Finite Group Databasebeta

← #127452

Subgroups — #127452

Order 512.

592 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 142 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#21373C211yesmaximal, characteristic
2#17361C212yesmaximal
3#7248C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5#2920—22no—
6C2×#431C411yescharacteristic
7C22×C4×C8C411yescharacteristic
8#1984C2212yes—
9#1954C412yes—
10#1869—22no—
11C4×C22⋊C8—22no—
12#1467C2214yes—
13#1466—22no—
14#1465—22no—
15#1465C414yes—
16#1458—22no—
17#1458C414yes—
18C22×#41Q811yescharacteristic
19C23×C8—21no—
20C4×#41—41no—
21C4×#41—41no—
22C4×#41—41no—
23C4×#41—41no—
24C2×C4×Q8Q811yescharacteristic
25C2×C4×Q8C2311yescharacteristic
26C2×C4×D4C2×C412yes—
27C2×C4×D4C2×C411yescharacteristic
28C2×C4×D4D411yescharacteristic
29C2×C4×D4D412yes—
30C2×#117C2×C412yes—
31C2×#117D411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D412yes—
34#432—24no—
35#432—42no—
36#431—24no—
37#431—42no—
38#431—41no—
39#431—41no—
40#423—42no—
41C2×C4⋊C8C2×C412yes—
42#407—24no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—24no—
45C4×M16—42no—
46C2×C4×C8—21no—
47C2×C4×C8—21no—
48C2×C4×C8—41no—
49C2×C4×C8—41no—
50C2×C4×C8C2×C412yes—
51#336—24no—
52#324—24no—
53#324C2×C418yes—
54#323—24no—
55#323—24no—
56C2×#41—24no—
57C2×#41—41no—
58C2×#41—41no—
59C2×#41—42no—
60C22×Q8C2×Q811yescharacteristic
61C22×D4C4⋊C412yes—
62C22×D4C4⋊C411yescharacteristic
63C23×C4C4211yescharacteristic
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C4⋊C412yes—
68C23×C4C4⋊C411yescharacteristic
69C2×M16—44no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—44no—
73C22×C8—42no—
74C4×Q8—41no—
75C4×Q8—41no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—41no—
81C4×D4—41no—
82C4×D4—41no—
83C4×D4—41no—
84C4×D4—42no—
85C4×D4—42no—
86#117—42no—
87#117—42no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4C4⋊C412yes—
91C2×C4⋊C4C4⋊C411yescharacteristic
92C2×C4⋊C4C22×C412yes—
93C2×C4⋊C4C2×D411yescharacteristic
94C2×C22⋊C4C4212yes—
95C2×C22⋊C4C22⋊C412yes—
96C2×C22⋊C4C22⋊C412yes—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42C4211yescharacteristic
106C2×C42C22⋊C412yes—
107C2×C42C22⋊C411yescharacteristic
108C2×C42C4⋊C412yes—
109C2×C42C4⋊C411yescharacteristic
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D412yes—
112C4⋊C8—42no—
113C4⋊C8—44no—
114C4⋊C8—44no—
115C22⋊C8—28no—
116C22⋊C8—48no—
117C22⋊C8—42no—
118C22⋊C8—44no—
119#97—44no—
120C4×C8—42no—
121C4×C8—44no—
122C4×C8—41no—
123C4×C8—41no—
124C4×C8—41no—
125C4×C8—41no—
126C24#9512yes—
127C24#9511yescharacteristic
128#41—48no—
129#41—48no—
130C2×Q8—22no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×D4—24no—
134C2×D4—22no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—41no—
138C2×D4—42no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4#9511yescharacteristic
159C22×C4#9511yescharacteristic
160C22×C4#9512yes—
161C22×C4#9511yescharacteristic
162C22×C4#10814yes—
163C22×C4C2×C4211yescharacteristic
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C2×C22⋊C412yes—
166C22×C4C2×C4⋊C412yes—
167C22×C4C2×C4⋊C411yescharacteristic
168M16—88no—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—48no—
172C2×C8—42no—
173C2×C8—48no—
174C2×C8—44no—
175C2×C8—42no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C42—22no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—21no—
191C42—21no—
192C42—22no—
193C42—21no—
194C42—22no—
195C42#9911yescharacteristic
196C42#9911yescharacteristic
197C42#9911yescharacteristic
198C42#9911yescharacteristic
199C23—21no—
200C23—21no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—41no—
205C23—41no—
206C23—42no—
207C23—42no—
208C23#34112yes—
209C23C2×#9511yescharacteristic
210Q8—44no—
211D4—44no—
212D4—48no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4#33812yes—
241C2×C4#34112yes—
242C2×C4#34211yescharacteristic
243C2×C4#34211yescharacteristic
244C2×C4#34211yescharacteristic
245C2×C4#34211yescharacteristic
246C2×C4C2×#9911yescharacteristic
247C2×C4C2×#9911yescharacteristic
248C2×C4C2×#10812yes—
249C8—44no—
250C8—88no—
251C8—82no—
252C22—21no—
253C22—21no—
254C22—22no—
255C22—42no—
256C22—42no—
257C22—44no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22C2×#33811yescharacteristic
265C22C2×#34111yescharacteristic
266C22C2×#34211yescharacteristic
267C4—21no—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—41no—
272C4—41no—
273C4—42no—
274C4—42no—
275C4#129312yes—
276C4#129312yes—
277C4#139112yes—
278C4#213212yes—
279C2—21no—
280C2—21no—
281C2—42no—
282C2—82no—
283C2C2×#129311yescharacteristic
284C2C2×#139111yescharacteristic
285C2#1258511yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C2×#342
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7248C2semidirect

Maximal subgroups

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

subgroupindex
#213732
#173612
#72482