Finite Group Databasebeta

← #376558

Subgroups — #376558

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 780 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23479C211yesmaximal, characteristic
2#21053C211yesmaximal, characteristic, complemented
3C2×#1896C211yesmaximal, characteristic
4#20899C211yesmaximal, characteristic
5C2×#1866C211yesmaximal, characteristic, complemented
6C2×#1765C211yesmaximal, characteristic, complemented
7C2×#1749C211yesmaximal, characteristic, complemented
8#11317C211yesmaximal, characteristic, complemented
9#11191C211yesmaximal, characteristic, complemented
10#11097C211yesmaximal, characteristic, complemented
11#10930C211yesmaximal, characteristic, complemented
12#10909C211yesmaximal, characteristic, complemented
13#10883C211yesmaximal, characteristic, complemented
14#10860C211yesmaximal, characteristic, complemented
15#10719C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2280C2211yescharacteristic
35#2036C412yescomplemented
36#2035C412yescomplemented
37#2034C412yescomplemented
38#2033C412yescomplemented
39#1924—22no—
40#1923—22no—
41#1896—24no—
42#1870—22no—
43#1869—22no—
44#1866—24no—
45#1765—24no—
46#1749—24no—
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
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C2×#131—22no—
67C2×#131—22no—
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—41no—
71C2×#131—41no—
72C22×M16—21no—
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C23×C8—21no—
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82#525—22no—
83#525—22no—
84#525—41no—
85#525—41no—
86C2×#123—21no—
87C2×C22⋊Q8C2×C412yes—
88C2×#121C2×C412yes—
89C2×#120—21no—
90C2×C4×D4—22no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C42—21no—
95C22×C42—22no—
96C22×C42—22no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42—41no—
100C22×C42—41no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C2×#108—22no—
104C2×#108—22no—
105C2×#108—41no—
106#427—41no—
107#427—41no—
108C2×#107C2×C412yescomplemented
109C2×#106C2×C412yescomplemented
110#424—22no—
111#424—22no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—22no—
118C2×#104—41no—
119C2×#104D414yes—
120#416—22no—
121#416—22no—
122#416—22no—
123#416—22no—
124C2×#103C2×C411yescharacteristic, complemented
125C2×#103C2×C412yescomplemented
126C2×#103C2×C411yescharacteristic, complemented
127C2×#102C2×C411yescharacteristic, complemented
128C2×#102C2×C412yescomplemented
129C2×#102C2×C411yescharacteristic, complemented
130#408—22no—
131#408—41no—
132#407—22no—
133#407—22no—
134#407—41no—
135#407—41no—
136C2×C22⋊C8—22no—
137C2×C22⋊C8—22no—
138C2×C22⋊C8—22no—
139C2×C22⋊C8—22no—
140C2×C22⋊C8—22no—
141C2×C22⋊C8—22no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8C2×C412yes—
144C2×C22⋊C8C2×C412yes—
145C2×C22⋊C8D412yes—
146C2×C22⋊C8D412yes—
147#405—22no—
148#405—41no—
149C2×#97—22no—
150C2×#97—22no—
151C2×#97—21no—
152C2×#97—22no—
153C2×#97—22no—
154C2×#97—41no—
155C2×#97—41no—
156C2×C4×C8—21no—
157#381—21no—
158#380C2×C412yes—
159#380C2×C412yes—
160#379—21no—
161C4×C22⋊C4—22no—
162#341—22no—
163#341—22no—
164#341—41no—
165#340C2×C412yescomplemented
166#340C2×C412yescomplemented
167#339—22no—
168#339—22no—
169#339—22no—
170#339—22no—
171#339—22no—
172#339—41no—
173#336—21no—
174#336—22no—
175#336—21no—
176#336—22no—
177#336—22no—
178#336—22no—
179#336D414yescomplemented
180C25#4111yescharacteristic
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41C2×D412yes—
192C2×#41#4112yes—
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4—22no—
196C22×D4C22×C411yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—24no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C22×C412yes—
