Finite Group Databasebeta

← #414930

Subgroups — #414930

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2C2×#1939C211yesmaximal, characteristic, complemented
3C2×#1803C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic, complemented
6#13906C211yesmaximal, characteristic
7#11257C212yesmaximal, complemented
8#11077C212yesmaximal, complemented
9#11076C212yesmaximal
10#11023C212yesmaximal
11#10841C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C4×SD16—24no—
16C2×C4×SD16—24no—
17C2×C4×SD16—24no—
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#380—21no—
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1939—24no—
34#1803—24no—
35#1802—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42#1439—22no—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C22×SD16—22no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16Q814yes—
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8D414yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C42—21no—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89C4×SD16—232no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—42no—
98C2×C4⋊C8—42no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—24no—
109C2×#102—42no—
110C2×#102—42no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8D414yes—
114#391—24no—
115#391—42no—
116#390—24no—
117#390—42no—
118#384—22no—
119#384—22no—
120#384—22no—
121#384—22no—
122#380—44no—
123#380—44no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D412yes—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—42no—
137#336—24no—
138C25—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8C2×D412yes—
151C22×Q8C2×Q812yes—
152C22×Q8#4112yes—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C2×D412yes—
165C22×D4C2×Q812yes—
166C22×D4#4112yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×SD16—28no—
185C2×SD16—28no—
186C2×SD16—28no—
187C2×SD16—28no—
188C2×SD16—28no—
189C2×SD16—28no—
190C2×SD16—28no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×Q812yes—
211C22×C8#4112yes—
212C4×Q8—28no—
213C4×Q8—28no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—44no—
219C4×D4—44no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—81no—
259C2×C4⋊C4—81no—
260C2×C4⋊C4—81no—
261C2×C4⋊C4—81no—
262C2×C4⋊C4SD1614yescomplemented
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42#4112yes—
287C2×C42#4112yes—
288C2×C42#4112yes—
289#106—28no—
290C4⋊C8—48no—
291C4⋊C8—44no—
292#103—28no—
293#103—48no—
294#102—28no—
295#102—48no—
296C4×C8—28no—
297#95—24no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—82no—
303#95—82no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24C22×D411yescharacteristic
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8C22⋊Q814yes—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4C22⋊Q814yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4#12112yes—
436C22×C4#12112yes—
437C22×C4#12112yes—
438C22×C4#12112yes—
439C22×C4#12112yes—
440C22×C4#12112yes—
441C22×C4C2×SD1614yes—
442C22×C4C2×SD1612yes—
443C22×C4#13512yes—
444C22×C4AGL(1,ℤ/8)12yes—
445C22×C4#13712yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×Q811yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4H(5,𝔽2)11yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459C22×C42−1+411yescharacteristic
460SD16—232no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8C22⋊Q814yes—
482C4⋊C4—28no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C22⋊C4—48no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—48no—
513C42#12014yes—
514C42#12314yes—
515C42#12314yes—
516C42#12414yes—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23—81no—
547C23—81no—
548C23—81no—
549C23—81no—
550C23—81no—
551C23—81no—
552C23C2×#12111yescharacteristic
553C23C2×#12111yescharacteristic
554C23C2×#12111yescharacteristic
555C23#54611yescharacteristic
556C23#54611yescharacteristic
557C23D4×Q811yescharacteristic
558C23#55011yescharacteristic
