Finite Group Databasebeta

← #163258

Subgroups — #163258

Order 512.

1254 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 234 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
2#20715C212yesmaximal
3#7727C218yesmaximal, complemented
4#7629C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊Q8C412yes—
7C23×C42C2211yescharacteristic
8C2×#432—22no—
9#2857—22no—
10#2857—22no—
11#2857C414yes—
12#2857C2214yes—
13C2×#391C2212yes—
14C2×#389—22no—
15C2×#384C411yescharacteristic
16C2×#380—22no—
17#1817C2214yes—
18#1797—22no—
19#1797C2214yes—
20#1779—22no—
21#1749—22no—
22#1749C2214yes—
23#1272C22116yescomplemented
24C23×Q8C2×C412yes—
25C24×C4C2×C411yescharacteristic
26C24×C4D412yes—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—44no—
30C2×C4⋊Q8C2×C414yes—
31C2×C22⋊Q8—24no—
32C2×C22⋊Q8—48no—
33C2×C22⋊Q8C2×C418yes—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4C2×C412yes—
38C22×C22⋊C4—22no—
39C22×C42—44no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C412yes—
42C22×C42D412yes—
43C22×C42D412yes—
44C22×C42D412yes—
45C22×C42C2312yes—
46C22×C42C2311yescharacteristic
47#432—48no—
48C2×C4⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—24no—
52C2×#97—24no—
53#391—24no—
54#391C2×C414yes—
55#391C2314yes—
56#389—24no—
57#389—24no—
58#389—24no—
59#389—24no—
60#384—21no—
61#384—21no—
62#384C2×C414yes—
63#380—24no—
64#380—24no—
65#380—24no—
66#380—24no—
67C2×#95—24no—
68#336—24no—
69#336C2318yes—
70C25C22⋊C411yescharacteristic
71C22×Q8—22no—
72C22×Q8—44no—
73C22×Q8—44no—
74C22×Q8—44no—
75C22×Q8C22×C414yes—
76C23×C4—24no—
77C23×C4—28no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—44no—
83C23×C4—44no—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C22×C8—28no—
95C4⋊Q8—44no—
96C4⋊Q8—48no—
97C4⋊Q8—44no—
98C22⋊Q8—416no—
99C22⋊Q8—416no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—48no—
108C2×C4⋊C4—48no—
109C2×C4⋊C4—48no—
110C2×C4⋊C4C22×C414yes—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—48no—
113C2×C42—24no—
114C2×C42—28no—
115C2×C42—44no—
116C2×C42—44no—
117C2×C42—44no—
118C2×C42—42no—
119C2×C42—44no—
120C2×C42C22⋊C411yescharacteristic
121C2×C42C22⋊C412yes—
122C2×C42C22⋊C412yes—
123C2×C42C22⋊C411yescharacteristic
124C2×C42C22×C412yes—
125C2×C42C22×C411yescharacteristic
126C2×C42C2×D412yes—
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42C2411yescharacteristic
130C4⋊C8—48no—
131C22⋊C8—48no—
132C22⋊C8—48no—
133C22⋊C8—48no—
134#97—48no—
135#95—28no—
136#95—28no—
137C24—22no—
138C24—24no—
139C24—24no—
140C24—41no—
141C24—41no—
142C24C2×C22⋊C412yes—
143C24C2×C22⋊C411yescharacteristic
144C2×Q8—28no—
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—816no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
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—42no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—432no—
182C22×C4—416no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—48no—
188C22×C4#10014yes—
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C412yes—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C2×C22⋊C412yes—
193C22×C4C2×C22⋊C412yes—
194C22×C4#13718yes—
195C22×C4C23×C411yescharacteristic
196C22×C4C22×D412yes—
197C2×C8—48no—
198C2×C8—48no—
199C2×C8—48no—
200C4⋊C4—44no—
201C4⋊C4—48no—
202C4⋊C4—48no—
203C4⋊C4—44no—
204C4⋊C4—48no—
205C4⋊C4—48no—
206C4⋊C4—416no—
207C22⋊C4—416no—
208C42—24no—
209C42—24no—
210C42—24no—
211C42—22no—
212C42—24no—
213C42—42no—
214C42—41no—
215C42—42no—
216C42—41no—
217C42#9914yes—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—28no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—48no—
229C23—48no—
230C23—44no—
231C23—44no—
232C23C2×#10012yes—
233C23#41914yes—
234C23#42114yes—
235C23C22×C22⋊C411yescharacteristic
236Q8—88no—
237Q8—88no—
238C2×C4—216no—
239C2×C4—28no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—48no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—48no—
268C2×C4—88no—
269C2×C4—88no—
270C2×C4C2×#9914yes—
271C2×C4C2×#9912yes—
272C2×C4C2×#10014yes—
273C2×C4#41914yes—
274C2×C4#56812yes—
275C2×C4C2×#13714yes—
276C8—88no—
277C22—22no—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—24no—
282C22—24no—
283C22—24no—
284C22—416no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22#153614yes—
290C22#287911yescharacteristic
291C22C22×#9911yescharacteristic
292C22C22×#10011yescharacteristic
293C22C2×#41912yes—
294C22C2×#42112yes—
295C4—24no—
296C4—24no—
297C4—24no—
298C4—22no—
299C4—22no—
300C4—24no—
301C4—42no—
302C4—84no—
303C4—84no—
304C2—24no—
305C2—44no—
306C2#1739611yescharacteristic
307C2#1749612yes—
308C2#1757812yes—
309C2C2×#153611yescharacteristic
310C2#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. #7727C2semidirect
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
#207152
#77272
#76292