Finite Group Databasebeta

← #384772

Subgroups — #384772

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21058C211yesmaximal, characteristic
2#20839C211yesmaximal, characteristic
3#20774C211yesmaximal, characteristic, complemented
4#11376C211yesmaximal, characteristic, complemented
5#11291C211yesmaximal, characteristic, complemented
6#11269C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11075C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic, complemented
12#10852C211yesmaximal, characteristic, complemented
13#10818C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2297C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1928C411yescharacteristic, complemented
30#1928C411yescharacteristic, complemented
31#1927C411yescharacteristic, complemented
32#1927C411yescharacteristic, complemented
33#1926C411yescharacteristic, complemented
34#1926C411yescharacteristic, complemented
35#1925C411yescharacteristic, complemented
36#1925C411yescharacteristic, complemented
37#1851—21no—
38#1851—21no—
39#1850—21no—
40#1850—21no—
41#1840—21no—
42#1840—21no—
43#1839—21no—
44#1839—21no—
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49#1735C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
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#427—41no—
92#427—41no—
93C2×#107C2×C411yescharacteristic
94C2×#107C2×C411yescharacteristic
95C2×#106C2×C411yescharacteristic
96C2×#106C2×C411yescharacteristic
97#424—41no—
98#424—41no—
99#423—22no—
100#423—41no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×#104—22no—
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—41no—
109C2×#104—41no—
110C2×#104—41no—
111#416—22no—
112#416—22no—
113#416—41no—
114#416—41no—
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123#408—22no—
124#408—41no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C4×C8—22no—
128C2×C4×C8—41no—
129C2×C4×C8—41no—
130#391C2×C411yescharacteristic
131#391C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#386—21no—
135#386—21no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—41no—
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146#376—21no—
147C2×#95—21no—
148C43—21no—
149#341—22no—
150#341—41no—
151#341—41no—
152#341—41no—
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#339—41no—
158#339—41no—
159#339—41no—
160#339—41no—
161#339—41no—
162#339—41no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—41no—
176#336—41no—
177#336—41no—
178#336C2×C411yescharacteristic
179#336C2×C411yescharacteristic
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182#336D411yescharacteristic
183#336D411yescharacteristic
184#336D411yescharacteristic
185#336D411yescharacteristic
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—21no—
234C2×M16—42no—
235C2×M16—42no—
236C2×M16—42no—
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—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—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×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42#4112yes—
332#107—41no—
333#107—41no—
334#106—41no—
335#106—41no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339C4⋊C8—42no—
340#104—44no—
341#104—44no—
342#103—22no—
343#103—22no—
344#103—41no—
345#103—41no—
346#102—22no—
347#102—22no—
348#102—41no—
349#102—41no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C4×C8—44no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—42no—
362C24—21no—
363C24C2×#4111yescharacteristic
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41—22no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8#11712yes—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4#11712yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—41no—
417C22×C4—41no—
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×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C42−1+411yescharacteristic
449C22×C42−1+411yescharacteristic
450C22×C42−1+411yescharacteristic
451M16—22no—
452M16—22no—
453M16—48no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
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—22no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—48no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
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—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
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—48no—
534C22⋊C4—42no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—22no—
553C42—22no—
554C42—42no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—42no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—81no—
569C42—81no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23C2×C4×D411yescharacteristic
582C23#54011yescharacteristic
583C23#54211yescharacteristic
584C23#55111yescharacteristic
585C23#55211yescharacteristic
586C23#55211yescharacteristic
587C23#55311yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592D4—22no—
593D4—22no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—41no—
599D4—41no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
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#44312yes—
655C2×C4C2×#11711yescharacteristic
656C2×C4C4×#4111yescharacteristic
657C2×C4#51811yescharacteristic
658C2×C4#52011yescharacteristic
659C2×C4#52011yescharacteristic
660C2×C4#52011yescharacteristic
661C2×C4#53511yescharacteristic
662C2×C4#53711yescharacteristic
663C2×C4#53811yescharacteristic
664C2×C4#53811yescharacteristic
665C2×C4#53811yescharacteristic
666C2×C4#53811yescharacteristic
667C2×C4#53911yescharacteristic
668C2×C4#53911yescharacteristic
669C2×C4#54211yescharacteristic
670C2×C4#54211yescharacteristic
671C2×C4#54611yescharacteristic
672C2×C4#54611yescharacteristic
673C2×C4#54711yescharacteristic
674C2×C4#54811yescharacteristic
675C2×C4#55111yescharacteristic
676C2×C4#55111yescharacteristic
677C2×C4#55211yescharacteristic
678C2×C4#55211yescharacteristic
679C2×C4#56011yescharacteristic
680C2×C4#56311yescharacteristic
681C2×C4#56311yescharacteristic
682C2×C4#56311yescharacteristic
683C8—41no—
684C8—41no—
685C8—41no—
686C8—41no—
687C8—41no—
688C8—41no—
689C8—42no—
690C8—41no—
691C8—41no—
692C8—81no—
693C8—81no—
694C8—81no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—81no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22#232411yescharacteristic
715C22#235211yescharacteristic
716C22#235911yescharacteristic
717C22#273811yescharacteristic
718C22#274911yescharacteristic
719C22#276411yescharacteristic
720C22#277011yescharacteristic
721C22C2×#44311yescharacteristic
722C22#295511yescharacteristic
723C22#295611yescharacteristic
724C22#295711yescharacteristic
725C4—22no—
726C4—44no—
727C4—42no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—82no—
732C4—81no—
733C4—81no—
734C4—81no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4#231811yescharacteristic
739C4#233711yescharacteristic
740C4#234311yescharacteristic
741C4#238311yescharacteristic
742C4#268511yescharacteristic
743C4#269511yescharacteristic
744C4#270211yescharacteristic
745C4#272311yescharacteristic
746C2—21no—
747C2—21no—
748C2—21no—
749C2—21no—
750C2—82no—
751C2#1438911yescharacteristic
752C2#2216511yescharacteristic
753C2#2216711yescharacteristic
754C2#2278311yescharacteristic
755C2#2278411yescharacteristic
756C2#2294811yescharacteristic
757C2#2295111yescharacteristic
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
3. #20774C2semidirect
4. #11376C2semidirect
5. #11291C2semidirect
6. #11269C2semidirect
7. #11221C2semidirect
11. #10874C2semidirect
12. #10852C2semidirect
13. #10818C2semidirect
29. #1928C4semidirect
30. #1928C4semidirect
31. #1927C4semidirect
32. #1927C4semidirect
33. #1926C4semidirect
34. #1926C4semidirect
35. #1925C4semidirect
36. #1925C4semidirect

Maximal subgroups

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

subgroupindex
#210582
#208392
#207742
#113762
#112912
#112692
#112212
#111912
#110752
#109292
#108742
#108522
#108182
#107912