Finite Group Databasebeta

← #392921

Subgroups — #392921

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic
2C2×#1944C211yesmaximal, characteristic, complemented
3C2×#1942C211yesmaximal, characteristic, complemented
4C2×#1809C211yesmaximal, characteristic
5C2×#1809C211yesmaximal, characteristic
6#14697C212yesmaximal
7#10881C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10863C211yesmaximal, characteristic, complemented
10#10861C211yesmaximal, characteristic, complemented
11#10722C212yesmaximal
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C2×C4×M16—24no—
20C2×C4×M16—24no—
21C2×C4×M16—24no—
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#385—22no—
29C2×#380C2212yes—
30C2×#380C2212yes—
31#1944—24no—
32#1942—24no—
33#1809—24no—
34#1809C418yescomplemented
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1436—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16C2×C414yes—
60C22×M16D414yes—
61C22×M16Q814yes—
62C22×M16Q814yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107C2×C414yescomplemented
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106C2×C414yescomplemented
97C4×M16—232no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97D414yes—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8D414yes—
110#389—24no—
111#389—24no—
112#389—24no—
113#389—24no—
114#385—48no—
115#380—24no—
116#380—24no—
117#380—24no—
118#380C2×C418yes—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340C2×C418yescomplemented
133C25C2×D411yescharacteristic
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
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—22no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C22×C412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×Q812yes—
164C23×C4C2×Q812yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4C2411yescharacteristic
168C2×M16—28no—
169C2×M16—28no—
170C2×M16—28no—
171C2×M16—28no—
172C2×M16—28no—
173C2×M16—28no—
174C2×M16—28no—
175C2×M16—28no—
176C2×M16—28no—
177C2×M16—28no—
178C2×M16—28no—
179C2×M16—28no—
180C2×M16—28no—
181C2×M16C4⋊C4116yescomplemented
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8C22×C412yes—
195C22×C8C22×C412yes—
196C22×C8C2×D412yes—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8C2×Q812yes—
203C22×C8C2×Q812yes—
204C22×C8C2×Q812yes—
205C22×C8C2×Q812yes—
206C22×C8#4112yes—
207C22×C8#4112yes—
208C22×C8#4112yes—
209C22×C8#4112yes—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
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—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C22×C414yes—
239C2×C4⋊C4C22×C414yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—28no—
247C2×C42—28no—
248C2×C42—28no—
249C2×C42—28no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268#107—28no—
269#107—44no—
270#106—28no—
271#106—44no—
272#97—28no—
273#97—28no—
274C4×C8—28no—
275C4×C8—28no—
276#95—28no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24C4×D412yes—
293C24#12112yes—
294C24C4⋊Q812yes—
295C24C22×D411yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4C2×C4⋊C414yes—
377C22×C4C4×D412yes—
378C22×C4C4×Q814yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4#12112yes—
382C22×C4#12112yes—
383C22×C4#12112yes—
384C22×C4C22⋊Q814yes—
385C22×C4#12514yes—
386C22×C4C4⋊Q812yes—
387C22×C4AGL(1,ℤ/8)14yes—
388C22×C4#13714yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×Q811yescharacteristic
394C22×C4C22×Q811yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403M16—232no—
404M16—232no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8C2×C4⋊C414yes—
424C2×C8C2×C4⋊C414yes—
425C2×C8C4×D414yes—
426C2×C8C4×D414yes—
427C2×C8#12014yes—
428C2×C8C22⋊Q814yes—
429C2×C8C22⋊Q814yes—
430C2×C8C22⋊Q814yes—
431C2×C8C22⋊Q814yes—
432C2×C8#12314yes—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C22⋊C4—416no—
452C22⋊C4—88no—
453C42—28no—
454C42—28no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23#38914yes—
486C23C2×C4×D411yescharacteristic
487C23C2×#12111yescharacteristic
488C23C2×#12111yescharacteristic
489C23C2×#12111yescharacteristic
490C23C2×C4⋊Q811yescharacteristic
491C23#53211yescharacteristic
492C23#53211yescharacteristic
493C23C2×AGL(1,ℤ/8)12yes—
494C23C2×#13712yes—
495C23#57512yes—
496C23#57512yes—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4#38914yes—
560C2×C4#46914yes—
561C2×C4#47014yes—
562C2×C4C22×C4⋊C411yescharacteristic
563C2×C4C2×C4×D411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×Q811yescharacteristic
566C2×C4C4×#4111yescharacteristic
567C2×C4C4×#4111yescharacteristic
568C2×C4C2×#12011yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4C2×C22⋊Q811yescharacteristic
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4C2×C22⋊Q811yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#52511yescharacteristic
577C2×C4#52511yescharacteristic
578C2×C4#52511yescharacteristic
579C2×C4#52511yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4C2×#12511yescharacteristic
582C2×C4#52911yescharacteristic
583C2×C4#52911yescharacteristic
584C2×C4#52911yescharacteristic
585C2×C4#52911yescharacteristic
586C2×C4#53311yescharacteristic
587C2×C4#53311yescharacteristic
588C2×C4#53311yescharacteristic
589C2×C4#53311yescharacteristic
590C8—24no—
591C8—24no—
592C8—24no—
593C8#38518yescomplemented
594C22—24no—
595C22—28no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22C2×#38911yescharacteristic
606C22C4×#12111yescharacteristic
607C22C4×#12111yescharacteristic
608C22#284711yescharacteristic
609C22#285411yescharacteristic
610C22#285711yescharacteristic
611C22#285711yescharacteristic
612C22C4×AGL(1,ℤ/8)12yes—
613C22C4×#13712yes—
614C22C2×#46911yescharacteristic
615C22C2×#47011yescharacteristic
616C22#305411yescharacteristic
617C22#305411yescharacteristic
618C22#305512yes—
619C22#305512yes—
620C22#305612yes—
621C22#305612yes—
622C22#316512yes—
623C22#316612yes—
624C22#317112yes—
625C22#317212yes—
626C22#318212yes—
627C22#318312yes—
628C4—24no—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—84no—
634C4—84no—
635C4—162no—
636C4—162no—
637C4C2×#38511yescharacteristic
638C4#229611yescharacteristic
639C4C4×C22⋊Q811yescharacteristic
640C4C4×C22⋊Q811yescharacteristic
641C4#284811yescharacteristic
642C4#285111yescharacteristic
643C4#285111yescharacteristic
644C4#285511yescharacteristic
645C2—44no—
646C2#1469711yescharacteristic
647C2#2115912yes—
648C2C4×#46911yescharacteristic
649C2C4×#47011yescharacteristic
650C2#2154012yes—
651C2#3293611yescharacteristic
652C2#3293711yescharacteristic
653C2#3314512yes—
654C2#3321111yescharacteristic
655C2#3321211yescharacteristic
656C2#3321812yes—
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
2. C2×#1944C2semidirect
3. C2×#1942C2semidirect
7. #10881C2semidirect
8. #10880C2semidirect
9. #10863C2semidirect
10. #10861C2semidirect
34. #1809C4semidirect
90. C2×#107C2×C4semidirect
96. C2×#106C2×C4semidirect
132. #340C2×C4semidirect
181. C2×M16C4⋊C4semidirect
593. C8#385semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
C2×#19442
C2×#19422
C2×#18092
#146972
#108812
#108802
#108632
#108612
#107222