Finite Group Databasebeta

← #377800

Subgroups — #377800

Order 512.

989 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23627C211yesmaximal, characteristic
2#21025C211yesmaximal, characteristic, complemented
3C2×#1866C211yesmaximal, characteristic
4C2×#1765C211yesmaximal, characteristic
5#20387C211yesmaximal, characteristic
6#11191C211yesmaximal, characteristic
7#11191C211yesmaximal, characteristic
8#11075C211yesmaximal, characteristic
9#10911C211yesmaximal, characteristic, complemented
10#10910C211yesmaximal, characteristic, complemented
11#10883C211yesmaximal, characteristic, complemented
12#10875C211yesmaximal, characteristic, complemented
13#10851C211yesmaximal, characteristic, complemented
14#10809C211yesmaximal, characteristic, complemented
15#10809C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C22×#97C2211yescharacteristic
28#2297C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2074C411yescharacteristic, complemented
32#2074C411yescharacteristic, complemented
33#2073C411yescharacteristic, complemented
34#2073C411yescharacteristic, complemented
35#2072C411yescharacteristic, complemented
36#2072C411yescharacteristic, complemented
37#2071C411yescharacteristic, complemented
38#2071C411yescharacteristic, complemented
39#1917—21no—
40#1917—21no—
41#1916—21no—
42#1916—21no—
43#1866—24no—
44#1765—24no—
45#1753—21no—
46#1753—21no—
47#1752—21no—
48C8×C22⋊C4—21no—
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51C2×#341C2211yescharacteristic
52#1735C2211yescharacteristic
53#1735C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C22×#41C2311yescharacteristic
70C2×#131—22no—
71C2×#131—22no—
72C2×#131—22no—
73C2×#131—41no—
74C2×#131—41no—
75C2×#131D414yes—
76C22×M16—21no—
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C23×C8—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C22×C4⋊C4—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#108—22no—
94C2×#108—22no—
95C2×#108—22no—
96#424—22no—
97#424—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—41no—
105C2×#104—41no—
106#416—22no—
107#416—22no—
108#416—22no—
109#416—22no—
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118#408—22no—
119#408—22no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122#405—22no—
123#405—22no—
124#405—22no—
125#405—41no—
126C4×M16—22no—
127C4×M16—41no—
128C2×#97—22no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—41no—
133C2×#97—41no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137#391C2×C411yescharacteristic
138#391C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#390C2×C411yescharacteristic
141#386—21no—
142#386—21no—
143#384C2×C411yescharacteristic
144#384C2×C411yescharacteristic
145#382C2×C411yescharacteristic
146#382C2×C411yescharacteristic
147#380—21no—
148#380—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—41no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C2×#95—21no—
156#341—22no—
157#341—22no—
158#341—41no—
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#340C2×C411yescharacteristic
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#339—22no—
168#339—22no—
169#339—22no—
170#339—22no—
171#339—22no—
172#339—41no—
173#336—22no—
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—22no—
183#336—41no—
184#336C2×C411yescharacteristic
185#336C2×C411yescharacteristic
186#336C2×C411yescharacteristic
187#336C2×C411yescharacteristic
188#336D411yescharacteristic
189#336D411yescharacteristic
190#336D411yescharacteristic
191#336D411yescharacteristic
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41C2×D412yes—
203C2×#41#4112yes—
204C22×Q8C22×C411yescharacteristic
205C22×D4—21no—
206C22×D4C22×C411yescharacteristic
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4#4111yescharacteristic
211C23×C4#4111yescharacteristic
212C23×C4C2411yescharacteristic
213#131—28no—
214#131—44no—
215#131—44no—
216#131—44no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—24no—
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—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—42no—
238C2×M16—42no—
239C2×M16C2×D412yes—
240C2×M16#4112yes—
241C2×M16#4112yes—
242C2×M16#4112yes—
243C2×M16#4112yes—
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—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—24no—
270C22×C8—21no—
271C22×C8—42no—
272C22×C8—42no—
273C22×C8C22×C411yescharacteristic
274C22×C8C22×C411yescharacteristic
275C22×C8C22×C411yescharacteristic
276C22×C8C22×C411yescharacteristic
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×D412yes—
280C22×C8C2×D411yescharacteristic
281C22×C8C2×D411yescharacteristic
282C22×C8#4112yes—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—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×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42C22×C411yescharacteristic
334C2×C42C22×C411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42#4112yes—
338#108—44no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341#104—28no—
342#104—44no—
343#103—22no—
344#103—22no—
345#103—22no—
346#103—22no—
347#102—22no—
348#102—22no—
