Finite Group Databasebeta

← #357795

Subgroups — #357795

Order 512.

2974 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic, complemented
2C22×#417C211yesmaximal, characteristic
3C22×#416C211yesmaximal, characteristic
4C2×#2311C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic, complemented
6C2×#1729C211yesmaximal, characteristic, complemented
7#13795C211yesmaximal, characteristic, complemented
8#11124C214yesmaximal, complemented
9#11123C214yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#107—24no—
18C22×#107—24no—
19C22×#107—24no—
20C22×#107C2214yes—
21C2×#417—24no—
22C2×#417—24no—
23C2×#417—24no—
24C2×#416—24no—
25C2×#416—24no—
26C2×#416C418yescomplemented
27C22×#103C2212yes—
28C22×#103C2212yes—
29C22×#102C2212yes—
30C22×#102C2212yes—
31#2311—24no—
32#2311—24no—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#386—22no—
36C2×#380—22no—
37#1736—24no—
38C2×#340C2214yes—
39#1729—24no—
40C2×#336C2214yes—
41C4⋊#95—22no—
42C4⋊#95C2214yes—
43C22⋊#95—22no—
44C22×#41—22no—
45C22×#41—22no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41—22no—
51C22×#41—22no—
52C22×#41—22no—
53C22×#41—22no—
54C22×#41—22no—
55C22×#41C2×C414yes—
56C22×#41D414yes—
57C22×#41Q814yes—
58C23×Q8C2311yescharacteristic
59C23×D4—21no—
60C23×D4C2311yescharacteristic
61C24×C4—21no—
62C24×C4—21no—
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C22×M16—22no—
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—22no—
70C22×M16—22no—
71C22×M16D414yes—
72C23×C8—22no—
73C23×C8—22no—
74C23×C8—22no—
75C23×C8—22no—
76C23×C8C2×C414yes—
77C23×C8D414yes—
78C23×C8D414yes—
79C23×C8C2312yes—
80C23×C8C2312yes—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—22no—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—22no—
92C2×#117C2×C414yes—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4D412yes—
106C22×C4⋊C4D414yes—
107C22×C4⋊C4D414yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4D412yes—
114C22×C42—21no—
115C22×C42C2312yes—
116C2×#107—232no—
117C2×#107—28no—
118C2×#107—28no—
119C2×#107—28no—
120#417—232no—
121#416—232no—
122C2×#103—24no—
123C2×#103—24no—
124C2×#103—24no—
125C2×#103—24no—
126C2×#103—24no—
127C2×#103C2×C418yescomplemented
128C2×#102—24no—
129C2×#102—24no—
130C2×#102—24no—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102C2×C418yescomplemented
134#391—24no—
135#391—24no—
136#390—24no—
137#390—24no—
138#386—216no—
139#380—216no—
140C2×#95—24no—
141C2×#95—28no—
142#340—28no—
143#336—28no—
144C25—21no—
145C25C2×D411yescharacteristic
146C2×#41—28no—
147C2×#41—28no—
148C2×#41—28no—
149C2×#41—28no—
150C2×#41—28no—
151C2×#41—28no—
152C2×#41—28no—
153C2×#41—28no—
154C2×#41—28no—
155C2×#41—28no—
156C2×#41—28no—
157C2×#41—28no—
158C2×#41—28no—
159C2×#41—28no—
160C2×#41—28no—
161C2×#41—44no—
162C2×#41—44no—
163C2×#41—44no—
164C2×#41—44no—
165C2×#41—44no—
166C2×#41—44no—
167C2×#41—44no—
168C2×#41—44no—
169C2×#41—44no—
170C2×#41—44no—
171C2×#41—44no—
172C2×#41—44no—
173C2×#41C4⋊C4116yes—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8C22×C412yes—
186C22×Q8C2×D412yes—
187C22×Q8C2×Q812yes—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4C22×C412yes—
207C22×D4C2×D412yes—
208C22×D4C2×Q812yes—
209C23×C4—21no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—84no—
227C23×C4—84no—
228C23×C4C22×C412yes—
229C23×C4C2×D414yes—
230C23×C4C2×D412yes—
231C23×C4C2×D412yes—
232C23×C4C2×D412yes—
233C23×C4C2×D412yes—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×Q812yes—
238C23×C4#4112yes—
239C23×C4#4112yes—
