Finite Group Databasebeta

← #108572

Subgroups — #108572

Order 512.

898 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C2×C4×C22⋊C8C211yesmaximal, characteristic
2#10719C212yesmaximal
3#8008C211yesmaximal, characteristic
4#7982C211yesmaximal, characteristic
5#7927C212yesmaximal
6C23×C42C2211yescharacteristic
7C2×#407C2212yes—
8C22×C22⋊C8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C4×C8C412yescomplemented
11#2280C2212yes—
12C4×C22⋊C8—24no—
13C2×#339—24no—
14C2×#336C412yescomplemented
15#1293C414yescomplemented
16#1289C414yescomplemented
17#1282—24no—
18C24×C4—21no—
19C24×C4C2311yescharacteristic
20C22×M16—22no—
21C23×C8C2×C412yes—
22C23×C8C2×C412yes—
23C2×#117—22no—
24C22×C42—22no—
25C22×C42—21no—
26C22×C42—44no—
27C22×C42C2×C411yescharacteristic
28C22×C42C2×C411yescharacteristic
29C22×C42D414yes—
30#407—22no—
31#407—22no—
32#407—22no—
33#407C2×C414yes—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8C2×C412yes—
41C2×C22⋊C8C2×C412yes—
42C2×C22⋊C8D412yes—
43C2×C22⋊C8Q812yes—
44C2×C4×C8—22no—
45C2×C4×C8—22no—
46C2×C4×C8—44no—
47#381—22no—
48#380—22no—
49#380C2×C414yes—
50#379—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53#339—48no—
54#336—24no—
55C25#4111yescharacteristic
56C23×C4—21no—
57C23×C4—28no—
58C23×C4—21no—
59C23×C4—24no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4—44no—
63C23×C4—44no—
64C23×C4C22×C411yescharacteristic
65C23×C4C22×C411yescharacteristic
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D412yes—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×Q811yescharacteristic
70C23×C4#4112yes—
71C23×C4#4112yes—
72C23×C4#4112yes—
73C23×C4#4111yescharacteristic
74C2×M16—24no—
75C2×M16—42no—
76C2×M16—44no—
77C2×M16—44no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8C4214yes—
88C22×C8C4⋊C414yes—
89#117—44no—
90#117—44no—
91C2×C4⋊C4—24no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C42—21no—
96C2×C42—24no—
97C2×C42—28no—
98C2×C42—21no—
99C2×C42—24no—
100C2×C42—42no—
101C2×C42—44no—
102C2×C42—48no—
103C2×C42—44no—
104C2×C42—42no—
105C2×C42C22⋊C412yes—
106C2×C42C22⋊C412yes—
107C22⋊C8—24no—
108C22⋊C8—24no—
109C22⋊C8—24no—
110C22⋊C8—24no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C4×C8—44no—
116C4×C8—44no—
117#95—24no—
118#95—42no—
119C24—24no—
120C24—41no—
121C24—41no—
122C24—42no—
123C24#11711yescharacteristic
124C24C4×D411yescharacteristic
125C24C4×Q811yescharacteristic
126C24#12312yes—
127C24#12612yes—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—216no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—48no—
150C22×C4—48no—
151C22×C4—432no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4C2×C4211yescharacteristic
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4C2×C4⋊C411yescharacteristic
158C22×C4#11711yescharacteristic
159C22×C4#11711yescharacteristic
160C22×C4#11711yescharacteristic
161C22×C4#11712yes—
162C22×C4#11712yes—
163C22×C4C4×D412yes—
164C22×C4C4×D412yes—
165C22×C4C4×D411yescharacteristic
166C22×C4C4×Q811yescharacteristic
167C22×C4#12112yes—
168C22×C4C22⋊Q812yes—
169C22×C4#12312yes—
170C22×C4#12412yes—
171C22×C4#12512yes—
172C22×C4#12612yes—
173M16—48no—
174M16—82no—
175M16—82no—
176C2×C8—24no—
177C2×C8—24no—
178C2×C8—44no—
179C2×C8—44no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—44no—
185C2×C8—44no—
186C2×C8—44no—
187C2×C8—84no—
188C4⋊C4—48no—
189C22⋊C4—48no—
190C22⋊C4—84no—
191C22⋊C4—82no—
192C22⋊C4—82no—
193C42—24no—
194C42—22no—
195C42—24no—
196C42—48no—
197C42—42no—
198C42—84no—
199C42#9914yes—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23—41no—
215C23—41no—
216C23—48no—
217C23—48no—
218C23C4×C4⋊C411yescharacteristic
219C23#38111yescharacteristic
220C23#38311yescharacteristic
221C23#38712yes—
222C23#38812yes—
223C23#44314yes—
224C2×C4—24no—
225C2×C4—216no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—432no—
249C2×C4—48no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4#37611yescharacteristic
256C2×C4C4×C22⋊C411yescharacteristic
257C2×C4#38011yescharacteristic
258C2×C4#38211yescharacteristic
259C2×C4#38712yes—
260C2×C4#38812yes—
261C2×C4C2×#9912yes—
262C2×C4#41012yes—
263C8—44no—
264C8—44no—
265C8—84no—
266C22—24no—
267C22—24no—
268C22—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—44no—
280C22—44no—
281C22—44no—
282C22—48no—
283C22—44no—
284C22#143411yescharacteristic
285C22#177212yes—
286C22#181911yescharacteristic
287C22#182011yescharacteristic
288C22#184112yes—
289C22#193714yes—
290C4—28no—
291C4—22no—
292C4—21no—
293C4—21no—
294C4—22no—
295C4—44no—
296C4—164no—
297C4C4×#9912yes—
298C4#179912yes—
299C2—21no—
300C2—24no—
301C2—21no—
302C2—22no—
303C2—22no—
304C2—42no—
305C2#1080111yescharacteristic
306C2#1081311yescharacteristic
307C2#1081811yescharacteristic
subgroupG/H
centerC2×C4#380
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22#1434
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C22×C4×C8C4semidirect
14. C2×#336C4semidirect
15. #1293C4semidirect
16. #1289C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊C82
#107192
#80082
#79822
#79272