Finite Group Databasebeta

← #222137

Subgroups — #222137

Order 512.

1084 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36150C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3#22648C211yesmaximal, characteristic, complemented
4C2×#1559C211yesmaximal, characteristic, complemented
5C2×#1551C211yesmaximal, characteristic, complemented
6#9704C211yesmaximal, characteristic, complemented
7#9698C211yesmaximal, characteristic, complemented
8#9142C211yesmaximal, characteristic, complemented
9#9141C211yesmaximal, characteristic, complemented
10#9138C211yesmaximal, characteristic, complemented
11#9131C211yesmaximal, characteristic, complemented
12#8587C211yesmaximal, characteristic, complemented
13#8344C211yesmaximal, characteristic, complemented
14#8343C211yesmaximal, characteristic, complemented
15#8235C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#435—21no—
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23#2858C2211yescharacteristic
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2540—21no—
27#2490—21no—
28#2454—21no—
29#2448C412yes—
30#2427—21no—
31#2426—21no—
32#2426C412yes—
33#2414—21no—
34#2407C412yes—
35#2367—21no—
36#2363—21no—
37#2341C412yes—
38C4×C4⋊Q8C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1954—21no—
41#1940—21no—
42#1924—21no—
43#1924—21no—
44#1851—21no—
45#1850—21no—
46#1842C2211yescharacteristic
47#1842C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1770—21no—
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55#1559—24no—
56#1551—24no—
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C2×#331C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#1296C2211yescharacteristic
72#1296C2211yescharacteristic
73C22×#41C2×C412yes—
74C22×M16—21no—
75C23×C8—21no—
76#533—22no—
77#533—22no—
78#533—41no—
79#533—41no—
80#532—22no—
81#532—22no—
82#532—41no—
83#532—41no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#125—21no—
88C2×#124—21no—
89C2×#124—21no—
90C2×#124C2×C412yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8C2×C412yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—21no—
98C2×#121C2×C412yes—
99C2×C4×Q8—21no—
100C2×C4×Q8C2×C412yes—
101C2×C4×Q8C2×C412yes—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4C2×C411yescharacteristic
105C2×C4×D4D411yescharacteristic
106C2×C4×D4D411yescharacteristic
107C2×#117—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#435—44no—
111#432—22no—
112#432—22no—
113#432—41no—
114#432—41no—
115#423—22no—
116#423—41no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—41no—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×C4×C8—21no—
143#399—22no—
144#397—22no—
145#396—22no—
146#394—22no—
147#394—22no—
148#392—22no—
149#391—21no—
150#391C2×C412yes—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390C2×C412yes—
159#390C2×C412yes—
160#390C2×C412yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389C2×C412yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#386—22no—
169#385—22no—
170#384C2311yescharacteristic
171#380—21no—
172#380—22no—
173#380C2×C412yes—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C22⋊C4—22no—
180C4×C22⋊C4C2×C411yescharacteristic
181C4×C22⋊C4C2×C411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C43C2311yescharacteristic
187#336—21no—
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#333—22no—
194#333—22no—
195#333—22no—
196#333D414yescomplemented
197#332—22no—
198#332—22no—
199#332—41no—
200#332—41no—
201#331—22no—
202#331—22no—
203#331—41no—
204#331—41no—
205C2×#41—24no—
206C2×#41—42no—
207C2×#41—44no—
208C2×#41—82no—
209C22×Q8—21no—
210C22×Q8C22×C412yes—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C22⋊C411yescharacteristic
216C22×D4C22×C412yes—
217C23×C4—22no—
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C2×M16—42no—
225C2×M16—42no—
226C2×M16—81no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8—81no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C2×D412yes—
245C4⋊Q8#4112yes—
246#127—21no—
247#127—21no—
248#127—21no—
249#127—21no—
250#125—42no—
251#125—42no—
252#124—24no—
253#124—42no—
254#124—42no—
255#124—42no—
256#124—81no—
257#124—81no—
258C22⋊Q8—24no—
259C22⋊Q8—42no—
260C22⋊Q8—42no—
261C22⋊Q8—42no—
262C22⋊Q8—81no—
263C22⋊Q8—81no—
264#121—42no—
265#121—42no—
266#121—42no—
267#121—42no—
268#121—42no—
269#121—42no—
270#121—81no—
271#121—81no—
272#121—82no—
273C4×Q8—24no—
274C4×Q8—42no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×Q8—82no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—42no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—41no—
285C4×D4—41no—
