Finite Group Databasebeta

← #124513

Subgroups — #124513

Order 512.

862 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 110 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×#1445C211yesmaximal, characteristic
2#8698C212yesmaximal
3#7104C214yesmaximal, complemented
4C23×C4⋊C4C411yescharacteristic
5C22×#95—22no—
6C22×#95C411yescharacteristic
7C22×#95C2211yescharacteristic
8#1779—22no—
9#1445—22no—
10#1445C412yes—
11#1445C2212yes—
12#1280—22no—
13#1280C2214yes—
14C24×C4—21no—
15C24×C4—21no—
16C24×C4C2×C411yescharacteristic
17C24×C4D412yes—
18C22×C4⋊C4—21no—
19C22×C4⋊C4—22no—
20C22×C4⋊C4—41no—
21C22×C4⋊C4—41no—
22C22×C4⋊C4—42no—
23C22×C4⋊C4C814yescomplemented
24C22×C4⋊C4C2×C412yes—
25C2×C22⋊C8—24no—
26C2×C22⋊C8—42no—
27C2×C22⋊C8—42no—
28C2×#95—21no—
29C2×#95—24no—
30C2×#95—24no—
31C2×#95—41no—
32C2×#95—41no—
33C2×#95—42no—
34C2×#95—48no—
35C2×#95C2×C412yes—
36C2×#95C2×C411yescharacteristic
37C2×#95D412yes—
38C2×#95C2311yescharacteristic
39C25C22⋊C411yescharacteristic
40C23×C4—21no—
41C23×C4—21no—
42C23×C4—22no—
43C23×C4—24no—
44C23×C4—24no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—24no—
50C23×C4—42no—
51C23×C4—44no—
52C23×C4—44no—
53C23×C4—44no—
54C23×C4—44no—
55C23×C4—42no—
56C23×C4—44no—
57C23×C4—44no—
58C23×C4—44no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4C22⋊C412yes—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C2×C812yes—
66C23×C4M1612yes—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C22×C8—44no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—44no—
73C2×C4⋊C4—44no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—44no—
77C2×C4⋊C4—44no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4C2×C814yes—
81C22⋊C8—44no—
82C22⋊C8—44no—
83C22⋊C8—84no—
84C22⋊C8—84no—
85#95—41no—
86#95—42no—
87#95—41no—
88#95—42no—
89#95—42no—
90#95—48no—
91#95—48no—
92C24—24no—
93C24—22no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24C22⋊C812yes—
98C24#9911yescharacteristic
99C24#9911yescharacteristic
100C24#10012yes—
101C24C2×C22⋊C411yescharacteristic
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—48no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—416no—
131C22×C4—416no—
132C22×C4—48no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—416no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4—88no—
149C22×C4—88no—
150C22×C4—84no—
151C22×C4—84no—
152C22×C4—84no—
153C22×C4—84no—
154C22×C4—84no—
155C22×C4—84no—
156C22×C4C22⋊C812yes—
157C22×C4C22×C811yescharacteristic
158C22×C4C2×M1611yescharacteristic
159C22×C4AGL(1,ℤ/8)14yes—
160C2×C8—84no—
161C2×C8—84no—
162C2×C8—84no—
163C4⋊C4—44no—
164C4⋊C4—48no—
165C4⋊C4—44no—
166C23—28no—
167C23—28no—
168C23—28no—
169C23—28no—
170C23—22no—
171C23—24no—
172C23—24no—
173C23—24no—
174C23—24no—
175C23—22no—
176C23—22no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—24no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23#32312yes—
186C23#32312yes—
187C23#35214yes—
188C23#35412yes—
189C23#35612yes—
190C23C2×C22⋊C811yescharacteristic
191C23C2×#9911yescharacteristic
192C23C2×#10011yescharacteristic
193C23#41812yes—
194C23#42112yes—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—44no—
213C2×C4—48no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—48no—
220C2×C4—42no—
221C2×C4—82no—
222C2×C4—88no—
223C2×C4—88no—
224C2×C4—816no—
225C2×C4—84no—
226C2×C4—82no—
227C2×C4—88no—
228C2×C4—88no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—84no—
232C2×C4—82no—
233C2×C4—82no—
234C8—164no—
235C22—28no—
236C22—216no—
237C22—28no—
238C22—22no—
239C22—24no—
240C22—24no—
241C22—22no—
242C22—22no—
243C22—24no—
244C22—24no—
245C22—22no—
246C22—21no—
247C22—22no—
248C22—21no—
249C22—21no—
250C22—22no—
251C22—21no—
252C22—22no—
253C22—22no—
254C22#131714yes—
255C22#131914yes—
256C22C2×#32311yescharacteristic
257C22#147812yes—
258C22#151012yes—
259C22#153311yescharacteristic
260C22#155511yescharacteristic
261C22C2×#35212yes—
262C22C2×#35411yescharacteristic
263C22C2×#35611yescharacteristic
264C4—44no—
265C4—42no—
266C4—41no—
267C4—41no—
268C4—84no—
269C4—82no—
270C4—84no—
271C2—21no—
272C2—24no—
273C2—24no—
274C2—22no—
275C2—21no—
276C2#759211yescharacteristic
277C2C2×#131711yescharacteristic
278C2C2×#131911yescharacteristic
279C2#1179712yes—
280C2#1180312yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7104C2semidirect
23. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#14452
#86982
#71042