Finite Group Databasebeta

← #128178

Subgroups — #128178

Order 512.

660 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 138 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#17365C212yesmaximal
2#13850C211yesmaximal, characteristic
3#7326C214yesmaximal
4C22×C4×Q8C2211yescharacteristic
5#2920—22no—
6C2×C4×C4⋊C4C411yescharacteristic
7C2×#376C411yescharacteristic
8#1467—22no—
9#1467C414yescomplemented
10#1463—22no—
11#1463C2214yes—
12#1461—22no—
13#1461C414yescomplemented
14#1446C412yes—
15#1445C2212yes—
16#1441—22no—
17C4×#95—22no—
18C23×Q8C2×C411yescharacteristic
19C2×C4×Q8—41no—
20C2×C4×Q8—41no—
21C2×C4×Q8—41no—
22C2×C4×Q8—41no—
23C2×C4×Q8C2×C412yes—
24C2×C4×Q8C2×C412yes—
25C2×C4×Q8D411yescharacteristic
26C2×C4×Q8D411yescharacteristic
27C2×C4×Q8Q811yescharacteristic
28C2×C4×Q8Q811yescharacteristic
29C22×C4⋊C4—21no—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4C2311yescharacteristic
32C22×C42—21no—
33C22×C42—21no—
34C22×C42C2×C411yescharacteristic
35C22×C42D412yes—
36#432—24no—
37#432—42no—
38#431—24no—
39#431—42no—
40#423—42no—
41#407—24no—
42C4×M16—42no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45#376—41no—
46#376—41no—
47C2×#95—24no—
48C2×#95—21no—
49C2×#95—24no—
50C2×#95C2×C412yes—
51C2×#95C2×C412yes—
52#324—24no—
53#324—24no—
54#324—24no—
55#324C2×C418yes—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—44no—
59C22×Q8C4212yes—
60C22×Q8C4⋊C411yescharacteristic
61C22×Q8C4⋊C411yescharacteristic
62C23×C4—22no—
63C23×C4—22no—
64C23×C4C22⋊C412yes—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C2×M16—44no—
70C4×Q8—42no—
71C4×Q8—42no—
72C4×Q8—42no—
73C4×Q8—42no—
74C4×Q8—41no—
75C4×Q8—41no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—41no—
81C4×Q8—41no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4C22⋊C412yes—
89C2×C4⋊C4C22⋊C412yes—
90C2×C4⋊C4C4⋊C412yes—
91C2×C4⋊C4C4⋊C412yes—
92C2×C4⋊C4C22×C412yes—
93C2×C4⋊C4C2×D411yescharacteristic
94C2×C4⋊C4C2×Q811yescharacteristic
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—24no—
104C2×C42—42no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—44no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42C4211yescharacteristic
111C2×C42C4211yescharacteristic
112C2×C42C22⋊C412yes—
113C2×C42C22⋊C412yes—
114C2×C42C4⋊C412yes—
115C2×C42C4⋊C412yes—
116C2×C42C4⋊C411yescharacteristic
117C2×C42C4⋊C411yescharacteristic
118C4⋊C8—44no—
119C4⋊C8—44no—
120C22⋊C8—28no—
121C22⋊C8—44no—
122#97—44no—
123C4×C8—44no—
124#95—42no—
125#95—42no—
126#95—48no—
127#95—48no—
128#95—42no—
129C24C2×C22⋊C411yescharacteristic
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—42no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—48no—
138C2×Q8—48no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4#9512yes—
165C22×C4#9511yescharacteristic
166C22×C4#9511yescharacteristic
167C22×C4#9512yes—
168C22×C4#9511yescharacteristic
169C22×C4#9511yescharacteristic
170C22×C4#9911yescharacteristic
171C22×C4#9911yescharacteristic
172C22×C4#9911yescharacteristic
173C22×C4#9911yescharacteristic
174C22×C4C2×C4211yescharacteristic
175C22×C4C2×C22⋊C412yes—
176C22×C4C2×C4⋊C411yescharacteristic
177C22×C4C2×C4⋊C412yes—
178M16—88no—
179C2×C8—48no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—48no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—22no—
195C42—22no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—22no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42—22no—
204C42—44no—
205C42—44no—
206C42—82no—
207C42—82no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23C2×#9511yescharacteristic
213C23C2×#9911yescharacteristic
214C23C2×#9911yescharacteristic
215Q8—44no—
216Q8—48no—
217Q8—44no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—48no—
251C2×C4—48no—
252C2×C4—82no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4#34211yescharacteristic
260C2×C4#34211yescharacteristic
261C2×C4#34211yescharacteristic
262C2×C4#34211yescharacteristic
263C2×C4#34414yes—
264C2×C4#35512yes—
265C2×C4#35612yes—
266C8—88no—
267C22—24no—
268C22—24no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22C2×#34211yescharacteristic
276C22C2×#34412yes—
277C22#212612yes—
278C22C2×#35511yescharacteristic
279C22C2×#35611yescharacteristic
280C4—21no—
281C4—21no—
282C4—24no—
283C4—42no—
284C4—41no—
285C4—41no—
286C4—42no—
287C4—42no—
288C4—41no—
289C4—41no—
290C4—84no—
291C4—84no—
292C4#139412yes—
293C4#139412yes—
294C2—21no—
295C2—22no—
296C2—21no—
297C2—22no—
298C2#811211yescharacteristic
299C2#1258111yescharacteristic
300C2C2×#139411yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
socleC22C2×#342
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1467C4semidirect
13. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#173652
#138502
#73262