Finite Group Databasebeta

← #363281

Subgroups — #363281

Order 512.

4006 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#550C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic, complemented
3C2×#1982C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic, complemented
7#13926C211yesmaximal, characteristic, complemented
8#11008C214yesmaximal, complemented
9#10788C214yesmaximal, complemented
10C23×#41—21no—
11C2×#550—24no—
12C2×#550—24no—
13C2×#550—24no—
14C2×#550—24no—
15C2×#550—24no—
16C2×#550C418yescomplemented
17C22×#127C2212yes—
18C22×#127C2211yescharacteristic
19C22×#121—21no—
20C22×#121—22no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2212yes—
23C22×C4×D4C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2214yes—
27C22×#102C2214yes—
28C22×#102C2212yes—
29C22×C4×C8C2211yescharacteristic
30C2×#391—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1984—24no—
34#1982—24no—
35#1957—24no—
36#1802—24no—
37C4×#102—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2214yes—
41#1441—22no—
42C4⋊#95—21no—
43C4⋊#95C2214yes—
44C22×#41—28no—
45C22×#41—28no—
46C22×#41—28no—
47C22×#41—44no—
48C22×#41—44no—
49C22×#41—44no—
50C22×#41—44no—
51C23×Q8D411yescharacteristic
52C23×D4—22no—
53C23×D4—21no—
54C23×D4—22no—
55C23×D4—21no—
56C23×D4C2312yes—
57C23×D4C2311yescharacteristic
58C24×C4—22no—
59C24×C4—21no—
60C23×C8C2312yes—
61C23×C8C2312yes—
62#550—232no—
63#550—232no—
64#550—232no—
65#550—232no—
66C2×#127—24no—
67C2×#127—24no—
68C2×#127—24no—
69C2×#127—24no—
70C2×#127—24no—
71C2×#127—22no—
72C2×#127—22no—
73C2×#127—22no—
74C2×#127C2×C418yes—
75C2×#127C2×C414yes—
76C2×#127D414yes—
77C2×#127D414yes—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—216no—
81C2×#121—216no—
82C2×#121—28no—
83C2×#121—216no—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×Q8C2×C414yes—
89C2×C4×Q8D414yes—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4C2×C418yes—
99C2×C4×D4C2×C414yes—
100C2×C4×D4D414yes—
101C2×C4×D4D414yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D412yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—22no—
110C22×C22⋊C4—21no—
111C22×C42—21no—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×#102—24no—
118C2×#102—24no—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—28no—
122C2×#102—28no—
123C2×#102—28no—
124C2×#102—28no—
125C2×#102—44no—
126C2×#102—44no—
127C2×#102—44no—
128C2×C4×C8—22no—
129C2×C4×C8—22no—
130C2×C4×C8—22no—
131#391—44no—
132#384—22no—
133#384—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C2×#95—24no—
137C2×#95—21no—
138C2×#95—24no—
139#340—24no—
140#340—24no—
141#336—28no—
142C25—22no—
143C25—21no—
144C2×#41—264no—
145C2×#41—432no—
146C2×#41—432no—
147C2×#41—432no—
148C2×#41—432no—
149C2×#41—432no—
150C2×#41—432no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8C22⋊C414yes—
156C22×Q8D814yes—
157C22×Q8SD1614yes—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—28no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—21no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—82no—
193C22×D4—82no—
194C22×D4—84no—
195C22×D4—84no—
196C22×D4C22×C414yes—
197C22×D4C22×C412yes—
198C22×D4C2×D414yes—
199C22×D4C2×D414yes—
200C22×D4C2×D412yes—
201C22×D4C2×D412yes—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—21no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D412yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2411yescharacteristic
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—24no—
234C22×C8—24no—
235C22×C8#4114yescomplemented
236#127—216no—
237#127—216no—
238#127—216no—
239#127—216no—
240#127—28no—
241#127—28no—
242#127—28no—
243#127C22⋊C4116yes—
244#121—264no—
245#121—2128no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—28no—
249C4×Q8—28no—
250C4×D4—216no—
251C4×D4—216no—
252C4×D4—216no—
253C4×D4—216no—
254C4×D4—28no—
255C4×D4—28no—
256C4×D4—28no—
257C4×D4C22⋊C4116yes—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4C22⋊C414yes—
280C2×C4⋊C4C22⋊C418yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4#4114yescomplemented
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C22×C414yes—
299C2×C42C22×C412yes—
300C2×C42C2×D414yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C4⋊C8—44no—
307#102—216no—
308#102—416no—
309#102—416no—
310#102—48no—
311C4×C8—44no—
312#95—42no—
313#95—48no—
314#95—84no—
315C24—24no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—42no—
322C24—44no—
323C24—42no—
324C24—44no—
325C24—42no—
326C24—41no—
327C24—42no—
328C24—41no—
329C24—42no—
330C24—41no—
331C24—42no—
332C24—41no—
333C24—44no—
334C24—42no—
335C24—44no—
336C24—42no—
337C24C22×D411yescharacteristic
338#41—4256no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8#102116yescomplemented
346C2×D4—232no—
