Finite Group Databasebeta

← #194862

Subgroups — #194862

Order 512.

1766 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1766 conjugacy classes of subgroups — too many to draw. Only its 266 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×#2561C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3#21765C212yesmaximal, complemented
4C2×#1462C211yesmaximal, characteristic
5#9028C214yesmaximal, complemented
6#9016C214yesmaximal, complemented
7#8019C212yesmaximal, complemented
8C22×C22⋊Q8C2211yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2561—24no—
12#2561C418yescomplemented
13C2×#397C2212yes—
14C2×#393C2212yes—
15#2312—24no—
16#2312—22no—
17#2312C2214yes—
18C2×#388—24no—
19C2×#388—22no—
20C2×#381—21no—
21C2×#380C2212yes—
22C2×#379—21no—
23C2×#379—21no—
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1894—24no—
27#1872—24no—
28#1823C2214yescomplemented
29#1787C2214yescomplemented
30#1462—24no—
31C2×#323C2214yes—
32#1294C2218yescomplemented
33C26D411yescharacteristic
34C23×Q8C2311yescharacteristic
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—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D414yes—
45C22×C4⋊C4C2311yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4C2×C414yes—
57C22×C22⋊C4D412yes—
58C22×C22⋊C4D414yes—
59C22×C22⋊C4C2312yes—
60C22×C42—22no—
61C22×C42—21no—
62C2×#103—28no—
63#407—24no—
64#407—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2314yes—
67#397—24no—
68#397C2×C418yescomplemented
69#393—24no—
70#393C2×C418yescomplemented
71#388—216no—
72#388—216no—
73#388—216no—
74#381—28no—
75#380—22no—
76#380—22no—
77#380D414yes—
78#380C2314yes—
79#379—24no—
80#379—28no—
81#379—24no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4C2×C414yes—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95C2312yes—
91#340—28no—
92#323—28no—
93C25—21no—
94C25—21no—
95C25—22no—
96C25—44no—
97C25—44no—
98C25—82no—
99C25C22⋊C414yes—
100C25C2×D411yescharacteristic
101C25C2×D412yes—
102C22×Q8—21no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—82no—
106C22×Q8C22×C412yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—44no—
115C23×C4—82no—
116C23×C4C22⋊C414yes—
117C23×C4C22×C412yes—
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—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—84no—
140C2×C4⋊C4C22×C412yes—
141C2×C4⋊C4C2×D418yes—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—48no—
160C2×C22⋊C4—48no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—84no—
165C2×C22⋊C4—84no—
166C2×C22⋊C4—84no—
167C2×C22⋊C4C22⋊C418yes—
168C2×C22⋊C4C22×C414yes—
169C2×C22⋊C4C2×D414yes—
170C2×C42—28no—
171C2×C42—24no—
172C2×C42—82no—
173C2×C42—84no—
174#103—88no—
175C22⋊C8—24no—
176C22⋊C8—24no—
177C22⋊C8—44no—
178#95—28no—
179#95—28no—
180#95—28no—
181#95—24no—
182#95—24no—
183#95—22no—
184#95C22×C414yes—
185C24—24no—
186C24—24no—
187C24—22no—
188C24—24no—
189C24—24no—
190C24—22no—
191C24—24no—
192C24—24no—
193C24—42no—
194C24—48no—
195C24—44no—
196C24—44no—
197C24—44no—
198C24—44no—
199C24—48no—
200C24—44no—
201C24—48no—
202C24—44no—
203C24—84no—
204C24—82no—
205C24—82no—
206C24—84no—
207C24—84no—
208C24—832no—
209C24C2×C22⋊C412yes—
210C24C2×C22⋊C414yes—
211C24#12011yescharacteristic
212C24#12011yescharacteristic
213C24#12312yes—
214C24C22×D411yescharacteristic
215C2×Q8—24no—
216C2×Q8—48no—
217C2×Q8—42no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8—84no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—48no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—44no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—84no—
251C22×C4—84no—
252C22×C4—82no—
253C22×C4C2×C22⋊C414yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C4×D414yes—
256C22×C4C4×D414yes—
257C22×C4#12012yes—
258C22×C4#12114yes—
259C22×C4#12114yes—
260C22×C4#12312yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D412yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4112yes—
265M16—88no—
266C2×C8—48no—
267C2×C8—44no—
268C4⋊C4—24no—
269C4⋊C4—48no—
270C4⋊C4—42no—
271C4⋊C4—48no—
272C4⋊C4—84no—
273C22⋊C4—28no—
274C22⋊C4—28no—
275C22⋊C4—216no—
276C22⋊C4—48no—
277C22⋊C4—416no—
278C22⋊C4—48no—
279C22⋊C4—48no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—88no—
283C22⋊C4—88no—
284C22⋊C4—88no—
285C42—84no—
286C42—88no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—24no—
292C23—24no—
293C23—24no—
294C23—24no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—24no—
299C23—24no—
300C23—24no—
301C23—42no—
302C23—42no—
303C23—416no—
304C23—44no—
305C23—44no—
306C23—48no—
307C23—48no—
308C23—416no—
309C23—48no—
310C23—44no—
311C23—48no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—82no—
316C23—82no—
317C23—84no—
318C23—84no—
319C23—88no—
320C23—816no—
321C23—832no—
322C23—832no—
323C23—816no—
324C23#38612yes—
325C23#38612yes—
326C23#45514yes—
327C23C22×C22⋊C411yescharacteristic
328C23C2×#12011yescharacteristic
329C23C2×#12311yescharacteristic
330C23#57514yes—
331Q8—48no—
332Q8—84no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—88no—
361C2×C4—816no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4#38614yes—
367C2×C4C2×C4×D412yes—
368C2×C4C2×#12112yes—
369C8—84no—
370C22—28no—
371C22—28no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—48no—
382C22—48no—
383C22—48no—
384C22—44no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—84no—
389C22—88no—
390C22—816no—
391C22—816no—
392C22—88no—
393C22#188514yes—
394C22C2×#38611yescharacteristic
395C22#289612yes—
396C22#300512yes—
397C22C2×#45512yes—
398C22#302612yes—
399C22#309312yes—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—44no—
404C4—84no—
405C4—88no—
406C4—82no—
407C4—82no—
408C2—22no—
409C2—22no—
410C2—24no—
411C2—42no—
412C2—84no—
413C2#1832914yes—
414C2#2091011yescharacteristic
415C2C2×#188511yescharacteristic
416C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
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
3. #21765C2semidirect
5. #9028C2semidirect
6. #9016C2semidirect
7. #8019C2semidirect
12. #2561C4semidirect
28. #1823C22semidirect
29. #1787C22semidirect
32. #1294C22semidirect
68. #397C2×C4semidirect
70. #393C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#23122
#217652
C2×#14622
#90282
#90162
#80192