Finite Group Databasebeta

← #302863

Subgroups — #302863

Order 512.

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

normal subgroups only

This group has 857 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
1#59864C211yesmaximal, characteristic
2#29208C212yesmaximal, complemented
3#28509C211yesmaximal, characteristic
4#10577C214yesmaximal, complemented
5#9864C212yesmaximal, complemented
6#8920C214yesmaximal
7C4×#331C211yesmaximal, characteristic
8C2×#476—22no—
9#2868C2211yescharacteristic
10#2804—22no—
11#2580—24no—
12#2571C2212yes—
13#2563—22no—
14#2487—22no—
15#2366C2212yes—
16D4×C4⋊C4—22no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2094—22no—
20#2057—24no—
21#1990—22no—
22#1957C2212yes—
23#1947C2218yes—
24#1845—22no—
25#1844C2212yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#331C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8D412yes—
34C2×#127C2311yescharacteristic
35C2×C4×Q8—22no—
36C2×C4×D4—24no—
37C2×C4×D4—22no—
38C2×C4×D4D412yes—
39C22×C4⋊C4—22no—
40#476—48no—
41C2×#107—28no—
42C2×#106—24no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8C2314yes—
45C2×#102—24no—
46C2×C4×C8—22no—
47C2×C4×C8C2312yes—
48#400—24no—
49#399—24no—
50#393—28no—
51#391—22no—
52#390—24no—
53#390—22no—
54#390D412yes—
55#389—24no—
56#389—22no—
57#389D414yes—
58#386—24no—
59#385—24no—
60#385—28no—
61#384—21no—
62#384—22no—
63#384D412yes—
64#384C2312yes—
65#384C2311yescharacteristic
66#382—24no—
67#380—22no—
68#380—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2314yes—
71C43C2311yescharacteristic
72#340—28no—
73#336—22no—
74#336—21no—
75#331Q818yescomplemented
76C22×Q8—22no—
77C22×Q8—41no—
78C22×D4—22no—
79C22×D4C2×D412yes—
80C23×C4—24no—
81C23×C4—22no—
82C22×C8—24no—
83C22×C8—22no—
84C4⋊Q8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—44no—
87C4⋊Q8—84no—
88#127—22no—
89#127C2×Q814yes—
90C4×Q8—84no—
91C4×D4—44no—
92C4×D4—44no—
93C4×D4—44no—
94C4×D4—88no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—84no—
106C2×C4⋊C4C2×D414yes—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D412yes—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C42—24no—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D412yes—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42C2411yescharacteristic
119#107—88no—
120#106—48no—
121C4⋊C8—24no—
122C4⋊C8—24no—
123C4⋊C8C2×Q818yes—
124#102—48no—
125C4×C8—24no—
126C4×C8—44no—
127#95—24no—
128#95—28no—
129#95—24no—
130#95—24no—
131C24—22no—
132C2×Q8—44no—
133C2×Q8—82no—
134C2×Q8—82no—
135C2×Q8—82no—
136C2×Q8—84no—
137C2×D4—22no—
138C2×D4—28no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—44no—
144C2×D4C4⋊Q814yes—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—48no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—84no—
152C22×C4—88no—
153C22×C4#12011yescharacteristic
154C22×C4#12012yes—
155C22×C4#12011yescharacteristic
156C22×C4C22×D411yescharacteristic
157C22×C4C22×D412yes—
158C2×C8—24no—
159C2×C8—28no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—44no—
163C2×C8—44no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—28no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—48no—
174C4⋊C4—82no—
175C4⋊C4—84no—
176C4⋊C4—84no—
177C4⋊C4—84no—
178C4⋊C4—82no—
179C22⋊C4—48no—
180C22⋊C4—82no—
181C22⋊C4—84no—
182C22⋊C4—88no—
183C42—24no—
184C42—22no—
185C42—41no—
186C42—48no—
187C42#13514yes—
188C42#13714yes—
189C42C22×Q812yes—
190C422−1+412yes—
191C23—24no—
192C23—44no—
193C23—44no—
194C23—42no—
195C23—42no—
196C23C2×#12011yescharacteristic
197Q8—88no—
198Q8—164no—
199D4—44no—
200D4—48no—
201D4—48no—
202D4—82no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—88no—
218C2×C4—84no—
219C2×C4—82no—
220C2×C4—168no—
221C2×C4#45214yes—
222C2×C4#45214yes—
223C2×C4#45312yes—
224C2×C4C2×C4⋊Q811yescharacteristic
225C2×C4#53611yescharacteristic
226C2×C4D4×Q814yes—
227C2×C4C2×#13512yes—
228C2×C4C2×#13712yes—
229C2×C4#57614yes—
230C8—44no—
231C8—48no—
232C22—44no—
233C22—48no—
234C22—82no—
235C22—82no—
236C22—84no—
237C22#243411yescharacteristic
238C22C2×#45212yes—
239C22#300812yes—
240C22C2×#45311yescharacteristic
241C22#302211yescharacteristic
242C4—21no—
243C4—21no—
244C4—24no—
245C4—41no—
246C4—84no—
247C4—82no—
248C4—84no—
249C4—162no—
250C4#163014yes—
251C4#338014yes—
252C4#338514yes—
253C2—84no—
254C2C2×#163011yescharacteristic
255C2#1871411yescharacteristic
256C2#1892712yes—
257C2#2582411yescharacteristic
258C2#2583912yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29208C2semidirect
4. #10577C2semidirect
5. #9864C2semidirect
75. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
#292082
#285092
#105772
#98642
#89202
C4×#3312