Finite Group Databasebeta

← #368535

Subgroups — #368535

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1872C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×C4×C22⋊C8C211yesmaximal, characteristic
4#14702C211yesmaximal, characteristic
5C2×C4⋊#95C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#10926C212yesmaximal, complemented
9#10820C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10730C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C4×C8C2212yes—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C22×#95—21no—
25C22×#95—21no—
26#1872—24no—
27#1872C418yescomplemented
28C4×C22⋊C8—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36#1436—21no—
37#1436—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C23×Q8C2311yescharacteristic
47C24×C4—21no—
48C24×C4—21no—
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C414yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4C2×C414yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—24no—
81C22×C42—42no—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103C2×C418yescomplemented
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2×C414yes—
97C2×C22⋊C8D414yes—
98C2×C4×C8—24no—
99C2×C4×C8—24no—
100C2×C4×C8—42no—
101#391—24no—
102#391—42no—
103#380—22no—
104#380—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—44no—
119C2×#95—44no—
120#340—24no—
121#340—24no—
122#340—42no—
123#340—42no—
124#340—42no—
125#340C2×C418yescomplemented
126#336—24no—
127C25C2×D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C22×C412yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4SD1614yes—
175C23×C4Q1614yes—
176C23×C4C22×C412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C22×C414yes—
201C22×C8C2×D414yes—
202C22⋊Q8—28no—
203C22⋊Q8—28no—
204C22⋊Q8—44no—
205C22⋊Q8—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4#4114yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—48no—
263C2×C42—48no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42#4114yes—
267#103—48no—
268#103—48no—
269C22⋊C8—28no—
270C22⋊C8—28no—
271C4×C8—48no—
272#95—42no—
273#95—42no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—44no—
284#95—42no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24C4×D412yes—
294C24#12112yes—
295C24#12412yes—
296C24C2×SD1612yes—
297C24C2×Q1612yes—
298C24#13512yes—
299C24#13512yes—
300C24C22×D411yescharacteristic
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4#11714yes—
424C22×C4C4×D414yes—
425C22×C4C4×D412yes—
426C22×C4#12112yes—
427C22×C4#12314yes—
428C22×C4#12312yes—
429C22×C4#12312yes—
430C22×C4#12412yes—
431C22×C4#12614yes—
432C22×C4C2×SD1612yes—
433C22×C4C2×SD1612yes—
434C22×C4C2×Q1612yes—
435C22×C4C2×Q1612yes—
436C22×C4AGL(1,ℤ/8)12yes—
437C22×C4#13712yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4111yescharacteristic
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)12yes—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C22⋊C4—48no—
482C22⋊C4—84no—
483C42—24no—
484C42—24no—
485C42—44no—
486C42—44no—
487C42—42no—
488C42—42no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23#38814yes—
518C23C4×SD1614yes—
519C23C4×Q1614yes—
520C23#46514yes—
521C23#46714yes—
522C23#48714yes—
523C23#48814yes—
524C23C2×C4×D411yescharacteristic
525C23C2×#12111yescharacteristic
526C23C2×#12311yescharacteristic
527C23#52511yescharacteristic
528C23C2×#12411yescharacteristic
529C23#53411yescharacteristic
530C23#53511yescharacteristic
531C23C22×SD1611yescharacteristic
532C23C22×Q1611yescharacteristic
533C23C2×#13511yescharacteristic
534C23C2×#13511yescharacteristic
535C23C2×AGL(1,ℤ/8)11yescharacteristic
536C23C2×#13711yescharacteristic
537C23#57511yescharacteristic
538C23#57511yescharacteristic
539Q8—44no—
540Q8—44no—
541Q8—82no—
542Q8—82no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—416no—
572C2×C4—416no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—816no—
594C2×C4—816no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4#38814yes—
626C2×C4#48314yes—
627C2×C4#48414yes—
628C2×C4C2×#11711yescharacteristic
629C2×C4C2×C4×D411yescharacteristic
630C2×C4C4×#4112yes—
631C2×C4#51812yes—
632C2×C4C2×#12311yescharacteristic
633C2×C4C2×#12611yescharacteristic
634C2×C4#52912yes—
635C2×C4#53812yes—
636C2×C4#54612yes—
637C2×C4#54612yes—
638C2×C4#55112yes—
639C2×C4#55112yes—
640C2×C4#55312yes—
641C2×C4#55312yes—
642C2×C4#55312yes—
643C2×C4#55312yes—
644C8—44no—
645C8—44no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22#193318yes—
661C22C2×#38811yescharacteristic
662C22C4×#12311yescharacteristic
663C22#232111yescharacteristic
664C22#241111yescharacteristic
665C22#241211yescharacteristic
666C22#250411yescharacteristic
667C22#259111yescharacteristic
668C22C2×C4×SD1611yescharacteristic
669C22C2×C4×Q1611yescharacteristic
670C22C4×AGL(1,ℤ/8)11yescharacteristic
671C22C4×#13711yescharacteristic
672C22C2×#46511yescharacteristic
673C22C2×#46711yescharacteristic
674C22#305511yescharacteristic
675C22#305611yescharacteristic
676C22C2×#48311yescharacteristic
677C22C2×#48411yescharacteristic
678C22#309111yescharacteristic
679C22#309111yescharacteristic
680C22#310111yescharacteristic
681C22#310411yescharacteristic
682C22#310511yescharacteristic
683C22#310811yescharacteristic
684C22C2×#48711yescharacteristic
685C22C2×#48811yescharacteristic
686C22#314111yescharacteristic
687C22#314211yescharacteristic
688C22#318711yescharacteristic
689C22#318811yescharacteristic
690C22#318911yescharacteristic
691C22#319011yescharacteristic
692C22#320811yescharacteristic
693C22#320911yescharacteristic
694C22#321111yescharacteristic
695C22#321111yescharacteristic
696C4—24no—
697C4—24no—
698C4—24no—
699C4—42no—
700C4—42no—
701C4—84no—
702C4—82no—
703C4—84no—
704C4—82no—
705C4—161no—
706C4—161no—
707C4#235312yes—
708C4#236412yes—
709C4#244412yes—
710C4#251012yes—
711C2—28no—
712C2#1402011yescharacteristic
713C2C2×#193311yescharacteristic
714C2#2109311yescharacteristic
715C2C4×#48311yescharacteristic
716C2C4×#48411yescharacteristic
717C2#2204311yescharacteristic
718C2#2204411yescharacteristic
719C2#2450111yescharacteristic
720C2#2450211yescharacteristic
721C2#2455111yescharacteristic
722C2#2455211yescharacteristic
723C2#2797211yescharacteristic
724C2#2797711yescharacteristic
725C2#2932911yescharacteristic
726C2#2933011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11316C2semidirect
7. #11190C2semidirect
8. #10926C2semidirect
9. #10820C2semidirect
27. #1872C4semidirect
91. C2×#103C2×C4semidirect
125. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18722
C2×C4×C22⋊C82
#147022
C2×C4⋊#952
#113162
#111902
#109262
#108202
#107302