286C4×D4—42no—
287#117—42no—
288#117—42no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4C22⋊C411yescharacteristic
299C2×C4⋊C4C22⋊C411yescharacteristic
300C2×C4⋊C4C22⋊C411yescharacteristic
301C2×C4⋊C4C22×C412yes—
302C2×C4⋊C4C22×C412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C22⋊C411yescharacteristic
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42C22×C411yescharacteristic
331C2×C42C22×C411yescharacteristic
332C2×C42C22×C411yescharacteristic
333C2×C42C22×C411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2411yescharacteristic
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—21no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—21no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—42no—
358C4⋊C8—42no—
359C4⋊C8—42no—
360C4⋊C8—42no—
361C4⋊C8—81no—
362C4⋊C8—81no—
363C4⋊C8C2×D412yes—
364C4⋊C8C2×D412yes—
365C22⋊C8—42no—
366C22⋊C8—42no—
367C22⋊C8—42no—
368C22⋊C8—42no—
369C22⋊C8—81no—
370C22⋊C8—81no—
371#97—42no—
372#97—42no—
373#97—42no—
374C4×C8—42no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385C24—22no—
386C24C2×C22⋊C411yescharacteristic
387C24C2×C22⋊C411yescharacteristic
388#41—88no—
389#41—88no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—41no—
394C2×Q8—41no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8C4×D414yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C2×C22⋊C411yescharacteristic
470C22×C4C2×C22⋊C411yescharacteristic
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4C2×C22⋊C411yescharacteristic
473C22×C4C23×C411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476M16—84no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—84no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—41no—
558C42—41no—
559C42—44no—
560C42—82no—
561C42AGL(1,ℤ/8)12yes—
562C42#13712yes—
563C42C22×D411yescharacteristic
564C42C2×#4111yescharacteristic
565C42H(5,𝔽2)11yescharacteristic
566C42H(5,𝔽2)11yescharacteristic
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23#41312yes—
588C23C22×C22⋊C411yescharacteristic
589Q8—44no—
590Q8—44no—
591Q8—81no—
592Q8—81no—
593Q8—81no—
594Q8—81no—
595D4—44no—
596D4—82no—
597D4—82no—
598D4—88no—
599D4—81no—
600D4—81no—
601D4—81no—
602D4—81no—
603D4—81no—
604D4—81no—
605D4—81no—
606D4—81no—
607D4—164no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—24no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—88no—
668C2×C4#41712yes—
669C2×C4#41712yes—
670C2×C4C2×C4×D411yescharacteristic
671C2×C4#51811yescharacteristic
672C2×C4D4211yescharacteristic
673C2×C4D4211yescharacteristic
674C2×C4#54611yescharacteristic
675C2×C4#54611yescharacteristic
676C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
677C2×C4C2×#13711yescharacteristic
678C2×C4#57511yescharacteristic
679C2×C4#57511yescharacteristic
680C2×C4#57611yescharacteristic
681C2×C4#57711yescharacteristic
682C2×C4#57711yescharacteristic
683C2×C4#57811yescharacteristic
684C8—42no—
685C8—42no—
686C8—81no—
687C8—81no—
688C8—81no—
689C8—81no—
690C8—81no—
691C8—81no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22D4×C22⋊C411yescharacteristic
714C22C2×#41311yescharacteristic
715C22#288811yescharacteristic
716C22C2×#41711yescharacteristic
717C22#289611yescharacteristic
718C22#289711yescharacteristic
719C22#289811yescharacteristic
720C4—22no—
721C4—22no—
722C4—21no—
723C4—21no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—44no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—82no—
734C4—82no—
735C4#326911yescharacteristic
736C4#327111yescharacteristic
737C4#327411yescharacteristic
738C4#327811yescharacteristic
739C4#331311yescharacteristic
740C4#332111yescharacteristic
741C4#333211yescharacteristic
742C4#333311yescharacteristic
743C2—42no—
744C2—81no—
745C2—162no—
746C2#1775311yescharacteristic
747C2#1776411yescharacteristic
748C2#1776611yescharacteristic
749C2#1777511yescharacteristic
750C2#2240811yescharacteristic
751C2#2242611yescharacteristic
752C2#2243211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36150C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22799C2semidirect
3. #22648C2semidirect
4. C2×#1559C2semidirect
5. C2×#1551C2semidirect
6. #9704C2semidirect
7. #9698C2semidirect
8. #9142C2semidirect
9. #9141C2semidirect
10. #9138C2semidirect
11. #9131C2semidirect
12. #8587C2semidirect
13. #8344C2semidirect
14. #8343C2semidirect
15. #8235C2semidirect
196. #333D4semidirect

Maximal subgroups

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

subgroupindex
#361502
#227992
#226482
C2×#15592
C2×#15512
#97042
#96982
#91422
#91412
#91382
#91312
#85872
#83442
#83432
#82352