Finite Group Databasebeta

← #371740

Subgroups — #371740

Order 512.

1007 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21058C211yesmaximal, characteristic
2C2×#1922C211yesmaximal, characteristic, complemented
3#20839C211yesmaximal, characteristic
4C2×#1764C211yesmaximal, characteristic
5#20368C211yesmaximal, characteristic, complemented
6#11317C211yesmaximal, characteristic, complemented
7#11269C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11075C211yesmaximal, characteristic
10#10934C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic, complemented
12#10882C211yesmaximal, characteristic, complemented
13#10874C211yesmaximal, characteristic, complemented
14#10852C211yesmaximal, characteristic, complemented
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2297C2211yescharacteristic
29C4×#117C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1928C411yescharacteristic, complemented
32#1928C411yescharacteristic, complemented
33#1927C411yescharacteristic, complemented
34#1927C411yescharacteristic, complemented
35#1926C411yescharacteristic, complemented
36#1926C411yescharacteristic, complemented
37#1925C411yescharacteristic, complemented
38#1925C411yescharacteristic, complemented
39#1922—24no—
40#1851—21no—
41#1851—21no—
42#1850—21no—
43#1850—21no—
44#1764—24no—
45#1746—21no—
46#1746—21no—
47#1745—21no—
48#1744—21no—
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
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—22no—
74C2×#131—41no—
75C2×#131—41no—
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
93#427—22no—
94#427—22no—
95C2×#107C2×C411yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99#424—22no—
100#424—22no—
101#423—22no—
102#423—41no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—22no—
111C2×#104—22no—
112C2×#104—41no—
113#416—22no—
114#416—22no—
115#416—22no—
116#416—22no—
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#103C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125#408—22no—
126#408—41no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129#405—22no—
130#405—22no—
131#405—22no—
132#405D414yes—
133C2×#97—21no—
134C2×#97—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—22no—
138C2×C4×C8—41no—
139C2×C4×C8—41no—
140#391C2×C411yescharacteristic
141#391C2×C411yescharacteristic
142#390C2×C411yescharacteristic
143#390C2×C411yescharacteristic
144#386—21no—
145#386—21no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—41no—
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4C2×C411yescharacteristic
150C4×C4⋊C4C2×C411yescharacteristic
151C4×C4⋊C4C2×C411yescharacteristic
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156#376—21no—
157C2×#95—21no—
158C43—21no—
159#341—22no—
160#341—22no—
161#341—22no—
162#341—41no—
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—41no—
172#339—41no—
173#336—21no—
174#336—21no—
175#336—22no—
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×#41#4112yes—
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—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—21no—
238C2×M16—42no—
239C2×M16—42no—
240C2×M16—42no—
241C2×M16C2×D412yes—
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—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×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#117C2×D412yes—
294#117#4112yes—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—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×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#107—22no—
339#107—22no—
340#106—22no—
341#106—22no—
342C4⋊C8—42no—
343C4⋊C8—42no—
344C4⋊C8—42no—
345C4⋊C8—42no—
346#104—28no—
347#104—44no—
348#103—22no—
349#103—22no—
350#103—22no—
351#103—22no—
352#102—22no—
353#102—22no—
354#102—22no—
355#102—22no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358#97—24no—
359#97—24no—
360C4×C8—24no—
361C4×C8—24no—
362C4×C8—44no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
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×Q8#11712yes—
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×D4#11712yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
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×D411yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4111yescharacteristic
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—22no—
466M16—22no—
467M16—48no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16—41no—