559C23C22×SD1611yescharacteristic
560C23C2×#13511yescharacteristic
561C23C2×AGL(1,ℤ/8)11yescharacteristic
562C23C2×#13711yescharacteristic
563C23#57611yescharacteristic
564C23#57711yescharacteristic
565C23#57711yescharacteristic
566C23#57811yescharacteristic
567Q8—24no—
568Q8—24no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575D4—24no—
576D4—24no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—816no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4#39414yes—
640C2×C4#39414yes—
641C2×C4#46014yes—
642C2×C4#46114yes—
643C2×C4#46314yes—
644C2×C4#46414yes—
645C2×C4#46514yes—
646C2×C4#46814yes—
647C2×C4C2×#12011yescharacteristic
648C2×C4C2×C22⋊Q811yescharacteristic
649C2×C4C2×C22⋊Q811yescharacteristic
650C2×C4C2×C22⋊Q811yescharacteristic
651C2×C4C2×#12311yescharacteristic
652C2×C4C2×#12311yescharacteristic
653C2×C4#52511yescharacteristic
654C2×C4#52511yescharacteristic
655C2×C4#52511yescharacteristic
656C2×C4C2×#12411yescharacteristic
657C2×C4#53511yescharacteristic
658C2×C4#53511yescharacteristic
659C2×C4#53511yescharacteristic
660C2×C4#53711yescharacteristic
661C2×C4#53711yescharacteristic
662C2×C4#53711yescharacteristic
663C2×C4#55211yescharacteristic
664C2×C4#55211yescharacteristic
665C2×C4#55311yescharacteristic
666C2×C4#55311yescharacteristic
667C2×C4#55311yescharacteristic
668C2×C4#55311yescharacteristic
669C2×C4#55411yescharacteristic
670C2×C4#55411yescharacteristic
671C2×C4#55411yescharacteristic
672C2×C4#55511yescharacteristic
673C2×C4#55511yescharacteristic
674C2×C4#55611yescharacteristic
675C8—24no—
676C8—42no—
677C8—82no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22C2×#39411yescharacteristic
694C22#241411yescharacteristic
695C22#241411yescharacteristic
696C22#241711yescharacteristic
697C22#241711yescharacteristic
698C22#253511yescharacteristic
699C22#253811yescharacteristic
700C22C2×#46111yescharacteristic
701C22C2×#46311yescharacteristic
702C22C2×#46011yescharacteristic
703C22C2×#46411yescharacteristic
704C22#304511yescharacteristic
705C22#304511yescharacteristic
706C22#304611yescharacteristic
707C22#304611yescharacteristic
708C22C2×#46511yescharacteristic
709C22C2×#46811yescharacteristic
710C22#306011yescharacteristic
711C22#306111yescharacteristic
712C22#329511yescharacteristic
713C22#329811yescharacteristic
714C22#330211yescharacteristic
715C22#330511yescharacteristic
716C22#332411yescharacteristic
717C22#332911yescharacteristic
718C22#333111yescharacteristic
719C22#333211yescharacteristic
720C22Q8×SD1611yescharacteristic
721C22#338111yescharacteristic
722C22#338511yescharacteristic
723C22#338611yescharacteristic
724C22#339211yescharacteristic
725C22#339411yescharacteristic
726C22#339811yescharacteristic
727C22#339911yescharacteristic
728C4—24no—
729C4—44no—
730C4—44no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—162no—
735C4—162no—
736C4#205518yes—
737C4#248511yescharacteristic
738C4#248511yescharacteristic
739C4#248711yescharacteristic
740C4#248711yescharacteristic
741C4#256211yescharacteristic
742C4#256311yescharacteristic
743C4#261511yescharacteristic
744C4#284811yescharacteristic
745C2—84no—
746C2#1518111yescharacteristic
747C2C2×#205511yescharacteristic
748C2#2354311yescharacteristic
749C2#2462811yescharacteristic
750C2#2462911yescharacteristic
751C2#2463011yescharacteristic
752C2#2463111yescharacteristic
753C2#2481311yescharacteristic
754C2#2481411yescharacteristic
755C2#2481511yescharacteristic
756C2#2481611yescharacteristic
757C2#2814111yescharacteristic
758C2#2815011yescharacteristic
759C2#2826411yescharacteristic
760C2#2827211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1939C2semidirect
3. C2×#1803C2semidirect
5. #13926C2semidirect
7. #11257C2semidirect
8. #11077C2semidirect
11. #10841C2semidirect
262. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
C2×#19392
C2×#18032
C2×#18022
#139262
#139062
#112572
#110772
#110762
#110232
#108412