Finite Group Databasebeta

← #386664

Subgroups — #386664

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21083C211yesmaximal, characteristic
2#21024C211yesmaximal, characteristic, complemented
3#20838C211yesmaximal, characteristic
4#11317C211yesmaximal, characteristic, complemented
5#11317C211yesmaximal, characteristic, complemented
6#11291C211yesmaximal, characteristic, complemented
7#11226C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11075C211yesmaximal, characteristic
10#10934C211yesmaximal, characteristic
11#10887C211yesmaximal, characteristic, complemented
12#10834C211yesmaximal, characteristic, complemented
13#10822C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2297C2211yescharacteristic
27#2296C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1934C411yescharacteristic, complemented
30#1934C411yescharacteristic, complemented
31#1934C411yescharacteristic, complemented
32#1934C411yescharacteristic, complemented
33#1933C411yescharacteristic, complemented
34#1933C411yescharacteristic, complemented
35#1933C411yescharacteristic, complemented
36#1933C411yescharacteristic, complemented
37#1917—21no—
38#1917—21no—
39#1916—21no—
40#1916—21no—
41#1851—21no—
42#1851—21no—
43#1850—21no—
44#1850—21no—
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49#1735C2211yescharacteristic
50#1735C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62#1728C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—41no—
71C2×#131—41no—
72C2×#131—41no—
73C2×#131—41no—
74C22×M16—21no—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C4⋊C4—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#424—22no—
92#424—41no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—41no—
96C2×C4⋊C8—41no—
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8D411yescharacteristic
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—41no—
105C2×#104—41no—
106C2×#104—41no—
107#416—22no—
108#416—22no—
109#416—41no—
110#416—41no—
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119#408—22no—
120#408—41no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123#405—41no—
124#405—41no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8—41no—
128C2×C4×C8—41no—
129C2×C4×C8C2×C411yescharacteristic
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C411yescharacteristic
132C2×C4×C8C2×C411yescharacteristic
133#391C2×C411yescharacteristic
134#391C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#389C2×C411yescharacteristic
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#386—21no—
142#386—21no—
143#385—21no—
144#385—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—41no—
148C4×C4⋊C4—41no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C2×#95—21no—
152#341—22no—
153#341—22no—
154#341—41no—
155#341—41no—
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#340C2×C411yescharacteristic
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#339—22no—
165#339—41no—
166#339—41no—
167#339—41no—
168#339—41no—
169#339—41no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—41no—
183#336—41no—
184#336—41no—
185#336—41no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41#4112yes—
197C2×#41#4112yes—
198C22×Q8C22×C411yescharacteristic
199C22×D4—21no—
200C22×D4C22×C411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4#4111yescharacteristic
205C23×C4#4111yescharacteristic
206C23×C4C2411yescharacteristic
207#131—44no—
208#131—44no—
209#131—44no—
210#131—44no—
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—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—42no—
234C2×M16—42no—
235C2×M16—42no—
236C2×M16#4112yes—
237C2×M16#4112yes—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
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—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—42no—
267C22×C8—42no—
268C22×C8—42no—
269C22×C8C22×C411yescharacteristic
270C22×C8C22×C411yescharacteristic
271C22×C8C22×C411yescharacteristic
272C22×C8C22×C411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C22×C8C2×D411yescharacteristic
275C22×C8C2×D411yescharacteristic
276C22×C8C2×D411yescharacteristic
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—21no—
287#117—21no—
288#117—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C42—21no—
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—21no—
319C2×C42—21no—
