Finite Group Databasebeta

← #124932

Subgroups — #124932

Order 512.

890 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 102 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#14154C211yesmaximal, characteristic
2#8560C212yesmaximal, complemented
3#7103C214yesmaximal, complemented
4C22×#127C411yescharacteristic
5C2×#390—22no—
6C2×#390—22no—
7C2×C4×C4⋊C4C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1797—22no—
10C2×#324—22no—
11#1438C412yes—
12#1438C2212yes—
13#1280C2214yes—
14C23×D4—21no—
15C23×D4—21no—
16C23×D4C2×C412yes—
17C2×#127—24no—
18C2×#127—41no—
19C2×#127—41no—
20C2×#127—42no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4D412yes—
23C22×C22⋊C4—24no—
24C22×C22⋊C4—24no—
25C22×C42—21no—
26C22×C42C2×C411yescharacteristic
27C22×C42D412yes—
28C2×C4⋊C8—42no—
29C2×C22⋊C8—24no—
30#390—48no—
31#390—44no—
32#390—44no—
33C4×C4⋊C4—41no—
34C4×C4⋊C4—41no—
35C4×C4⋊C4—42no—
36C2×#95C2×C412yes—
37C2×#95C2×C411yescharacteristic
38C2×#95C2311yescharacteristic
39#336—42no—
40#324—44no—
41C25—22no—
42C22×D4—24no—
43C22×D4—24no—
44C22×D4—22no—
45C22×D4—42no—
46C22×D4—42no—
47C22×D4—42no—
48C22×D4—42no—
49C22×D4—44no—
50C22×D4—44no—
51C22×D4—44no—
52C22×D4—41no—
53C22×D4—41no—
54C22×D4—42no—
55C22×D4—44no—
56C22×D4—84no—
57C22×D4M1614yes—
58C23×C4—22no—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22×C411yescharacteristic
63C23×C4C2×D412yes—
64C22×C8—44no—
65#127—48no—
66#127—48no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4D814yes—
73C2×C4⋊C4SD1614yes—
74C2×C22⋊C4—48no—
75C2×C22⋊C4—48no—
76C2×C22⋊C4—48no—
77C2×C22⋊C4—44no—
78C2×C22⋊C4—44no—
79C2×C22⋊C4—84no—
80C2×C22⋊C4—84no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—44no—
88C2×C42—42no—
89C2×C42—44no—
90C4⋊C8—84no—
91C22⋊C8—44no—
92C22⋊C8—44no—
93#95—41no—
94#95—41no—
95#95—42no—
96C24—24no—
97C24—44no—
98C24—44no—
99C24—44no—
100C24—44no—
101C24—42no—
102C24—42no—
103C24—44no—
104C24—44no—
105C24C2×C22⋊C411yescharacteristic
106C2×D4—44no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4—48no—
110C2×D4—44no—
111C2×D4—416no—
112C2×D4—416no—
113C2×D4—88no—
114C2×D4—82no—
115C2×D4—82no—
116C2×D4—88no—
117C2×D4—84no—
118C2×D4—84no—
119C2×D4—88no—
120C2×D4—88no—
121C2×D4—84no—
122C2×D4—84no—
123C2×D4—82no—
124C2×D4—82no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4#9911yescharacteristic
149C22×C4#9911yescharacteristic
150C22×C4#10012yes—
151C22×C4#10214yes—
152C22×C4#10214yes—
153C22×C4C2×M1612yes—
154C22×C4C2×D812yes—
155C22×C4C2×SD1612yes—
156C2×C8—84no—
157C2×C8—84no—
158C2×C8—84no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—48no—
162C22⋊C4—816no—
163C22⋊C4—88no—
164C22⋊C4—88no—
165C42—24no—
166C42—24no—
167C42—44no—
168C42—82no—
169C42—84no—
170C23—22no—
171C23—21no—
172C23—21no—
173C23—48no—
174C23—44no—
175C23—44no—
176C23—44no—
177C23—48no—
178C23—84no—
179C23—84no—
180C23—84no—
181C23—84no—
182C23—84no—
183C23—84no—
184C23—84no—
185C23—84no—
186C23—84no—
187C23—84no—
188C23—82no—
189C23—84no—
190C23—82no—
191C23—82no—
192C23—84no—
193C23—82no—
194C23#40711yescharacteristic
195C23C2×#9911yescharacteristic
196C23C2×#10011yescharacteristic
197C23C2×#10212yes—
198C23#42112yes—
199D4—816no—
200D4—88no—
201D4—88no—
202D4—88no—
203D4—88no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—84no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4#33114yes—
236C2×C4#35312yes—
237C2×C4#35512yes—
238C8—164no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—88no—
248C22—88no—
249C22—84no—
250C22—84no—
251C22—84no—
252C22—164no—
253C22—164no—
254C22—164no—
255C22—164no—
256C22#146211yescharacteristic
257C22#148612yes—
258C22#151912yes—
259C22#153311yescharacteristic
260C22C2×#33111yescharacteristic
261C22#212212yes—
262C22C2×#35311yescharacteristic
263C22C2×#35511yescharacteristic
264C4—24no—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—44no—
269C4—42no—
270C4—42no—
271C4—84no—
272C2—21no—
273C2—22no—
274C2—21no—
275C2—164no—
276C2#764011yescharacteristic
277C2#1150311yescharacteristic
278C2#1150511yescharacteristic
279C2#1187712yes—
280C2#1188212yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC22×#127C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8560C2semidirect
3. #7103C2semidirect

Maximal subgroups

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

subgroupindex
#141542
#85602
#71032