Finite Group Databasebeta

← #395256

Subgroups — #395256

Order 512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic, complemented
2C2×#2361C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic, complemented
4C2×#1852C212yesmaximal, complemented
5#15125C211yesmaximal, characteristic, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11123C212yesmaximal, complemented
8#11042C211yesmaximal, characteristic, complemented
9#10969C212yesmaximal
10#10907C212yesmaximal, complemented
11#10788C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#474—28no—
18C2×#474—24no—
19C22×#107C2212yes—
20C22×#107C2212yes—
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2212yes—
25C2×#393—21no—
26#2361—24no—
27#2361—24no—
28C2×#390C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#388—21no—
31C2×#386—21no—
32C2×#385—21no—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#2057—24no—
37#1852—28no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1442—21no—
46#1439—21no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C24×C4C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121Q814yescomplemented
67C2×C4×D4—28no—
68C2×C4×D4—24no—
69C2×C4×D4—22no—
70C2×C4×D4—42no—
71C2×C4×D4—44no—
72C2×C4×D4—44no—
73C2×C4×D4D414yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4D411yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#474—232no—
97C2×#107—24no—
98C2×#107—24no—
99C2×#107—24no—
100C2×#107—24no—
101C2×#107—42no—
102C2×#107D418yescomplemented
103C2×C4⋊C8—24no—
104C2×C4⋊C8—22no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—42no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—24no—
114#393—44no—
115#390—22no—
116#390—22no—
117#389—22no—
118#389D414yes—
119#388—28no—
120#386—28no—
121#385—28no—
122#380—24no—
123#380—42no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#336—24no—
142#336—22no—
143C25—21no—
144C25C2×D411yescharacteristic
145C22×Q8—44no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—82no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4C2×D412yes—
165C22×D4C2×Q812yes—
166C22×D4#4112yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—82no—
189C23×C4—82no—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×Q812yes—
196C23×C4#4112yes—
197C23×C4C2411yescharacteristic
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8C2×D414yes—
211C22×C8#4114yes—
212C4⋊Q8—28no—
213C4⋊Q8—48no—
214C4⋊Q8—44no—
215#121—28no—
216#121—28no—
217#121—28no—
218#121—28no—
219C4×D4—28no—
220C4×D4—44no—
221C4×D4—48no—
222C4×D4—432no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×Q812yes—
258C2×C4⋊C4#4112yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4—82no—
277C2×C22⋊C4D814yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289#107—216no—
290#107—48no—
291C4⋊C8—28no—
292#102—216no—
293#102—48no—
294C22⋊C8—216no—
295#95—24no—
296#95—24no—
297#95—22no—
298#95—22no—
299#95—44no—
300#95—42no—
301#95—44no—
302#95—42no—
303#95—82no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—42no—
316C24—42no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—42no—
322C24—42no—
323C24C22⋊Q812yes—
324C24C2×D812yes—
325C24#13712yes—
326C24C22×D411yescharacteristic
327C2×Q8—48no—
328C2×Q8—88no—
329C2×Q8—88no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×D4—216no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4C22⋊Q814yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—48no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—84no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—84no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4#12012yes—
436C22×C4#12012yes—
437C22×C4C22⋊Q812yes—
438C22×C4C2×D812yes—
439C22×C4C2×D812yes—
440C22×C4AGL(1,ℤ/8)12yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q811yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+412yes—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4#12014yes—
485C4⋊C4C22⋊Q814yes—
486C22⋊C4—216no—
487C22⋊C4—28no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—48no—
497C42—41no—
498C42—41no—
499C42—42no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23#47414yescomplemented
546C23C2×#12011yescharacteristic
547C23C2×C22⋊Q811yescharacteristic
548C23#52511yescharacteristic
549C23D4212yes—
550C23#54712yes—
551C23C22×D811yescharacteristic
552C23C2×AGL(1,ℤ/8)11yescharacteristic
553C23C2×#13711yescharacteristic
554C23#57511yescharacteristic
555C23#57612yes—
556C23#57812yes—
557Q8—816no—
558D4—24no—
559D4—24no—
560D4—432no—
561D4—42no—
562D4—42no—
563D4—44no—
564D4—44no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—816no—
614C2×C4—816no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4#44714yes—
629C2×C4C2×#12011yescharacteristic
630C2×C4C2×C22⋊Q811yescharacteristic
631C2×C4C2×C22⋊Q811yescharacteristic
632C2×C4#52511yescharacteristic
633C2×C4#52512yes—
634C2×C4#52511yescharacteristic
635C2×C4#53211yescharacteristic
636C2×C4#53212yes—
637C2×C4#53411yescharacteristic
638C2×C4#53611yescharacteristic
639C2×C4#53912yes—
640C2×C4#54212yes—
641C2×C4D4212yes—
642C2×C4#54812yes—
643C2×C4#54812yes—
644C2×C4D4×Q812yes—
645C2×C4#55112yes—
646C2×C4#55412yes—
647C8—44no—
648C8—44no—
649C22—24no—
650C22—28no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22#240311yescharacteristic
672C22#243211yescharacteristic
673C22#244912yes—
674C22#246112yes—
675C22#284811yescharacteristic
676C22C2×#44711yescharacteristic
677C22#300711yescharacteristic
678C22#300811yescharacteristic
679C22#301111yescharacteristic
680C22C2×#47411yescharacteristic
681C22#307711yescharacteristic
682C22#308312yes—
683C22#310011yescharacteristic
684C22#311011yescharacteristic
685C22#312412yes—
686C22#326412yes—
687C22#327012yes—
688C22#327712yes—
689C22D4×D812yes—
690C22#333412yes—
691C22#334612yes—
692C22#334712yes—
693C22#335312yes—
694C4—24no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4—82no—
701C4—84no—
702C4—81no—
703C4—162no—
704C4#238811yescharacteristic
705C4#240911yescharacteristic
706C4#245212yes—
707C4#246312yes—
708C4#256411yescharacteristic
709C4#263811yescharacteristic
710C2—44no—
711C2—84no—
712C2#1472311yescharacteristic
713C2#2384811yescharacteristic
714C2#2386711yescharacteristic
715C2#2570311yescharacteristic
716C2#2571311yescharacteristic
717C2#2633612yes—
718C2#2635212yes—
719C2#2678412yes—
720C2#2680312yes—
721C2#3295311yescharacteristic
722C2#3296311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#474C2semidirect
3. C2×#2057C2semidirect
4. C2×#1852C2semidirect
5. #15125C2semidirect
6. #11227C2semidirect
7. #11123C2semidirect
8. #11042C2semidirect
10. #10907C2semidirect
11. #10788C2semidirect
66. C2×#121Q8semidirect
102. C2×#107D4semidirect
545. C23#474semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#23612
C2×#20572
C2×#18522
#151252
#112272
#111232
#110422
#109692
#109072
#107882