Finite Group Databasebeta

← #192756

Subgroups — #192756

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21365C211yesmaximal, characteristic, complemented
4#20737C211yesmaximal, characteristic
5C2×#1488C211yesmaximal, characteristic
6#8896C211yesmaximal, characteristic, complemented
7#8894C211yesmaximal, characteristic, complemented
8#8892C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic, complemented
10#8888C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic, complemented
12#8884C211yesmaximal, characteristic, complemented
13#8882C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic, complemented
15#7892C211yesmaximal, characteristic, complemented
16C2×C4×#41—21no—
17C2×C4×#41C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2351—21no—
22Q8×C4⋊C4C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24#2349—22no—
25#2349—21no—
26D4×C4⋊C4—21no—
27D4×C4⋊C4—21no—
28D4×C4⋊C4C2211yescharacteristic
29D4×C4⋊C4C2211yescharacteristic
30#2296—22no—
31#2296—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1949—22no—
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1854—22no—
43#1853—21no—
44#1852—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1825—21no—
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1802C2211yescharacteristic
53#1802C2211yescharacteristic
54#1788—21no—
55#1776—21no—
56#1776—21no—
57C4×#103—21no—
58C4×#102—21no—
59#1488C418yescomplemented
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72C22×#41D411yescharacteristic
73C4×#41—22no—
74C4×#41—22no—
75C4×#41—44no—
76C4×#41—44no—
77C4×#41C2×C414yes—
78C4×#41Q814yes—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8D411yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—22no—
86C2×C4×D4—22no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4D411yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C2×#117—22no—
92C2×#117—21no—
93C2×#117—21no—
94C2×#117—21no—
95C2×#117D411yescharacteristic
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4D411yescharacteristic
100C22×C42—21no—
101C22×C42—22no—
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104#432—22no—
105#432C2×C414yescomplemented
106#431—22no—
107#431—22no—
108#431—22no—
109#431C2×C414yescomplemented
110C2×#107—21no—
111C2×#107—21no—
112C2×#107—22no—
113C2×#106—21no—
114C2×#106—21no—
115C2×#106—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×#103—21no—
122C2×#103—21no—
123C2×#102—21no—
124C2×#102—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×#97—21no—
129C2×C4×C8—21no—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#389—22no—
133#389—22no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—22no—
139#389—22no—
140#385—22no—
141#385—22no—
142#385—22no—
143#385—22no—
144#385—22no—
145#385—22no—
146#384—21no—
147#384—21no—
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#382—22no—
153#382—21no—
154#382—21no—
155#380—22no—
156#380—21no—
157#380—21no—
158#380—21no—
159#380—21no—
160#380—21no—
161#380D412yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4C2311yescharacteristic
176C4×C4⋊C4C2311yescharacteristic
177C2×#95—22no—
178C2×#95—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182#340—21no—
183#340—22no—
184#340—22no—
185#340—21no—
186#340—21no—
187#340—21no—
188#340—21no—
189Q8⋊C8C2×C414yescomplemented
190Q8⋊C8C2×C414yescomplemented
191D4⋊C8C2×C414yescomplemented
192D4⋊C8C2×C414yescomplemented
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—22no—
197C2×#41—22no—
198C2×#41C4⋊C414yes—
199C22×Q8C2×D411yescharacteristic
200C22×D4—21no—
201C22×D4C2×D411yescharacteristic
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C4×Q8—22no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225C4×Q8C4⋊C414yes—
226C4×Q8C22×C412yes—
227C4×Q8C2×Q812yes—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—22no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—48no—
236C4×D4—48no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4C4⋊C414yes—
242C4×D4C22×C412yes—
243C4×D4C2×Q812yes—
244#117—22no—
245#117—24no—
246#117—24no—
247#117—48no—
248#117—42no—
249#117—42no—
250#117—82no—
251#117C4⋊C414yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—41no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—48no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—41no—
311C2×C42C22×C412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×Q812yes—
316C2×C42#4112yes—
317C2×C42#4112yes—
318C2×C42C2411yescharacteristic
319#107—42no—
