Finite Group Databasebeta

← #142149

Subgroups — #142149

Order 512.

413 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 98 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#33208C211yesmaximal, characteristic
2#12788C211yesmaximal, characteristic
3#10706C211yesmaximal, characteristic
4#7263C212yesmaximal
5#7262C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C2×#405—21no—
8C22×C4×C8C2211yescharacteristic
9#2851—21no—
10C2×#364—21no—
11#2112C2212yes—
12#1991—21no—
13#1942—21no—
14#1941C412yescomplemented
15#1846C411yescharacteristic, complemented
16#1808—21no—
17#1807C2212yes—
18C2×#341C2211yescharacteristic
19#1735C2211yescharacteristic
20C2×#339—21no—
21#1376—21no—
22C22×M16—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#125—21no—
26C2×#117—21no—
27C22×C42D411yescharacteristic
28C2×M32—22no—
29#499—42no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#106C2×C412yes—
33C2×#106C2×C412yes—
34#405—24no—
35#405—42no—
36C2×#97—41no—
37C2×C4×C8—42no—
38C2×C4×C8—41no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#389—22no—
44#387—42no—
45#385—42no—
46#382—21no—
47#380—22no—
48#364—42no—
49C22⋊C16—22no—
50C22⋊C16—22no—
51#341—21no—
52#341Q812yes—
53#340—22no—
54#340—22no—
55#340—21no—
56#340Q812yes—
57#339—24no—
58C23×C4—21no—
59C23×C4C2×D411yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—22no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8SD1611yescharacteristic
74C22×C8SD1611yescharacteristic
75C22×C8Q1611yescharacteristic
76C22×C8Q1611yescharacteristic
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C22×C8C2×Q811yescharacteristic
80C22×C8C2×Q811yescharacteristic
81C22×C8#4111yescharacteristic
82C22×C8#4111yescharacteristic
83#125—42no—
84#125—42no—
85#117—22no—
86#117—22no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—42no—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C22⋊C411yescharacteristic
102M32—42no—
103M32—42no—
104C2×C16—42no—
105#107—42no—
106#107—42no—
107#106—42no—
108#106—42no—
109#97—44no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114C4×C8—44no—
115#95—42no—
116#95—42no—
117#95—42no—
118C24AGL(1,ℤ/8)11yescharacteristic
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C22⋊Q811yescharacteristic
133C22×C4C22⋊Q811yescharacteristic
134C22×C4C2×SD1611yescharacteristic
135C22×C4C2×Q1611yescharacteristic
136C22×C4AGL(1,ℤ/8)11yescharacteristic
137M16—24no—
138M16—42no—
139M16—42no—
140M16—42no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8#9912yes—
159C2×C8#10311yescharacteristic
160C2×C8#10311yescharacteristic
161C2×C8#10311yescharacteristic
162C2×C8#10311yescharacteristic
163C2×C8#10412yes—
164C2×C8C4×Q811yescharacteristic
165C2×C8C4×Q811yescharacteristic
166C2×C8#12411yescharacteristic
167C2×C8C4⋊Q811yescharacteristic
168C4⋊C4—24no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—84no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—42no—
183C42—42no—
184C42—42no—
185C42#10412yes—
186C16—82no—
187C23—22no—
188C23—21no—
189C23—21no—
190C23—41no—
191C23#41811yescharacteristic
192C23#47611yescharacteristic
193C23#47711yescharacteristic
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—48no—
210C2×C4—41no—
211C2×C4—44no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4#39111yescharacteristic
221C2×C4C2×#9911yescharacteristic
222C2×C4#41011yescharacteristic
223C2×C4C2×#10311yescharacteristic
224C2×C4C2×#10411yescharacteristic
225C2×C4C2×#10411yescharacteristic
226C2×C4#47611yescharacteristic
227C2×C4#47711yescharacteristic
228C8—21no—
229C8—21no—
230C8—21no—
231C8—21no—
232C8—21no—
233C8—21no—
234C8—44no—
235C8#32812yes—
236C8#32812yes—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—42no—
245C22—42no—
246C22#150511yescharacteristic
247C22#150711yescharacteristic
248C22#198711yescharacteristic
249C22#223112yes—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—42no—
254C4—84no—
255C4—162no—
256C4—162no—
257C4C2×#32811yescharacteristic
258C4#150311yescharacteristic
259C4#199411yescharacteristic
260C4#199611yescharacteristic
261C2—21no—
262C2—22no—
263C2—21no—
264C2—41no—
265C2#990811yescharacteristic
266C2#1327111yescharacteristic
267C2#1327311yescharacteristic
subgroupG/H
center, socleC22#1987
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24AGL(1,ℤ/8)
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
6. C2×#499C4semidirect
14. #1941C4semidirect
15. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332082
#127882
#107062
#72632
#72622