240C23×C4#4112yes—
241C23×C4#4112yes—
242C23×C4#4112yes—
243C23×C4#4112yes—
244C23×C4C2411yescharacteristic
245C2×M16—28no—
246C2×M16—28no—
247C2×M16—28no—
248C2×M16—44no—
249C2×M16—44no—
250C2×M16—44no—
251C2×M16—44no—
252C22×C8—28no—
253C22×C8—28no—
254C22×C8—28no—
255C22×C8—28no—
256C22×C8—28no—
257C22×C8—28no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8C22⋊C4116yescomplemented
269C22×C8C22×C414yes—
270C22×C8C2×D414yes—
271C22×C8C2×D414yes—
272C22×C8C2×D414yes—
273#117—28no—
274#117—28no—
275#117—44no—
276#117—44no—
277#117—44no—
278#117—44no—
279#117—44no—
280#117—44no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4C22×C414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—216no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—88no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—42no—
332C2×C42—42no—
333#107—232no—
334#103—216no—
335#103—216no—
336#102—216no—
337#102—216no—
338#95—216no—
339#95—88no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—44no—
350C24—44no—
351C24C4×D412yes—
352C24#12012yes—
353C24#12112yes—
354C24#12112yes—
355C24C22⋊Q812yes—
356C24#12312yes—
357C24#12712yes—
358C24C22×D411yescharacteristic
359#41—232no—
360#41—232no—
361#41—416no—
362#41—416no—
363#41—416no—
364#41—416no—
365#41—416no—
366#41—416no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8C2×C4⋊C414yes—
395C2×D4—216no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4C2×C4⋊C414yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4C2×C22⋊C414yes—
471C22×C4C2×C4⋊C414yes—
472C22×C4#11714yes—
473C22×C4C4×D414yes—
474C22×C4C4×D414yes—
475C22×C4C4×D414yes—
476C22×C4C4×D412yes—
477C22×C4#12014yes—
478C22×C4#12014yes—
479C22×C4#12012yes—
480C22×C4#12114yes—
481C22×C4#12114yes—
482C22×C4#12114yes—
483C22×C4#12114yes—
484C22×C4#12114yes—
485C22×C4#12114yes—
486C22×C4#12112yes—
487C22×C4#12112yes—
488C22×C4C22⋊Q814yes—
489C22×C4C22⋊Q814yes—
490C22×C4C22⋊Q814yes—
491C22×C4C22⋊Q814yes—
492C22×C4C22⋊Q814yes—
493C22×C4C22⋊Q812yes—
494C22×C4#12314yes—
495C22×C4#12314yes—
496C22×C4#12314yes—
497C22×C4#12314yes—
498C22×C4#12312yes—
499C22×C4#12414yes—
500C22×C4#12414yes—
501C22×C4#12414yes—
502C22×C4#12614yes—
503C22×C4#12614yes—
504C22×C4#12712yes—
505C22×C4C23×C411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×Q811yescharacteristic
513C22×C4C2×#4111yescharacteristic
514C22×C4C2×#4111yescharacteristic
515C22×C4C2×#4111yescharacteristic
516C22×C4C2×#4111yescharacteristic
517C22×C4C2×#4111yescharacteristic
518C22×C4C2×#4111yescharacteristic
519M16—416no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—232no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—42no—
531C2×C8—42no—
532C2×C8—42no—
533C2×C8—42no—
534C2×C8C2×C22⋊C418yes—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—816no—
553C42—42no—
554C42—42no—
555C42—42no—
556C42—42no—
557C42—48no—
558C23—28no—
559C23—28no—
560C23—24no—
561C23—24no—
562C23—28no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—28no—
568C23—24no—
569C23—48no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—48no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—84no—
580C23—82no—
581C23—82no—
582C23—82no—
583C23—82no—
584C23—84no—
585C23#38514yes—
586C23#38814yes—
587C23#39014yes—
588C23#39214yes—
589C23#39414yes—
590C23#39614yes—
591C23#39914yes—
592C23C2×C4×D411yescharacteristic
593C23C2×#12011yescharacteristic
594C23C2×#12111yescharacteristic
595C23C2×#12111yescharacteristic
596C23C2×C22⋊Q811yescharacteristic
597C23C2×#12311yescharacteristic
