Finite Group Databasebeta

← #390855

Subgroups — #390855

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2#21024C211yesmaximal, characteristic, complemented
3#20838C211yesmaximal, characteristic
4#11376C211yesmaximal, characteristic, complemented
5#11258C211yesmaximal, characteristic, complemented
6#11233C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11125C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10924C211yesmaximal, characteristic
11#10887C211yesmaximal, characteristic, complemented
12#10879C211yesmaximal, characteristic, complemented
13#10833C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#405C2211yescharacteristic
28#2297C2211yescharacteristic
29#2296C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#2068C411yescharacteristic, complemented
32#2068C411yescharacteristic, complemented
33#2067C411yescharacteristic, complemented
34#2067C411yescharacteristic, complemented
35#2066C411yescharacteristic, complemented
36#2066C411yescharacteristic, complemented
37#2065C411yescharacteristic, complemented
38#2065C411yescharacteristic, complemented
39#1917—21no—
40#1917—21no—
41#1916—21no—
42#1916—21no—
43#1851—21no—
44#1851—21no—
45#1850—21no—
46#1850—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
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×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#108—41no—
91C2×#108—41no—
92C2×#108—41no—
93#427—41no—
94#427—41no—
95C2×#107C2×C411yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99#424—22no—
100#424—41no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—41no—
104C2×C4⋊C8—41no—
105C2×C4⋊C8—41no—
106C2×C4⋊C8—41no—
107C2×C4⋊C8C2×C411yescharacteristic
108C2×C4⋊C8C2×C411yescharacteristic
109C2×C4⋊C8C2×C411yescharacteristic
110C2×C4⋊C8C2×C411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D411yescharacteristic
113C2×C4⋊C8D411yescharacteristic
114C2×C4⋊C8D411yescharacteristic
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—41no—
118C2×#104—41no—
119C2×#104—41no—
120C2×#104Q814yes—
121#416—41no—
122#416—41no—
123#416—41no—
124#416—41no—
125C2×#103C2×C411yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133#408—22no—
134#408—41no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137#405—41no—
138#405—41no—
139C2×#97—21no—
140C2×C4×C8—21no—
141#391C2×C411yescharacteristic
142#391C2×C411yescharacteristic
143#390C2×C411yescharacteristic
144#390C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#386—21no—
150#386—21no—
151#385—21no—
152#385—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157#376—41no—
158#376—41no—
159C2×#95—21no—
160C2×#95—21no—
161#341—41no—
162#341—41no—
163#341—41no—
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#339—22no—
169#339—22no—
170#339—22no—
171#339—41no—
172#339—41no—
173#339—41no—
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—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#336—41no—
187#336—41no—
188#336—41no—
189#336—41no—
190C2×#41—21no—
191C2×#41—21no—
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×#41C2×Q812yes—
201C2×#41#4112yes—
202C22×Q8C22×C411yescharacteristic
203C22×D4—21no—
204C22×D4C22×C411yescharacteristic
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4#4111yescharacteristic
209C23×C4#4111yescharacteristic
210C23×C4C2411yescharacteristic
211#131—44no—
212#131—44no—
213#131—44no—
214#131—44no—
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—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—42no—
238C2×M16—42no—
239C2×M16—42no—
240C2×M16C2×Q812yes—
241C2×M16#4112yes—
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—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—42no—
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×D411yescharacteristic
280C22×C8C2×D411yescharacteristic
281#117—21no—
282#117—21no—
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—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—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×C42C2×Q812yes—
332C2×C42#4112yes—
333#108—44no—
334#107—41no—
335#107—41no—
336#106—41no—
337#106—41no—
338C4⋊C8—41no—
339C4⋊C8—41no—
340C4⋊C8—41no—
341C4⋊C8—41no—
342C4⋊C8—41no—
343C4⋊C8—41no—
344C4⋊C8—41no—
345C4⋊C8—41no—
346C4⋊C8—42no—
347C4⋊C8—42no—
348#104—44no—
349#104—44no—
350#103—41no—
351#103—41no—
352#103—41no—
353#103—41no—
354#102—41no—
355#102—41no—
356#102—41no—
357#102—41no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360#97—42no—
361C4×C8—42no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—42no—
371#95—42no—
372C24—21no—
373C24C2×#4111yescharacteristic
