Finite Group Databasebeta

← #405822

Subgroups — #405822

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1823C211yesmaximal, characteristic
2C22×#340C211yesmaximal, characteristic
3#13904C211yesmaximal, characteristic
4#13881C211yesmaximal, characteristic
5#11358C212yesmaximal, complemented
6#11316C212yesmaximal, complemented
7#11316C212yesmaximal, complemented
8#11268C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C2×#382C2212yes—
17C2×#381—21no—
18C2×#381—21no—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#1823—24no—
23C2×#340—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1445—22no—
37#1442—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C24×C4—21no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—24no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—42no—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4D414yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42C2312yes—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80#382—24no—
81#382—42no—
82#381—44no—
83#381—44no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D412yes—
101#340—24no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#336—24no—
112#336—42no—
113C25C2×D411yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—44no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—82no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C22×C8—28no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—48no—
164C22×C8—48no—
165C22×C8—44no—
166C22×C8—44no—
167C22×C8#4114yes—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4#4114yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C22⋊C8—44no—
231#95—24no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—82no—
241#95—82no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24#12112yes—
247C24#12312yes—
248C24#12412yes—
249C24#13512yes—
250C24#13512yes—
251C24AGL(1,ℤ/8)12yes—
252C24AGL(1,ℤ/8)12yes—
253C24C22×D411yescharacteristic
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—48no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—84no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4#12112yes—
335C22×C4#12314yes—
336C22×C4#12314yes—
337C22×C4#12312yes—
338C22×C4#12412yes—
339C22×C4#12614yes—
340C22×C4#12614yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4H(5,𝔽2)12yes—
350C22×C42−1+412yes—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—416no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—48no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—48no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23#39714yes—
418C23#46314yes—
419C23#46314yes—
420C23#48214yes—
421C23#48214yes—
422C23#48814yes—
423C23#49014yes—
424C23C2×#12111yescharacteristic
425C23C2×#12311yescharacteristic
426C23C2×#12411yescharacteristic
427C23#54612yes—
428C23#54712yes—
429C23C2×#13511yescharacteristic
430C23C2×#13511yescharacteristic
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432C23C2×AGL(1,ℤ/8)11yescharacteristic
433C23#57712yes—
434C23#57812yes—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—816no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—816no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4#39714yes—
500C2×C4C2×#12311yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C2×C4#52511yescharacteristic
503C2×C4#52511yescharacteristic
504C2×C4C2×#12611yescharacteristic
505C2×C4C2×#12611yescharacteristic
506C2×C4#52912yes—
507C2×C4#52912yes—
508C2×C4#53511yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#53912yes—
511C2×C4#53912yes—
512C2×C4#55112yes—
513C2×C4#55112yes—
514C2×C4#55112yes—
515C2×C4#55212yes—
516C2×C4#55212yes—
517C2×C4#55212yes—
518C8—44no—
519C8—81no—
520C8—81no—
521C22—24no—
522C22—24no—
523C22—28no—
524C22—28no—
525C22—24no—
526C22—24no—
527C22—28no—
528C22—28no—
529C22—24no—
530C22—24no—
531C22#207318yes—
532C22C2×#39711yescharacteristic
533C22#241111yescharacteristic
534C22#241311yescharacteristic
535C22#241912yes—
536C22#248812yes—
537C22C2×#46311yescharacteristic
538C22C2×#46311yescharacteristic
539C22#304311yescharacteristic
540C22#304411yescharacteristic
541C22C2×#48211yescharacteristic
542C22C2×#48211yescharacteristic
543C22#309712yes—
544C22C2×#48811yescharacteristic
545C22C2×#49011yescharacteristic
546C22#313712yes—
547C22#329612yes—
548C22#330012yes—
549C22#330612yes—
550C22#330812yes—
551C22#333612yes—
552C22#333712yes—
553C22#334312yes—
554C22#334512yes—
555C4—24no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—84no—
561C4—161no—
562C4—162no—
563C4—161no—
564C4#248211yescharacteristic
565C4#248411yescharacteristic
566C4#249312yes—
567C4#249312yes—
568C4#262311yescharacteristic
569C4#285311yescharacteristic
570C2—28no—
571C2#1491911yescharacteristic
572C2C2×#207311yescharacteristic
573C2#2363311yescharacteristic
574C2#2447711yescharacteristic
575C2#2448111yescharacteristic
576C2#2457711yescharacteristic
577C2#2458111yescharacteristic
578C2#2494112yes—
579C2#2494512yes—
580C2#2769912yes—
581C2#2771712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11358C2semidirect
6. #11316C2semidirect
7. #11316C2semidirect
8. #11268C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18232
C22×#3402
#139042
#138812
#113582
#113162
#112682
#107302
#107212
#107142