Finite Group Databasebeta

← #127458

Subgroups — #127458

Order 512.

571 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 140 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#21847C211yesmaximal, characteristic
2#17361C212yesmaximal
3#7249C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5#2920—22no—
6C2×#431C411yescharacteristic
7C22×C4⋊C8C411yescharacteristic
8#1984C412yes—
9#1954C2212yes—
10#1869—22no—
11#1797—22no—
12#1467—22no—
13#1466C414yes—
14#1465—22no—
15#1465C2214yes—
16#1458—22no—
17#1458C414yes—
18C22×#41Q811yescharacteristic
19C23×C8—21no—
20C4×#41—42no—
21C4×#41—42no—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×Q8D411yescharacteristic
24C2×C4×D4C2×C412yes—
25C2×C4×D4D412yes—
26C2×C4×D4Q811yescharacteristic
27C2×C4×D4C2311yescharacteristic
28C2×#117C2×C412yes—
29C2×#117D411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32#432—24no—
33#432—42no—
34#431—24no—
35#431—42no—
36#431—42no—
37#423—42no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—42no—
41C2×C4⋊C8C2×C412yes—
42#407—24no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—24no—
45C4×M16—42no—
46C2×C4×C8C2×C412yes—
47#336—24no—
48#324—24no—
49#324—24no—
50#323—24no—
51#323C2×C418yes—
52C2×#41—24no—
53C2×#41—42no—
54C2×#41—42no—
55C22×Q8C4⋊C411yescharacteristic
56C22×D4C4⋊C412yes—
57C22×D4C2×Q811yescharacteristic
58C23×C4C4211yescharacteristic
59C23×C4C22⋊C411yescharacteristic
60C23×C4C4⋊C412yes—
61C23×C4C4⋊C411yescharacteristic
62C23×C4C4⋊C412yes—
63C23×C4C2×D412yes—
64C2×M16—44no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—42no—
68C22×C8—82no—
69C4×Q8—41no—
70C4×Q8—41no—
71C4×Q8—42no—
72C4×D4—42no—
73C4×D4—42no—
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—41no—
78C4×D4—41no—
79#117—42no—
80#117—42no—
81#117—42no—
82C2×C4⋊C4C4212yes—
83C2×C4⋊C4C22⋊C412yes—
84C2×C4⋊C4C4⋊C411yescharacteristic
85C2×C4⋊C4C2×D411yescharacteristic
86C2×C22⋊C4C22⋊C412yes—
87C2×C22⋊C4C4⋊C412yes—
88C2×C22⋊C4C22×C412yes—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42C4211yescharacteristic
96C2×C42C22⋊C412yes—
97C2×C42C22⋊C412yes—
98C2×C42C22⋊C412yes—
99C2×C42C22⋊C411yescharacteristic
100C2×C42C4⋊C411yescharacteristic
101C2×C42C22×C411yescharacteristic
102C4⋊C8—42no—
103C4⋊C8—44no—
104C4⋊C8—44no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C22⋊C8—28no—
108C22⋊C8—42no—
109C22⋊C8—44no—
110C22⋊C8—84no—
111#97—44no—
112C4×C8—42no—
113C4×C8—44no—
114C24#9511yescharacteristic
115C24C2×C4⋊C412yes—
116#41—416no—
117C2×Q8—22no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×D4—24no—
121C2×D4—22no—
122C2×D4—42no—
123C2×D4—41no—
124C2×D4—42no—
125C2×D4—41no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—82no—
144C22×C4—81no—
145C22×C4#9512yes—
146C22×C4#9511yescharacteristic
147C22×C4#9512yes—
148C22×C4#9511yescharacteristic
149C22×C4#9511yescharacteristic
150C22×C4#10814yes—
151C22×C4C2×C4211yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×C22⋊C412yes—
154C22×C4C2×C4⋊C411yescharacteristic
155M16—84no—
156M16—84no—
157C2×C8—48no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—84no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C42—22no—
172C42—22no—
173C42—22no—
174C42—22no—
175C42—22no—
176C42—22no—
177C42—21no—
178C42—21no—
179C42—22no—
180C42—22no—
181C42#9912yes—
182C42#9912yes—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—41no—
189C23—41no—
190C23—42no—
191C23—42no—
192C23#34112yes—
193C23C2×#9511yescharacteristic
194Q8—44no—
195D4—44no—
196D4—48no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—28no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4#33812yes—
226C2×C4#34112yes—
227C2×C4#34212yes—
228C2×C4#34212yes—
229C2×C4C2×#9911yescharacteristic
230C2×C4C2×#9911yescharacteristic
231C2×C4C2×#10812yes—
232C8—82no—
233C8—88no—
234C8—82no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—42no—
239C22—42no—
240C22—44no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—82no—
247C22C2×#33811yescharacteristic
248C22C2×#34111yescharacteristic
249C22C2×#34211yescharacteristic
250C4—22no—
251C4—21no—
252C4—22no—
253C4—42no—
254C4—41no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4#129314yes—
259C4#212811yescharacteristic
260C4#212911yescharacteristic
261C2—21no—
262C2—21no—
263C2—42no—
264C2—82no—
265C2C2×#129311yescharacteristic
266C2#1257911yescharacteristic
267C2#1258311yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ 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
#218472
#173612
#72492