Finite Group Databasebeta

← #370190

Subgroups — #370190

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1855C211yesmaximal, characteristic
2#20843C211yesmaximal, characteristic
3#20835C211yesmaximal, characteristic
4C2×#1753C211yesmaximal, characteristic
5#20335C211yesmaximal, characteristic
6#11258C211yesmaximal, characteristic, complemented
7#11258C211yesmaximal, characteristic, complemented
8#11221C211yesmaximal, characteristic, complemented
9#11221C211yesmaximal, characteristic, complemented
10#10883C211yesmaximal, characteristic, complemented
11#10882C211yesmaximal, characteristic, complemented
12#10860C211yesmaximal, characteristic, complemented
13#10859C211yesmaximal, characteristic, complemented
14#10733C211yesmaximal, characteristic
15#10732C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C23×M16C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#407C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2280C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1855—24no—
32#1854C412yescomplemented
33#1854C412yescomplemented
34#1853C412yescomplemented
35#1852C412yescomplemented
36#1851—22no—
37#1851—22no—
38#1753—24no—
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51#1729—22no—
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1728—22no—
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C24×C4C2311yescharacteristic
62C22×M16—21no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—22no—
70C22×M16—22no—
71C22×M16—41no—
72C22×M16—41no—
73C22×M16—41no—
74C22×M16—41no—
75C22×M16D412yes—
76C22×M16D412yes—
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C23×C8—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#117—21no—
87C2×#117—21no—
88C2×#117—41no—
89C2×#117—41no—
90C2×#117—41no—
91C2×#117—41no—
92C2×#117C2×C412yes—
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4C2×C412yes—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C2×#108—22no—
104C2×#108—22no—
105C2×#108—22no—
106C2×#108—22no—
107C2×#108—41no—
108C2×#108—41no—
109#427—41no—
110#427—41no—
111#427—41no—
112#427—41no—
113C2×#107C2×C412yes—
114C2×#107C2×C412yes—
115C2×#106C2×C412yes—
116C2×#106C2×C412yes—
117#424—22no—
118#424—22no—
119#424—41no—
120#424—41no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—22no—
123#407—22no—
124#407—22no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8C2×C411yescharacteristic
128C2×C22⋊C8C2×C411yescharacteristic
129C2×C22⋊C8C2×C411yescharacteristic
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8D414yes—
132C2×C22⋊C8Q814yes—
133C2×#97—22no—
134C2×#97—22no—
135C2×#97—41no—
136C2×C4×C8—22no—
137C2×C4×C8—22no—
138C2×C4×C8—41no—
139#381—21no—
140#380C2×C412yes—
141#380C2×C412yes—
142#379—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4Q814yes—
147C2×#95—21no—
148#341—22no—
149#341—22no—
150#340C2×C412yes—
151#340C2×C412yes—
152#340C2×C412yes—
153#340C2×C412yes—
154#339—22no—
155#339—22no—
156#339—41no—
157#339—41no—
158#339—41no—
159#339—41no—
160#339—41no—
161#339—41no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170C25#4111yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×Q812yes—
188C23×C4C2×Q812yes—
189C23×C4#4111yescharacteristic
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—42no—
220C2×M16—42no—
221C2×M16—44no—
222C2×M16—44no—
223C2×M16—44no—
224C2×M16—44no—
225C2×M16—44no—
226C2×M16—44no—
227C2×M16—44no—
228C2×M16—44no—
229C2×M16—42no—
230C2×M16—42no—
231C2×M16#4112yes—
232C2×M16#4112yes—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—24no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—42no—
257C22×C8—42no—
258C22×C8C22×C411yescharacteristic
259C22×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C22×C411yescharacteristic
262C22×C8C2×D412yes—
263C22×C8C2×D412yes—
264C22×C8C2×D412yes—
265C22×C8C2×D412yes—
266C22×C8C2×Q812yes—
267C22×C8C2×Q812yes—
268C22×C8#4112yes—
269C22×C8#4112yes—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—42no—
287#117—42no—
288#117—42no—
289#117—42no—
290#117—42no—
291#117—42no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4C22×C412yes—
296C2×C4⋊C4C22×C412yes—
297C2×C4⋊C4C22×C412yes—
