Finite Group Databasebeta

← #137231

Subgroups — #137231

Order 512.

490 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 88 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#23638C211yesmaximal, characteristic
2#21939C211yesmaximal, characteristic
3#12870C212yesmaximal, complemented
4#11418C211yesmaximal, characteristic
5#7262C212yesmaximal, complemented
6C2×#427C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2280—21no—
9#2280—21no—
10#2112C2212yes—
11#2077—22no—
12#2076C412yescomplemented
13#2075C412yescomplemented
14#1854—24no—
15#1808—21no—
16#1807C2212yes—
17#1735—21no—
18#1735C2211yescharacteristic
19#1386—24no—
20#1314—22no—
21C24×C4D411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×#117—21no—
25C2×#117—21no—
26C2×#117—22no—
27C2×M32—22no—
28#427—22no—
29#427—22no—
30C2×#107—22no—
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×#106—21no—
34C2×#106D414yes—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8C2×C412yes—
38C2×C22⋊C8D412yes—
39#381—41no—
40#381—41no—
41#380—22no—
42#380—22no—
43#380—22no—
44#380—22no—
45#379—41no—
46#379—41no—
47C4×C22⋊C4—41no—
48C4×C22⋊C4—41no—
49C4×C22⋊C4—42no—
50C22⋊C16—22no—
51C22⋊C16—22no—
52#340—22no—
53#340—22no—
54#340—22no—
55#340—22no—
56#340—21no—
57#340C2×C412yes—
58C25D811yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4C22⋊C412yes—
66C23×C4D811yescharacteristic
67C23×C4SD1612yes—
68C23×C4C2×D411yescharacteristic
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—42no—
72C22×C8—42no—
73C22×C8C22⋊C412yes—
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D412yes—
76C22×C8C2×D411yescharacteristic
77C22×C8#4112yes—
78#117—42no—
79#117—42no—
80#117—41no—
81#117—41no—
82#117—41no—
83#117—41no—
84#117—82no—
85#117—82no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—24no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—42no—
97C2×C42—42no—
98C2×C42—41no—
99C2×C42—41no—
100M32—44no—
101C2×C16—42no—
102#107—22no—
103#107—22no—
104#107—82no—
105#106—42no—
106#106—44no—
107C22⋊C8—24no—
108C22⋊C8—24no—
109#95—42no—
110#95—42no—
111#95—42no—
112#95—42no—
113C24—21no—
114C24—21no—
115C24—42no—
116C24—42no—
117C24—81no—
118C24#10212yes—
119C24C2×D811yescharacteristic
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—44no—
138C22×C4—82no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4—82no—
142C22×C4#10212yes—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4#12011yescharacteristic
145C22×C4#12311yescharacteristic
146C22×C4C2×SD1611yescharacteristic
147C22×C4AGL(1,ℤ/8)11yescharacteristic
148C22×C4#13711yescharacteristic
149C2×C8—24no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—81no—
157C2×C8C4×D412yes—
158C2×C8#12112yes—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C22⋊C4—84no—
168C22⋊C4—84no—
169C22⋊C4—82no—
170C22⋊C4—84no—
171C22⋊C4—84no—
172C22⋊C4—84no—
173C22⋊C4—81no—
174C22⋊C4—81no—
175C42—82no—
176C42—84no—
177C42—81no—
178C42—81no—
179C16—82no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—81no—
193C23—81no—
194C23—88no—
195C23C2×#10211yescharacteristic
196C23#44711yescharacteristic
197C23#48011yescharacteristic
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4—82no—
219C2×C4—84no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4#38611yescharacteristic
225C2×C4#41711yescharacteristic
226C2×C4#44811yescharacteristic
227C2×C4#45512yes—
228C2×C4#45812yes—
229C2×C4#48111yescharacteristic
230C8—22no—
231C8—22no—
232C8—42no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22—84no—
247C22—84no—
248C22#160712yes—
249C22#187411yescharacteristic
250C22#214512yes—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—164no—
255C4—162no—
256C4—161no—
257C4—161no—
258C4#160812yes—
259C4#188511yescharacteristic
260C4#189011yescharacteristic
261C2—22no—
262C2—22no—
263C2—41no—
264C2—82no—
265C2#902511yescharacteristic
266C2#1300211yescharacteristic
267C2#1300811yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12870C2semidirect
5. #7262C2semidirect
12. #2076C4semidirect
13. #2075C4semidirect

Maximal subgroups

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

subgroupindex
#236382
#219392
#128702
#114182
#72622