Finite Group Databasebeta

← #124867

Subgroups — #124867

Order 512.

718 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 82 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#13959C211yesmaximal, characteristic
2#7910C212yesmaximal, complemented
3#7132C212yesmaximal, complemented
4#7103C212yesmaximal
5C22×C4×D4C411yescharacteristic
6C2×#432—22no—
7C2×#383C411yescharacteristic
8C2×#376C2211yescharacteristic
9#1445C411yescharacteristic
10#1445C2211yescharacteristic
11#1438C411yescharacteristic
12#1438C2211yescharacteristic
13#1437—22no—
14C22⋊#95—22no—
15#1280—22no—
16#1280C2214yes—
17C23×D4C2×C411yescharacteristic
18C24×C4—21no—
19C2×C4×D4—22no—
20C2×C4×D4—22no—
21C2×C4×D4—42no—
22C2×C4×D4—42no—
23C22×C4⋊C4C2×C411yescharacteristic
24C22×C22⋊C4—21no—
25C22×C22⋊C4—22no—
26C22×C22⋊C4—22no—
27C22×C42C2×C411yescharacteristic
28C22×C42D412yes—
29#432—28no—
30#432—44no—
31C2×C4⋊C8—42no—
32C2×C22⋊C8—24no—
33C2×#97—42no—
34#383—42no—
35#376—42no—
36C2×#95—21no—
37C2×#95—21no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95C2×C411yescharacteristic
41C2×#95C2×C411yescharacteristic
42C2×#95C2×C411yescharacteristic
43C2×#95D412yes—
44C2×#95C2311yescharacteristic
45C25—21no—
46C22×D4—21no—
47C22×D4—41no—
48C22×D4—41no—
49C22×D4—44no—
50C22×D4M1612yes—
51C23×C4—21no—
52C23×C4—24no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—42no—
56C23×C4—42no—
57C23×C4—82no—
58C23×C4—82no—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C412yes—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22×C411yescharacteristic
63C23×C4C2×D412yes—
64C22×C8—44no—
65C4×D4—48no—
66C4×D4—48no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4M1612yes—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—44no—
75C2×C22⋊C4—44no—
76C2×C22⋊C4—44no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—82no—
80C2×C22⋊C4—82no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—24no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—22no—
89C2×C42—22no—
90C4⋊C8—48no—
91C22⋊C8—28no—
92C22⋊C8—44no—
93#97—48no—
94#95—41no—
95#95—42no—
96#95—41no—
97#95—41no—
98#95—41no—
99#95—42no—
100#95—42no—
101#95—82no—
102#95—82no—
103C24—24no—
104C24—22no—
105C24—42no—
106C24—42no—
107C24—42no—
108C24—41no—
109C24—41no—
110C24—42no—
111C24—42no—
112C24C2×C22⋊C411yescharacteristic
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—44no—
116C2×D4—44no—
117C2×D4—48no—
118C2×D4—44no—
119C2×D4—44no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—48no—
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—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—88no—
156C22×C4—84no—
157C22×C4—84no—
158C22×C4—88no—
159C22×C4#9912yes—
160C22×C4#10012yes—
161C22×C4#10414yes—
162C22×C4C2×M1611yescharacteristic
163C22×C4C2×M1611yescharacteristic
164C22×C4AGL(1,ℤ/8)12yes—
165C22×C4#13712yes—
166C2×C8—48no—
167C2×C8—84no—
168C2×C8—84no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C22⋊C4—48no—
172C22⋊C4—88no—
173C22⋊C4—84no—
174C22⋊C4—84no—
175C42—24no—
176C42—24no—
177C42—24no—
178C42—24no—
179C42—24no—
180C42—24no—
181C23—22no—
182C23—21no—
183C23—21no—
184C23—24no—
185C23—44no—
186C23—44no—
187C23—44no—
188C23—44no—
189C23—42no—
190C23—44no—
191C23—44no—
192C23—44no—
193C23—42no—
194C23—44no—
195C23—82no—
196C23—82no—
197C23—82no—
198C23—82no—
199C23—82no—
200C23—81no—
201C23—81no—
202C23—81no—
203C23—82no—
204C23—81no—
205C23#40711yescharacteristic
206C23C2×#9911yescharacteristic
207C23C2×#10011yescharacteristic
208C23#41712yes—
209C23C2×#10412yes—
210D4—44no—
211D4—44no—
212D4—48no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—816no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C8—88no—
259C22—22no—
260C22—22no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—22no—
265C22—21no—
266C22—21no—
267C22—48no—
268C22—44no—
269C22—84no—
270C22—84no—
271C22—84no—
272C22—82no—
273C22—84no—
274C22—84no—
275C22—82no—
276C22#146211yescharacteristic
277C22#148912yes—
278C22#150412yes—
279C22#152511yescharacteristic
280C22#156011yescharacteristic
281C22#212012yes—
282C22#212811yescharacteristic
283C22#213411yescharacteristic
284C4—42no—
285C4—42no—
286C4—42no—
287C4—41no—
288C4—41no—
289C4—44no—
290C4—42no—
291C4—82no—
292C4—161no—
293C4—161no—
294C2—21no—
295C2—21no—
296C2—22no—
297C2—84no—
298C2#763111yescharacteristic
299C2#1150111yescharacteristic
300C2#1150911yescharacteristic
301C2#1175412yes—
302C2#1176112yes—
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. #7910C2semidirect
3. #7132C2semidirect

Maximal subgroups

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

subgroupindex
#139592
#79102
#71322
#71032