Finite Group Databasebeta

← #194860

Subgroups — #194860

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#21765C212yesmaximal
4C2×#1463C211yesmaximal, characteristic
5#9027C214yesmaximal, complemented
6#9015C214yesmaximal, complemented
7#8019C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117—21no—
10C23×C4⋊C4C2211yescharacteristic
11C2×#407C2212yes—
12#2583—24no—
13#2583C418yescomplemented
14C2×#398C2212yes—
15C2×#395C2212yes—
16#2316—24no—
17#2316—22no—
18#2316C2214yes—
19C2×#389—24no—
20C2×#389—22no—
21C2×#385—21no—
22C2×#385—21no—
23C2×#380—21no—
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1894—24no—
27#1872—24no—
28#1823C2214yes—
29#1787C2214yes—
30#1463—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8C2311yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8C2×C414yes—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—48no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2×C414yes—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4D414yes—
57C22×C4⋊C4D414yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C42—22no—
64C22×C42—21no—
65C2×#103—28no—
66#407—24no—
67#407—24no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2314yes—
70#398—24no—
71#398C2×C418yescomplemented
72#395—24no—
73#395C2×C418yescomplemented
74#389—216no—
75#389—216no—
76#389—216no—
77#385—28no—
78#385—28no—
79#380—24no—
80#380—22no—
81#380—22no—
82#380—24no—
83#380D414yes—
84#380C2314yes—
85C4×C22⋊C4—22no—
86C4×C22⋊C4C2×C414yes—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95C2312yes—
92#340—28no—
93#324—28no—
94C25C2×D411yescharacteristic
95C22×Q8—21no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—82no—
99C22×Q8C22×C412yes—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—82no—
113C23×C4—82no—
114C23×C4C22⋊C414yes—
115C23×C4C22⋊C414yes—
116C23×C4C22×C412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4#4114yes—
124C23×C4C2411yescharacteristic
125C2×M16—44no—
126C2×M16—84no—
127C22×C8—24no—
128C22⋊Q8—28no—
129C22⋊Q8—28no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132#117—416no—
133#117—48no—
134#117—48no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—48no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—84no—
158C2×C4⋊C4—84no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4C22⋊C418yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C414yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×D418yes—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—48no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—42no—
174C2×C42—28no—
175C2×C42—24no—
176C2×C42—48no—
177C2×C42—82no—
178#103—88no—
179C22⋊C8—24no—
180C22⋊C8—24no—
181C22⋊C8—44no—
182#95—28no—
183#95—28no—
184#95—28no—
185#95—24no—
186#95—24no—
187#95—22no—
188#95C22×C414yes—
189C24—22no—
190C24—22no—
191C24—24no—
192C24—42no—
193C24C2×C22⋊C412yes—
194C24#12011yescharacteristic
195C24#12011yescharacteristic
196C24#12312yes—
197C24C22×D411yescharacteristic
198C2×Q8—24no—
199C2×Q8—48no—
200C2×Q8—42no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8—84no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—216no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—216no—
216C22×C4—28no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—28no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—48no—
235C22×C4—48no—
236C22×C4—44no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4—82no—
249C22×C4—84no—
250C22×C4C2×C22⋊C414yes—
251C22×C4C2×C22⋊C414yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C4×D414yes—
254C22×C4C4×D414yes—
255C22×C4#12012yes—
256C22×C4#12114yes—
257C22×C4#12114yes—
258C22×C4#12312yes—
259C22×C4AGL(1,ℤ/8)14yes—
260C22×C4#13714yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D412yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4112yes—
265M16—88no—
266C2×C8—48no—
267C2×C8—44no—
268C4⋊C4—28no—
269C4⋊C4—28no—
270C4⋊C4—216no—
271C4⋊C4—24no—
272C4⋊C4—48no—
273C4⋊C4—416no—
274C4⋊C4—42no—
275C4⋊C4—48no—
276C4⋊C4—416no—
277C4⋊C4—84no—
278C4⋊C4—88no—
279C4⋊C4—88no—
280C22⋊C4—216no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C42—416no—
286C42—84no—
287C23—21no—
288C23—21no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—24no—
294C23—24no—
295C23—24no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—42no—
301C23—42no—
302C23—48no—
303C23—48no—
304C23#38612yes—
305C23#38612yes—
306C23#45614yes—
307C23C22×C22⋊C411yescharacteristic
308C23C2×#12011yescharacteristic
309C23C2×#12311yescharacteristic
310C23C2×AGL(1,ℤ/8)12yes—
311C23C2×#13712yes—
312Q8—48no—
313Q8—84no—
314C2×C4—216no—
315C2×C4—216no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4—816no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4#38614yes—
358C2×C4#41718yes—
359C2×C4C2×C4×D412yes—
360C2×C4C2×#12112yes—
361C8—84no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—21no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—48no—
376C22—48no—
377C22—42no—
378C22—42no—
379C22#188614yes—
380C22C2×#38611yescharacteristic
381C22C2×#41712yes—
382C22#300311yescharacteristic
383C22#300411yescharacteristic
384C22C2×#45612yes—
385C22#302612yes—
386C22#309212yes—
387C4—28no—
388C4—22no—
389C4—22no—
390C4—24no—
391C4—48no—
392C4—44no—
393C4—88no—
394C4—82no—
395C4—82no—
396C2—22no—
397C2—22no—
398C2—24no—
399C2—42no—
400C2#1832512yes—
401C2#1832712yes—
402C2#2090911yescharacteristic
403C2C2×#188611yescharacteristic
404C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9027C2semidirect
6. #9015C2semidirect
13. #2583C4semidirect
71. #398C2×C4semidirect
73. #395C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#23162
#217652
C2×#14632
#90272
#90152
#80192