Finite Group Databasebeta

← #367051

Subgroups — #367051

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#470C211yesmaximal, characteristic
2C2×#1971C211yesmaximal, characteristic, complemented
3C2×#1969C211yesmaximal, characteristic, complemented
4C2×#1944C211yesmaximal, characteristic, complemented
5C2×#1809C211yesmaximal, characteristic
6C2×C4×#106C211yesmaximal, characteristic, complemented
7C2×#1753C211yesmaximal, characteristic
8#13997C212yesmaximal
9#11067C212yesmaximal, complemented
10#10970C212yesmaximal
11#10932C212yesmaximal, complemented
12C23×Q16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#470—24no—
16C2×#470—24no—
17C2×#470C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1971—24no—
34#1969—24no—
35#1944—24no—
36#1809—24no—
37C4×#106—24no—
38#1753—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44#1445—22no—
45#1439—22no—
46#1437—22no—
47C4⋊#95—22no—
48C23×Q8C2312yes—
49C24×C4C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16C2×C414yes—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—22no—
61C22×M16—22no—
62C22×M16—22no—
63C22×M16C2×C414yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—42no—
72C2×C22⋊Q8—42no—
73C2×C22⋊Q8C2×C418yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—22no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#470—232no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106C2×C414yes—
93C2×#106D414yes—
94C2×#106D414yes—
95C2×#106D414yes—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103C2×C418yes—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8D414yes—
114#391—24no—
115#391—24no—
116#391—24no—
117#391—42no—
118#389—24no—
119#389—42no—
120#380—24no—
121#380—42no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—42no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135#340—24no—
136#340—24no—
137#340—24no—
138#336—22no—
139#336—22no—
140C25C2×D411yescharacteristic
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22×C414yes—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4C22×C412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C2×Q16—28no—
180C2×Q16—28no—
181C2×Q16—28no—
182C2×Q16—44no—
183C2×Q16—44no—
184C2×Q16—44no—
185C2×Q16—44no—
186C2×M16—28no—
187C2×M16—28no—
188C2×M16—28no—
189C2×M16—44no—
190C2×M16—44no—
191C2×M16—44no—
192C2×M16—44no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8C22×C412yes—
212C22×C8C22×C412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8#4112yes—
219C22×C8#4112yes—
220C22×C8#4112yes—
221C22⋊Q8—216no—
222C22⋊Q8—48no—
223C22⋊Q8—48no—
224C22⋊Q8—48no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—82no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—82no—
272C2×C42C2×D412yes—
273C2×C42#4112yes—
274#107—44no—
275#106—28no—
276#106C22⋊C4116yescomplemented
277#103—216no—
278#103—48no—
279#103—48no—
280C22⋊C8—28no—
281#97—44no—
282C4×C8—28no—
283#95—22no—
284#95—22no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24C4×D412yes—
303C24C22×D411yescharacteristic
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
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—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4C4×D412yes—
412C22×C4#12112yes—
413C22×C4#12112yes—
414C22×C4#13714yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C4H(5,𝔽2)11yescharacteristic
427C22×C4H(5,𝔽2)11yescharacteristic
428C22×C42−1+411yescharacteristic
429Q16—416no—
430M16—416no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—41no—
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×C8C2×C22⋊C414yes—
453C2×C8C4×D414yes—
454C2×C8C4×D414yes—
455C2×C8#12114yes—
456C2×C8#12114yes—
457C2×C8#12414yes—
458C2×C8#12714yes—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—416no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—88no—
472C4⋊C4C2×C22⋊C418yes—
473C22⋊C4—416no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—88no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—88no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×C4×D411yescharacteristic
510C23C2×#12111yescharacteristic
511C23#53211yescharacteristic
512C23D4211yescharacteristic
513C23D4211yescharacteristic
514C23D4211yescharacteristic
515C23#54711yescharacteristic
516C23C2×#13712yes—
517C23#57512yes—
518C23#57712yes—
519C23#57812yes—
520Q8—44no—
521Q8—44no—
522Q8—44no—
523Q8—44no—
524Q8—82no—
525Q8—82no—
526Q8—82no—
527Q8—82no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4#47014yescomplemented
591C2×C4C22×C22⋊C411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C4×#4111yescharacteristic
595C2×C4#51811yescharacteristic
596C2×C4#52011yescharacteristic
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4#52511yescharacteristic
600C2×C4C2×#12411yescharacteristic
601C2×C4#52911yescharacteristic
602C2×C4C2×#12711yescharacteristic
603C2×C4#53511yescharacteristic
604C2×C4#53711yescharacteristic
605C2×C4#54011yescharacteristic
606C2×C4#54211yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54611yescharacteristic
610C2×C4#54811yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4#54811yescharacteristic
613C2×C4#55011yescharacteristic
614C2×C4#55011yescharacteristic
615C2×C4#55011yescharacteristic
616C2×C4#55311yescharacteristic
617C2×C4#55311yescharacteristic
618C2×C4#55511yescharacteristic
619C8—42no—
620C8—42no—
621C8—42no—
622C8#39018yes—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22—81no—
629C22—82no—
630C22—81no—
631C22—82no—
632C22—81no—
633C22—81no—
634C22C4×#12111yescharacteristic
635C22D4×C22⋊C411yescharacteristic
636C22#236111yescharacteristic
637C22#241611yescharacteristic
638C22#242811yescharacteristic
639C22#249111yescharacteristic
640C22#254111yescharacteristic
641C22C4×#13712yes—
642C22#294712yes—
643C22C2×#47011yescharacteristic
644C22#305411yescharacteristic
645C22#305911yescharacteristic
646C22#306011yescharacteristic
647C22#317212yes—
648C22#317812yes—
649C22#326612yes—
650C22#326912yes—
651C22#327512yes—
652C22#327512yes—
653C22#327612yes—
654C22#327612yes—
655C22#334412yes—
656C22#334912yes—
657C4—24no—
658C4—48no—
659C4—42no—
660C4—42no—
661C4—81no—
662C4—84no—
663C4—81no—
664C4—84no—
665C4—162no—
666C4C2×#39011yescharacteristic
667C4#232411yescharacteristic
668C4#234111yescharacteristic
669C4#236311yescharacteristic
670C4#241411yescharacteristic
671C4#242611yescharacteristic
672C4#249011yescharacteristic
673C4#254011yescharacteristic
674C2—82no—
675C2#1399811yescharacteristic
676C2C4×#47011yescharacteristic
677C2#2153711yescharacteristic
678C2#2244412yes—
679C2#2303812yes—
680C2#2478311yescharacteristic
681C2#2478511yescharacteristic
682C2#2541712yes—
683C2#2779312yes—
684C2#2848711yescharacteristic
685C2#2848911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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×#1971C2semidirect
3. C2×#1969C2semidirect
4. C2×#1944C2semidirect
6. C2×C4×#106C2semidirect
9. #11067C2semidirect
11. #10932C2semidirect
17. C2×#470C4semidirect
276. #106C22⋊C4semidirect
590. C2×C4#470semidirect

Maximal subgroups

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

subgroupindex
C22×#4702
C2×#19712
C2×#19692
C2×#19442
C2×#18092
C2×C4×#1062
C2×#17532
#139972
#110672
#109702
#109322