Finite Group Databasebeta

← #124993

Subgroups — #124993

Order 512.

608 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 78 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#14169C211yesmaximal, characteristic
2#8699C212yesmaximal, complemented
3#7131C212yesmaximal, complemented
4#7103C212yesmaximal
5C22×#124C411yescharacteristic
6C2×#376C411yescharacteristic
7C2×#376C2211yescharacteristic
8#1817—22no—
9#1446C411yescharacteristic
10#1446C2211yescharacteristic
11#1438C411yescharacteristic
12#1438C2211yescharacteristic
13#1437—22no—
14#1437—22no—
15#1280—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C23×D4C2×C411yescharacteristic
19C2×#124—22no—
20C2×#124—22no—
21C2×#124—42no—
22C2×#124—42no—
23C22×C22⋊C4—21no—
24C22×C22⋊C4—21no—
25C22×C22⋊C4—22no—
26C22×C22⋊C4—22no—
27C22×C42—21no—
28C22×C42C2×C411yescharacteristic
29C22×C42D412yes—
30C2×C22⋊C8—24no—
31#376—42no—
32#376—42no—
33C2×#95—21no—
34C2×#95—22no—
35C2×#95—22no—
36C2×#95C2×C411yescharacteristic
37C2×#95C2×C411yescharacteristic
38C2×#95C2×C411yescharacteristic
39C2×#95D412yes—
40C2×#95C2311yescharacteristic
41#336—44no—
42C25—21no—
43C22×Q8—21no—
44C22×Q8—41no—
45C22×Q8—41no—
46C22×Q8—82no—
47C22×Q8M1612yes—
48C22×D4—21no—
49C22×D4—41no—
50C22×D4—41no—
51C22×D4—82no—
52C22×D4M1612yes—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22×C411yescharacteristic
60C23×C4C2×D412yes—
61C22×C8—44no—
62#124—48no—
63#124—48no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—42no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—44no—
71C2×C22⋊C4—44no—
72C2×C22⋊C4—44no—
73C2×C22⋊C4—44no—
74C2×C22⋊C4—82no—
75C2×C22⋊C4—82no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—24no—
81C2×C42—24no—
82C2×C42—42no—
83C2×C42—42no—
84C2×C42—42no—
85C2×C42—42no—
86C22⋊C8—44no—
87C22⋊C8—44no—
88#95—41no—
89#95—42no—
90#95—41no—
91#95—41no—
92#95—41no—
93#95—42no—
94#95—82no—
95#95—82no—
96C24—22no—
97C24—42no—
98C24—42no—
99C24—42no—
100C24—42no—
101C24—41no—
102C24—41no—
103C24—42no—
104C24—42no—
105C24C2×C22⋊C411yescharacteristic
106C2×Q8—42no—
107C2×Q8—42no—
108C2×Q8—82no—
109C2×Q8—82no—
110C2×Q8—84no—
111C2×Q8—84no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—82no—
115C2×D4—84no—
116C2×D4—84no—
117C2×D4—82no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4#9912yes—
154C22×C4#10012yes—
155C22×C4C2×M1611yescharacteristic
156C22×C4C2×M1611yescharacteristic
157C22×C4#13514yes—
158C2×C8—84no—
159C2×C8—84no—
160C2×C8—84no—
161C22⋊C4—48no—
162C22⋊C4—48no—
163C22⋊C4—88no—
164C22⋊C4—88no—
165C42—24no—
166C42—24no—
167C42—48no—
168C42—48no—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—42no—
173C23—42no—
174C23—44no—
175C23—42no—
176C23—44no—
177C23—82no—
178C23—82no—
179C23—82no—
180C23—82no—
181C23—82no—
182C23—82no—
183C23—82no—
184C23—82no—
185C23—82no—
186C23—82no—
187C23—81no—
188C23—81no—
189C23—82no—
190C23—81no—
191C23—81no—
192C23—82no—
193C23#40711yescharacteristic
194C23C2×#9911yescharacteristic
195C23C2×#10011yescharacteristic
196C23#41612yes—
197C23#42112yes—
198Q8—84no—
199Q8—84no—
200D4—84no—
201D4—84no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—82no—
243C8—164no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—84no—
253C22—84no—
254C22—82no—
255C22—82no—
256C22—82no—
257C22—162no—
258C22—162no—
259C22—162no—
260C22—162no—
261C22#146211yescharacteristic
262C22#148812yes—
263C22#151612yes—
264C22#153611yescharacteristic
265C22#154311yescharacteristic
266C22#212212yes—
267C22#212611yescharacteristic
268C22#213211yescharacteristic
269C4—44no—
270C4—42no—
271C4—41no—
272C4—41no—
273C4—42no—
274C4—82no—
275C4—82no—
276C4—82no—
277C4—82no—
278C2—21no—
279C2—21no—
280C2—22no—
281C2—162no—
282C2#764511yescharacteristic
283C2#1150311yescharacteristic
284C2#1150711yescharacteristic
285C2#1184412yes—
286C2#1185112yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#376 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8699C2semidirect
3. #7131C2semidirect

Maximal subgroups

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

subgroupindex
#141692
#86992
#71312
#71032