Finite Group Databasebeta

← #373500

Subgroups — #373500

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1938C211yesmaximal, characteristic
2#21058C211yesmaximal, characteristic
3#21051C211yesmaximal, characteristic
4C2×#1922C211yesmaximal, characteristic, complemented
5#20430C211yesmaximal, characteristic, complemented
6#11269C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11075C211yesmaximal, characteristic
9#11068C211yesmaximal, characteristic
10#10934C211yesmaximal, characteristic
11#10934C211yesmaximal, characteristic
12#10875C211yesmaximal, characteristic, complemented
13#10852C211yesmaximal, characteristic, complemented
14#10809C211yesmaximal, characteristic, complemented
15#10707C211yesmaximal, characteristic, complemented
16C2×#427C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C22×#97C2211yescharacteristic
30#2297C2211yescharacteristic
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1938—24no—
34#1928C411yescharacteristic, complemented
35#1928C411yescharacteristic, complemented
36#1927C411yescharacteristic, complemented
37#1927C411yescharacteristic, complemented
38#1926C411yescharacteristic, complemented
39#1926C411yescharacteristic, complemented
40#1925C411yescharacteristic, complemented
41#1925C411yescharacteristic, complemented
42#1924—21no—
43#1924—21no—
44#1923—21no—
45#1923—21no—
46#1922—24no—
47#1768—21no—
48#1768—21no—
49#1767—21no—
50C4×C4⋊C8—21no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55#1735C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C22×#41C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#427—22no—
86#427—22no—
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91#424—22no—
92#424—22no—
93#424—22no—
94#424D414yes—
95#423—22no—
96#423—41no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—22no—
106C2×#104—41no—
107C2×#104—41no—
108C2×#104—41no—
109#416—22no—
110#416—22no—
111#416—22no—
112#416—22no—
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121#408—22no—
122#408—22no—
123#408—22no—
124#408—41no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129#405—22no—
130#405—22no—
131#405—22no—
132#405—22no—
133#405—41no—
134#405—41no—
135C2×#97—22no—
136C2×#97—21no—
137C2×#97—21no—
138C2×#97—21no—
139C2×#97—22no—
140C2×#97—41no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144#391C2×C411yescharacteristic
145#391C2×C411yescharacteristic
146#390C2×C411yescharacteristic
147#390C2×C411yescharacteristic
148#386—21no—
149#386—21no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—41no—
152C4×C4⋊C4C2×C411yescharacteristic
153C4×C4⋊C4C2×C411yescharacteristic
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160#376—21no—
161C2×#95—21no—
162C43—21no—
163#341—22no—
164#341—22no—
165#341—22no—
166#341—41no—
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#340C2×C411yescharacteristic
170#340C2×C411yescharacteristic
171#339—22no—
172#339—22no—
173#339—22no—
174#339—22no—
175#339—22no—
176#339—22no—
177#336—22no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—41no—
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×#41—21no—
203C2×#41—21no—
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
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—21no—
234C2×M16C2×D412yes—
235C2×M16#4112yes—
236C2×M16#4112yes—
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—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×C8C22×C411yescharacteristic
264C22×C8C22×C411yescharacteristic
265C22×C8C22×C411yescharacteristic
266C22×C8C22×C411yescharacteristic
267C22×C8C2×D411yescharacteristic
268C22×C8C2×D411yescharacteristic
269C22×C8C2×D412yes—
270C22×C8C2×D411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8#4112yes—
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#117C2×D412yes—
282#117#4112yes—
283#117#4112yes—
284#117#4112yes—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—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×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42#4112yes—
328#107—22no—
329#107—22no—
330#106—22no—
331#106—22no—
332C4⋊C8—24no—
333C4⋊C8—24no—
334C4⋊C8—24no—
335C4⋊C8—24no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338#104—44no—
339#104—44no—
340#103—22no—
341#103—22no—
342#103—22no—
343#103—22no—
344#102—22no—
345#102—22no—
346#102—22no—
347#102—22no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352#97—44no—
353#97—42no—
354#97—42no—
355#97—42no—
356C4×C8—42no—
357C4×C8—42no—
358C4×C8—42no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—42no—
368C24—21no—
369C24C2×#4111yescharacteristic