320#107—42no—
321#107—82no—
322#106—42no—
323#106—42no—
324#106—82no—
325C4⋊C8—24no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8C22×C412yes—
333C4⋊C8C22×C412yes—
334#103—42no—
335#103—42no—
336#102—42no—
337#102—42no—
338C22⋊C8—24no—
339C22⋊C8—24no—
340C22⋊C8—42no—
341#97—24no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—42no—
345C4×C8C22×C412yes—
346C4×C8C22×C412yes—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—21no—
352#95—22no—
353#95—21no—
354#95—22no—
355#95—21no—
356#95—21no—
357#95—41no—
358#95—41no—
359C24—21no—
360C24#12011yescharacteristic
361#41—28no—
362#41—28no—
363#41—28no—
364#41—28no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8C2×C4⋊C412yes—
369C2×Q8AGL(1,ℤ/8)12yes—
370C2×Q8#13712yes—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4C2×C4⋊C412yes—
378C2×D4AGL(1,ℤ/8)12yes—
379C2×D4#13712yes—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C2×C4⋊C412yes—
426C22×C4C4×D412yes—
427C22×C4C4×D412yes—
428C22×C4#12011yescharacteristic
429C22×C4#12011yescharacteristic
430C22×C4#12011yescharacteristic
431C22×C4C22⋊Q812yes—
432C22×C4C22⋊Q812yes—
433C22×C4#12312yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×C4⋊C412yes—
495C4⋊C4C2×C4⋊C412yes—
496C4⋊C4#13512yes—
497C4⋊C4#13512yes—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—24no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—24no—
510C42—42no—
511C42—44no—
512C42—42no—
513C42—44no—
514C42—44no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—82no—
520C42C2×C4⋊C412yes—
521C42C4×D412yes—
522C42C4×D412yes—
523C42C22⋊Q812yes—
524C42C22⋊Q812yes—
525C42#12312yes—
526C42C23×C411yescharacteristic
527C42C22×Q811yescharacteristic
528C42C2×#4111yescharacteristic
529C42C2×#4111yescharacteristic
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23#38512yes—
544C23C2×#12011yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8#42814yes—
549D4—216no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4#42814yes—
554C2×C4—216no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—84no—
608C2×C4—88no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4#38512yes—
616C2×C4#38512yes—
617C2×C4#38512yes—
618C2×C4C22×C4⋊C411yescharacteristic
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C2×C22⋊Q811yescharacteristic
622C2×C4C2×C22⋊Q811yescharacteristic
623C2×C4C2×#12311yescharacteristic
624C2×C4C2×#13511yescharacteristic
625C2×C4C2×#13511yescharacteristic
626C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
627C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
628C2×C4C2×#13711yescharacteristic
629C2×C4C2×#13711yescharacteristic
630C2×C4#57511yescharacteristic
631C2×C4#57511yescharacteristic
632C8—42no—
633C8—42no—
634C8—42no—
635C8—42no—
636C8—81no—
637C8—81no—
638C22—22no—
639C22—24no—
640C22—22no—
641C22—22no—
642C22—41no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—41no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22C2×#38511yescharacteristic
653C22#300211yescharacteristic
654C22#300211yescharacteristic
655C22#300311yescharacteristic
656C22#300411yescharacteristic
657C22#302111yescharacteristic
658C22#302211yescharacteristic
659C4—22no—
660C4—24no—
661C4—21no—
662C4—22no—
663C4—21no—
664C4—22no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—81no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—161no—
674C4—161no—
675C4C2×#42811yescharacteristic
676C4C2×#42811yescharacteristic
677C4C4×#13511yescharacteristic
678C4#294311yescharacteristic
679C4#307611yescharacteristic
680C4#307611yescharacteristic
681C4#309211yescharacteristic
682C4#309211yescharacteristic
683C2—42no—
684C2—44no—
685C2#1879411yescharacteristic
686C2#1879711yescharacteristic
687C2#1880611yescharacteristic
688C2#1880911yescharacteristic
689C2#2084411yescharacteristic
690C2#2086511yescharacteristic
691C2#2087211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21984C2semidirect
3. #21365C2semidirect
6. #8896C2semidirect
7. #8894C2semidirect
8. #8892C2semidirect
9. #8890C2semidirect
10. #8888C2semidirect
11. #8886C2semidirect
12. #8884C2semidirect
13. #8882C2semidirect
14. #8046C2semidirect
15. #7892C2semidirect
59. #1488C4semidirect
105. #432C2×C4semidirect
109. #431C2×C4semidirect
189. Q8⋊C8C2×C4semidirect
190. Q8⋊C8C2×C4semidirect
191. D4⋊C8C2×C4semidirect
192. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#219842
#213652
#207372
C2×#14882
#88962
#88942
#88922
#88902
#88882
#88862
#88842
#88822
#80462
#78922