Finite Group Databasebeta

← #193066

Subgroups — #193066

Order 512.

1232 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 274 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#21200C211yesmaximal, characteristic
3#20800C212yesmaximal, complemented
4C2×#1525C211yesmaximal, characteristic
5#8910C218yesmaximal, complemented
6#8028C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#428—24no—
9C2×#417—22no—
10C2×C4×M16C2211yescharacteristic
11C2×C4×M16C2212yes—
12#2379C2214yes—
13#2369—21no—
14#2366—21no—
15#2354—22no—
16D4×C4⋊C4—22no—
17#2324—24no—
18#2311—22no—
19#2285C2212yes—
20C2×#384C2211yescharacteristic
21#1959C2214yes—
22#1957—21no—
23#1956—21no—
24#1847C2218yes—
25#1845—22no—
26#1844—22no—
27#1525C418yescomplemented
28C2×#329C2214yes—
29#1295C2218yes—
30C22×#41D412yes—
31C22×M16—22no—
32C22×M16—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×#117D414yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C42C2311yescharacteristic
48#428—416no—
49C2×#107—28no—
50C2×#106—28no—
51#417—48no—
52C2×#103—24no—
53C2×#102—24no—
54C4×M16—24no—
55C4×M16—24no—
56C4×M16C2×C418yescomplemented
57C2×#97C2312yes—
58C2×#97C2314yes—
59C2×C4×C8—22no—
60C2×C4×C8—21no—
61#391—24no—
62#390—24no—
63#390—24no—
64#389—24no—
65#388—28no—
66#386—24no—
67#386—24no—
68#385—28no—
69#384—22no—
70#384—21no—
71#384D412yes—
72#384D412yes—
73#384C2312yes—
74#382C2314yes—
75#380—24no—
76#380—22no—
77#380—22no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81#329C2×C4116yescomplemented
82C2×#41—24no—
83C2×#41—44no—
84C2×#41—44no—
85C2×#41C4⋊C418yes—
86C22×Q8C2×D412yes—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4C2×D412yes—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—41no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C2×M16—28no—
96C2×M16—28no—
97C2×M16—28no—
98C2×M16—24no—
99C2×M16—42no—
100C2×M16—42no—
101C22×C8—24no—
102C22×C8—22no—
103C4⋊Q8—22no—
104C4⋊Q8C4⋊C414yes—
105#127—22no—
106#127C4⋊C414yes—
107#124—24no—
108#124C22×C414yes—
109#124C2×Q814yes—
110#121—28no—
111#121—28no—
112C4×D4—28no—
113C4×D4—28no—
114C4×D4—88no—
115#117—48no—
116#117—84no—
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—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—84no—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D414yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—24no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42C22×C412yes—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141C2×C42C2×Q812yes—
142C2×C42#4114yes—
143C2×C42C2411yescharacteristic
144#107—416no—
145#106—416no—
146#103—48no—
147#102—48no—
148#97—24no—
149#97—24no—
150#97C22×C418yes—
151C4×C8—28no—
152C4×C8—42no—
153#95—28no—
154#95—24no—
155#95—24no—
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—42no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—82no—
172C2×D4—82no—
173C2×D4C2×C4⋊C414yes—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—84no—
186C22×C4—88no—
187C22×C4—84no—
188C22×C4—82no—
189C22×C4C2×C4⋊C412yes—
190C22×C4C4×D414yes—
191C22×C4#12011yescharacteristic
192C22×C4#12012yes—
193C22×C4C22⋊Q814yes—
194C22×C4#12312yes—
195C22×C4C22×D411yescharacteristic
196C22×C4C22×D412yes—
197M16—232no—
198M16—48no—
199C2×C8—28no—
200C2×C8—28no—
201C2×C8—28no—
202C2×C8—24no—
203C2×C8—42no—
204C2×C8—42no—
205C4⋊C4—28no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—84no—
215C4⋊C4—82no—
216C4⋊C4—84no—
217C22⋊C4—28no—
218C22⋊C4—28no—
219C22⋊C4—84no—
220C22⋊C4—84no—
221C22⋊C4—88no—
222C42—48no—
223C42—84no—
224C42C2×C4⋊C412yes—
225C42C4×D414yes—
226C42C22⋊Q814yes—
227C42#12312yes—
228C42C23×C411yescharacteristic
229C42C22×Q811yescharacteristic
230C42C2×#4112yes—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—24no—
236C23—44no—
237C23—42no—
238C23—42no—
239C23—44no—
240C23#38512yes—
241C23C2×#12011yescharacteristic
242Q8—44no—
243Q8—82no—
244D4—44no—
245D4—416no—
246D4—82no—
247D4—88no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—28no—
255C2×C4—28no—
256C2×C4—24no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—416no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—82no—
267C2×C4—84no—
268C2×C4—168no—
269C2×C4—164no—
270C2×C4#38512yes—
271C2×C4#38514yes—
272C2×C4#45312yes—
273C2×C4#45314yes—
274C2×C4#45512yes—
275C2×C4#45614yes—
276C2×C4C22×C4⋊C411yescharacteristic
277C2×C4C2×C4×D412yes—
278C2×C4C2×C22⋊Q812yes—
279C2×C4C2×#12311yescharacteristic
280C8—216no—
281C8—44no—
282C22—22no—
283C22—24no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—44no—
288C22—48no—
289C22—82no—
290C22—84no—
291C22—82no—
292C22C2×#38511yescharacteristic
293C22C2×#45311yescharacteristic
294C22C2×#45312yes—
295C22C2×#45511yescharacteristic
296C22C2×#45612yes—
297C4—21no—
298C4—21no—
299C4—22no—
300C4—22no—
301C4—44no—
302C4—84no—
303C4—84no—
304C4—162no—
305C4#185614yes—
306C4#185918yes—
307C2—42no—
308C2—84no—
309C2#1907311yescharacteristic
310C2#1907612yes—
311C2#1907811yescharacteristic
312C2C2×#185611yescharacteristic
313C2C2×#185912yes—
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
3. #20800C2semidirect
5. #8910C2semidirect
6. #8028C2semidirect
27. #1525C4semidirect
56. C4×M16C2×C4semidirect
81. #329C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343682
#212002
#208002
C2×#15252
#89102
#80282