Finite Group Databasebeta

← #138532

Subgroups — #138532

Order 512.

417 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 96 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#29771C211yesmaximal, characteristic
2#21961C211yesmaximal, characteristic
3#11887C212yesmaximal
4C2×#1322C211yesmaximal, characteristic
5#7230C212yesmaximal
6C2×#467C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8#2855—21no—
9#2285—21no—
10C2×C22⋊C16C2212yes—
11#2087C412yescomplemented
12#2068—22no—
13#2056C412yescomplemented
14#1947—24no—
15#1892—21no—
16#1808—21no—
17#1808C2211yescharacteristic
18#1808C2212yes—
19#1348—24no—
20#1322—22no—
21C22×Q16C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×C22⋊Q8—21no—
25C2×#117—41no—
26C22×C42D411yescharacteristic
27C22×C16—22no—
28#467—22no—
29#467—22no—
30C2×#107—22no—
31C2×#107—21no—
32C2×#107—22no—
33C2×#107C2×C411yescharacteristic
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8C2×C412yes—
37C2×C4⋊C8D412yes—
38C2×#103—21no—
39C2×#103—22no—
40#391—22no—
41#389—22no—
42#387—42no—
43#385—41no—
44#384—22no—
45#382—41no—
46#381—41no—
47#380—22no—
48#380—21no—
49#380—41no—
50C4×C4⋊C4—22no—
51C4×C4⋊C4—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—
77#117—82no—
78#117—82no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—42no—
94C2×C42C22⋊C412yes—
95C2×C42SD1612yes—
96C2×C16—42no—
97C2×C16—82no—
98#107—22no—
99#107—81no—
100#107—82no—
101#107—82no—
102C4⋊C8—24no—
103C4⋊C8—24no—
104#103—42no—
105#103—82no—
106#95—42no—
107#95—42no—
108#95—42no—
109#95—42no—
110#95—41no—
111#95—42no—
112C24C2×D811yescharacteristic
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—41no—
116C2×Q8—41no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—44no—
129C22×C4—81no—
130C22×C4#10212yes—
131C22×C4#10212yes—
132C22×C4SD3212yes—
133C22×C4Q3212yes—
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4#12011yescharacteristic
136C22×C4#12311yescharacteristic
137C22×C4C2×SD1611yescharacteristic
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139C22×C4#13711yescharacteristic
140Q16—82no—
141C2×C8—22no—
142C2×C8—41no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8C4×D412yes—
150C2×C8#12112yes—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—84no—
158C4⋊C4—84no—
159C4⋊C4—84no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C22⋊C4—81no—
163C22⋊C4—81no—
164C22⋊C4—82no—
165C22⋊C4—81no—
166C22⋊C4—82no—
167C22⋊C4—81no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—84no—
173C16—82no—
174C23—41no—
175C23—41no—
176C23C2×#10211yescharacteristic
177C23#44711yescharacteristic
178C23#48011yescharacteristic
179C23C2×SD3211yescharacteristic
180C23C2×Q3211yescharacteristic
181C23#50911yescharacteristic
182C23#51011yescharacteristic
183Q8—82no—
184Q8—81no—
185Q8—81no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—81no—
205C2×C4—82no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4—84no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—84no—
215C2×C4#35814yes—
216C2×C4#38611yescharacteristic
217C2×C4#41711yescharacteristic
218C2×C4#44811yescharacteristic
219C2×C4#48111yescharacteristic
220C8—22no—
221C8—22no—
222C8—42no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22#187411yescharacteristic
234C22C2×#35811yescharacteristic
235C22#213911yescharacteristic
236C22#218711yescharacteristic
237C22#218811yescharacteristic
238C22#223511yescharacteristic
239C22#223611yescharacteristic
240C4—24no—
241C4—21no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—164no—
246C4—162no—
247C4—161no—
248C4—161no—
249C4#191211yescharacteristic
250C4#191311yescharacteristic
251C2—21no—
252C2—21no—
253C2—21no—
254C2—21no—
255C2—21no—
256C2—21no—
257C2#920411yescharacteristic
258C2#1299311yescharacteristic
259C2#1300811yescharacteristic
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. #2087C4semidirect
13. #2056C4semidirect

Maximal subgroups

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

subgroupindex
#297712
#219612
#118872
C2×#13222
#72302