Finite Group Databasebeta

← #138542

Subgroups — #138542

Order 512.

391 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 82 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#29773C211yesmaximal, characteristic
2#21960C211yesmaximal, characteristic
3#11889C212yesmaximal
4#11440C211yesmaximal, characteristic
5#7233C212yesmaximal
6C2×#467C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8#2855—21no—
9#2283—21no—
10#2112C2212yes—
11#2086C412yescomplemented
12#2066—22no—
13#2056C412yescomplemented
14#1949—24no—
15#1892—21no—
16#1808—21no—
17#1807C2211yescharacteristic
18#1807C2212yes—
19#1348—24no—
20#1322—22no—
21C22×Q16C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×C22⋊Q8—21no—
25C22×C42D411yescharacteristic
26C2×M32—22no—
27#467—22no—
28#467—22no—
29C2×#107—22no—
30C2×#107C2×C411yescharacteristic
31C2×#106—21no—
32C2×#106—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8C2×C412yes—
36C2×C4⋊C8D412yes—
37C2×#103—21no—
38C2×#103—22no—
39#391—22no—
40#389—22no—
41#387—42no—
42#385—41no—
43#383—41no—
44#382—22no—
45#381—41no—
46#381—41no—
47#380—22no—
48#380—21no—
49C4×C4⋊C4—22no—
50C4×C22⋊C4—41no—
51#376—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#340—22no—
55#340—22no—
56#340C2×C412yes—
57#340D414yes—
58C22×Q8—21no—
59C23×C4D811yescharacteristic
60C23×C4D811yescharacteristic
61C23×C4C2×D411yescharacteristic
62C2×Q16—22no—
63C2×Q16—81no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—81no—
67C22×C8—81no—
68C22×C8C22⋊C412yes—
69C22×C8C22×C411yescharacteristic
70C22×C8C2×D412yes—
71C22×C8C2×D411yescharacteristic
72C22×C8#4112yes—
73C22⋊Q8—41no—
74C22⋊Q8—41no—
75C22⋊Q8—41no—
76C22⋊Q8—41no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—41no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—42no—
90C2×C42C22⋊C412yes—
91C2×C42SD1612yes—
92M32—82no—
93C2×C16—42no—
94#107—22no—
95#107—82no—
96#106—81no—
97#106—82no—
98C4⋊C8—24no—
99C4⋊C8—24no—
100#103—42no—
101#103—82no—
102#95—42no—
103#95—42no—
104#95—42no—
105#95—42no—
106#95—42no—
107#95—42no—
108#95—41no—
109#95—42no—
110C24C2×D811yescharacteristic
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—41no—
114C2×Q8—41no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—44no—
127C22×C4#10212yes—
128C22×C4#10212yes—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4#12011yescharacteristic
131C22×C4#12311yescharacteristic
132C22×C4C2×SD1611yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134C22×C4#13711yescharacteristic
135Q16—82no—
136C2×C8—22no—
137C2×C8—41no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—41no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—81no—
144C2×C8C4×D412yes—
145C2×C8#12112yes—
146C4⋊C4—42no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—84no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C22⋊C4—84no—
156C22⋊C4—82no—
157C22⋊C4—81no—
158C22⋊C4—81no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42—22no—
163C42—84no—
164C16—82no—
165C23—22no—
166C23—22no—
167C23—41no—
168C23C2×#10211yescharacteristic
169C23#44711yescharacteristic
170C23#48011yescharacteristic
171Q8—82no—
172Q8—81no—
173Q8—81no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—81no—
192C2×C4—82no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4—84no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—84no—
202C2×C4#38611yescharacteristic
203C2×C4#41711yescharacteristic
204C2×C4#44811yescharacteristic
205C2×C4#48111yescharacteristic
206C8—22no—
207C8—22no—
208C8—42no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—44no—
217C22#187411yescharacteristic
218C22#214512yes—
219C22#219112yes—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—164no—
226C4—162no—
227C4—161no—
228C4—161no—
229C4#191211yescharacteristic
230C4#191311yescharacteristic
231C2—22no—
232C2—22no—
233C2—41no—
234C2#920411yescharacteristic
235C2#1300011yescharacteristic
236C2#1300211yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2086C4semidirect
13. #2056C4semidirect

Maximal subgroups

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

subgroupindex
#297732
#219602
#118892
#114402
#72332