Finite Group Databasebeta

← #142151

Subgroups — #142151

Order 512.

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

normal subgroups only

This group has 391 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#33216C211yesmaximal, characteristic
2#12790C211yesmaximal, characteristic
3#10699C211yesmaximal, characteristic
4#7255C212yesmaximal
5#7254C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C2×#405—21no—
8C2×#405C2211yescharacteristic
9#2629—21no—
10C2×#364—21no—
11C2×#349C2212yes—
12#1992—21no—
13#1943C412yescomplemented
14#1942—21no—
15#1845C411yescharacteristic, complemented
16#1809C2212yes—
17#1808—21no—
18C2×#338C2211yescharacteristic
19C2×H(3,ℤ/4)—21no—
20C2×H(3,ℤ/4)C2211yescharacteristic
21#1375—21no—
22C22×M16—21no—
23C22×M16C2311yescharacteristic
24C23×C8D411yescharacteristic
25C2×#125—21no—
26C2×#117—21no—
27C2×#117D411yescharacteristic
28C2×M32—22no—
29#499—42no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107C2×C412yes—
33C2×#106C2×C412yes—
34#405—42no—
35#405—42no—
36#405—42no—
37C2×#97—22no—
38C2×#97C2×C411yescharacteristic
39C2×#97D411yescharacteristic
40C2×C4×C8—22no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#398—42no—
44#396—42no—
45#389—22no—
46#384—21no—
47#380—22no—
48#364—42no—
49#349—22no—
50#349—22no—
51#340—22no—
52#340—22no—
53#338—21no—
54#338Q812yes—
55H(3,ℤ/4)—24no—
56H(3,ℤ/4)—21no—
57H(3,ℤ/4)Q812yes—
58C23×C4C2×D411yescharacteristic
59C2×M16—22no—
60C2×M16—22no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16C2×Q811yescharacteristic
68C2×M16C2×Q811yescharacteristic
69C2×M16#4111yescharacteristic
70C2×M16#4111yescharacteristic
71C22×C8—22no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—22no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8SD1612yes—
78C22×C8Q1612yes—
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#125—42no—
82#125—42no—
83#117—22no—
84#117—22no—
85#117—22no—
86#117—41no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—42no—
99C2×C42—22no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C22⋊C411yescharacteristic
102M32—42no—
103M32—42no—
104C2×C16—42no—
105#107—42no—
106#107—41no—
107#107—41no—
108#107—42no—
109#106—42no—
110#97—41no—
111#97—41no—
112#97—44no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—44no—
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×C4C2×C22⋊C411yescharacteristic
129C22×C4C22⋊Q811yescharacteristic
130C22×C4C22⋊Q811yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132C22×C4C2×Q1611yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134M16—41no—
135M16—41no—
136M16—41no—
137M16—41no—
138M16—42no—
139M16—42no—
140M16—42no—
141M16—42no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—24no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8#9912yes—
156C2×C8#10312yes—
157C2×C8#10312yes—
158C2×C8#10411yescharacteristic
159C2×C8#10411yescharacteristic
160C2×C8C4×Q811yescharacteristic
161C2×C8C4×Q811yescharacteristic
162C2×C8#12411yescharacteristic
163C2×C8C4⋊Q811yescharacteristic
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—84no—
172C4⋊C4—162no—
173C4⋊C4—162no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—162no—
178C42—21no—
179C42—24no—
180C42—21no—
181C42—21no—
182C42—42no—
183C42#10412yes—
184C16—82no—
185C23—21no—
186C23—21no—
187C23—41no—
188C23—41no—
189C23#41811yescharacteristic
190C23#47611yescharacteristic
191C23#47711yescharacteristic
192C2×C4—21no—
193C2×C4—21no—
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—22no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—44no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#39111yescharacteristic
218C2×C4C2×#9911yescharacteristic
219C2×C4#41011yescharacteristic
220C2×C4C2×#10311yescharacteristic
221C2×C4C2×#10411yescharacteristic
222C2×C4C2×#10411yescharacteristic
223C2×C4#47611yescharacteristic
224C2×C4#47711yescharacteristic
225C8—21no—
226C8—21no—
227C8—21no—
228C8—21no—
229C8—22no—
230C8—44no—
231C8#32812yes—
232C8#32812yes—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—81no—
242C22#150511yescharacteristic
243C22#150711yescharacteristic
244C22#198711yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—81no—
249C4—84no—
250C4—162no—
251C4—162no—
252C4C2×#32811yescharacteristic
253C4#150311yescharacteristic
254C4#199411yescharacteristic
255C4#199611yescharacteristic
256C2—21no—
257C2—21no—
258C2—41no—
259C2—41no—
260C2#990811yescharacteristic
261C2#1177011yescharacteristic
262C2#1177111yescharacteristic
subgroupG/H
center, socleC22#1505
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#499C4semidirect
13. #1943C4semidirect
15. #1845C4semidirect

Maximal subgroups

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

subgroupindex
#332162
#127902
#106992
#72552
#72542