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
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×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C23×C411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C4H(5,𝔽2)11yescharacteristic
455C22×C42−1+411yescharacteristic
456C22×C42−1+411yescharacteristic
457M16—22no—
458M16—22no—
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—21no—
476C2×C8—22no—
477C2×C8—21no—
478C2×C8—22no—
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—21no—
492C2×C8—21no—
493C2×C8—22no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—21no—
501C2×C8—21no—
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#12112yes—
524C2×C8#12112yes—
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—48no—
534C4⋊C4#11712yes—
535C4⋊C4#11712yes—
536C4⋊C4C4×D412yes—
537C4⋊C4C4×D412yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—42no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—42no—
546C22⋊C4—81no—
547C22⋊C4—81no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—42no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—42no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—81no—
573C42—81no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23C2×C4×D411yescharacteristic
586C23#52911yescharacteristic
587C23#53211yescharacteristic
588C23#55111yescharacteristic
589C23#55211yescharacteristic
590C23#55211yescharacteristic
591C23#55311yescharacteristic
592Q8—41no—
593Q8—41no—
594Q8—41no—
595Q8—41no—
596D4—41no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—81no—
601D4—81no—
602D4—81no—
603D4—81no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
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—22no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
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—81no—
657C2×C4—81no—
658C2×C4#44412yes—
659C2×C4C2×#11711yescharacteristic
660C2×C4C2×C4×D411yescharacteristic
661C2×C4C4×#4111yescharacteristic
662C2×C4C4×#4111yescharacteristic
663C2×C4#52011yescharacteristic
664C2×C4#52011yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C2×C4#52511yescharacteristic
667C2×C4#52911yescharacteristic
668C2×C4#52911yescharacteristic
669C2×C4#52911yescharacteristic
670C2×C4#52911yescharacteristic
671C2×C4#53811yescharacteristic
672C2×C4#53911yescharacteristic
673C2×C4#54111yescharacteristic
674C2×C4#54211yescharacteristic
675C2×C4#54611yescharacteristic
676C2×C4#54611yescharacteristic
677C2×C4#54611yescharacteristic
678C2×C4#54711yescharacteristic
679C2×C4#54711yescharacteristic
680C2×C4#54811yescharacteristic
681C2×C4#54811yescharacteristic
682C2×C4#55011yescharacteristic
683C2×C4#55511yescharacteristic
684C2×C4#55611yescharacteristic
685C2×C4#55611yescharacteristic
686C2×C4#55911yescharacteristic
687C8—22no—
688C8—22no—
689C8—41no—
690C8—41no—
691C8—41no—
692C8—41no—
693C8—42no—
694C8—42no—
695C8—41no—
696C8—41no—
697C8—42no—
698C8—81no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22C4×#12111yescharacteristic
719C22#235211yescharacteristic
720C22#235911yescharacteristic
721C22#251911yescharacteristic
722C22#253211yescharacteristic
723C22#255411yescharacteristic
724C22#259611yescharacteristic
725C22C2×#44411yescharacteristic
726C22#295511yescharacteristic
727C22#295611yescharacteristic
728C22#295711yescharacteristic
729C4—22no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—42no—
737C4—82no—
738C4—82no—
739C4—81no—
740C4—81no—
741C4—81no—
742C4#229811yescharacteristic
743C4#233411yescharacteristic
744C4#235411yescharacteristic
745C4#236711yescharacteristic
746C4#242511yescharacteristic
747C4#242611yescharacteristic
748C4#245911yescharacteristic
749C4#246911yescharacteristic
750C2—21no—
751C2—21no—
752C2—21no—
753C2—21no—
754C2—161no—
755C2#1411911yescharacteristic
756C2#2151811yescharacteristic
757C2#2151911yescharacteristic
758C2#2278211yescharacteristic
759C2#2278411yescharacteristic
760C2#2294911yescharacteristic
761C2#2295011yescharacteristic
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
4. C2×#1922C2semidirect
5. #20430C2semidirect
6. #11269C2semidirect
7. #11221C2semidirect
12. #10875C2semidirect
13. #10852C2semidirect
14. #10809C2semidirect
15. #10707C2semidirect
34. #1928C4semidirect
35. #1928C4semidirect
36. #1927C4semidirect
37. #1927C4semidirect
38. #1926C4semidirect
39. #1926C4semidirect
40. #1925C4semidirect
41. #1925C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19382
#210582
#210512
C2×#19222
#204302
#112692
#112212
#110752
#110682
#109342
#108752
#108522
#108092
#107072