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—22no—
499C2×C8—21no—
500C2×C8—22no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—22no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—48no—
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#12112yes—
530C2×C8#12112yes—
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—48no—
542C4⋊C4C4×D412yes—
543C4⋊C4C4×D412yes—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—42no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—42no—
552C22⋊C4—81no—
553C22⋊C4—81no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—22no—
565C42—22no—
566C42—42no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—42no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—81no—
579C42—81no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23C2×C4×D411yescharacteristic
592C23#52911yescharacteristic
593C23#53211yescharacteristic
594C23#55111yescharacteristic
595C23#55211yescharacteristic
596C23#55211yescharacteristic
597C23#55311yescharacteristic
598Q8—22no—
599Q8—22no—
600Q8—41no—
601Q8—41no—
602D4—22no—
603D4—22no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—41no—
609D4—41no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—24no—
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—21no—
623C2×C4—22no—
624C2×C4—22no—
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—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
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—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4#44312yes—
665C2×C4C2×#11711yescharacteristic
666C2×C4C2×C4×D411yescharacteristic
667C2×C4C4×#4111yescharacteristic
668C2×C4C4×#4111yescharacteristic
669C2×C4#52011yescharacteristic
670C2×C4#52011yescharacteristic
671C2×C4C2×#12111yescharacteristic
672C2×C4#52511yescharacteristic
673C2×C4#52511yescharacteristic
674C2×C4#52511yescharacteristic
675C2×C4#52911yescharacteristic
676C2×C4#52911yescharacteristic
677C2×C4#53511yescharacteristic
678C2×C4#53611yescharacteristic
679C2×C4#54211yescharacteristic
680C2×C4#54211yescharacteristic
681C2×C4#54611yescharacteristic
682C2×C4#54611yescharacteristic
683C2×C4#54711yescharacteristic
684C2×C4#54811yescharacteristic
685C2×C4#55111yescharacteristic
686C2×C4#55111yescharacteristic
687C2×C4#55111yescharacteristic
688C2×C4#55211yescharacteristic
689C2×C4#55211yescharacteristic
690C2×C4#55211yescharacteristic
691C2×C4#55311yescharacteristic
692C2×C4#55611yescharacteristic
693C8—22no—
694C8—22no—
695C8—41no—
696C8—41no—
697C8—42no—
698C8—42no—
699C8—41no—
700C8—41no—
701C8—42no—
702C8—41no—
703C8—41no—
704C8—81no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—21no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—81no—
721C22—81no—
722C22—81no—
723C22—81no—
724C22C4×#12111yescharacteristic
725C22#235211yescharacteristic
726C22#235911yescharacteristic
727C22#252711yescharacteristic
728C22#253011yescharacteristic
729C22#260311yescharacteristic
730C22#261111yescharacteristic
731C22C2×#44311yescharacteristic
732C22#295511yescharacteristic
733C22#295611yescharacteristic
734C22#295711yescharacteristic
735C4—22no—
736C4—44no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—82no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4—81no—
748C4#229011yescharacteristic
749C4#232611yescharacteristic
750C4#234311yescharacteristic
751C4#237311yescharacteristic
752C4#241111yescharacteristic
753C4#241511yescharacteristic
754C4#247011yescharacteristic
755C4#247711yescharacteristic
756C2—21no—
757C2—21no—
758C2—21no—
759C2—21no—
760C2—82no—
761C2#1408011yescharacteristic
762C2#2151711yescharacteristic
763C2#2151911yescharacteristic
764C2#2278311yescharacteristic
765C2#2278411yescharacteristic
766C2#2294811yescharacteristic
767C2#2294911yescharacteristic
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. C2×#1922C2semidirect
5. #20368C2semidirect
6. #11317C2semidirect
7. #11269C2semidirect
11. #10909C2semidirect
12. #10882C2semidirect
13. #10874C2semidirect
14. #10852C2semidirect
31. #1928C4semidirect
32. #1928C4semidirect
33. #1927C4semidirect
34. #1927C4semidirect
35. #1926C4semidirect
36. #1926C4semidirect
37. #1925C4semidirect
38. #1925C4semidirect

Maximal subgroups

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

subgroupindex
#210582
C2×#19222
#208392
C2×#17642
#203682
#113172
#112692
#111912
#110752
#109342
#109092
#108822
#108742
#108522
#107912