Finite Group Databasebeta

← #119982

Subgroups — #119982

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1803C211yesmaximal, characteristic
2C22×Q8⋊C8C211yesmaximal, characteristic
3#13926C211yesmaximal, characteristic
4#10849C211yesmaximal, characteristic
5#10841C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic
7C2×#1279C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C22×#103—21no—
12C22×C4×C8C2211yescharacteristic
13C2×#391—21no—
14C2×#384C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1803—22no—
18#1803C414yescomplemented
19C2×#340—21no—
20C2×#340—22no—
21C2×#340—21no—
22C2×#340—21no—
23C2×Q8⋊C8—22no—
24C2×Q8⋊C8—22no—
25C2×Q8⋊C8—22no—
26#1441—22no—
27C4⋊#95—21no—
28C4⋊#95—21no—
29C4⋊#95—21no—
30#1296—22no—
31#1279—22no—
32C23×Q8D411yescharacteristic
33C23×C8—21no—
34C23×C8—21no—
35C23×C8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8D412yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2×C412yes—
72C2×C4⋊C8D412yes—
73C2×#103—24no—
74C2×#103—42no—
75C2×#103—42no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8D412yes—
82#391—42no—
83#391—42no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384C2×C412yes—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101#340—24no—
102#340—42no—
103#340—44no—
104#340—42no—
105Q8⋊C8—28no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8C22⋊C412yes—
118C22×Q8D812yes—
119C22×Q8SD1612yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—41no—
147C22×C8—41no—
148C22×C8—41no—
149C22×C8—41no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—41no—
159C4×Q8—41no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
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—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
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—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4SD1612yes—
221C2×C4⋊C4Q1612yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254#103—48no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—41no—
258C4×C8—41no—
259#95—44no—
260#95—44no—
261#95—41no—
262#95—41no—
263#95—82no—
264#95—82no—
265C24#12011yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8#10214yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
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—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12311yescharacteristic
352C22×C4#12311yescharacteristic
353C22×C4#12311yescharacteristic
354C22×C4#12311yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4#12411yescharacteristic
357C22×C4#12411yescharacteristic
358C22×C4C2×D811yescharacteristic
359C22×C4C2×SD1611yescharacteristic
360C22×C4C2×SD1611yescharacteristic
361C22×C4C2×Q1611yescharacteristic
362C22×C4#13511yescharacteristic
363C22×C4#13511yescharacteristic
364C22×C4#13511yescharacteristic
365C22×C4#13511yescharacteristic
366C22×C4AGL(1,ℤ/8)11yescharacteristic
367C22×C4AGL(1,ℤ/8)11yescharacteristic
368C22×C4#13711yescharacteristic
369C22×C4#13711yescharacteristic
370C22×C4#13711yescharacteristic
371C22×C4#13711yescharacteristic
372C22×C4#13711yescharacteristic
373C22×C4#13711yescharacteristic
374C2×C8—24no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—81no—
388C2×C8—81no—
389C2×C8—81no—
390C2×C8—81no—
391C2×C8—81no—
392C2×C8—81no—
393C2×C8—81no—
394C2×C8—81no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—81no—
436C42—81no—
437C42—81no—
438C42—81no—
439C42#11711yescharacteristic
440C42#11711yescharacteristic
441C42C4×D411yescharacteristic
442C42C4×D411yescharacteristic
443C42#12011yescharacteristic
444C42#12311yescharacteristic
445C42#12311yescharacteristic
446C42#12311yescharacteristic
447C42#12311yescharacteristic
448C42#12311yescharacteristic
449C42#12411yescharacteristic
450C42#12411yescharacteristic
451C42#12611yescharacteristic
452C42#12611yescharacteristic
453C42#12611yescharacteristic
454C42#12611yescharacteristic
455C23#38611yescharacteristic
456C23#39411yescharacteristic
457C23#39411yescharacteristic
458C23#44711yescharacteristic
459C23#44811yescharacteristic
460C23#44811yescharacteristic
461C23#44911yescharacteristic
462C23#45111yescharacteristic
463C23#45211yescharacteristic
464C23#45211yescharacteristic
465C23#45211yescharacteristic
466C23#45311yescharacteristic
467C23#45411yescharacteristic
468C23#45511yescharacteristic
469C23#45611yescharacteristic
470Q8—24no—
471Q8—24no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
