Finite Group Databasebeta

← #162908

Subgroups — #162908

Order 512.

1390 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1390 conjugacy classes of subgroups — too many to draw. Only its 258 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×#2857C211yesmaximal, characteristic
2C4×#407C212yesmaximal
3#7687C218yesmaximal, complemented
4#7629C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊Q8C412yes—
7C23×C42C2211yescharacteristic
8C2×#407—22no—
9C2×C4×M16—22no—
10#2857—22no—
11#2857—22no—
12#2857C414yes—
13#2857C2214yes—
14C2×#391C2212yes—
15C2×#389—22no—
16C2×#384C411yescharacteristic
17C2×#380—22no—
18#1749—22no—
19#1749C2214yes—
20C4×C22⋊C8—22no—
21C4×C22⋊C8C2214yes—
22#1728C2214yes—
23#1272C22116yescomplemented
24C23×Q8C2×C412yes—
25C24×C4C2×C411yescharacteristic
26C24×C4D412yes—
27C22×M16—24no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—44no—
31C2×C4⋊Q8C2×C414yes—
32C2×C22⋊Q8—24no—
33C2×C22⋊Q8—48no—
34C2×C22⋊Q8C2×C418yes—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2×C412yes—
39C22×C22⋊C4—22no—
40C22×C42—44no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C412yes—
43C22×C42D412yes—
44C22×C42D412yes—
45C22×C42D412yes—
46C22×C42C2312yes—
47C22×C42C2311yescharacteristic
48#407—216no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C4×M16—48no—
52C2×#97—24no—
53C2×C4×C8—24no—
54#391—24no—
55#391C2×C414yes—
56#391C2314yes—
57#389—24no—
58#389—24no—
59#389—24no—
60#389—24no—
61#384—21no—
62#384—21no—
63#384C2×C414yes—
64#380—24no—
65#380—24no—
66#380—24no—
67#380—24no—
68C2×#95—24no—
69#336—24no—
70#336C2318yes—
71C25C22⋊C411yescharacteristic
72C22×Q8—22no—
73C22×Q8—44no—
74C22×Q8—44no—
75C22×Q8—44no—
76C22×Q8C22×C414yes—
77C23×C4—24no—
78C23×C4—28no—
79C23×C4—28no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—44no—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C2×M16—216no—
96C2×M16—48no—
97C2×M16—48no—
98C22×C8—28no—
99C4⋊Q8—44no—
100C4⋊Q8—48no—
101C4⋊Q8—44no—
102C22⋊Q8—416no—
103C22⋊Q8—416no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—48no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—48no—
114C2×C4⋊C4C22×C414yes—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—48no—
117C2×C42—24no—
118C2×C42—28no—
119C2×C42—44no—
120C2×C42—44no—
121C2×C42—44no—
122C2×C42—44no—
123C2×C42—42no—
124C2×C42C22⋊C411yescharacteristic
125C2×C42C22⋊C412yes—
126C2×C42C22⋊C412yes—
127C2×C42C22⋊C411yescharacteristic
128C2×C42C22×C412yes—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D412yes—
133C2×C42C2411yescharacteristic
134C22⋊C8—216no—
135C22⋊C8—216no—
136#97—48no—
137C4×C8—48no—
138#95—28no—
139#95—28no—
140C24—22no—
141C24—24no—
142C24—24no—
143C24—41no—
144C24—41no—
145C24C2×C22⋊C412yes—
146C24C2×C22⋊C411yescharacteristic
147C2×Q8—28no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—816no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—432no—
184C22×C4—416no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—48no—
190C22×C4#10014yes—
191C22×C4#10418yes—
192C22×C4C2×C22⋊C412yes—
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C2×C22⋊C412yes—
195C22×C4C2×C22⋊C412yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4#13718yes—
198C22×C4C23×C411yescharacteristic
199C22×C4C22×D412yes—
200M16—432no—
201C2×C8—216no—
202C2×C8—48no—
203C2×C8—48no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—48no—
208C4⋊C4—416no—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C22⋊C4—416no—
212C42—24no—
213C42—24no—
214C42—24no—
215C42—22no—
216C42—24no—
217C42—42no—
218C42—41no—
219C42—42no—
220C42—41no—
221C42#9914yes—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—28no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—48no—
233C23—48no—
234C23—44no—
235C23—44no—
236C23C2×#10012yes—
237C23#41914yes—
238C23C2×#10414yes—
239C23C22×C22⋊C411yescharacteristic
240Q8—88no—
241Q8—88no—
242C2×C4—28no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—216no—
254C2×C4—216no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—48no—
263C2×C4—48no—
264C2×C4—48no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—48no—
270C2×C4—88no—
271C2×C4—88no—
272C2×C4C2×#9912yes—
273C2×C4C2×#9914yes—
274C2×C4C2×#10014yes—
275C2×C4#41914yes—
276C2×C4C2×#10414yes—
277C2×C4C2×#10414yes—
278C2×C4#56812yes—
279C2×C4C2×#13714yes—
280C8—416no—
281C22—22no—
282C22—24no—
283C22—24no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—24no—
288C22—416no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22#152514yes—
294C22#287911yescharacteristic
295C22C22×#9911yescharacteristic
296C22C22×#10011yescharacteristic
297C22C2×#41912yes—
298C22C22×#10412yes—
299C4—24no—
300C4—24no—
301C4—22no—
302C4—22no—
303C4—24no—
304C4—42no—
305C4—84no—
306C4—84no—
307C4#150818yes—
308C2—24no—
309C2—44no—
310C2#1739611yescharacteristic
311C2#1750112yes—
312C2C2×#150812yes—
313C2C2×#152511yescharacteristic
314C2#1774611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7687C2semidirect
4. #7629C2semidirect
23. #1272C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C4×#4072
#76872
#76292