Finite Group Databasebeta

← #368158

Subgroups — #368158

Order 512.

2116 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2361C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1874C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic, complemented
5#14130C211yesmaximal, characteristic, complemented
6#11293C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#10885C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21#2361—24no—
22#2361—24no—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#388—21no—
27C2×#386—21no—
28C2×#385—21no—
29C2×#385—21no—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1893C418yescomplemented
34#1874—24no—
35C2×#340—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1442—21no—
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C23×D4—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—42no—
57C23×C8—42no—
58C23×C8C2×C414yescomplemented
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121C2×C414yes—
65C2×#121D414yes—
66C2×#121Q814yes—
67C2×C4×D4—28no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—24no—
71C2×C4×D4—42no—
72C2×C4×D4—44no—
73C2×C4×D4—44no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4D414yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—42no—
104C2×#102—42no—
105C2×#102—42no—
106C2×#102C2×C418yescomplemented
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110#390—22no—
111#390—22no—
112#389—22no—
113#389—22no—
114#389—24no—
115#389—42no—
116#388—28no—
117#386—28no—
118#385—28no—
119#385—44no—
120#380—22no—
121#380C2×C414yescomplemented
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—42no—
139#340C2×C418yescomplemented
140#336—42no—
141#336—42no—
142C25—21no—
143C25C2×D411yescharacteristic
144C22×D4—24no—
145C22×D4—24no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4C22×C412yes—
159C22×D4C2×D412yes—
160C22×D4C2×Q812yes—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—82no—
183C23×C4—82no—
184C23×C4—82no—
185C23×C4C22×C412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×Q812yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C22×C8—28no—
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×C8—22no—
211C22×C8—48no—
212C22×C8—44no—
213C22×C8—44no—
214C22×C8—48no—
215C22×C8C22×C414yes—
216#121—28no—
217#121—28no—
218#121—28no—
219#121C4⋊C4116yescomplemented
220C4×D4—432no—
221C4×D4—48no—
222C4×D4—44no—
223C4×D4—44no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×Q812yes—
264C2×C4⋊C4#4114yes—
265C2×C4⋊C4#4114yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—82no—
284C2×C22⋊C4SD1614yes—
285C2×C42—24no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—42no—
295C2×C42—42no—
296#102—48no—
297#102—48no—
298C22⋊C8—44no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—22no—
304#95—44no—
305#95—44no—
306#95—82no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—42no—
315C24—42no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—42no—
321C24—42no—
322C24C4×D412yes—
323C24#12012yes—
324C24C22⋊Q812yes—
325C24C2×SD1612yes—
326C24#13512yes—
327C24AGL(1,ℤ/8)12yes—
328C24AGL(1,ℤ/8)12yes—
329C24C22×D411yescharacteristic
330C2×D4—216no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4C2×C4⋊C414yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—88no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C2×C4⋊C414yes—
436C22×C4C4×D414yes—
437C22×C4C4×D412yes—
438C22×C4#12012yes—
439C22×C4C22⋊Q814yes—
440C22×C4C22⋊Q812yes—
441C22×C4#12314yes—
442C22×C4C2×SD1612yes—
443C22×C4C2×SD1612yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q811yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4111yescharacteristic
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—416no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—88no—
489C4⋊C4C2×C4⋊C414yes—
490C4⋊C4#12314yes—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—84no—
498C42—48no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—42no—
504C42—42no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23#38514yes—
545C23C4×SD1614yes—
546C23#44214yes—
547C23#44914yes—
548C23#45014yes—
549C23#47614yes—
550C23#47814yes—
551C23C2×C4×D411yescharacteristic
552C23C2×#12011yescharacteristic
553C23C2×C22⋊Q811yescharacteristic
554C23#54612yes—
555C23#54612yes—
556C23C22×SD1611yescharacteristic
557C23C2×#13511yescharacteristic
558C23C2×AGL(1,ℤ/8)11yescharacteristic
559C23C2×AGL(1,ℤ/8)11yescharacteristic
560C23#57612yes—
561C23#57812yes—
562D4—28no—
563D4—432no—
564D4—42no—
565D4—42no—
566D4—44no—
567D4—42no—
568D4—42no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—816no—
621C2×C4—816no—
622C2×C4—816no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4#38514yes—
634C2×C4C22×C4⋊C411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C4×#4112yes—
637C2×C4#51812yes—
638C2×C4C2×C22⋊Q811yescharacteristic
639C2×C4C2×#12311yescharacteristic
640C2×C4C2×#12311yescharacteristic
641C2×C4#52511yescharacteristic
642C2×C4#52912yes—
643C2×C4#53411yescharacteristic
644C2×C4#53511yescharacteristic
645C2×C4#53812yes—
646C2×C4#54612yes—
647C2×C4#54612yes—
648C2×C4#55312yes—
649C2×C4#55312yes—
650C2×C4#55412yes—
651C2×C4#55412yes—
652C8—44no—
653C8—82no—
654C22—28no—
655C22—28no—
656C22—28no—
657C22—28no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22#186418yes—
676C22C2×#38511yescharacteristic
677C22#236412yes—
678C22#240411yescharacteristic
679C22#240511yescharacteristic
680C22#245012yes—
681C22C2×C4×SD1611yescharacteristic
682C22C2×#44211yescharacteristic
683C22#295112yes—
684C22C2×#45011yescharacteristic
685C22C2×#44911yescharacteristic
686C22#300811yescharacteristic
687C22#301111yescharacteristic
688C22C2×#47611yescharacteristic
689C22C2×#47811yescharacteristic
690C22#308312yes—
691C22#329512yes—
692C22#329712yes—
693C22#329812yes—
694C22#330012yes—
695C22#330812yes—
696C22#330812yes—
697C22#331812yes—
698C22#331812yes—
699C4—24no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—84no—
705C4—81no—
706C4—84no—
707C4—82no—
708C4—81no—
709C4C4×#12311yescharacteristic
710C4#232111yescharacteristic
711C4#234912yes—
712C4#244412yes—
713C4#248711yescharacteristic
714C4#257911yescharacteristic
715C2—28no—
716C2—84no—
717C2#1401611yescharacteristic
718C2C2×#186411yescharacteristic
719C2#2088111yescharacteristic
720C2#2312612yes—
721C2#2313212yes—
722C2#2396111yescharacteristic
723C2#2396311yescharacteristic
724C2#2403311yescharacteristic
725C2#2403511yescharacteristic
726C2#2641012yes—
727C2#2642012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×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
4. C22×#340C2semidirect
5. #14130C2semidirect
6. #11293C2semidirect
7. #11066C2semidirect
8. #10885C2semidirect
9. #10788C2semidirect
10. #10730C2semidirect
11. #10714C2semidirect
33. #1893C4semidirect
58. C23×C8C2×C4semidirect
106. C2×#102C2×C4semidirect
121. #380C2×C4semidirect
139. #340C2×C4semidirect
219. #121C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23612
C2×#18932
C2×#18742
C22×#3402
#141302
#112932
#110662
#108852
#107882
#107302
#107142