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—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
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—161no—
529C2×C4—161no—
530C2×C4—161no—
531C2×C4—161no—
532C2×C4#38111yescharacteristic
533C2×C4#38611yescharacteristic
534C2×C4#38811yescharacteristic
535C2×C4#38811yescharacteristic
536C2×C4#38811yescharacteristic
537C2×C4#38811yescharacteristic
538C2×C4#39411yescharacteristic
539C2×C4#39411yescharacteristic
540C2×C4#39711yescharacteristic
541C2×C4#39711yescharacteristic
542C2×C4#39711yescharacteristic
543C2×C4#39711yescharacteristic
544C2×C4C2×#10211yescharacteristic
545C2×C4#41611yescharacteristic
546C2×C4#41711yescharacteristic
547C2×C4#41911yescharacteristic
548C2×C4C4×SD1611yescharacteristic
549C2×C4C4×Q1611yescharacteristic
550C2×C4#44011yescharacteristic
551C2×C4#44111yescharacteristic
552C2×C4#46011yescharacteristic
553C2×C4#46111yescharacteristic
554C2×C4#46211yescharacteristic
555C2×C4#46211yescharacteristic
556C2×C4#46311yescharacteristic
557C2×C4#46411yescharacteristic
558C2×C4#46411yescharacteristic
559C2×C4#46411yescharacteristic
560C2×C4#46511yescharacteristic
561C2×C4#46511yescharacteristic
562C2×C4#46511yescharacteristic
563C2×C4#46611yescharacteristic
564C2×C4#46611yescharacteristic
565C2×C4#46711yescharacteristic
566C2×C4#46811yescharacteristic
567C2×C4#46811yescharacteristic
568C2×C4#46811yescharacteristic
569C2×C4#47011yescharacteristic
570C2×C4#47011yescharacteristic
571C2×C4#47011yescharacteristic
572C2×C4#48011yescharacteristic
573C2×C4#48011yescharacteristic
574C2×C4#48011yescharacteristic
575C2×C4#48111yescharacteristic
576C2×C4#48211yescharacteristic
577C2×C4#48211yescharacteristic
578C2×C4#48311yescharacteristic
579C2×C4#48311yescharacteristic
580C2×C4#48311yescharacteristic
581C2×C4#48411yescharacteristic
582C2×C4#48511yescharacteristic
583C2×C4#48511yescharacteristic
584C2×C4#48511yescharacteristic
585C2×C4#48511yescharacteristic
586C2×C4#48511yescharacteristic
587C2×C4#48511yescharacteristic
588C2×C4#48711yescharacteristic
589C2×C4#48811yescharacteristic
590C2×C4#48911yescharacteristic
591C2×C4#49111yescharacteristic
592C8—44no—
593C8—82no—
594C8—82no—
595C22#144011yescharacteristic
596C22#162111yescharacteristic
597C22#162311yescharacteristic
598C22#162611yescharacteristic
599C22#162711yescharacteristic
600C22#163011yescharacteristic
601C22#163211yescharacteristic
602C22#163611yescharacteristic
603C22#163811yescharacteristic
604C22#164011yescharacteristic
605C22#164111yescharacteristic
606C22#164311yescharacteristic
607C22#164511yescharacteristic
608C22#164911yescharacteristic
609C22#165011yescharacteristic
610C22#165311yescharacteristic
611C22#165311yescharacteristic
612C22#187311yescharacteristic
613C22#187411yescharacteristic
614C22#187811yescharacteristic
615C22#188011yescharacteristic
616C22#188211yescharacteristic
617C22#188311yescharacteristic
618C22#203311yescharacteristic
619C22#203311yescharacteristic
620C22#203511yescharacteristic
621C22#203511yescharacteristic
622C22#203511yescharacteristic
623C22#203611yescharacteristic
624C22#203711yescharacteristic
625C22#203811yescharacteristic
626C22#203911yescharacteristic
627C22#204011yescharacteristic
628C22#204211yescharacteristic
629C22#204311yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—44no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—162no—
639C4—162no—
640C4#182311yescharacteristic
641C4#182511yescharacteristic
642C4#189311yescharacteristic
643C4#189411yescharacteristic
644C4#192711yescharacteristic
645C4#192811yescharacteristic
646C4#193311yescharacteristic
647C4#193511yescharacteristic
648C4#205511yescharacteristic
649C4#205611yescharacteristic
650C4#207211yescharacteristic
651C4#207411yescharacteristic
652C4#207511yescharacteristic
653C4#207511yescharacteristic
654C4#207711yescharacteristic
655C4#207711yescharacteristic
656C2#897411yescharacteristic
657C2#897911yescharacteristic
658C2#898211yescharacteristic
659C2#898711yescharacteristic
660C2#1041111yescharacteristic
661C2#1041511yescharacteristic
662C2#1041911yescharacteristic
663C2#1042311yescharacteristic
664C2#1043311yescharacteristic
665C2#1043411yescharacteristic
666C2#1043711yescharacteristic
667C2#1043811yescharacteristic
668C2#1119011yescharacteristic
669C2#1119511yescharacteristic
670C2#1119911yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1803C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18032
C22×Q8⋊C82
#139262
#108492
#108412
C2×#12962
C2×#12792