349#102—22no—
350#102—22no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353#97—42no—
354#97—42no—
355#97—44no—
356#97—42no—
357C4×C8—42no—
358C4×C8—42no—
359C4×C8—42no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—42no—
369C24—21no—
370C24C2×#4111yescharacteristic
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8C4×D412yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4C4×D412yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
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×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C4×D411yescharacteristic
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4H(5,𝔽2)11yescharacteristic
455C22×C4H(5,𝔽2)11yescharacteristic
456C22×C42−1+411yescharacteristic
457C22×C42−1+411yescharacteristic
458M16—22no—
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—48no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16#12314yes—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—22no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—22no—
506C2×C8—48no—
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—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8#12112yes—
527C2×C8#12112yes—
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—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—42no—
541C4⋊C4—48no—
542C4⋊C4—42no—
543C4⋊C4—81no—
544C4⋊C4—81no—
545C22⋊C4—42no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—42no—
553C22⋊C4—81no—
554C22⋊C4—81no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—22no—
564C42—22no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—22no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23C2×C4×D411yescharacteristic
589C23#52911yescharacteristic
590C23#53211yescharacteristic
591C23#55111yescharacteristic
592C23#55211yescharacteristic
593C23#55611yescharacteristic
594C23#55911yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599D4—22no—
600D4—22no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—41no—
606D4—41no—
607C2×C4—21no—
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—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
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—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4#44412yes—
662C2×C4C2×C4×D411yescharacteristic
663C2×C4C4×#4111yescharacteristic
664C2×C4C4×#4111yescharacteristic
665C2×C4C4×#4111yescharacteristic
666C2×C4#52011yescharacteristic
667C2×C4#52011yescharacteristic
668C2×C4C2×#12111yescharacteristic
669C2×C4C2×#12311yescharacteristic
670C2×C4#52511yescharacteristic
671C2×C4#52511yescharacteristic
672C2×C4#52911yescharacteristic
673C2×C4#52911yescharacteristic
674C2×C4#53411yescharacteristic
675C2×C4#53611yescharacteristic
676C2×C4#53911yescharacteristic
677C2×C4#53911yescharacteristic
678C2×C4#54611yescharacteristic
679C2×C4#54611yescharacteristic
680C2×C4#54711yescharacteristic
681C2×C4#54811yescharacteristic
682C2×C4#55111yescharacteristic
683C2×C4#55111yescharacteristic
684C2×C4#55211yescharacteristic
685C2×C4#55211yescharacteristic
686C2×C4#55311yescharacteristic
687C2×C4#55311yescharacteristic
688C2×C4#55611yescharacteristic
689C2×C4#55611yescharacteristic
690C8—22no—
691C8—22no—
692C8—24no—
693C8—41no—
694C8—41no—
695C8—42no—
696C8—42no—
697C8—41no—
698C8—41no—
699C8—41no—
700C8—41no—
701C8—81no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22C4×#12111yescharacteristic
722C22#235911yescharacteristic
723C22#236711yescharacteristic
724C22#249911yescharacteristic
725C22#251711yescharacteristic
726C22#252311yescharacteristic
727C22#260211yescharacteristic
728C22C2×#44411yescharacteristic
729C22#295511yescharacteristic
730C22#295611yescharacteristic
731C22#295711yescharacteristic
732C4—22no—
733C4—42no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—82no—
738C4—82no—
739C4—81no—
740C4—81no—
741C4—81no—
742C4—81no—
743C4—81no—
744C4—81no—
745C4C4×#12311yescharacteristic
746C4#232211yescharacteristic
747C4#237311yescharacteristic
748C4#237311yescharacteristic
749C4#241211yescharacteristic
750C4#241511yescharacteristic
751C4#246011yescharacteristic
752C4#246611yescharacteristic
753C2—21no—
754C2—21no—
755C2—21no—
756C2—21no—
757C2—82no—
758C2#1420211yescharacteristic
759C2#2151811yescharacteristic
760C2#2151911yescharacteristic
761C2#2295011yescharacteristic
762C2#2295111yescharacteristic
763C2#2323511yescharacteristic
764C2#2323611yescharacteristic
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 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21025C2semidirect
9. #10911C2semidirect
10. #10910C2semidirect
11. #10883C2semidirect
12. #10875C2semidirect
13. #10851C2semidirect
14. #10809C2semidirect
15. #10809C2semidirect
31. #2074C4semidirect
32. #2074C4semidirect
33. #2073C4semidirect
34. #2073C4semidirect
35. #2072C4semidirect
36. #2072C4semidirect
37. #2071C4semidirect
38. #2071C4semidirect

Maximal subgroups

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

subgroupindex
#236272
#210252
C2×#18662
C2×#17652
#203872
#111912
#110752
#109112
#109102
#108832
#108752
#108512
#108092