298C2×C4⋊C4C22×C412yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42C2×Q812yes—
328C2×C42C2×Q812yes—
329#108—44no—
330#108—44no—
331#107—42no—
332#107—42no—
333#106—42no—
334#106—42no—
335C4⋊C8—44no—
336C4⋊C8—44no—
337C22⋊C8—42no—
338C22⋊C8—42no—
339C22⋊C8C4⋊C414yes—
340C22⋊C8C4⋊C414yes—
341C22⋊C8C4⋊C414yes—
342C22⋊C8C4⋊C414yes—
343#97—44no—
344C4×C8—44no—
345#95—22no—
346#95—22no—
347#95—24no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—42no—
353C24—42no—
354C24C4×D411yescharacteristic
355C24C4×D411yescharacteristic
356C24C22⋊Q812yes—
357C24#12512yes—
358C24C2×#4111yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4C2×C4⋊C412yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C4×Q812yes—
417C22×C4C4×Q812yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4C22⋊Q812yes—
421C22×C4#12512yes—
422C22×C4C4⋊Q812yes—
423C22×C4C4⋊Q812yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×Q811yescharacteristic
428C22×C4C22×Q811yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438M16—22no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—48no—
443M16—48no—
444M16—48no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—48no—
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×C8C2×C4⋊C412yes—
500C2×C8C2×C4⋊C412yes—
501C2×C8C2×C4⋊C412yes—
502C2×C8C2×C4⋊C412yes—
503C2×C8#12314yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C23—22no—
543C23—22no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—82no—
564C23#38912yes—
565C23#38912yes—
566C23C2×C4×D411yescharacteristic
567C23C2×C22⋊Q811yescharacteristic
568C23C2×#12511yescharacteristic
569C23#52911yescharacteristic
570C23#52911yescharacteristic
571C23#53811yescharacteristic
572C23#53811yescharacteristic
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—88no—
620C2×C4#38912yes—
621C2×C4#38912yes—
622C2×C4C22×C4⋊C411yescharacteristic
623C2×C4C2×C4×Q811yescharacteristic
624C2×C4C4×#4111yescharacteristic
625C2×C4C4×#4111yescharacteristic
626C2×C4#51811yescharacteristic
627C2×C4#51811yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×#12311yescharacteristic
630C2×C4#52511yescharacteristic
631C2×C4C2×C4⋊Q811yescharacteristic
632C2×C4#53411yescharacteristic
633C2×C4#53511yescharacteristic
634C2×C4#54611yescharacteristic
635C2×C4#54611yescharacteristic
636C2×C4#54611yescharacteristic
637C2×C4#54611yescharacteristic
638C2×C4#55111yescharacteristic
639C2×C4#55111yescharacteristic
640C2×C4#55111yescharacteristic
641C2×C4#55111yescharacteristic
642C2×C4#55411yescharacteristic
643C2×C4#55411yescharacteristic
644C2×C4#55411yescharacteristic
645C2×C4#55411yescharacteristic
646C2×C4#55411yescharacteristic
647C2×C4#55411yescharacteristic
648C2×C4#55411yescharacteristic
649C2×C4#55411yescharacteristic
650C8—22no—
651C8—22no—
652C8—22no—
653C8—22no—
654C8—24no—
655C8—42no—
656C8—42no—
657C8—42no—
658C8—81no—
659C8—81no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22C2×#38911yescharacteristic
683C22C4×#12311yescharacteristic
684C22#232111yescharacteristic
685C22#245011yescharacteristic
686C22#245011yescharacteristic
687C22#250511yescharacteristic
688C22#250511yescharacteristic
689C22#295612yes—
690C22#295712yes—
691C4—22no—
692C4—22no—
693C4—44no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#234911yescharacteristic
700C4#234911yescharacteristic
701C4#236811yescharacteristic
702C4#236811yescharacteristic
703C4#241111yescharacteristic
704C4#241211yescharacteristic
705C4#250211yescharacteristic
706C4#258311yescharacteristic
707C2—21no—
708C2—21no—
709C2—21no—
710C2—21no—
711C2—42no—
712C2—42no—
713C2#1404411yescharacteristic
714C2#2116311yescharacteristic
715C2#2116511yescharacteristic
716C2#2159412yes—
717C2#2206212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11258C2semidirect
7. #11258C2semidirect
8. #11221C2semidirect
9. #11221C2semidirect
10. #10883C2semidirect
11. #10882C2semidirect
12. #10860C2semidirect
13. #10859C2semidirect
32. #1854C4semidirect
33. #1854C4semidirect
34. #1853C4semidirect
35. #1852C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18552
#208432
#208352
C2×#17532
#203352
#112582
#112212
#108832
#108822
#108602
#108592
#107332
#107322