Finite Group Databasebeta

← #374082

Subgroups — #374082

Order 512.

2700 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#451C211yesmaximal, characteristic
2C23×#103C211yesmaximal, characteristic
3C2×Q8×C22⋊C4C211yesmaximal, characteristic
4C2×#1892C211yesmaximal, characteristic
5C2×#1878C212yesmaximal, complemented
6#14139C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11024C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C23×Q16C2211yescharacteristic
14C23×Q16C2211yescharacteristic
15C24×C8C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C2×#451—24no—
20C2×#451—24no—
21C2×#451C418yescomplemented
22C22×#103—28no—
23C22×#103—24no—
24C22×#103C2212yes—
25C22×#103C2212yes—
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2212yes—
29C22×C22⋊C8C2211yescharacteristic
30Q8×C22⋊C4—24no—
31Q8×C22⋊C4—24no—
32C2×#391—22no—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#380—21no—
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4—21no—
38C2×C4×C22⋊C4C2211yescharacteristic
39#1892—24no—
40#1878—28no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1441—22no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C22⋊#95—21no—
50C23×Q8—22no—
51C23×Q8—21no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—24no—
55C23×Q8—24no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8C2×C414yes—
59C23×Q8D414yes—
60C23×Q8D414yes—
61C23×Q8C2311yescharacteristic
62C23×Q8C2311yescharacteristic
63C23×Q8C2311yescharacteristic
64C24×C4—21no—
65C24×C4—21no—
66C24×C4C2311yescharacteristic
67C22×Q16—24no—
68C22×Q16—22no—
69C22×Q16—22no—
70C22×Q16—22no—
71C22×Q16—22no—
72C22×Q16—24no—
73C22×Q16—42no—
74C22×Q16—42no—
75C22×Q16C2×C414yescomplemented
76C22×Q16C2×C414yescomplemented
77C23×C8—22no—
78C23×C8—24no—
79C23×C8—22no—
80C23×C8—22no—
81C23×C8—42no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2312yes—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8C2×C414yes—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—42no—
96C2×C4×Q8—42no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4D411yescharacteristic
110C22×C22⋊C4D411yescharacteristic
111C22×C42—22no—
112C22×C42C2312yes—
113#451—232no—
114C2×#103—232no—
115C2×#103—24no—
116C2×#103—24no—
117C2×#103—24no—
118C2×#103—22no—
119C2×#103—22no—
120C2×#103—22no—
121C2×#103—22no—
122C2×#103—24no—
123C2×#103—24no—
124C2×#103—24no—
125C2×#103—24no—
126C2×#103—42no—
127C2×#103C2×C414yescomplemented
128C2×#103C2×C414yescomplemented
129C2×#103D418yes—
130C2×C22⋊C8—22no—
131C2×C22⋊C8—22no—
132C2×C22⋊C8C2×C414yes—
133#391—216no—
134#391—24no—
135#391—24no—
136#391—24no—
137#391—42no—
138#380—28no—
139#380—22no—
140#380D414yes—
141C4×C22⋊C4—28no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95D412yes—
152#340—24no—
153#340—24no—
154#340—42no—
155#336—24no—
156#336—42no—
157C25C2×D411yescharacteristic
158C22×Q8—24no—
159C22×Q8—28no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—28no—
163C22×Q8—28no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—216no—
174C22×Q8—28no—
175C22×Q8—216no—
176C22×Q8—28no—
177C22×Q8—22no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—21no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×Q8—48no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—48no—
195C22×Q8—48no—
196C22×Q8—48no—
197C22×Q8—48no—
198C22×Q8—48no—
199C22×Q8—48no—
200C22×Q8—48no—
201C22×Q8C22⋊C4116yes—
202C22×Q8C22×C412yes—
203C22×Q8C22×C412yes—
204C22×Q8C22×C412yes—
205C22×Q8C2×D412yes—
206C22×Q8C2×D412yes—
207C22×Q8C2×D412yes—
208C22×Q8C2×D412yes—
209C22×Q8C2×D414yes—
210C22×Q8#4114yes—
211C23×C4—21no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—82no—
231C23×C4C22×C412yes—
232C23×C4C2×D412yes—
233C23×C4C2×D412yes—
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D412yes—
238C23×C4#4112yes—
239C23×C4#4112yes—
240C23×C4C2411yescharacteristic
241C2×Q16—28no—
242C2×Q16—28no—
243C2×Q16—216no—
244C2×Q16—48no—
245C2×Q16—44no—
246C2×Q16—48no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—48no—
250C2×Q16—44no—
251C22×C8—28no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—48no—
267C22×C8—48no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8C22×C412yes—
271C22×C8C22×C412yes—
272C22×C8C2×D414yes—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C4×Q8—48no—
278C4×Q8—48no—
279C4×Q8—48no—
280C4×Q8—48no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4C22×C412yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4Q1614yescomplemented
322C2×C42—28no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—44no—
330C2×C42—44no—
331#103—216no—
332#103—28no—
333#103—28no—
334#103—216no—
335#103—48no—
336C22⋊C8—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—44no—
342#95—44no—
343#95—82no—
344#95—82no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24C4×D412yes—
350C24#12012yes—
351C24#12112yes—
352C24C2×Q1612yes—
353C24#13512yes—
354C24#13712yes—
355C24#13712yes—
356C24C22×D411yescharacteristic
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—216no—
374C2×Q8—232no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—416no—
