Finite Group Databasebeta

← #395887

Subgroups — #395887

Order 512.

1982 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#2066C212yesmaximal, complemented
3C2×#1872C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1853C211yesmaximal, characteristic
6C2×#1797C211yesmaximal, characteristic
7#14702C211yesmaximal, characteristic
8#11227C212yesmaximal, complemented
9#11220C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4—21no—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#106C2212yes—
18C22×C4⋊C8C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389—21no—
26C2×#389—21no—
27C2×#389—28no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#380C2211yescharacteristic
31C22×#95—21no—
32#2066—28no—
33#1872—24no—
34#1872—24no—
35#1853—24no—
36#1797—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C22⋊#95—21no—
48C22⋊#95—21no—
49C23×Q8C2311yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8Q814yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4D414yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—24no—
89C22×C42—42no—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#106—24no—
95C2×#106—24no—
96C2×#106—24no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—42no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8D414yes—
112#391—24no—
113#391—42no—
114#389—28no—
115#389—28no—
116#389—24no—
117#389—24no—
118#389—24no—
119#389D418yes—
120#380—22no—
121#380—22no—
122C2×#95—28no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—42no—
139#336—24no—
140C25C2×D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8C2×Q812yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4SD1614yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×Q812yes—
197C23×C4#4112yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C23×C4C2411yescharacteristic
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8C2×D414yes—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217C22⋊Q8—44no—
218C22⋊Q8—44no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—216no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—432no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×Q812yes—
264C2×C4⋊C4#4114yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—48no—
283C2×C42—48no—
284C2×C42—41no—
285C2×C42—41no—
286C2×C42C2×D414yes—
287#106—48no—
288C4⋊C8—48no—
289#103—48no—
290#103—48no—
291C22⋊C8—28no—
292C22⋊C8—44no—
293#95—28no—
294#95—28no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—42no—
303#95—42no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24#12112yes—
313C24#12112yes—
314C24C22⋊Q812yes—
315C24C2×SD1612yes—
316C24#13512yes—
317C24#13712yes—
318C24#13712yes—
319C24C22×D411yescharacteristic
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—216no—
357C22×C4—28no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4#12012yes—
445C22×C4#12012yes—
446C22×C4#12114yes—
447C22×C4#12112yes—
448C22×C4#12112yes—
449C22×C4C22⋊Q814yes—
450C22×C4C22⋊Q814yes—
451C22×C4C22⋊Q812yes—
452C22×C4#12614yes—
453C22×C4C2×SD1612yes—
454C22×C4C2×SD1612yes—
455C22×C4AGL(1,ℤ/8)12yes—
456C22×C4#13712yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×Q811yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4111yescharacteristic
465C22×C4H(5,𝔽2)12yes—
466C22×C42−1+412yes—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C4⋊C4—216no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4#12118yes—
502C22⋊C4—48no—
503C22⋊C4—84no—
504C42—24no—
505C42—24no—
506C42—44no—
507C42—44no—
508C42—42no—
509C42—42no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23#39614yes—
539C23#46114yes—
540C23#46414yes—
541C23#46514yes—
542C23#47014yes—
543C23#47514yes—
544C23#47914yes—
545C23C2×#12011yescharacteristic
546C23C2×#12111yescharacteristic
547C23C2×#12111yescharacteristic
548C23C2×C22⋊Q811yescharacteristic
549C23#52511yescharacteristic
550C23#53411yescharacteristic
551C23#53611yescharacteristic
552C23C22×SD1611yescharacteristic
553C23C2×#13511yescharacteristic
554C23C2×AGL(1,ℤ/8)11yescharacteristic
555C23C2×#13711yescharacteristic
556C23C2×#13711yescharacteristic
557C23C2×#13711yescharacteristic
558C23#57511yescharacteristic
559C23#57511yescharacteristic
560Q8—44no—
561Q8—44no—
562Q8—82no—
563Q8—82no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—416no—
591C2×C4—416no—
592C2×C4—432no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—816no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—82no—
630C2×C4—81no—
631C2×C4—82no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4#39614yes—
648C2×C4#45014yes—
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×#12112yes—
651C2×C4C2×C22⋊Q811yescharacteristic
652C2×C4C2×C22⋊Q811yescharacteristic
653C2×C4#52512yes—
654C2×C4C2×#12611yescharacteristic
655C2×C4#53712yes—
656C2×C4#53912yes—
657C2×C4D4212yes—
658C2×C4#54612yes—
659C2×C4#54712yes—
660C2×C4#54812yes—
661C2×C4D4×Q812yes—
662C2×C4#55212yes—
663C2×C4#55312yes—
664C2×C4#55412yes—
665C8—44no—
666C8—82no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22#206618yescomplemented
682C22C2×#39611yescharacteristic
683C22#240911yescharacteristic
684C22#241011yescharacteristic
685C22#241311yescharacteristic
686C22#257911yescharacteristic
687C22#261811yescharacteristic
688C22#284711yescharacteristic
689C22C2×#45011yescharacteristic
690C22#300211yescharacteristic
691C22#300411yescharacteristic
692C22#300711yescharacteristic
693C22C2×#46111yescharacteristic
694C22C2×#46411yescharacteristic
695C22#303611yescharacteristic
696C22#303711yescharacteristic
697C22C2×#46511yescharacteristic
698C22C2×#47011yescharacteristic
699C22#305511yescharacteristic
700C22#305611yescharacteristic
701C22C2×#47511yescharacteristic
702C22C2×#47911yescharacteristic
703C22#307911yescharacteristic
704C22#308011yescharacteristic
705C22#310111yescharacteristic
706C22#310811yescharacteristic
707C22#311011yescharacteristic
708C22#311711yescharacteristic
709C22#318711yescharacteristic
710C22#318811yescharacteristic
711C22#319511yescharacteristic
712C22#319611yescharacteristic
713C22#322911yescharacteristic
714C22#323211yescharacteristic
715C22#323611yescharacteristic
716C22#323911yescharacteristic
717C4—24no—
718C4—24no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—48no—
723C4—84no—
724C4—82no—
725C4—82no—
726C4—161no—
727C4—161no—
728C4#241612yes—
729C4#241712yes—
730C4#244912yes—
731C4#248112yes—
732C2—28no—
733C2#1472911yescharacteristic
734C2C2×#206611yescharacteristic
735C2#2361711yescharacteristic
736C2#2430111yescharacteristic
737C2#2430711yescharacteristic
738C2#2437911yescharacteristic
739C2#2438611yescharacteristic
740C2#2460111yescharacteristic
741C2#2460611yescharacteristic
742C2#2924011yescharacteristic
743C2#2924411yescharacteristic
744C2#2975611yescharacteristic
745C2#2976111yescharacteristic
746C2#3290511yescharacteristic
747C2#3290911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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×#2066C2semidirect
8. #11227C2semidirect
9. #11220C2semidirect
10. #11067C2semidirect
681. C22#2066semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#20662
C2×#18722
C2×#18532
C2×#17972
#147022
#112272
#112202
#110672
#107302