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×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C4⋊C8—41no—
333C4⋊C8—41no—
334C4⋊C8—41no—
335C4⋊C8—41no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338#104—44no—
339#104—44no—
340#103—22no—
341#103—22no—
342#103—41no—
343#103—41no—
344#102—22no—
345#102—22no—
346#102—41no—
347#102—41no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350#97—42no—
351C4×C8—41no—
352C4×C8—41no—
353C4×C8—41no—
354C4×C8—41no—
355C4×C8—42no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—42no—
365C24—21no—
366C24C2×#4111yescharacteristic
367#41—22no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8#11712yes—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4#11712yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C42−1+411yescharacteristic
452C22×C42−1+411yescharacteristic
453C22×C42−1+411yescharacteristic
454M16—22no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16—48no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—22no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—48no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C2×C8—41no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—81no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23C2×C4×D411yescharacteristic
585C23#53811yescharacteristic
586C23#53911yescharacteristic
587C23#55211yescharacteristic
588C23#55311yescharacteristic
589C23#55611yescharacteristic
590C23#55911yescharacteristic
591Q8—22no—
592Q8—22no—
593Q8—41no—
594Q8—41no—
595D4—22no—
596D4—22no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—41no—
602D4—41no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4#44312yes—
657C2×C4#44412yes—
658C2×C4C2×#11711yescharacteristic
659C2×C4C4×#4111yescharacteristic
660C2×C4#51811yescharacteristic
661C2×C4#52011yescharacteristic
662C2×C4#52011yescharacteristic
663C2×C4#52011yescharacteristic
664C2×C4#53411yescharacteristic
665C2×C4#53711yescharacteristic
666C2×C4#53811yescharacteristic
667C2×C4#53811yescharacteristic
668C2×C4#53811yescharacteristic
669C2×C4#53911yescharacteristic
670C2×C4#53911yescharacteristic
671C2×C4#53911yescharacteristic
672C2×C4#54011yescharacteristic
673C2×C4#54211yescharacteristic
674C2×C4#54611yescharacteristic
675C2×C4#54611yescharacteristic
676C2×C4#54811yescharacteristic
677C2×C4#54811yescharacteristic
678C2×C4#55111yescharacteristic
679C2×C4#55111yescharacteristic
680C2×C4#55211yescharacteristic
681C2×C4#55211yescharacteristic
682C2×C4#56011yescharacteristic
683C2×C4#56311yescharacteristic
684C2×C4#56311yescharacteristic
685C2×C4#56311yescharacteristic
686C8—41no—
687C8—41no—
688C8—41no—
689C8—41no—
690C8—41no—
691C8—41no—
692C8—41no—
693C8—41no—
694C8—42no—
695C8—42no—
696C8—81no—
697C8—81no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22#232511yescharacteristic
718C22#235211yescharacteristic
719C22#236711yescharacteristic
720C22#274511yescharacteristic
721C22#277111yescharacteristic
722C22#278711yescharacteristic
723C22#279711yescharacteristic
724C22C2×#44311yescharacteristic
725C22C2×#44411yescharacteristic
726C22#295711yescharacteristic
727C22#295711yescharacteristic
728C4—22no—
729C4—22no—
730C4—42no—
731C4—41no—
732C4—41no—
733C4—82no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—81no—
739C4—81no—
740C4—81no—
741C4#231811yescharacteristic
742C4#233511yescharacteristic
743C4#234311yescharacteristic
744C4#238311yescharacteristic
745C4#269711yescharacteristic
746C4#270211yescharacteristic
747C4#270711yescharacteristic
748C4#272811yescharacteristic
749C2—21no—
750C2—21no—
751C2—21no—
752C2—21no—
753C2—82no—
754C2#1443611yescharacteristic
755C2#2220411yescharacteristic
756C2#2220611yescharacteristic
757C2#2278211yescharacteristic
758C2#2278311yescharacteristic
759C2#2323411yescharacteristic
760C2#2323511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21024C2semidirect
4. #11317C2semidirect
5. #11317C2semidirect
6. #11291C2semidirect
7. #11226C2semidirect
11. #10887C2semidirect
12. #10834C2semidirect
13. #10822C2semidirect
29. #1934C4semidirect
30. #1934C4semidirect
31. #1934C4semidirect
32. #1934C4semidirect
33. #1933C4semidirect
34. #1933C4semidirect
35. #1933C4semidirect
36. #1933C4semidirect

Maximal subgroups

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

subgroupindex
#210832
#210242
#208382
#113172
#112912
#112262
#111912
#110752
#109342
#108872
#108342
#108222
#107912