Finite Group Databasebeta

← #192137

Subgroups — #192137

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34057C211yesmaximal, characteristic
2#21944C212yesmaximal, complemented
3C2×#1465C211yesmaximal, characteristic
4#13960C211yesmaximal, characteristic
5#8855C214yesmaximal, complemented
6#8854C212yesmaximal, complemented
7#8853C212yesmaximal, complemented
8#8020C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#407C2212yes—
11#2318—24no—
12#2315C2211yescharacteristic
13#2315C2211yescharacteristic
14#2313C2212yes—
15#2312—21no—
16#2308—21no—
17#2290—24no—
18C2×#386—21no—
19#2283—21no—
20#2280C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4—21no—
23C22×#95C2211yescharacteristic
24#1854—24no—
25#1853C2214yes—
26#1852C2214yes—
27#1809—24no—
28#1465C418yescomplemented
29C2×#324C2212yes—
30C2×#323C2212yes—
31#1441C2212yes—
32#1439C2212yes—
33#1434—21no—
34C22⋊#95—21no—
35C22⋊#95—21no—
36#1294C2214yes—
37#1294C2214yes—
38C23×D4D411yescharacteristic
39C24×C4—21no—
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4Q814yes—
48C2×#117D414yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4D411yescharacteristic
56C22×C22⋊C4C2311yescharacteristic
57C22×C42—22no—
58C22×C42D411yescharacteristic
59C2×#107—24no—
60C2×#106—24no—
61#407—24no—
62#407C2×C418yescomplemented
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2314yes—
65#388—24no—
66#388—24no—
67#387—24no—
68#387—24no—
69#386—24no—
70#386—24no—
71#386—44no—
72#385—24no—
73#385—24no—
74#383—22no—
75#382—22no—
76#381—22no—
77#381—22no—
78#380—21no—
79#380—21no—
80#380—22no—
81#380—22no—
82#380D414yes—
83#380C2312yes—
84#380C2312yes—
85#379—22no—
86C4×C4⋊C4—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92#376—22no—
93C2×#95—22no—
94C2×#95—24no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2312yes—
101#340—24no—
102#340—24no—
103#324C2×C418yescomplemented
104#323C2×C418yescomplemented
105C25C2×D411yescharacteristic
106C25C2×D411yescharacteristic
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4C4⋊C414yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—44no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4—82no—
128C23×C4C4⋊C414yes—
129C23×C4C22×C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×Q812yes—
137C23×C4#4114yes—
138C23×C4C2411yescharacteristic
139C2×M16—28no—
140C2×M16—84no—
141C22×C8—24no—
142C4×D4—28no—
143C4×D4—28no—
144C4×D4—28no—
145C4×D4—28no—
146#117—84no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C2×D414yes—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×Q812yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4C4⋊C414yes—
178C2×C22⋊C4C22×C412yes—
179C2×C22⋊C4C2×Q812yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—84no—
186C2×C42C4⋊C414yes—
187#107—84no—
188#106—84no—
189C22⋊C8—28no—
190C22⋊C8—24no—
191C22⋊C8C22×C418yes—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95—22no—
200#95—24no—
201#95—44no—
202#95—44no—
203#95—41no—
204#95—41no—
205#95—42no—
206#95—82no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24C2×C4⋊C412yes—
220C24C2×C4⋊C412yes—
221C24#12012yes—
222C24#12312yes—
223C24AGL(1,ℤ/8)12yes—
224C24#13712yes—
225C24C22×D411yescharacteristic
226C2×D4—28no—
227C2×D4—28no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—88no—
282C22×C4—88no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4C2×C4⋊C412yes—
286C22×C4C2×C4⋊C412yes—
287C22×C4C2×C4⋊C412yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C4×D414yes—
290C22×C4C4×D414yes—
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4C22⋊Q814yes—
294C22×C4C22⋊Q814yes—
295C22×C4#12312yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×Q811yescharacteristic
299C22×C4C2×#4112yes—
300M16—88no—
301C2×C8—28no—
302C2×C8—44no—
303C2×C8—44no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C22⋊C4—28no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C22⋊C4—88no—
324C22⋊C4—84no—
325C42—24no—
326C42—24no—
327C42—48no—
328C42—84no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—24no—
340C23—24no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23—42no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23—82no—
362C23—82no—
363C23#38514yes—
364C23#42814yes—
365C23C22×C4⋊C411yescharacteristic
366C23C2×#12011yescharacteristic
367C23C2×#12311yescharacteristic
368C23C2×AGL(1,ℤ/8)11yescharacteristic
369C23C2×#13711yescharacteristic
370C23#57511yescharacteristic
371C23#57511yescharacteristic
372D4—416no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—48no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—84no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—88no—
415C2×C4—88no—
416C2×C4—816no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4#38512yes—
426C2×C4#38512yes—
427C2×C4C2×C4×D412yes—
428C2×C4C2×C22⋊Q812yes—
429C8—48no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—44no—
452C22—42no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22C2×#38511yescharacteristic
457C22C2×#42811yescharacteristic
458C22#291111yescharacteristic
459C22#300611yescharacteristic
460C22#300711yescharacteristic
461C22#301812yes—
462C22#302612yes—
463C22#309411yescharacteristic
464C22#309511yescharacteristic
465C4—22no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—84no—
473C4—84no—
474C4—84no—
475C4—162no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—44no—
481C2—44no—
482C2#1832212yes—
483C2#1832312yes—
484C2#2084611yescharacteristic
485C2#2085511yescharacteristic
486C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
2. #21944C2semidirect
5. #8855C2semidirect
6. #8854C2semidirect
7. #8853C2semidirect
8. #8020C2semidirect
28. #1465C4semidirect
62. #407C2×C4semidirect
103. #324C2×C4semidirect
104. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340572
#219442
C2×#14652
#139602
#88552
#88542
#88532
#80202