Finite Group Databasebeta

← #404935

Subgroups — #404935

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal
5#15195C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11199C212yesmaximal
9#11042C211yesmaximal, characteristic
10#10907C212yesmaximal, complemented
11#10807C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17#2347—24no—
18#2347—24no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#388—21no—
22C2×#388—21no—
23C2×#384C2211yescharacteristic
24C2×#383—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2084—24no—
32#1947—24no—
33#1854—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1442—21no—
43#1442—21no—
44#1439—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389D414yes—
92#388—28no—
93#388—28no—
94#384—22no—
95#384—22no—
96#383—28no—
97#380—22no—
98#380—22no—
99#380—24no—
100#380—24no—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D414yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#336—24no—
127#336—22no—
128C25C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4112yes—
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8#4114yes—
164C22×C8#4114yes—
165#117—28no—
166#117—28no—
167#117—28no—
168#117—28no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4#4112yes—
210C2×C4⋊C4#4112yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—82no—
237C2×C42—82no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240#107—48no—
241#106—48no—
242C4⋊C8—28no—
243C22⋊C8—216no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—42no—
252#95—44no—
253#95—42no—
254#95—82no—
255#95—82no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24#12112yes—
267C24C22×D411yescharacteristic
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4#12012yes—
342C22×C4#12012yes—
343C22×C4#12112yes—
344C22×C4#13512yes—
345C22×C4#13712yes—
346C22×C4#13712yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4112yes—
355C22×C42−1+412yes—
356C22×C42−1+412yes—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—48no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4#12114yes—
396C4⋊C4#12114yes—
397C22⋊C4—216no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—84no—
401C22⋊C4—88no—
402C42—216no—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—84no—
410C42—84no—
411C23—22no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23C2×#12011yescharacteristic
435C23C2×#12111yescharacteristic
436C23#52511yescharacteristic
437C23#54712yes—
438C23#54712yes—
439C23C2×#13511yescharacteristic
440C23C2×#13711yescharacteristic
441C23C2×#13711yescharacteristic
442C23#57511yescharacteristic
443C23#57612yes—
444C23#57712yes—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4#45214yes—
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4#52511yescharacteristic
515C2×C4#52511yescharacteristic
516C2×C4#52512yes—
517C2×C4#52511yescharacteristic
518C2×C4#52912yes—
519C2×C4#53211yescharacteristic
520C2×C4#53611yescharacteristic
521C2×C4#53611yescharacteristic
522C2×C4#53612yes—
523C2×C4#54112yes—
524C2×C4#54712yes—
525C2×C4#54812yes—
526C2×C4#55212yes—
527C2×C4#55212yes—
528C2×C4#55212yes—
529C2×C4#55612yes—
530C8—44no—
531C8—44no—
532C22—44no—
533C22—44no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—81no—
538C22—81no—
539C22—82no—
540C22—82no—
541C22—81no—
542C22—81no—
543C22#240611yescharacteristic
544C22#241312yes—
545C22#243211yescharacteristic
546C22#247812yes—
547C22#284711yescharacteristic
548C22C2×#45211yescharacteristic
549C22#300711yescharacteristic
550C22#300811yescharacteristic
551C22#301011yescharacteristic
552C22#303511yescharacteristic
553C22#303911yescharacteristic
554C22#304512yes—
555C22#310711yescharacteristic
556C22#311611yescharacteristic
557C22#311912yes—
558C22#333912yes—
559C22#334012yes—
560C22#334612yes—
561C22#334612yes—
562C22#334712yes—
563C22#335012yes—
564C22#335612yes—
565C22#335612yes—
566C4—24no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—44no—
571C4—44no—
572C4—82no—
573C4—81no—
574C4—82no—
575C4—162no—
576C4—162no—
577C4#239211yescharacteristic
578C4#241512yes—
579C4#247512yes—
580C4#259911yescharacteristic
581C4#261111yescharacteristic
582C4#262111yescharacteristic
583C2—82no—
584C2#1490711yescharacteristic
585C2#2409211yescharacteristic
586C2#2410811yescharacteristic
587C2#2458512yes—
588C2#2459612yes—
589C2#2570111yescharacteristic
590C2#2571311yescharacteristic
591C2#2746512yes—
592C2#2748412yes—
593C2#3290711yescharacteristic
594C2#3291311yescharacteristic
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
2. C2×#2084C2semidirect
3. C2×#1947C2semidirect
6. #11227C2semidirect
7. #11220C2semidirect
10. #10907C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
C2×#20842
C2×#19472
C2×#18542
#151952
#112272
#112202
#111992
#110422
#109072
#108072