Finite Group Databasebeta

← #360396

Subgroups — #360396

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1802C211yesmaximal, characteristic
2C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C4×#102C211yesmaximal, characteristic
4#13911C211yesmaximal, characteristic, complemented
5#13906C211yesmaximal, characteristic
6#11315C212yesmaximal, complemented
7#11189C212yesmaximal
8#10925C212yesmaximal
9#10849C211yesmaximal, characteristic, complemented
10#10841C211yesmaximal, characteristic, complemented
11#10820C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×C4×C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#390C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382—21no—
23C2×#380—21no—
24C2×#380—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1802—24no—
28C8×C4⋊C4—24no—
29C4×#102C418yescomplemented
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#336C2212yes—
38#1442—22no—
39#1439—22no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8D414yes—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—42no—
82C2×#102C2×C414yes—
83C2×#102C2×C414yes—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—24no—
87C2×C4×C8—24no—
88C2×C4×C8—42no—
89C2×C4×C8C2×C414yes—
90#390—24no—
91#390—42no—
92#384—22no—
93#384—22no—
94#384—22no—
95#384—22no—
96#382—44no—
97#380—44no—
98#380—44no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D412yes—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340C2×C414yes—
117#340C2×C414yes—
118#336—24no—
119C25—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C22×C412yes—
132C22×D4#4112yes—
133C22×D4#4112yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8C22×C412yes—
168C22×C8C22×C412yes—
169C22×C8C2×D414yes—
170C22×C8#4112yes—
171C22×C8#4112yes—
172C4×D4—28no—
173C4×D4—28no—
174C4×D4—44no—
175C4×D4—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4D814yes—
220C2×C4⋊C4SD1614yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C22×C412yes—
243C2×C42C2×D412yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248C4⋊C8—28no—
249#102—28no—
250#102—28no—
251#102—48no—
252C4×C8—28no—
253C4×C8—48no—
254#95—24no—
255#95—24no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—82no—
261#95—82no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24C22×D411yescharacteristic
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4#11714yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
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—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
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—84no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4C4×D412yes—
365C22×C4C4×D412yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4#12412yes—
369C22×C4#12412yes—
370C22×C4C2×D814yes—
371C22×C4C2×D812yes—
372C22×C4C2×SD1614yes—
373C22×C4C2×SD1612yes—
374C22×C4#13512yes—
375C22×C4#13512yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C42−1+411yescharacteristic
389C22×C42−1+411yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8#11714yes—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C22⋊C4—48no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—48no—
445C42#11714yes—
446C42C4×D414yes—
447C42#12314yes—
448C42#12614yes—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23C2×C4×D411yescharacteristic
485C23C2×#12111yescharacteristic
486C23C2×#12411yescharacteristic
487C23#54611yescharacteristic
488C23#54711yescharacteristic
489C23#54811yescharacteristic
490C23#55011yescharacteristic
491C23C22×D811yescharacteristic
492C23C22×SD1611yescharacteristic
493C23C2×#13511yescharacteristic
494C23C2×#13511yescharacteristic
495C23#57611yescharacteristic
496C23#57711yescharacteristic
497C23#57711yescharacteristic
498C23#57811yescharacteristic
499D4—24no—
500D4—24no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—816no—
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—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4#38814yes—
564C2×C4#38814yes—
565C2×C4C4×D814yes—
566C2×C4C4×SD1614yes—
567C2×C4#46514yes—
568C2×C4#46614yes—
569C2×C4#48614yes—
570C2×C4#48814yes—
571C2×C4C2×#11711yescharacteristic
572C2×C4C2×#11711yescharacteristic
573C2×C4C2×#11711yescharacteristic
574C2×C4C2×C4×D411yescharacteristic
575C2×C4#52011yescharacteristic
576C2×C4#52011yescharacteristic
577C2×C4C2×#12311yescharacteristic
578C2×C4C2×#12611yescharacteristic
579C2×C4#52911yescharacteristic
580C2×C4#52911yescharacteristic
581C2×C4#52911yescharacteristic
582C2×C4#52911yescharacteristic
583C2×C4#54011yescharacteristic
584C2×C4#54111yescharacteristic
585C2×C4#54211yescharacteristic
586C2×C4#54211yescharacteristic
587C2×C4#54611yescharacteristic
588C2×C4#54611yescharacteristic
589C2×C4#54811yescharacteristic
590C2×C4#54811yescharacteristic
591C2×C4#55211yescharacteristic
592C2×C4#55211yescharacteristic
593C2×C4#55211yescharacteristic
594C2×C4#55311yescharacteristic
595C2×C4#55311yescharacteristic
596C2×C4#55311yescharacteristic
597C2×C4#55511yescharacteristic
598C2×C4#55611yescharacteristic
599C8—24no—
600C8—42no—
601C8—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22C2×#38811yescharacteristic
618C22#235211yescharacteristic
619C22#235411yescharacteristic
620C22#242611yescharacteristic
621C22#242711yescharacteristic
622C22#248911yescharacteristic
623C22#249011yescharacteristic
624C22C2×C4×D811yescharacteristic
625C22C2×C4×SD1611yescharacteristic
626C22#295111yescharacteristic
627C22#295211yescharacteristic
628C22C2×#46511yescharacteristic
629C22C2×#46611yescharacteristic
630C22#306211yescharacteristic
631C22#306311yescharacteristic
632C22C2×#48611yescharacteristic
633C22C2×#48811yescharacteristic
634C22#313611yescharacteristic
635C22#313711yescharacteristic
636C22#329411yescharacteristic
637C22#329511yescharacteristic
638C22#330211yescharacteristic
639C22#330311yescharacteristic
640C22#333411yescharacteristic
641C22#333511yescharacteristic
642C22#334011yescharacteristic
643C22#334111yescharacteristic
644C22#335811yescharacteristic
645C22#335911yescharacteristic
646C22#336011yescharacteristic
647C22#336111yescharacteristic
648C22#339111yescharacteristic
649C22#339211yescharacteristic
650C22#339311yescharacteristic
651C22#339411yescharacteristic
652C4—24no—
653C4—44no—
654C4—44no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—162no—
659C4—162no—
660C4#193418yes—
661C4#228311yescharacteristic
662C4#231411yescharacteristic
663C4#235211yescharacteristic
664C4#235211yescharacteristic
665C4#242011yescharacteristic
666C4#242111yescharacteristic
667C4#249411yescharacteristic
668C4#249511yescharacteristic
669C2—84no—
670C2#1387511yescharacteristic
671C2C2×#193411yescharacteristic
672C2#2108711yescharacteristic
673C2#2276611yescharacteristic
674C2#2277011yescharacteristic
675C2#2280511yescharacteristic
676C2#2280711yescharacteristic
677C2#2529211yescharacteristic
678C2#2529311yescharacteristic
679C2#2533811yescharacteristic
680C2#2533911yescharacteristic
681C2#2772711yescharacteristic
682C2#2772811yescharacteristic
683C2#2774711yescharacteristic
684C2#2774811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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×C8×C4⋊C4C2semidirect
4. #13911C2semidirect
6. #11315C2semidirect
9. #10849C2semidirect
10. #10841C2semidirect
11. #10820C2semidirect
29. C4×#102C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18022
C2×C8×C4⋊C42
C2×C4×#1022
#139112
#139062
#113152
#111892
#109252
#108492
#108412
#108202