Finite Group Databasebeta

← #195642

Subgroups — #195642

Order 512.

1262 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1262 conjugacy classes of subgroups — too many to draw. Only its 282 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×#3139C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3#21394C212yesmaximal
4C2×#1558C211yesmaximal, characteristic
5#9060C218yesmaximal, complemented
6#7904C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4⋊Q8C2211yescharacteristic
9#3139—24no—
10#3139C418yescomplemented
11C2×#491—22no—
12C2×#487C2214yes—
13C2×#432C2212yes—
14C2×#419—22no—
15C2×C4×M16C2211yescharacteristic
16#2369—24no—
17#2369—22no—
18#2369C2214yes—
19C2×#391—22no—
20C2×#389—22no—
21C2×C4×C4⋊C4—22no—
22C2×C4×C4⋊C4C2212yes—
23#1988C2214yes—
24#1981C2214yes—
25#1872—22no—
26#1872—22no—
27#1763—24no—
28#1558—24no—
29C2×#333C2214yes—
30#1279C2218yes—
31C23×Q8D412yes—
32C22×M16—21no—
33#533—22no—
34#533—24no—
35#533C2×C414yes—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—24no—
38C2×C4⋊Q8C2×C414yes—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8D414yes—
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2312yes—
43C2×#125—21no—
44C2×C22⋊Q8—22no—
45C2×C4×Q8—22no—
46C2×#117—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4D412yes—
51C22×C42—22no—
52C22×C42—22no—
53C22×C42C2311yescharacteristic
54#491—216no—
55#487—28no—
56#487C2×C4116yescomplemented
57#432—24no—
58#432—24no—
59C2×C4⋊C8C2314yes—
60#419—216no—
61C2×#103—28no—
62C2×#103—24no—
63C2×#103—24no—
64C2×C22⋊C8—22no—
65C4×M16—22no—
66C4×M16C2×C414yes—
67C2×#97—21no—
68C2×#97—22no—
69C2×C4×C8C2312yes—
70#391—28no—
71#391—28no—
72#389—28no—
73#389—28no—
74C4×C4⋊C4—28no—
75C4×C4⋊C4—28no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D414yes—
79C4×C4⋊C4C2314yes—
80C2×#95—24no—
81C2×#95—22no—
82#340—24no—
83#340—24no—
84#333—28no—
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—22no—
88C22×Q8—44no—
89C22×Q8—44no—
90C22×Q8C22⋊C418yes—
91C22×Q8C2×D414yes—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C2×M16—24no—
98C2×M16—42no—
99C2×M16—42no—
100C22×C8—22no—
101C22×C8—24no—
102C4⋊Q8—28no—
103C4⋊Q8—28no—
104C4⋊Q8—24no—
105C4⋊Q8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8C22⋊C418yes—
110C4⋊Q8C22×C414yes—
111C4⋊Q8C22×C414yes—
112C4⋊Q8C2×D414yes—
113#125—24no—
114#125—42no—
115C22⋊Q8—28no—
116C22⋊Q8—44no—
117C4×Q8—28no—
118C4×Q8—44no—
119#117—24no—
120#117—42no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—84no—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4C2×D414yes—
136C2×C22⋊C4—41no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—28no—
140C2×C42—48no—
141C2×C42—41no—
142C2×C42—84no—
143C2×C42C22×C412yes—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42#4114yes—
148C2×C42C2411yescharacteristic
149C4⋊C8—24no—
150C4⋊C8—24no—
151#103—232no—
152#103—48no—
153#103—48no—
154C22⋊C8—44no—
155#97—24no—
156#97—44no—
157C4×C8—22no—
158C4×C8C22×C414yes—
159#95—216no—
160#95—24no—
161#95—24no—
162C24#12011yescharacteristic
163C2×Q8—24no—
164C2×Q8—28no—
165C2×Q8—24no—
166C2×Q8—416no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—88no—
173C2×Q8—82no—
174C2×Q8—82no—
175C2×Q8C2×C22⋊C418yes—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—28no—
184C22×C4—24no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4C2×C22⋊C412yes—
194C22×C4C4×D414yes—
195C22×C4#12011yescharacteristic
196C22×C4#12012yes—
197C22×C4#12114yes—
198C22×C4#12312yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D412yes—
201M16—48no—
202C2×C8—24no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—44no—
206C2×C8—44no—
207C2×C8—44no—
208C4⋊C4—28no—
209C4⋊C4—28no—
210C4⋊C4—28no—
211C4⋊C4—24no—
212C4⋊C4—216no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—48no—
221C4⋊C4—88no—
222C4⋊C4#13718yes—
223C22⋊C4—44no—
224C42—48no—
225C42—48no—
226C42—44no—
227C42—88no—
228C42C2×C22⋊C412yes—
229C42C4×D414yes—
230C42#12114yes—
231C42#12312yes—
232C42C23×C411yescharacteristic
233C42C22×D411yescharacteristic
234C42C2×#4112yes—
235C23—22no—
236C23—22no—
237C23#38612yes—
238C23#45512yes—
239C23#45512yes—
240C23C2×#12011yescharacteristic
241Q8—48no—
242Q8—48no—
243Q8—84no—
244Q8—88no—
245Q8—84no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—28no—
255C2×C4—28no—
256C2×C4—28no—
257C2×C4—28no—
258C2×C4—416no—
259C2×C4—416no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—88no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#38612yes—
272C2×C4#38614yes—
273C2×C4C22×C22⋊C411yescharacteristic
274C2×C4C2×C4×D412yes—
275C2×C4C2×#12112yes—
276C2×C4C2×#12311yescharacteristic
277C2×C4C2×#13714yes—
278C2×C4#57514yes—
279C8—44no—
280C8—84no—
281C22—24no—
282C22—22no—
283C22—22no—
284C22—24no—
285C22#188514yes—
286C22C2×#38611yescharacteristic
287C22#300412yes—
288C22#300512yes—
289C22C2×#45511yescharacteristic
290C22C2×#45511yescharacteristic
291C4—24no—
292C4—21no—
293C4—21no—
294C4—22no—
295C4—48no—
296C4—48no—
297C4—84no—
298C4—84no—
299C4C4×#13714yes—
300C4#303714yes—
301C2—24no—
302C2#1853012yes—
303C2#1855111yescharacteristic
304C2#1855211yescharacteristic
305C2#2092612yes—
306C2C2×#188511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9060C2semidirect
10. #3139C4semidirect
56. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
C2×#23692
#213942
C2×#15582
#90602
#79042