218C23×C4C2×D412yes—
219C23×C4#4111yescharacteristic
220C23×C4C2411yescharacteristic
221#131—28no—
222#131—44no—
223#131—44no—
224#131—44no—
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—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—42no—
248C2×M16—42no—
249C2×M16—42no—
250C2×M16C2×D412yes—
251C2×M16#4112yes—
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—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—21no—
276C22×C8—42no—
277C22×C8—42no—
278C22×C8—42no—
279C22×C8C22×C412yes—
280C22×C8C22×C411yescharacteristic
281C22×C8C22×C411yescharacteristic
282C22×C8C2×D412yes—
283C22×C8C2×D412yes—
284C22×C8C2×D412yes—
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8#4112yes—
288C22×C8#4112yes—
289#123—42no—
290#123—42no—
291C22⋊Q8—24no—
292C22⋊Q8—42no—
293#121—24no—
294#121—42no—
295#120—42no—
296#120—42no—
297C4×D4—44no—
298C4×D4—44no—
299#117—21no—
300#117—21no—
301#117—21no—
302#117—21no—
303#117—21no—
304#117—21no—
305#117—21no—
306#117—21no—
307#117—21no—
308#117—21no—
309#117—21no—
310#117—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C22×C412yes—
313C2×C4⋊C4C22×C411yescharacteristic
314C2×C4⋊C4C22×C411yescharacteristic
315C2×C4⋊C4C22×C412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—44no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42#4112yes—
344C2×C42#4112yes—
345#108—44no—
346#107—42no—
347#106—42no—
348C4⋊C8—42no—
349C4⋊C8—42no—
350#104—28no—
351#104—44no—
352#103—24no—
353#103—22no—
354#103—22no—
355#102—24no—
356#102—22no—
357#102—22no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362C22⋊C8—44no—
363C22⋊C8—44no—
364C22⋊C8—44no—
365#97—44no—
366#97—44no—
367#97—42no—
368C4×C8—42no—
369#95—22no—
370#95—22no—
371C24—21no—
372C24—21no—
373C24—24no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—42no—
380C24C4×D412yes—
381C24C2×#4111yescharacteristic
382#41—22no—
383#41—22no—
384#41—22no—
385#41—22no—
386#41—22no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8C4×D412yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—82no—
418C2×D4C4×D412yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4C4×D412yes—
480C22×C4C23×C411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C22×C4C2×#4111yescharacteristic
486C22×C4C2×#4111yescharacteristic
487C22×C4C2×#4111yescharacteristic
488C22×C4C2×#4111yescharacteristic
489C22×C4C2×#4111yescharacteristic
490C22×C4H(5,𝔽2)11yescharacteristic
491C22×C4H(5,𝔽2)11yescharacteristic
492C22×C4H(5,𝔽2)11yescharacteristic
493C22×C42−1+411yescharacteristic
494M16—22no—
495M16—22no—
496M16—22no—
497M16—22no—
498M16—48no—
499M16—41no—
500M16—41no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505M16—41no—
506M16—41no—
507M16—41no—
508M16—41no—
509M16—41no—
510M16—41no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—48no—
536C2×C8—41no—
537C2×C8—41no—
538C2×C8—41no—
539C2×C8—41no—
540C2×C8—41no—
541C2×C8—41no—
542C2×C8—41no—
543C2×C8—41no—
544C2×C8—41no—
545C2×C8—41no—
546C2×C8—41no—
547C2×C8—41no—
548C2×C8—41no—
549C2×C8—41no—
550C2×C8—41no—
551C2×C8—41no—
552C2×C8—41no—
553C2×C8—41no—
554C2×C8C4×D412yes—
555C2×C8C4×D412yes—
556C2×C8#12112yes—
557C2×C8#12112yes—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—41no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C22⋊C4—41no—
573C22⋊C4—41no—
574C22⋊C4—41no—
575C22⋊C4—44no—
576C22⋊C4—82no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—48no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42#12314yes—
592C23—22no—
593C23—22no—
594C23—24no—
595C23—24no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—42no—
608C23—42no—
609C23—41no—
610C23—41no—
611C23—41no—
612C23—41no—
613C23—41no—
