
7#7875C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2212yes—
10C22×C4×C8C2211yescharacteristic
11C2×#384C2212yes—
12#2283C2211yescharacteristic
13#1948C2214yes—
14#1947C2214yes—
15#1939C2212yes—
16#1939C2212yes—
17#1853—22no—
18#1852—22no—
19#1850—21no—
20#1850—21no—
21#1847—22no—
22#1836—22no—
23#1834—22no—
24#1834C2214yes—
25#1768—22no—
26C2×#340—21no—
27C2×#340—21no—
28#1296C2214yes—
29#1296C2214yes—
30#1277C2214yes—
31C23×C8D411yescharacteristic
32C23×C8D411yescharacteristic
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C42C2311yescharacteristic
37#432—24no—
38#432—24no—
39#432Q818yes—
40C2×#107—24no—
41C2×#106—24no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8C2311yescharacteristic
45C2×C4⋊C8C2311yescharacteristic
46C2×C4⋊C8C2314yes—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—21no—
49C2×#97—22no—
50C2×#97—21no—
51C2×C4×C8—24no—
52C2×C4×C8D412yes—
53C2×C4×C8C2312yes—
54#384—22no—
55#384—22no—
56#384—22no—
57#384C2314yes—
58#383—21no—
59#382D412yes—
60#381—21no—
61#380D411yescharacteristic
62#380D411yescharacteristic
63C4×C4⋊C4C2312yes—
64C4×C22⋊C4—21no—
65#376—21no—
66#340—24no—
67#340—24no—
68#340—24no—
69#340—24no—
70#336—22no—
71#336—22no—
72C23×C4—42no—
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4#4111yescharacteristic
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8C2×D412yes—
84C22×C8C2×D412yes—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—84no—
94C2×C4⋊C4—84no—
95C2×C4⋊C4C2×D412yes—
96C2×C4⋊C4C2×D412yes—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—22no—
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102C2×C42C2×Q814yes—
103C2×C42#4114yes—
104C2×C42#4111yescharacteristic
105C2×C42C2411yescharacteristic
106#107—84no—
107#106—84no—
108C4⋊C8—24no—
109C4⋊C8—22no—
110C4⋊C8—22no—
111C4⋊C8—44no—
112C4⋊C8—44no—
113C4⋊C8C2×Q818yes—
114C22⋊C8—28no—
115C22⋊C8—42no—
116#97—28no—
117#97—42no—
118C4×C8—24no—
119C4×C8—24no—
120#95—21no—
121#95—21no—
122#95—22no—
123#95—22no—
124#95—22no—
125C24#12111yescharacteristic
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—44no—
134C22×C4—84no—
135C22×C4—84no—
136C22×C4—84no—
137C22×C4#12111yescharacteristic
138C22×C4#12111yescharacteristic
139C22×C4#12111yescharacteristic
140C22×C4C22⋊Q814yes—
141C22×C4#12312yes—
142C22×C4#12514yes—
143C22×C4C4⋊Q812yes—
144C22×C4C22×D411yescharacteristic
145C22×C4C22×D411yescharacteristic
146C22×C4C2×#4111yescharacteristic
147C2×C8—28no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—44no—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8—42no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—84no—
169C4⋊C4—84no—
170C4⋊C4—84no—
171C4⋊C4—84no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C42—42no—
175C42—42no—
176C42C22⋊Q814yes—
177C42#12312yes—
178C42#12514yes—
179C42C4⋊Q812yes—
180C42C22×Q812yes—
181C42C2×#4112yes—
182C23—22no—
183C23—22no—
184C23#39812yes—
185C23#47212yes—
186C23#47312yes—
187C23C2×#12111yescharacteristic
188C2×C4—28no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—28no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—84no—
206C2×C4—84no—
207C2×C4—84no—
208C2×C4—168no—
209C2×C4#39812yes—
210C2×C4#39812yes—
211C2×C4#39812yes—
212C2×C4C2×C22⋊Q812yes—
213C2×C4C2×#12311yescharacteristic
214C2×C4C2×#12512yes—
215C2×C4C2×C4⋊Q811yescharacteristic
216C2×C4#57512yes—
217C2×C4#57512yes—
218C2×C4#57714yes—
219C8—48no—
220C8—42no—
221C8—42no—
222C8—81no—
223C8—81no—
224C22—22no—
225C22—22no—
226C22—24no—
227C22—24no—
228C22#208914yes—
229C22C2×#39811yescharacteristic
230C22#304012yes—
231C22#305712yes—
232C22C2×#47211yescharacteristic
233C22C2×#47311yescharacteristic
234C4—22no—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—82no—
239C4—82no—
240C4—164no—
241C4#308414yes—
242C4#309311yescharacteristic
243C4#309311yescharacteristic
244C4#316612yes—
245C2—24no—
246C2#1936212yes—
247C2#1936312yes—
248C2#2365911yescharacteristic
249C2#2366811yescharacteristic
250C2C2×#208911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10658C2semidirect
5. #10657C2semidirect

Maximal subgroups

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

subgroupindex
#219872
#213512
C2×#18342
#106582
#106572
#80402
#78752