Finite Group Databasebeta

← #127470

Subgroups — #127470

Order 512.

713 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 136 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#21786C211yesmaximal, characteristic
2#17362C212yesmaximal
3#7249C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#431C411yescharacteristic
6C2×#431C411yescharacteristic
7C2×#407—22no—
8#1954—22no—
9#1954C412yes—
10#1954C2212yes—
11#1779—22no—
12#1465—22no—
13#1465C2214yes—
14#1463—22no—
15#1462C414yes—
16#1459—22no—
17#1459C414yes—
18C23×D4Q811yescharacteristic
19C24×C4D412yes—
20C22×M16—42no—
21C2×C4×D4—42no—
22C2×C4×D4—42no—
23C2×C4×D4C2×C412yes—
24C2×C4×D4C2×C411yescharacteristic
25C2×C4×D4D412yes—
26C2×C4×D4D411yescharacteristic
27C2×C4×D4Q811yescharacteristic
28C2×C4×D4C2311yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C22⋊C4C2×C412yes—
31C22×C42C2×C411yescharacteristic
32#431—42no—
33#431—42no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8C2×C412yes—
36#407—24no—
37#407—24no—
38#407—42no—
39#407—42no—
40C2×C22⋊C8—24no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—24no—
45C2×C22⋊C8—42no—
46C2×C4×C8—21no—
47C2×C4×C8C2×C412yes—
48#324—24no—
49#324—24no—
50#323—24no—
51#323C2×C418yes—
52C25C4⋊C412yes—
53C22×D4—24no—
54C22×D4—42no—
55C22×D4—42no—
56C22×D4C4⋊C412yes—
57C22×D4C4⋊C411yescharacteristic
58C22×D4C2×Q811yescharacteristic
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—84no—
64C23×C4C4211yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C412yes—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C4⋊C411yescharacteristic
70C23×C4C4⋊C412yes—
71C23×C4C2×D412yes—
72C2×M16—44no—
73C2×M16—44no—
74C2×M16—88no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—44no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—41no—
81C4×D4—41no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—42no—
85C4×D4—42no—
86C4×D4—41no—
87C4×D4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C22⋊C412yes—
90C2×C4⋊C4C4⋊C411yescharacteristic
91C2×C4⋊C4C2×D411yescharacteristic
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4C4212yes—
95C2×C22⋊C4C22⋊C412yes—
96C2×C22⋊C4C4⋊C412yes—
97C2×C22⋊C4C22×C412yes—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42C4211yescharacteristic
101C2×C42C22⋊C411yescharacteristic
102C2×C42C4⋊C411yescharacteristic
103C2×C42C22×C411yescharacteristic
104C4⋊C8—42no—
105C4⋊C8—42no—
106C22⋊C8—28no—
107C22⋊C8—44no—
108C22⋊C8—44no—
109C22⋊C8—44no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—84no—
113C22⋊C8—84no—
114C4×C8—42no—
115C4×C8—42no—
116C24—22no—
117C24—22no—
118C24—24no—
119C24—42no—
120C24—82no—
121C24—82no—
122C24#9512yes—
123C24#9511yescharacteristic
124C24C2×C4⋊C412yes—
125C2×D4—24no—
126C2×D4—24no—
127C2×D4—22no—
128C2×D4—42no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—416no—
134C2×D4#10814yes—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—48no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—84no—
154C22×C4—82no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4—84no—
158C22×C4—84no—
159C22×C4—81no—
160C22×C4#9511yescharacteristic
161C22×C4#9511yescharacteristic
162C22×C4#9512yes—
163C22×C4#9511yescharacteristic
164C22×C4#9912yes—
165C22×C4#9912yes—
166C22×C4C2×C4211yescharacteristic
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×C22⋊C412yes—
169C22×C4C2×C4⋊C411yescharacteristic
170M16—88no—
171M16—88no—
172C2×C8—48no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—88no—
179C4⋊C4—42no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—22no—
187C42—22no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—41no—
200C23—41no—
201C23—44no—
202C23—44no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—44no—
208C23—44no—
209C23—82no—
210C23—82no—
211C23—88no—
212C23—88no—
213C23#34114yes—
214C23#34212yes—
215C23C2×#9511yescharacteristic
216D4—44no—
217D4—48no—
218D4—44no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4#33812yes—
248C2×C4#34212yes—
249C2×C4C2×#9911yescharacteristic
250C2×C4C2×#9911yescharacteristic
251C2×C4C2×#10812yes—
252C8—82no—
253C8—88no—
254C8—82no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—24no—
261C22—44no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22—82no—
272C22—88no—
273C22—88no—
274C22C2×#33811yescharacteristic
275C22C2×#34111yescharacteristic
276C22C2×#34211yescharacteristic
277C4—21no—
278C4—22no—
279C4—22no—
280C4—22no—
281C4—41no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4#212712yes—
286C2—21no—
287C2—21no—
288C2—42no—
289C2—44no—
290C2—82no—
291C2#800811yescharacteristic
292C2#1257911yescharacteristic
293C2#1258211yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7249C2semidirect

Maximal subgroups

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

subgroupindex
#217862
#173622
#72492