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C4×Q812yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4C4×Q812yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
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×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4C4×D411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C4×D411yescharacteristic
446C22×C4C4×D411yescharacteristic
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×Q811yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C4H(5,𝔽2)11yescharacteristic
455C22×C4H(5,𝔽2)11yescharacteristic
456C22×C4H(5,𝔽2)11yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459C22×C42−1+411yescharacteristic
460C22×C42−1+411yescharacteristic
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—41no—
475M16—41no—
476M16—41no—
477M16—41no—
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—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—22no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
504C2×C8—48no—
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—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8—41no—
530C2×C8—41no—
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—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—81no—
546C4⋊C4—81no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—81no—
556C22⋊C4—81no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23C2×C4×D411yescharacteristic
591C23#53811yescharacteristic
592C23#53911yescharacteristic
593C23#55211yescharacteristic
594C23#55311yescharacteristic
595C23#55411yescharacteristic
596C23#55511yescharacteristic
597Q8—22no—
598Q8—22no—
599Q8—41no—
600Q8—41no—
601D4—22no—
602D4—22no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—41no—
608D4—41no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
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—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4C2×C4×Q811yescharacteristic
665C2×C4C4×#4111yescharacteristic
666C2×C4#51811yescharacteristic
667C2×C4#51811yescharacteristic
668C2×C4#52011yescharacteristic
669C2×C4#52011yescharacteristic
670C2×C4#53411yescharacteristic
671C2×C4#53711yescharacteristic
672C2×C4#53811yescharacteristic
673C2×C4#53811yescharacteristic
674C2×C4#53911yescharacteristic
675C2×C4#53911yescharacteristic
676C2×C4#54111yescharacteristic
677C2×C4#54211yescharacteristic
678C2×C4#54311yescharacteristic
679C2×C4#54411yescharacteristic
680C2×C4#54611yescharacteristic
681C2×C4#54611yescharacteristic
682C2×C4D4×Q811yescharacteristic
683C2×C4#55011yescharacteristic
684C2×C4#55111yescharacteristic
685C2×C4#55111yescharacteristic
686C2×C4#55411yescharacteristic
687C2×C4#55411yescharacteristic
688C2×C4#56011yescharacteristic
689C2×C4#56011yescharacteristic
690C2×C4#56211yescharacteristic
691C2×C4#56211yescharacteristic
692C8—41no—
693C8—41no—
694C8—41no—
695C8—41no—
696C8—41no—
697C8—41no—
698C8—41no—
699C8—41no—
700C8—42no—
701C8—42no—
702C8—81no—
703C8—81no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—81no—
720C22—81no—
721C22—81no—
722C22—81no—
723C22#232511yescharacteristic
724C22#236011yescharacteristic
725C22#236311yescharacteristic
726C22#274411yescharacteristic
727C22#275011yescharacteristic
728C22#278411yescharacteristic
729C22#279011yescharacteristic
730C22#295511yescharacteristic
731C22#295511yescharacteristic
732C22#295711yescharacteristic
733C22#295711yescharacteristic
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—42no—
739C4—82no—
740C4—82no—
741C4—81no—
742C4—81no—
743C4—81no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4#232911yescharacteristic
748C4#233411yescharacteristic
749C4#236811yescharacteristic
750C4#237711yescharacteristic
751C4#269711yescharacteristic
752C4#270411yescharacteristic
753C4#270511yescharacteristic
754C4#273211yescharacteristic
755C2—21no—
756C2—21no—
757C2—21no—
758C2—21no—
759C2—82no—
760C2#1458611yescharacteristic
761C2#2220511yescharacteristic
762C2#2220611yescharacteristic
763C2#2298411yescharacteristic
764C2#2298411yescharacteristic
765C2#2311511yescharacteristic
766C2#2311511yescharacteristic
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. #11376C2semidirect
5. #11258C2semidirect
6. #11233C2semidirect
7. #11221C2semidirect
11. #10887C2semidirect
12. #10879C2semidirect
13. #10833C2semidirect
31. #2068C4semidirect
32. #2068C4semidirect
33. #2067C4semidirect
34. #2067C4semidirect
35. #2066C4semidirect
36. #2066C4semidirect
37. #2065C4semidirect
38. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
#210242
#208382
#113762
#112582
#112332
#112212
#111252
#109342
#109242
#108872
#108792
#108332
#107912