598C23C2×#12711yescharacteristic
599C23#57614yes—
600C23#57814yes—
601Q8—24no—
602Q8—24no—
603Q8—42no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609D4—24no—
610D4—24no—
611D4—432no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618C2×C4—24no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—432no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—48no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—816no—
638C2×C4—816no—
639C2×C4—82no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—82no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4#38018yes—
647C2×C4#38514yes—
648C2×C4#38518yes—
649C2×C4#38618yes—
650C2×C4#38814yes—
651C2×C4#38818yes—
652C2×C4#39014yes—
653C2×C4#39214yes—
654C2×C4#39318yes—
655C2×C4#39418yes—
656C2×C4#39414yes—
657C2×C4#39618yes—
658C2×C4#39614yes—
659C2×C4#39718yes—
660C2×C4#39914yes—
661C2×C4C22×C22⋊C411yescharacteristic
662C2×C4C22×C4⋊C411yescharacteristic
663C2×C4C2×#11711yescharacteristic
664C2×C4C2×C4×D411yescharacteristic
665C2×C4C2×C4×D411yescharacteristic
666C2×C4C2×C4×D411yescharacteristic
667C2×C4C2×#12011yescharacteristic
668C2×C4C2×#12011yescharacteristic
669C2×C4C2×#12111yescharacteristic
670C2×C4C2×#12111yescharacteristic
671C2×C4C2×#12111yescharacteristic
672C2×C4C2×#12111yescharacteristic
673C2×C4C2×#12111yescharacteristic
674C2×C4C2×#12111yescharacteristic
675C2×C4C2×C22⋊Q811yescharacteristic
676C2×C4C2×C22⋊Q811yescharacteristic
677C2×C4C2×C22⋊Q811yescharacteristic
678C2×C4C2×C22⋊Q811yescharacteristic
679C2×C4C2×C22⋊Q811yescharacteristic
680C2×C4C2×#12311yescharacteristic
681C2×C4C2×#12311yescharacteristic
682C2×C4C2×#12311yescharacteristic
683C2×C4C2×#12311yescharacteristic
684C2×C4C2×#12411yescharacteristic
685C2×C4C2×#12411yescharacteristic
686C2×C4C2×#12411yescharacteristic
687C2×C4C2×#12611yescharacteristic
688C2×C4C2×#12611yescharacteristic
689C8—28no—
690C8—44no—
691C22—28no—
692C22—28no—
693C22—24no—
694C22—24no—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—28no—
699C22—44no—
700C22—48no—
701C22—84no—
702C22—84no—
703C22—82no—
704C22—82no—
705C22—84no—
706C22—84no—
707C22—84no—
708C22#143918yes—
709C22C2×#38511yescharacteristic
710C22C2×#38811yescharacteristic
711C22C2×#39011yescharacteristic
712C22C2×#39211yescharacteristic
713C22C2×#39411yescharacteristic
714C22C2×#39611yescharacteristic
715C22C2×#39911yescharacteristic
716C22#295114yes—
717C22#300812yes—
718C22#301112yes—
719C22#304112yes—
720C22#304412yes—
721C22#306114yes—
722C22#308314yes—
723C22#309814yes—
724C22#315512yes—
725C22#315612yes—
726C4—44no—
727C4—84no—
728C4—84no—
729C4—84no—
730C4—84no—
731C4#143918yes—
732C4C2×#38011yescharacteristic
733C4C2×#38511yescharacteristic
734C4C2×#38611yescharacteristic
735C4C2×#38811yescharacteristic
736C4C2×#39311yescharacteristic
737C4C2×#39411yescharacteristic
738C4C2×#39611yescharacteristic
739C4C2×#39711yescharacteristic
740C2—28no—
741C2—84no—
742C2C2×#143911yescharacteristic
743C2#2088112yes—
744C2#2106712yes—
745C2#2123012yes—
746C2#2342211yescharacteristic
747C2#2342511yescharacteristic
748C2#2348812yes—
749C2#2361112yes—
750C2#2373712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#107C2semidirect
5. C2×#1736C2semidirect
6. C2×#1729C2semidirect
7. #13795C2semidirect
8. #11124C2semidirect
9. #11123C2semidirect
26. C2×#416C4semidirect
127. C2×#103C2×C4semidirect
133. C2×#102C2×C4semidirect
268. C22×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×#1072
C22×#4172
C22×#4162
C2×#23112
C2×#17362
C2×#17292
#137952
#111242
#111232