Finite Group Databasebeta

← #126972

Subgroups — #126972

Order 512.

986 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C2×#1464C212yesmaximal
2#13756C211yesmaximal, characteristic
3#7187C214yesmaximal
4C22×#117C411yescharacteristic
5C22×#117C2211yescharacteristic
6C23×C42C411yescharacteristic
7C2×#432—22no—
8#1464—24no—
9#1464C418yescomplemented
10C2×#324—22no—
11C2×#323C2214yes—
12C4⋊#95—22no—
13C22⋊#95C412yes—
14C22⋊#95C2212yes—
15C4×#95—22no—
16C24×C4C2×C411yescharacteristic
17C24×C4C2×C412yes—
18C2×#117—24no—
19C2×#117—42no—
20C2×#117—42no—
21C2×#117—42no—
22C2×#117—42no—
23C2×#117C2×C414yes—
24C2×#117D412yes—
25C2×#117Q812yes—
26C22×C4⋊C4—21no—
27C22×C4⋊C4D412yes—
28C22×C22⋊C4C2×C412yes—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2311yescharacteristic
31C22×C42—21no—
32C22×C42—21no—
33C22×C42—21no—
34C22×C42—24no—
35C22×C42—42no—
36C22×C42—42no—
37C22×C42D412yes—
38#432—28no—
39#432—44no—
40C2×C4⋊C8—42no—
41C2×C22⋊C8—24no—
42C2×#97—42no—
43C2×#95—24no—
44C2×#95—24no—
45#324—28no—
46#323—24no—
47#323C2×C418yes—
48C25C22×C411yescharacteristic
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—48no—
59C23×C4—42no—
60C23×C4—42no—
61C23×C4C4212yes—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C412yes—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C4⋊C412yes—
66C23×C4M1614yes—
67C23×C4C2×D412yes—
68C22×C8—44no—
69#117—44no—
70#117—44no—
71#117—44no—
72#117—44no—
73#117—416no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4C22⋊C414yes—
80C2×C4⋊C4C4⋊C414yes—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4C4212yes—
89C2×C22⋊C4C4⋊C412yes—
90C2×C22⋊C4M1614yes—
91C2×C22⋊C4C22×C412yes—
92C2×C22⋊C4C2×D411yescharacteristic
93C2×C22⋊C4C2×Q811yescharacteristic
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—24no—
97C2×C42—24no—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—42no—
101C2×C42—416no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42C22⋊C414yes—
108C2×C42C4⋊C414yes—
109C4⋊C8—48no—
110C22⋊C8—28no—
111C22⋊C8—44no—
112#97—48no—
113#95—48no—
114#95—48no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24—21no—
119C24—44no—
120C24C2×C4211yescharacteristic
121C24C2×C22⋊C411yescharacteristic
122C24C2×C4⋊C411yescharacteristic
123C24C2×M1612yes—
124C24C2×M1612yes—
125C22×C4—28no—
126C22×C4—28no—
127C22×C4—28no—
128C22×C4—28no—
129C22×C4—24no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—28no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—48no—
151C22×C4—48no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—416no—
155C22×C4—48no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—48no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4#9512yes—
167C22×C4#9512yes—
168C22×C4#9514yes—
169C22×C4#9912yes—
170C22×C4#9912yes—
171C22×C4#10012yes—
172C22×C4#10112yes—
173C22×C4C2×C22⋊C412yes—
174C22×C4C2×C4⋊C412yes—
175C2×C8—48no—
176C2×C8—88no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—48no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C42—24no—
188C42—24no—
189C42—24no—
190C42—24no—
191C42—48no—
192C42—48no—
193C42—48no—
194C23—24no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—44no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23C2×#9511yescharacteristic
213C23C4×M1612yes—
214C23#40711yescharacteristic
215C23#40711yescharacteristic
216C23C2×#9911yescharacteristic
217C23C2×#9911yescharacteristic
218C23C2×#10011yescharacteristic
219C23C2×#10111yescharacteristic
220C23#42312yes—
221C2×C4—216no—
222C2×C4—216no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—28no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—48no—
241C2×C4—416no—
242C2×C4—48no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—48no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—48no—
251C2×C4—48no—
252C2×C4—44no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#34214yes—
259C2×C4#34314yes—
260C8—88no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—42no—
270C22—42no—
271C22—44no—
272C22—44no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—44no—
280C22#146211yescharacteristic
281C22#146211yescharacteristic
282C22#146311yescharacteristic
283C22#146311yescharacteristic
284C22#172811yescharacteristic
285C22C2×#34211yescharacteristic
286C22C2×#34311yescharacteristic
287C22#212014yes—
288C4—24no—
289C4—24no—
290C4—42no—
291C4—44no—
292C4—42no—
293C4—48no—
294C4—84no—
295C2—21no—
296C2—21no—
297C2—21no—
298C2—21no—
299C2—42no—
300C2—42no—
301C2#790911yescharacteristic
302C2#1150112yes—
303C2#1151812yes—
304C2#1257312yes—
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1464C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14642
#137562
#71872