Finite Group Databasebeta

← #327997

Subgroups — #327997

Order 512.

1111 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 192 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×#2551C211yesmaximal, characteristic
2#33149C212yesmaximal
3#27910C211yesmaximal, characteristic
4#10573C214yesmaximal, complemented
5#10349C214yesmaximal, complemented
6#8725C211yesmaximal, characteristic
7#8707C212yesmaximal
8C22×C4⋊Q8—22no—
9C2×#484—22no—
10#2857C2211yescharacteristic
11#2551—22no—
12#2551—24no—
13#2551C2214yes—
14C2×#395—22no—
15C2×#389—22no—
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2212yes—
18#2095—22no—
19#1987C2214yes—
20#1956C2212yes—
21#1947C2214yes—
22#1921—24no—
23#1852—22no—
24#1845C2212yes—
25#1824—22no—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28#1296C2214yes—
29#1279C2218yes—
30C23×Q8—22no—
31C2×C4⋊Q8—28no—
32C2×C4⋊Q8—28no—
33C2×C4⋊Q8—416no—
34C2×C4⋊Q8C2311yescharacteristic
35C2×C22⋊Q8—21no—
36C22×C4⋊C4—24no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4D412yes—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D412yes—
41C22×C42—22no—
42C22×C42C2311yescharacteristic
43#484—48no—
44C2×#107—24no—
45C2×#107—24no—
46C2×C4⋊C8C2314yes—
47C2×#103—24no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—22no—
50C2×C4×C8C2312yes—
51#395—28no—
52#395—28no—
53#391—21no—
54#389—28no—
55#389—28no—
56#389—22no—
57#384—22no—
58#384D412yes—
59#384C2311yescharacteristic
60#384C2312yes—
61#380—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4D414yes—
65C4×C4⋊C4C2314yes—
66C2×#95—22no—
67C2×#95—22no—
68#340—24no—
69#340—24no—
70#340—24no—
71#336—22no—
72#336—21no—
73C22×Q8—28no—
74C22×Q8—21no—
75C22×Q8—416no—
76C22×Q8—84no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D412yes—
82C22×C8—22no—
83C22×C8—24no—
84C4⋊Q8—24no—
85C4⋊Q8—432no—
86C4⋊Q8—416no—
87C4⋊Q8—416no—
88C22⋊Q8—44no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—216no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—416no—
101C2×C4⋊C4—48no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4Q1618yes—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D414yes—
107C2×C4⋊C4C2×D414yes—
108C2×C22⋊C4—41no—
109C2×C42—24no—
110C2×C42—24no—
111C2×C42—24no—
112C2×C42—48no—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2411yescharacteristic
116#107—48no—
117#107—48no—
118C4⋊C8—28no—
119#103—48no—
120C22⋊C8—24no—
121C22⋊C8—44no—
122C4×C8—24no—
123#95—28no—
124#95—24no—
125#95—24no—
126#95—42no—
127C24#12011yescharacteristic
128C2×Q8—432no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—88no—
133C2×Q8—832no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—28no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—28no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—41no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—48no—
149C22×C4—44no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4#12011yescharacteristic
153C22×C4#12012yes—
154C22×C4C2×Q1614yes—
155C22×C4#13714yes—
156C22×C4C22×D411yescharacteristic
157C22×C4C22×D412yes—
158C2×C8—24no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—44no—
162C2×C8—44no—
163C2×C8—44no—
164C4⋊C4—28no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—28no—
168C4⋊C4—44no—
169C4⋊C4—48no—
170C4⋊C4—432no—
171C4⋊C4—42no—
172C4⋊C4—48no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4C2×Q1618yes—
178C4⋊C4AGL(1,ℤ/8)18yes—
179C22⋊C4—82no—
180C42—22no—
181C42—24no—
182C42—48no—
183C42—48no—
184C422−1+414yes—
185C23—21no—
186C23—22no—
187C23—21no—
188C23#45114yes—
189C23#45312yes—
190C23C2×#12011yescharacteristic
191Q8—84no—
192Q8—832no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—28no—
198C2×C4—28no—
199C2×C4—21no—
200C2×C4—28no—
201C2×C4—28no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—44no—
205C2×C4—416no—
206C2×C4—416no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—48no—
212C2×C4—44no—
213C2×C4—88no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4#45114yes—
220C2×C4#53612yes—
221C2×C4#53614yes—
222C2×C4C22×Q1612yes—
223C2×C4C2×AGL(1,ℤ/8)14yes—
224C2×C4C2×#13712yes—
225C8—44no—
226C8—84no—
227C22—24no—
228C22—22no—
229C22—24no—
230C22—22no—
231C22#163314yes—
232C22#261411yescharacteristic
233C22C2×#45112yes—
234C22#300312yes—
235C22C2×#45311yescharacteristic
236C22#302011yescharacteristic
237C4—24no—
238C4—22no—
239C4—48no—
240C4—48no—
241C4—41no—
242C4—84no—
243C4—161no—
244C4—161no—
245C4#323014yes—
246C4#323914yes—
247C2—24no—
248C2C2×#163311yescharacteristic
249C2#1852411yescharacteristic
250C2#1875912yes—
251C2#2964212yes—
252C2#2965011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #10573C2semidirect
5. #10349C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#331492
#279102
#105732
#103492
#87252
#87072