Finite Group Databasebeta

← #366608

Subgroups — #366608

Order 512.

2578 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#103C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic
3C2×#1981C212yesmaximal, complemented
4C2×#1892C211yesmaximal, characteristic
5C2×#1872C211yesmaximal, characteristic
6C2×#1850C211yesmaximal, characteristic
7C2×#1787C211yesmaximal, characteristic
8C22⋊C4×C4⋊C4C211yesmaximal, characteristic
9#11077C212yesmaximal, complemented
10#11010C212yesmaximal, complemented
11#10932C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4⋊Q8C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×#103—28no—
19C22×#103—24no—
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2212yes—
26C22×#103C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28#2434—24no—
29#2434C418yescomplemented
30C2×#395—22no—
31C2×#391C2212yes—
32C2×#385—21no—
33C2×#385—21no—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1981—28no—
39#1892—24no—
40#1872—24no—
41#1850—24no—
42#1787—24no—
43C2×#340C2211yescharacteristic
44C2×#340C2212yes—
45C2×#340C2211yescharacteristic
46C2×#336C2212yes—
47#1436—22no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C23×Q8—21no—
51C23×Q8—22no—
52C23×Q8—24no—
53C23×Q8—24no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—22no—
57C23×Q8—22no—
58C23×Q8—42no—
59C23×Q8—42no—
60C23×Q8C2×C414yes—
61C23×Q8C2311yescharacteristic
62C23×Q8C2312yes—
63C24×C4—21no—
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C23×C8—22no—
67C23×C8—24no—
68C23×C8—22no—
69C23×C8—22no—
70C23×C8—42no—
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—42no—
79C2×C4⋊Q8C2×C418yescomplemented
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8C2×C414yes—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4D414yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4D411yescharacteristic
101C22×C22⋊C4D411yescharacteristic
102C22×C42—22no—
103C22×C42C2312yes—
104C2×C4⋊C8—24no—
105C2×C4⋊C8—24no—
106C2×C4⋊C8—42no—
107C2×#103—232no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—24no—
116C2×#103—22no—
117C2×#103—22no—
118C2×#103—24no—
119C2×#103—24no—
120C2×#103—24no—
121C2×#103—24no—
122C2×#103—42no—
123C2×#103—42no—
124C2×#103—42no—
125C2×#103—42no—
126C2×#103D418yes—
127C2×C22⋊C8—22no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—22no—
130#395—216no—
131#391—24no—
132#391C2×C418yescomplemented
133#385—28no—
134#385—28no—
135#380—22no—
136#380D414yes—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—42no—
139C4×C22⋊C4—44no—
140C4×C22⋊C4C2×C414yes—
141C4×C22⋊C4D414yes—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95D412yes—
149#340—22no—
150#340—22no—
151#340—24no—
152#336—24no—
153C25C2×D411yescharacteristic
154C22×Q8—24no—
155C22×Q8—22no—
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—28no—
160C22×Q8—216no—
161C22×Q8—28no—
162C22×Q8—216no—
163C22×Q8—28no—
164C22×Q8—28no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—48no—
184C22×Q8—48no—
185C22×Q8—48no—
186C22×Q8—48no—
187C22×Q8—48no—
188C22×Q8—48no—
189C22×Q8—48no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—48no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8C22×C412yes—
198C22×Q8C22×C414yes—
199C22×Q8C2×D414yes—
200C22×Q8#4114yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—21no—
207C23×C4—24no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—21no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—82no—
221C23×C4C22×C412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4#4112yes—
229C23×C4#4112yes—
230C23×C4C2411yescharacteristic
231C22×C8—28no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—48no—
249C22×C8—48no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2×D414yes—
253C4⋊Q8—216no—
254C4⋊Q8—48no—
255C4⋊Q8—48no—
256C4⋊Q8—48no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4C22×C412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D412yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4C22⋊C414yes—
304C2×C22⋊C4C22⋊C414yes—
305C2×C22⋊C4SD1614yes—
306C2×C22⋊C4Q1614yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—82no—
314C2×C42C22×C414yes—
315C2×C42C2×D414yes—
316C4⋊C8—48no—
317#103—216no—
318#103—216no—
319#103—48no—
320#103—48no—
321#103—48no—
322#103—44no—
323#103—44no—
324C22⋊C8—44no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—44no—
329#95—44no—
330#95—82no—
331#95C22⋊C418yes—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24C2×C22⋊C412yes—
337C24#12112yes—
338C24#12112yes—
339C24C2×SD1612yes—
340C24C2×Q1612yes—
341C24#13712yes—
342C24#13712yes—
343C24C22×D411yescharacteristic
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—232no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—416no—
365C2×Q8—432no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—416no—