614C23—42no—
615C23—42no—
616C23—41no—
617C23#44414yes—
618C23C2×C4×D411yescharacteristic
619C23#52511yescharacteristic
620C23#52911yescharacteristic
621C23#54611yescharacteristic
622C23#55111yescharacteristic
623C23#55111yescharacteristic
624C23#55211yescharacteristic
625Q8—22no—
626Q8—22no—
627Q8—41no—
628Q8—41no—
629D4—22no—
630D4—22no—
631D4—42no—
632D4—42no—
633D4—42no—
634D4—42no—
635D4—41no—
636D4—41no—
637D4—88no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—22no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—432no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4C2×C4×D411yescharacteristic
682C2×C4C2×C4×D411yescharacteristic
683C2×C4C4×#4111yescharacteristic
684C2×C4C4×#4111yescharacteristic
685C2×C4#51811yescharacteristic
686C2×C4#52011yescharacteristic
687C2×C4C2×#12111yescharacteristic
688C2×C4C2×#12311yescharacteristic
689C2×C4#52511yescharacteristic
690C2×C4#52911yescharacteristic
691C2×C4#52911yescharacteristic
692C2×C4#53211yescharacteristic
693C2×C4#53811yescharacteristic
694C2×C4#53811yescharacteristic
695C2×C4#54211yescharacteristic
696C2×C4#54211yescharacteristic
697C2×C4D4211yescharacteristic
698C2×C4D4211yescharacteristic
699C2×C4#54611yescharacteristic
700C2×C4#54611yescharacteristic
701C2×C4#54611yescharacteristic
702C2×C4#54611yescharacteristic
703C2×C4#54711yescharacteristic
704C2×C4#54711yescharacteristic
705C2×C4#54811yescharacteristic
706C2×C4#55311yescharacteristic
707C2×C4#55311yescharacteristic
708C2×C4#55311yescharacteristic
709C8—22no—
710C8—22no—
711C8—22no—
712C8—22no—
713C8—42no—
714C8—42no—
715C8—42no—
716C8—42no—
717C8—81no—
718C8—81no—
719C22—21no—
720C22—21no—
721C22—21no—
722C22—21no—
723C22—22no—
724C22—22no—
725C22—22no—
726C22—22no—
727C22—24no—
728C22—24no—
729C22—24no—
730C22—44no—
731C22—44no—
732C22—44no—
733C22—44no—
734C22—42no—
735C22—42no—
736C22—42no—
737C22—42no—
738C22—42no—
739C22—44no—
740C22—81no—
741C22—81no—
742C22—82no—
743C22C4×#12111yescharacteristic
744C22#235811yescharacteristic
745C22#235911yescharacteristic
746C22#241811yescharacteristic
747C22#247011yescharacteristic
748C22#247711yescharacteristic
749C22#257611yescharacteristic
750C22C2×#44412yes—
751C22#295612yes—
752C4—44no—
753C4—44no—
754C4—42no—
755C4—42no—
756C4—81no—
757C4—81no—
758C4—82no—
759C4—82no—
760C4C4×#12311yescharacteristic
761C4#233711yescharacteristic
762C4#236111yescharacteristic
763C4#236411yescharacteristic
764C4#242211yescharacteristic
765C4#242411yescharacteristic
766C4#242711yescharacteristic
767C4#255911yescharacteristic
768C2—21no—
769C2—21no—
770C2—21no—
771C2—21no—
772C2—44no—
773C2—82no—
774C2#1418211yescharacteristic
775C2#2154611yescharacteristic
776C2#2154911yescharacteristic
777C2#2288612yes—
778C2#2295012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21053C2semidirect
5. C2×#1866C2semidirect
6. C2×#1765C2semidirect
7. C2×#1749C2semidirect
8. #11317C2semidirect
9. #11191C2semidirect
10. #11097C2semidirect
11. #10930C2semidirect
12. #10909C2semidirect
13. #10883C2semidirect
14. #10860C2semidirect
15. #10719C2semidirect
35. #2036C4semidirect
36. #2035C4semidirect
37. #2034C4semidirect
38. #2033C4semidirect
108. C2×#107C2×C4semidirect
109. C2×#106C2×C4semidirect
124. C2×#103C2×C4semidirect
125. C2×#103C2×C4semidirect
126. C2×#103C2×C4semidirect
127. C2×#102C2×C4semidirect
128. C2×#102C2×C4semidirect
129. C2×#102C2×C4semidirect
165. #340C2×C4semidirect
166. #340C2×C4semidirect
179. #336D4semidirect

Maximal subgroups

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

subgroupindex
#234792
#210532
C2×#18962
#208992
C2×#18662
C2×#17652
C2×#17492
#113172
#111912
#110972
#109302
#109092
#108832
#108602
#107192