Finite Group Databasebeta

← #327978

Subgroups — #327978

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#33148C212yesmaximal
3#27905C211yesmaximal, characteristic
4#10574C214yesmaximal, complemented
5#10349C214yesmaximal, complemented
6#8723C211yesmaximal, characteristic
7#8706C212yesmaximal
8C22×C4⋊Q8—22no—
9C2×#483—22no—
10#2857C2211yescharacteristic
11#2551—22no—
12#2551—24no—
13#2551C2214yes—
14C2×#395—22no—
15C2×#389—22no—
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2212yes—
18#2096—22no—
19#1987C2214yes—
20#1956C2212yes—
21#1948C2214yes—
22#1920—24no—
23#1853—22no—
24#1844C2212yes—
25#1823—22no—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28#1296C2214yes—
29#1279C2218yes—
30C23×Q8—22no—
31C2×C4⋊Q8—28no—
32C2×C4⋊Q8—28no—
33C2×C4⋊Q8—48no—
34C2×C4⋊Q8—48no—
35C2×C4⋊Q8C2311yescharacteristic
36C2×C22⋊Q8—21no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D412yes—
42C22×C42—22no—
43C22×C42C2311yescharacteristic
44#483—48no—
45C2×#106—24no—
46C2×#106—24no—
47C2×C4⋊C8C2314yes—
48C2×#103—24no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—22no—
51C2×C4×C8C2312yes—
52#395—28no—
53#395—28no—
54#391—21no—
55#389—28no—
56#389—28no—
57#389—22no—
58#384—22no—
59#384D412yes—
60#384C2311yescharacteristic
61#384C2312yes—
62#380—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4D414yes—
66C4×C4⋊C4C2314yes—
67C2×#95—22no—
68C2×#95—22no—
69#340—24no—
70#340—24no—
71#340—24no—
72#336—22no—
73#336—21no—
74C22×Q8—28no—
75C22×Q8—21no—
76C22×Q8—416no—
77C22×Q8—84no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D412yes—
83C22×C8—22no—
84C22×C8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—416no—
87C4⋊Q8—432no—
88C4⋊Q8—416no—
89C22⋊Q8—44no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—216no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—416no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4SD1618yes—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D414yes—
109C2×C4⋊C4C2×D414yes—
110C2×C22⋊C4—41no—
111C2×C42—24no—
112C2×C42—24no—
113C2×C42—24no—
114C2×C42—44no—
115C2×C42—44no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42C2411yescharacteristic
119#106—48no—
120#106—48no—
121C4⋊C8—28no—
122#103—48no—
123C22⋊C8—24no—
124C22⋊C8—44no—
125C4×C8—24no—
126#95—28no—
127#95—24no—
128#95—24no—
129#95—42no—
130C24#12011yescharacteristic
131C2×Q8—432no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—88no—
136C2×Q8—832no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—28no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—28no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—41no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—48no—
152C22×C4—44no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4#12011yescharacteristic
156C22×C4#12012yes—
157C22×C4C2×SD1614yes—
158C22×C4AGL(1,ℤ/8)14yes—
159C22×C4C22×D411yescharacteristic
160C22×C4C22×D412yes—
161C2×C8—24no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—44no—
165C2×C8—44no—
166C2×C8—44no—
167C4⋊C4—28no—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—28no—
171C4⋊C4—44no—
172C4⋊C4—432no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—48no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4C2×SD1618yes—
182C4⋊C4#13718yes—
183C22⋊C4—82no—
184C42—22no—
185C42—24no—
186C42—44no—
187C42—44no—
188C42—48no—
189C422−1+414yes—
190C23—21no—
191C23—22no—
192C23—21no—
193C23#45014yes—
194C23#45312yes—
195C23C2×#12011yescharacteristic
196Q8—84no—
197Q8—832no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—28no—
203C2×C4—28no—
204C2×C4—21no—
205C2×C4—28no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—44no—
210C2×C4—416no—
211C2×C4—416no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—48no—
217C2×C4—44no—
218C2×C4—88no—
219C2×C4—81no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4#45014yes—
225C2×C4#53612yes—
226C2×C4#53614yes—
227C2×C4C22×SD1612yes—
228C2×C4C2×AGL(1,ℤ/8)12yes—
229C2×C4C2×#13714yes—
230C8—44no—
231C8—84no—
232C22—24no—
233C22—22no—
234C22—24no—
235C22—22no—
236C22#162914yes—
237C22#261411yescharacteristic
238C22C2×#45012yes—
239C22#300412yes—
240C22C2×#45311yescharacteristic
241C22#302011yescharacteristic
242C4—24no—
243C4—22no—
244C4—48no—
245C4—48no—
246C4—41no—
247C4—84no—
248C4—161no—
249C4—161no—
250C4#322914yes—
251C4#323814yes—
252C2—24no—
253C2C2×#162911yescharacteristic
254C2#1854611yescharacteristic
255C2#1877712yes—
256C2#2964112yes—
257C2#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. #10574C2semidirect
5. #10349C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#331482
#279052
#105742
#103492
#87232
#87062