371C2×Q8—432no—
372C2×Q8—416no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—42no—
376C2×Q8—41no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—48no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—42no—
393C2×Q8—41no—
394C2×Q8—41no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8C4×D418yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—48no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—48no—
468C22×C4—84no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4C2×C22⋊C414yes—
472C22×C4C2×C22⋊C412yes—
473C22×C4C2×C22⋊C414yes—
474C22×C4C4×D414yes—
475C22×C4C4×D414yes—
476C22×C4#12112yes—
477C22×C4#12112yes—
478C22×C4#12414yes—
479C22×C4#12714yes—
480C22×C4C2×SD1614yes—
481C22×C4C2×Q1614yes—
482C22×C4C23×C411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4111yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4111yescharacteristic
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—416no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—42no—
507C2×C8—42no—
508C2×C8—42no—
509C2×C8—42no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—48no—
519C4⋊C4—416no—
520C4⋊C4—416no—
521C4⋊C4—42no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—48no—
527C4⋊C4#12014yes—
528C22⋊C4—28no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4#103116yescomplemented
532C42—24no—
533C42—24no—
534C42—24no—
535C42—24no—
536C42—88no—
537C23—24no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23#39014yes—
554C23C2×#10314yes—
555C23#41914yes—
556C23#46114yes—
557C23#46114yes—
558C23#46214yes—
559C23#46214yes—
560C23C22×C22⋊C411yescharacteristic
561C23C2×#12111yescharacteristic
562C23C2×#12111yescharacteristic
563C23D4212yes—
564C23#54612yes—
565C23C22×SD1611yescharacteristic
566C23C22×Q1611yescharacteristic
567C23C2×#13711yescharacteristic
568C23C2×#13711yescharacteristic
569C23#57712yes—
570C23#57812yes—
571Q8—28no—
572Q8—28no—
573Q8—432no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—44no—
578Q8—44no—
579Q8—44no—
580Q8—42no—
581Q8—44no—
582Q8—42no—
583Q8—81no—
584Q8—82no—
585Q8—81no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—432no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—816no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—81no—
641C2×C4—82no—
642C2×C4—81no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4#39014yes—
646C2×C4C2×#10318yescomplemented
647C2×C4C2×C4×D411yescharacteristic
648C2×C4C2×C4×D411yescharacteristic
649C2×C4C2×C4×D412yes—
650C2×C4#51812yes—
651C2×C4C2×#12011yescharacteristic
652C2×C4#52512yes—
653C2×C4#52511yescharacteristic
654C2×C4C2×#12411yescharacteristic
655C2×C4C2×#12711yescharacteristic
656C2×C4#53411yescharacteristic
657C2×C4#53511yescharacteristic
658C2×C4#53812yes—
659C2×C4D4212yes—
660C2×C4D4212yes—
661C2×C4#54612yes—
662C2×C4#54612yes—
663C2×C4#54612yes—
664C2×C4#55112yes—
665C8—44no—
666C8—81no—
667C8—81no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—28no—
672C22—28no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—28no—
677C22—28no—
678C22#198118yescomplemented
679C22C2×#39011yescharacteristic
680C22D4×C22⋊C412yes—
681C22#241011yescharacteristic
682C22#241111yescharacteristic
683C22#242412yes—
684C22C22×#10311yescharacteristic
685C22C2×#41911yescharacteristic
686C22#289812yes—
687C22C2×#46111yescharacteristic
688C22C2×#46111yescharacteristic
689C22C2×#46211yescharacteristic
690C22C2×#46211yescharacteristic
691C22#304211yescharacteristic
692C22#304411yescharacteristic
693C22#304612yes—
694C22#326612yes—
695C22#327012yes—
696C22D4×SD1612yes—
697C22D4×Q1612yes—
698C22#329812yes—
699C22#329912yes—
700C22#331212yes—
701C22#332012yes—
702C4—28no—
703C4—24no—
704C4—44no—
705C4—44no—
706C4—48no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—84no—
711C4C4×#12011yescharacteristic
712C4#232111yescharacteristic
713C4#235812yes—
714C4#243311yescharacteristic
715C4#247212yes—
716C4#253911yescharacteristic
717C2—28no—
718C2#1399511yescharacteristic
719C2C2×#198111yescharacteristic
720C2#2126211yescharacteristic
721C2D4×#10312yes—
722C2#2242512yes—
723C2#2435811yescharacteristic
724C2#2436011yescharacteristic
725C2#2448011yescharacteristic
726C2#2448211yescharacteristic
727C2#2507812yes—
728C2#2508212yes—
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
3. C2×#1981C2semidirect
9. #11077C2semidirect
10. #11010C2semidirect
11. #10932C2semidirect
29. #2434C4semidirect
79. C2×C4⋊Q8C2×C4semidirect
132. #391C2×C4semidirect
531. C22⋊C4#103semidirect
646. C2×C4C2×#103semidirect
678. C22#1981semidirect

Maximal subgroups

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

subgroupindex
C23×#1032
C2×#24342
C2×#19812
C2×#18922
C2×#18722
C2×#18502
C2×#17872
C22⋊C4×C4⋊C42
#110772
#110102
#109322