Finite Group Databasebeta

← #163627

Subgroups — #163627

Order 512.

1174 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1174 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×#2851C211yesmaximal, characteristic
2#20626C212yesmaximal
3#7711C218yesmaximal, complemented
4#7641C214yesmaximal, complemented
5C22×#125C411yescharacteristic
6C23×C42C2211yescharacteristic
7C2×#423—22no—
8C2×#407—22no—
9#2851—22no—
10#2851—22no—
11#2851C414yes—
12#2851C2214yes—
13C2×#389C2212yes—
14C2×#387—22no—
15C2×#385—22no—
16C2×#382C411yescharacteristic
17C2×#380C412yes—
18#1797—22no—
19#1797—22no—
20#1797C2214yes—
21#1797C2214yes—
22#1728C2214yes—
23#1271C22116yescomplemented
24C24×C4C2×C411yescharacteristic
25C24×C4D412yes—
26C22×M16—24no—
27C2×#125—21no—
28C2×#125—21no—
29C2×#125—44no—
30C2×#125C2×C414yes—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C412yes—
35C22×C22⋊C4—22no—
36C22×C42—44no—
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C412yes—
39C22×C42D412yes—
40C22×C42D412yes—
41C22×C42D412yes—
42C22×C42C2311yescharacteristic
43C22×C42C2312yes—
44#423—48no—
45C2×C4⋊C8—24no—
46C2×C4⋊C8—24no—
47#407—48no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—24no—
50#389—22no—
51#389—22no—
52#389C2×C414yes—
53#389C2314yes—
54#387—24no—
55#387—24no—
56#387—24no—
57#387—24no—
58#385—24no—
59#385—24no—
60#385—24no—
61#385—24no—
62#382—21no—
63#382—21no—
64#382C2×C414yes—
65#380—22no—
66#380—22no—
67#380C2×C414yes—
68#380C2×C414yes—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72#336—24no—
73#336C2318yes—
74C25C22⋊C411yescharacteristic
75C23×C4—24no—
76C23×C4—24no—
77C23×C4—24no—
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—
94C2×M16—48no—
95C2×M16—48no—
96C2×M16—48no—
97C22×C8—28no—
98#125—48no—
99#125—44no—
100#125—44no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—48no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4C22×C414yes—
113C2×C4⋊C4C22×C414yes—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—48no—
117C2×C42—24no—
118C2×C42—28no—
119C2×C42—44no—
120C2×C42—44no—
121C2×C42—44no—
122C2×C42—42no—
123C2×C42—44no—
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
134C4⋊C8—48no—
135C4⋊C8—48no—
136C22⋊C8—48no—
137C22⋊C8—48no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144C24—22no—
145C24—24no—
146C24—24no—
147C24—41no—
148C24—41no—
149C24C2×C22⋊C412yes—
150C24C2×C22⋊C411yescharacteristic
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
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—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—416no—
184C22×C4—416no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4#10114yes—
193C22×C4C2×C22⋊C412yes—
194C22×C4C2×C22⋊C412yes—
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4AGL(1,ℤ/8)14yes—
199C22×C4#13714yes—
200C22×C4C23×C411yescharacteristic
201C22×C4C22×D412yes—
202M16—816no—
203C2×C8—48no—
204C2×C8—48no—
205C2×C8—48no—
206C4⋊C4—28no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—48no—
213C4⋊C4—44no—
214C4⋊C4—48no—
215C22⋊C4—48no—
216C22⋊C4—48no—
217C42—24no—
218C42—24no—
219C42—24no—
220C42—22no—
221C42—24no—
222C42—42no—
223C42—41no—
224C42—42no—
225C42—41no—
226C42#9914yes—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—28no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—48no—
238C23—48no—
239C23—44no—
240C23—44no—
241C23C2×#10112yes—
242C23#41714yes—
243C23#42114yes—
244C23C22×C22⋊C411yescharacteristic
245C2×C4—28no—
246C2×C4—28no—
247C2×C4—28no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—48no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—88no—
280C2×C4—88no—
281C2×C4C2×#9914yes—
282C2×C4C2×#9912yes—
283C2×C4C2×#10114yes—
284C2×C4#41714yes—
285C2×C4#56812yes—
286C2×C4C2×AGL(1,ℤ/8)12yes—
287C2×C4C2×#13712yes—
288C8—88no—
289C22—22no—
290C22—24no—
291C22—24no—
292C22—22no—
293C22—24no—
294C22—24no—
295C22—24no—
296C22—416no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22#156114yes—
302C22#287911yescharacteristic
303C22C22×#9911yescharacteristic
304C22C22×#10111yescharacteristic
305C22C2×#41712yes—
306C22C2×#42112yes—
307C4—22no—
308C4—22no—
309C4—24no—
310C4—24no—
311C4—22no—
312C4—22no—
313C4—24no—
314C4—42no—
315C4—84no—
316C4—84no—
317C2—24no—
318C2—44no—
319C2#1739711yescharacteristic
320C2#1749212yes—
321C2#1759212yes—
322C2#1766711yescharacteristic
323C2C2×#156111yescharacteristic
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. #7711C2semidirect
4. #7641C2semidirect
23. #1271C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#206262
#77112
#76412