388C2×Q8—416no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—416no—
392C2×Q8—432no—
393C2×Q8—416no—
394C2×Q8—416no—
395C2×Q8—432no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—41no—
407C2×Q8—42no—
408C2×Q8—41no—
409C2×Q8—42no—
410C2×Q8—41no—
411C2×Q8—41no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8C2×C22⋊C414yes—
417C2×Q8C2×C22⋊C414yes—
418C2×Q8C4×D418yes—
419C2×Q8#12014yes—
420C2×Q8#12118yes—
421C2×Q8#12314yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—216no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—48no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—88no—
492C22×C4—88no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4C2×C22⋊C414yes—
496C22×C4C4×D414yes—
497C22×C4C4×D412yes—
498C22×C4#12012yes—
499C22×C4#12114yes—
500C22×C4#12112yes—
501C22×C4#12314yes—
502C22×C4C2×Q1614yes—
503C22×C4C23×C411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4112yes—
509C22×C4C2×#4111yescharacteristic
510C22×C4C2×#4111yescharacteristic
511C22×C4H(5,𝔽2)12yes—
512C22×C4H(5,𝔽2)12yes—
513Q16—416no—
514Q16—416no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—416no—
523C2×C8—41no—
524C2×C8—42no—
525C2×C8—42no—
526C2×C8—41no—
527C2×C8—42no—
528C2×C8—41no—
529C2×C8—42no—
530C2×C8—41no—
531C2×C8—42no—
532C2×C8—42no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—48no—
540C4⋊C4—416no—
541C4⋊C4—42no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—48no—
548C22⋊C4—28no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C42—416no—
553C42—42no—
554C42—42no—
555C42—42no—
556C42—42no—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—22no—
568C23—24no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23#38614yes—
574C23C4×Q1614yes—
575C23#44114yes—
576C23#45114yes—
577C23#45214yes—
578C23#46214yes—
579C23#46414yes—
580C23C2×C4×D411yescharacteristic
581C23C2×#12011yescharacteristic
582C23C2×#12111yescharacteristic
583C23D4212yes—
584C23#54612yes—
585C23C22×Q1611yescharacteristic
586C23C2×#13511yescharacteristic
587C23C2×#13711yescharacteristic
588C23C2×#13711yescharacteristic
589C23#57712yes—
590C23#57812yes—
591Q8—24no—
592Q8—28no—
593Q8—24no—
594Q8—28no—
595Q8—24no—
596Q8—24no—
597Q8—24no—
598Q8—432no—
599Q8—44no—
600Q8—42no—
601Q8—42no—
602Q8—44no—
603Q8—42no—
604Q8—44no—
605Q8—42no—
606Q8—81no—
607Q8—81no—
608Q8—82no—
609Q8#38618yes—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—432no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—816no—
656C2×C4—81no—
657C2×C4—82no—
658C2×C4—81no—
659C2×C4—82no—
660C2×C4—82no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—81no—
666C2×C4—82no—
667C2×C4—81no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4#38614yes—
671C2×C4C22×C22⋊C411yescharacteristic
672C2×C4C2×C4×D411yescharacteristic
673C2×C4C2×C4×D412yes—
674C2×C4#51812yes—
675C2×C4C2×#12011yescharacteristic
676C2×C4C2×#12111yescharacteristic
677C2×C4C2×#12112yes—
678C2×C4C2×#12311yescharacteristic
679C2×C4C2×#12311yescharacteristic
680C2×C4#53411yescharacteristic
681C2×C4#53412yes—
682C2×C4#53411yescharacteristic
683C2×C4D4212yes—
684C2×C4#54612yes—
685C2×C4#54612yes—
686C2×C4#54612yes—
687C2×C4#55112yes—
688C2×C4#55112yes—
689C8—42no—
690C8—44no—
691C8—42no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—28no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22#187818yescomplemented
703C22C2×#38611yescharacteristic
704C22#235812yes—
705C22#241011yescharacteristic
706C22#241012yes—
707C22#241211yescharacteristic
708C22C2×C4×Q1611yescharacteristic
709C22C2×#44111yescharacteristic
710C22#294712yes—
711C22C2×#45111yescharacteristic
712C22C2×#45211yescharacteristic
713C22#301312yes—
714C22C2×#46211yescharacteristic
715C22C2×#46411yescharacteristic
716C22#304311yescharacteristic
717C22#304411yescharacteristic
718C22#326612yes—
719C22#327012yes—
720C22#328212yes—
721C22D4×Q1612yes—
722C22#329912yes—
723C22#330012yes—
724C22#331212yes—
725C22#331612yes—
726C4—24no—
727C4—28no—
728C4—44no—
729C4—44no—
730C4—84no—
731C4—82no—
732C4—82no—
733C4—84no—
734C4—82no—
735C4C2×#38611yescharacteristic
736C4#230811yescharacteristic
737C4#235812yes—
738C4#241011yescharacteristic
739C4#241211yescharacteristic
740C4#241212yes—
741C2—28no—
742C2#1413411yescharacteristic
743C2C2×#187811yescharacteristic
744C2#2093011yescharacteristic
745C2#2285912yes—
746C2#2287112yes—
747C2#2435711yescharacteristic
748C2#2436011yescharacteristic
749C2#2443112yes—
750C2#2443712yes—
751C2#2452911yescharacteristic
752C2#2453211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
5. C2×#1878C2semidirect
7. #11190C2semidirect
8. #11024C2semidirect
9. #10789C2semidirect
21. C2×#451C4semidirect
75. C22×Q16C2×C4semidirect
76. C22×Q16C2×C4semidirect
127. C2×#103C2×C4semidirect
128. C2×#103C2×C4semidirect
321. C2×C22⋊C4Q16semidirect
702. C22#1878semidirect

Maximal subgroups

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

subgroupindex
C22×#4512
C23×#1032
C2×Q8×C22⋊C42
C2×#18922
C2×#18782
#141392
#111902
#110242
#107892
#107302
#107142