Finite Group Databasebeta

← #93502

Subgroups — #93502

Order 512.

500 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 86 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#10715C211yesmaximal, characteristic
2#8557C212yesmaximal
3C23×M16C411yescharacteristic
4C22×C22⋊C8C411yescharacteristic
5C2×C4×C22⋊C4C2211yescharacteristic
6#1797—22no—
7C2×#336—22no—
8C2×#323C414yes—
9#1280—22no—
10C24×C4C2×C411yescharacteristic
11C22×M16—21no—
12C22×M16—44no—
13C22×M16—41no—
14C22×M16—41no—
15C22×M16C812yescomplemented
16C23×C8—21no—
17C23×C8—21no—
18C22×C22⋊C4C2×C412yes—
19C22×C42D412yes—
20C2×C4⋊C8—42no—
21C2×C22⋊C8—21no—
22C2×C22⋊C8—24no—
23C2×C22⋊C8—41no—
24C2×C22⋊C8—41no—
25C2×C22⋊C8C812yescomplemented
26C4×C22⋊C4—41no—
27C4×C22⋊C4—41no—
28C2×#95D411yescharacteristic
29C2×#95Q811yescharacteristic
30#336—42no—
31#336—44no—
32#323—44no—
33C25C4211yescharacteristic
34C23×C4—21no—
35C23×C4—21no—
36C23×C4—21no—
37C23×C4—21no—
38C23×C4—42no—
39C23×C4—42no—
40C23×C4C22⋊C412yes—
41C23×C4C22⋊C411yescharacteristic
42C23×C4C4⋊C412yes—
43C23×C4C4⋊C411yescharacteristic
44C23×C4C2×C811yescharacteristic
45C23×C4C2×C811yescharacteristic
46C23×C4M1611yescharacteristic
47C23×C4M1611yescharacteristic
48C2×M16—44no—
49C2×M16—44no—
50C2×M16—42no—
51C2×M16—48no—
52C2×M16—44no—
53C2×M16—42no—
54C2×M16—44no—
55C22×C8—24no—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—42no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—42no—
62C22×C8—42no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—44no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—42no—
73C2×C42—42no—
74C4⋊C8—82no—
75C4⋊C8—82no—
76C22⋊C8—42no—
77C22⋊C8—42no—
78C22⋊C8—44no—
79C22⋊C8—44no—
80#95—41no—
81#95—41no—
82C24—21no—
83C24—21no—
84C24—22no—
85C24—22no—
86C24—42no—
87C24—41no—
88C24—41no—
89C24#9511yescharacteristic
90C24C4×C811yescharacteristic
91C24#9711yescharacteristic
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—42no—
110C22×C4—44no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—42no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C22⋊C811yescharacteristic
127C22×C4C22⋊C811yescharacteristic
128C22×C4#9911yescharacteristic
129C22×C4#9911yescharacteristic
130C22×C4#10011yescharacteristic
131C22×C4#10012yes—
132C22×C4#10111yescharacteristic
133C22×C4#10112yes—
134C22×C4C4⋊C811yescharacteristic
135C22×C4C4⋊C811yescharacteristic
136M16—48no—
137M16—44no—
138M16—44no—
139C2×C8—24no—
140C2×C8—42no—
141C2×C8—44no—
142C2×C8—42no—
143C2×C8—44no—
144C2×C8—44no—
145C2×C8—42no—
146C2×C8—82no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—84no—
150C2×C8—84no—
151C2×C8—84no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C42—42no—
155C42—42no—
156C42—82no—
157C23—22no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—22no—
163C23—22no—
164C23—42no—
165C23—42no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—41no—
172C23—41no—
173C23—42no—
174C23—41no—
175C23—41no—
176C23—82no—
177C23—82no—
178C23—82no—
179C23—82no—
180C23#33611yescharacteristic
181C23#34211yescharacteristic
182C23#34311yescharacteristic
183C23#34312yes—
184C23#34412yes—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4#32311yescharacteristic
221C2×C4#32311yescharacteristic
222C2×C4#32411yescharacteristic
223C2×C4#32411yescharacteristic
224C2×C4AΣL(1,𝔽16)12yes—
225C2×C4#35212yes—
226C8—44no—
227C8—82no—
228C8—164no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22#128011yescharacteristic
250C22#128312yes—
251C22#130512yes—
252C22#130711yescharacteristic
253C22#130911yescharacteristic
254C22#139012yes—
255C22#139111yescharacteristic
256C22#139211yescharacteristic
257C4—21no—
258C4—21no—
259C4—41no—
260C4—41no—
261C4—42no—
262C4—42no—
263C4—82no—
264C4#131611yescharacteristic
265C4#131711yescharacteristic
266C4#132612yes—
267C2—22no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—81no—
272C2#703411yescharacteristic
273C2#718711yescharacteristic
274C2#718811yescharacteristic
275C2#732012yes—
276C2#732112yes—
subgroupG/H
center, socleC23#342
commutatorC24C4×C8
Frattini, cosocleC2×C4×C22⋊C4C22
prime radicalC25C42
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×C4×C22⋊C4 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C22×M16C8semidirect
25. C2×C22⋊C8C8semidirect

Maximal subgroups

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

subgroupindex
#107152
#85572