Finite Group Databasebeta

← #404932

Subgroups — #404932

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1853C212yesmaximal
5#15195C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11200C212yesmaximal
9#11042C211yesmaximal, characteristic
10#10908C212yesmaximal, complemented
11#10806C211yesmaximal, characteristic, complemented
12C22×#117—21no—
13C23×C4⋊C4C2211yescharacteristic
14C22×#106C2212yes—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18#2344—24no—
19#2344—24no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#381—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2084—24no—
32#1948—24no—
33#1853—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×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#117—28no—
53C2×#117—44no—
54C2×#117—44no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D414yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389D414yes—
95#387—28no—
96#387—28no—
97#384—22no—
98#384—22no—
99#381—28no—
100#380—22no—
101#380—22no—
102#380—24no—
103#380—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4D414yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#336—24no—
129#336—22no—
130C25C2×D411yescharacteristic
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8#4114yes—
173C22×C8#4114yes—
174#117—432no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
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—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4#4112yes—
223C2×C4⋊C4#4112yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4SD1614yes—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250#106—48no—
251#106—48no—
252C4⋊C8—28no—
253C22⋊C8—216no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—42no—
262#95—44no—
263#95—42no—
264#95—82no—
265#95—82no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24#12112yes—
275C24C2×SD1612yes—
276C24#13712yes—
277C24C22×D411yescharacteristic
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—216no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—48no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4#12112yes—
363C22×C4C2×SD1612yes—
364C22×C4C2×SD1612yes—
365C22×C4AGL(1,ℤ/8)12yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4112yes—
374C22×C42−1+412yes—
375C22×C42−1+412yes—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—216no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—48no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—48no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4#12114yes—
416C4⋊C4#12114yes—
417C22⋊C4—28no—
418C22⋊C4—44no—
419C22⋊C4—48no—
420C22⋊C4—88no—
421C42—22no—
422C42—22no—
423C42—24no—
424C42—48no—
425C42—41no—
426C42—41no—
427C42—42no—
428C42—48no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23#46114yes—
452C23C2×#12011yescharacteristic
453C23C2×#12111yescharacteristic
454C23#52511yescharacteristic
455C23#54712yes—
456C23#54712yes—
457C23C22×SD1611yescharacteristic
458C23C2×AGL(1,ℤ/8)11yescharacteristic
459C23C2×#13711yescharacteristic
460C23#57511yescharacteristic
461C23#57712yes—
462C23#57712yes—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—432no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4#45014yes—
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×#12111yescharacteristic
535C2×C4#52511yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4#52512yes—
538C2×C4#52511yescharacteristic
539C2×C4#52912yes—
540C2×C4#53211yescharacteristic
541C2×C4#53611yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53612yes—
544C2×C4#54112yes—
545C2×C4#54712yes—
546C2×C4#54812yes—
547C2×C4#55212yes—
548C2×C4#55212yes—
549C2×C4#55212yes—
550C2×C4#55612yes—
551C8—44no—
552C8—44no—
553C22—28no—
554C22—24no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22#240611yescharacteristic
565C22#241312yes—
566C22#243211yescharacteristic
567C22#247812yes—
568C22#284711yescharacteristic
569C22C2×#45011yescharacteristic
570C22#300711yescharacteristic
571C22#301011yescharacteristic
572C22#301011yescharacteristic
573C22C2×#46111yescharacteristic
574C22#303611yescharacteristic
575C22#304312yes—
576C22#310111yescharacteristic
577C22#311011yescharacteristic
578C22#312612yes—
579C22#333512yes—
580C22#333512yes—
581C22#334412yes—
582C22#334412yes—
583C22#334512yes—
584C22#334812yes—
585C22#335112yes—
586C22#335412yes—
587C4—24no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—44no—
592C4—44no—
593C4—82no—
594C4—82no—
595C4—81no—
596C4—162no—
597C4—162no—
598C4#239211yescharacteristic
599C4#241512yes—
600C4#247512yes—
601C4#259911yescharacteristic
602C4#261111yescharacteristic
603C4#262111yescharacteristic
604C2—44no—
605C2#1490711yescharacteristic
606C2#2408911yescharacteristic
607C2#2411111yescharacteristic
608C2#2458012yes—
609C2#2458912yes—
610C2#2569911yescharacteristic
611C2#2571511yescharacteristic
612C2#2745612yes—
613C2#2747712yes—
614C2#3290511yescharacteristic
615C2#3291211yescharacteristic
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×#1948C2semidirect
6. #11227C2semidirect
7. #11220C2semidirect
10. #10908C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C2×#20842
C2×#19482
C2×#18532
#151952
#112272
#112202
#112002
#110422
#109082
#108062