Finite Group Databasebeta

← #223317

Subgroups — #223317

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2353C211yesmaximal, characteristic
2#22554C212yesmaximal, complemented
3#9881C212yesmaximal, complemented
4#9084C212yesmaximal, complemented
5#9079C212yesmaximal
6#8699C211yesmaximal, characteristic, complemented
7#8698C211yesmaximal, characteristic, complemented
8#8559C211yesmaximal, characteristic, complemented
9#8152C211yesmaximal, characteristic, complemented
10#7910C211yesmaximal, characteristic, complemented
11C2×#1280C211yesmaximal, characteristic, complemented
12C22×C4×D4C411yescharacteristic
13C2×#432C2212yes—
14C22×C22⋊C8C2212yes—
15#2353—22no—
16#2353—22no—
17#2353—22no—
18#2353—22no—
19C2×#388C411yescharacteristic
20C2×#388C2211yescharacteristic
21C2×#387C411yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#386C411yescharacteristic
24C2×#386C2211yescharacteristic
25C2×#383C411yescharacteristic
26C2×#381C411yescharacteristic
27C2×#380C411yescharacteristic
28C2×C4×C22⋊C4C411yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×#376C2211yescharacteristic
32C22×#95C2211yescharacteristic
33#1924—22no—
34#1924—22no—
35#1850—22no—
36#1817C2212yes—
37#1797C2212yes—
38#1779C2212yes—
39#1752—22no—
40#1749C2212yes—
41C2×#323C2212yes—
42C2×#323C2212yes—
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45#1280C2212yes—
46#1280C2212yes—
47#1280C2212yes—
48#1280C2212yes—
49#1280C2211yescharacteristic
50#1280C2211yescharacteristic
51C23×D4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C24×C4C2311yescharacteristic
56C23×C8—22no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2311yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#432—24no—
80#432—42no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2312yes—
92C2×#97—22no—
93C2×#97—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#383—22no—
107#383—22no—
108#381—22no—
109#381—22no—
110#380—22no—
111#380—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4D412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121C4×C22⋊C4D412yes—
122#376—22no—
123#376—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95C2×C411yescharacteristic
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C411yescharacteristic
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C411yescharacteristic
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C411yescharacteristic
133C2×#95C2×C411yescharacteristic
134C2×#95C2×C411yescharacteristic
135C2×#95C2×C411yescharacteristic
136C2×#95C2×C411yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#323—24no—
145#323—42no—
146C25C22×C411yescharacteristic
147C25C22×C411yescharacteristic
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4M1614yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—41no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—81no—
171C23×C4—81no—
172C23×C4—81no—
173C23×C4—81no—
174C23×C4—81no—
175C23×C4—81no—
176C23×C4M1614yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2411yescharacteristic
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8—82no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—28no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—81no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4—81no—
232C2×C22⋊C4C22⋊C411yescharacteristic
233C2×C22⋊C4C22⋊C411yescharacteristic
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—41no—
251C2×C42—81no—
252C2×C42—81no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C4⋊C8—44no—
258C4⋊C8—82no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—44no—
267C22⋊C8—44no—
268C22⋊C8—82no—
269C22⋊C8#4114yes—
270#97—44no—
271#97—82no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95C22⋊C412yes—
287#95C22⋊C412yes—
288#95C22⋊C412yes—
289#95C22⋊C412yes—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24C2×C22⋊C411yescharacteristic
309C24C2×C22⋊C411yescharacteristic
310C24C2×M1612yes—
311C24C2×M1612yes—
312C24C2×M1612yes—
313C24C23×C411yescharacteristic
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—48no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—48no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—84no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C412yes—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×M1612yes—
420C22×C4C2×M1612yes—
421C22×C4C2×M1612yes—
422C22×C4#13112yes—
423C22×C4C22×D412yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C42−1+411yescharacteristic
427C2×C8—24no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—88no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C42—24no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—81no—
470C42—81no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23#40712yes—
525C23#40712yes—
526C23#41012yes—
527C23#41312yes—
528C23C22×C22⋊C411yescharacteristic
529C23C4×#4111yescharacteristic
530C23#52011yescharacteristic
531C23C22×M1611yescharacteristic
532C23C22×M1611yescharacteristic
533C23C2×#13111yescharacteristic
534C23#56811yescharacteristic
535D4—48no—
536D4—48no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4#54612yes—
621C2×C4#54712yes—
622C8—44no—
623C8—82no—
624C8—82no—
625C8—82no—
626C22—21no—
627C22—21no—
628C22—22no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—22no—
641C22—21no—
642C22—21no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22#146414yes—
667C22#233911yescharacteristic
668C22C2×#40711yescharacteristic
669C22#288011yescharacteristic
670C22C2×#41011yescharacteristic
671C22#288311yescharacteristic
672C22C2×#41311yescharacteristic
673C22#288811yescharacteristic
674C22#296511yescharacteristic
675C22#297011yescharacteristic
676C22#298911yescharacteristic
677C22#299011yescharacteristic
678C4—22no—
679C4—22no—
680C4—42no—
681C4—42no—
682C4—41no—
683C4—41no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—41no—
688C4—41no—
689C4—42no—
690C4—42no—
691C4—81no—
692C4—81no—
693C4—81no—
694C2—21no—
695C2—21no—
696C2—21no—
697C2—21no—
698C2—24no—
699C2—42no—
700C2—82no—
701C2C2×#146411yescharacteristic
702C2#1736711yescharacteristic
703C2#1741211yescharacteristic
704C2#1742711yescharacteristic
705C2#2247111yescharacteristic
706C2#2247711yescharacteristic
707C2#2248211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22554C2semidirect
3. #9881C2semidirect
4. #9084C2semidirect
6. #8699C2semidirect
7. #8698C2semidirect
8. #8559C2semidirect
9. #8152C2semidirect
10. #7910C2semidirect
11. C2×#1280C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23532
#225542
#98812
#90842
#90792
#86992
#86982
#85592
#81522
#79102
C2×#12802