347C2×D4—232no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—28no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—44no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—416no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—88no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4—816no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—816no—
404C2×D4C2×C22⋊C418yes—
405C2×D4C2×C22⋊C414yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—232no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—216no—
421C22×C4—24no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4C2×C22⋊C414yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C414yes—
452C22×C4C2×C22⋊C412yes—
453C22×C4#12012yes—
454C22×C4#12012yes—
455C22×C4#12012yes—
456C22×C4#12012yes—
457C22×C4C2×D814yes—
458C22×C4C2×D812yes—
459C22×C4C2×SD1614yes—
460C22×C4C2×SD1612yes—
461C22×C4AGL(1,ℤ/8)12yes—
462C22×C4AGL(1,ℤ/8)12yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—44no—
484C2×C8—44no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—216no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4C2×C22⋊C414yes—
500C22⋊C4—232no—
501C22⋊C4—216no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—84no—
507C42C2×C22⋊C414yes—
508C42C2×C22⋊C414yes—
509C42#12014yes—
510C42#12014yes—
511C23—28no—
512C23—24no—
513C23—44no—
514C23—48no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—48no—
520C23—44no—
521C23—44no—
522C23—48no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—42no—
539C23—48no—
540C23—44no—
541C23—82no—
542C23—84no—
543C23—82no—
544C23—84no—
545C23—84no—
546C23—82no—
547C23—84no—
548C23—82no—
549C23—82no—
550C23—84no—
551C23—82no—
552C23—84no—
553C23—84no—
554C23—82no—
555C23—84no—
556C23—82no—
557C23C22×C22⋊C411yescharacteristic
558C23C2×#12011yescharacteristic
559C23C2×#12011yescharacteristic
560C23#54812yes—
561C23#55012yes—
562C23C22×D811yescharacteristic
563C23C22×SD1611yescharacteristic
564C23C2×AGL(1,ℤ/8)11yescharacteristic
565C23C2×AGL(1,ℤ/8)11yescharacteristic
566C23#57612yes—
567C23#57712yes—
568Q8—232no—
569D4—24no—
570D4—216no—
571D4—24no—
572D4—464no—
573D4—44no—
574D4—44no—
575D4—44no—
576D4—48no—
577D4—42no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—432no—
583D4—816no—
584D4—832no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—464no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—432no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4#37914yes—
622C2×C4#37914yes—
623C2×C4C2×#10218yes—
624C2×C4C2×#10214yes—
625C2×C4#41814yes—
626C2×C4#44714yes—
627C2×C4#44714yes—
628C2×C4#45014yes—
629C2×C4#45014yes—
630C2×C4C22×C22⋊C411yescharacteristic
631C2×C4C22×C22⋊C411yescharacteristic
632C2×C4C4×#4112yes—
633C2×C4#52012yes—
634C2×C4C2×#12011yescharacteristic
635C2×C4C2×#12011yescharacteristic
636C2×C4#52512yes—
637C2×C4#53211yescharacteristic
638C2×C4#53211yescharacteristic
639C2×C4#53511yescharacteristic
640C2×C4#53611yescharacteristic
641C2×C4#53712yes—
642C2×C4#54612yes—
643C2×C4#54612yes—
644C2×C4#54712yes—
645C2×C4#54712yes—
646C2×C4#54812yes—
647C2×C4#55012yes—
648C8—44no—
649C8—82no—
650C22—44no—
651C22—48no—
652C22—48no—
653C22—44no—
654C22—48no—
655C22—44no—
656C22—84no—
657C22—88no—
658C22—88no—
659C22—84no—
660C22—82no—
661C22—84no—
662C22—82no—
663C22—84no—
664C22—82no—
665C22—84no—
666C22—82no—
667C22—84no—
668C22—84no—
669C22—88no—
670C22—84no—
671C22—88no—
672C22C2×#37911yescharacteristic
673C22#234112yes—
674C22#240712yes—
675C22#243111yescharacteristic
676C22#243211yescharacteristic
677C22C22×#10211yescharacteristic
678C22C2×#41811yescharacteristic
679C22#289712yes—
680C22C2×#44711yescharacteristic
681C22C2×#44711yescharacteristic
682C22C2×#45011yescharacteristic
683C22C2×#45011yescharacteristic
684C22#300811yescharacteristic
685C22#300911yescharacteristic
686C22#301212yes—
687C22#335812yes—
688C22#336212yes—
689C22#336612yes—
690C22#336812yes—
691C22#339112yes—
692C22#339212yes—
693C22#339712yes—
694C22#339912yes—
695C4—28no—
696C4—24no—
697C4—44no—
698C4—44no—
699C4—84no—
700C4—162no—
701C4#178618yes—
702C4#229711yescharacteristic
703C4#231111yescharacteristic
704C4#233912yes—
705C4#240712yes—
706C4#243111yescharacteristic
707C4#243211yescharacteristic
708C2—84no—
709C2—88no—
710C2#1393711yescharacteristic
711C2C2×#178611yescharacteristic
712C2#2060211yescharacteristic
713C2#2254012yes—
714C2#2254512yes—
715C2#2418412yes—
716C2#2418812yes—
717C2#2562711yescharacteristic
718C2#2563111yescharacteristic
719C2#2564511yescharacteristic
720C2#2564911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1984C2semidirect
3. C2×#1982C2semidirect
4. C2×#1957C2semidirect
5. C2×#1802C2semidirect
6. C2×C4×#102C2semidirect
7. #13926C2semidirect
8. #11008C2semidirect
9. #10788C2semidirect
16. C2×#550C4semidirect
235. C22×C8#41semidirect
284. C2×C4⋊C4#41semidirect
345. C2×Q8#102semidirect

Maximal subgroups

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

subgroupindex
C22×#5502
C2×#19842
C2×#19822
C2×#19572
C2×#18022
C2×C4×#1022
#139262
#110082
#107882