Finite Group Databasebeta

← #193101

Subgroups — #193101

Order 512.

1084 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1084 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
1#34368C211yesmaximal, characteristic
2#21988C212yesmaximal, complemented
3#21199C211yesmaximal, characteristic
4C2×#1536C211yesmaximal, characteristic
5#8911C218yesmaximal, complemented
6#8034C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#432C2212yes—
9C2×#416—22no—
10C22×#97C2211yescharacteristic
11#2379C2214yes—
12#2369—21no—
13#2366—21no—
14#2354—22no—
15D4×C4⋊C4—22no—
16#2324—24no—
17#2311—22no—
18#2285C2211yescharacteristic
19#2285C2212yes—
20#1961—21no—
21#1960—21no—
22#1959C2214yes—
23#1948—22no—
24#1947—22no—
25#1854—24no—
26#1847C2214yes—
27#1836C2214yes—
28#1536C418yescomplemented
29C2×#329C2214yes—
30#1295C2218yes—
31C22×#41D412yes—
32C23×C8—21no—
33#532—24no—
34#532C2×C414yes—
35#532Q814yes—
36C2×C4⋊Q8D411yescharacteristic
37C2×#127D411yescharacteristic
38C2×#124C2312yes—
39C2×#121—22no—
40C2×C4×D4—24no—
41C2×C4×D4—22no—
42C2×#117—22no—
43C2×#117—21no—
44C2×#117D414yes—
45C22×C4⋊C4—22no—
46C22×C42C2311yescharacteristic
47#432—24no—
48#432C2×C418yescomplemented
49C2×#107—24no—
50C2×#106—24no—
51C2×C4⋊C8—22no—
52#416—48no—
53C2×#103—24no—
54C2×#102—24no—
55C2×C22⋊C8—22no—
56C2×#97—24no—
57C2×#97—21no—
58C2×#97C2312yes—
59C2×#97C2314yes—
60#391—24no—
61#390—24no—
62#390—24no—
63#389—24no—
64#388—28no—
65#386—24no—
66#386—24no—
67#385—28no—
68#384D412yes—
69#384D412yes—
70#384C2312yes—
71#382—21no—
72#382C2314yes—
73#380—24no—
74#380—22no—
75#380—22no—
76#380—22no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81#340—24no—
82#340—24no—
83#329C2×C4116yescomplemented
84C2×#41—24no—
85C2×#41—44no—
86C2×#41—44no—
87C2×#41C4⋊C418yes—
88C22×Q8C2×D412yes—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4C2×D412yes—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C22×C8—24no—
97C22×C8—22no—
98C22×C8—24no—
99C22×C8—42no—
100C22×C8—42no—
101C4⋊Q8—22no—
102C4⋊Q8C4⋊C414yes—
103#127—22no—
104#127C4⋊C414yes—
105#124—24no—
106#124C22×C414yes—
107#124C2×Q814yes—
108#121—28no—
109#121—28no—
110C4×D4—28no—
111C4×D4—28no—
112C4×D4—88no—
113#117—44no—
114#117—84no—
115#117—84no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—84no—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4C2×D414yes—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—41no—
133C2×C42—24no—
134C2×C42—24no—
135C2×C42—41no—
136C2×C42C22×C412yes—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D412yes—
139C2×C42C2×Q812yes—
140C2×C42#4114yes—
141C2×C42C2411yescharacteristic
142#107—84no—
143#106—84no—
144C4⋊C8—28no—
145#103—48no—
146#102—48no—
147C22⋊C8—28no—
148#97—24no—
149#97—24no—
150#97—42no—
151#97C22×C418yes—
152#95—28no—
153#95—24no—
154#95—24no—
155#95—42no—
156C24—22no—
157C24#12011yescharacteristic
158#41—416no—
159#41—88no—
160C2×Q8—22no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8C2×C4⋊C414yes—
164C2×D4—28no—
165C2×D4—22no—
166C2×D4—28no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—82no—
172C2×D4—82no—
173C2×D4C2×C4⋊C414yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—28no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—84no—
186C22×C4—88no—
187C22×C4C2×C4⋊C412yes—
188C22×C4C4×D414yes—
189C22×C4#12012yes—
190C22×C4#12011yescharacteristic
191C22×C4C22⋊Q814yes—
192C22×C4#12312yes—
193C22×C4C22×D411yescharacteristic
194C22×C4C22×D412yes—
195C2×C8—24no—
196C2×C8—28no—
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—42no—
200C2×C8—48no—
201C2×C8—42no—
202C4⋊C4—28no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—84no—
214C22⋊C4—28no—
215C22⋊C4—28no—
216C22⋊C4—84no—
217C22⋊C4—84no—
218C22⋊C4—88no—
219C22⋊C4—82no—
220C42—48no—
221C42—84no—
222C42—82no—
223C42C2×C4⋊C412yes—
224C42C4×D414yes—
225C42C22⋊Q814yes—
226C42#12312yes—
227C42C23×C411yescharacteristic
228C42C22×Q811yescharacteristic
229C42C2×#4112yes—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—24no—
235C23—42no—
236C23—42no—
237C23—44no—
238C23—44no—
239C23#38512yes—
240C23C2×#12011yescharacteristic
241Q8—44no—
242Q8—82no—
243D4—44no—
244D4—416no—
245D4—82no—
246D4—88no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—416no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—168no—
271C2×C4#38514yes—
272C2×C4#38512yes—
273C2×C4#45312yes—
274C2×C4#45512yes—
275C2×C4C22×C4⋊C411yescharacteristic
276C2×C4C2×C4×D412yes—
277C2×C4C2×C22⋊Q812yes—
278C2×C4C2×#12311yescharacteristic
279C8—48no—
280C8—44no—
281C22—22no—
282C22—24no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—44no—
287C22—48no—
288C22—84no—
289C22—82no—
290C22—82no—
291C22C2×#38511yescharacteristic
292C22C2×#45311yescharacteristic
293C22C2×#45511yescharacteristic
294C22#301912yes—
295C22#302012yes—
296C4—21no—
297C4—21no—
298C4—22no—
299C4—22no—
300C4—24no—
301C4—44no—
302C4—84no—
303C4—84no—
304C4—162no—
305C4#185614yes—
306C2—42no—
307C2—84no—
308C2#1905312yes—
309C2#1905611yescharacteristic
310C2#1905811yescharacteristic
311C2C2×#185611yescharacteristic
312C2#2086412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
2. #21988C2semidirect
5. #8911C2semidirect
6. #8034C2semidirect
28. #1536C4semidirect
48. #432C2×C4semidirect
83. #329C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343682
#219882
#211992
C2×#15362
#89112
#80342