Finite Group Databasebeta

← #414594

Subgroups — #414594

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#441C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic
3C2×#2581C211yesmaximal, characteristic
4C2×#1985C211yesmaximal, characteristic
5C2×#1805C211yesmaximal, characteristic
6C2×#1803C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11067C212yesmaximal
9#11067C212yesmaximal
10#11024C212yesmaximal
11#10843C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#441—24no—
17C2×#441—24no—
18C2×#441—24no—
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#97C2211yescharacteristic
26#2582—24no—
27#2582—24no—
28#2581—24no—
29#2581—24no—
30C2×#400—21no—
31C2×#398—22no—
32C2×#395—21no—
33C2×#391—21no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#389—21no—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#387—22no—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42#1985—24no—
43#1805—24no—
44#1803—24no—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C23×Q8D411yescharacteristic
52C23×Q8C2311yescharacteristic
53C23×Q8C2311yescharacteristic
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—22no—
57C22×Q16—22no—
58C22×Q16—22no—
59C22×Q16—22no—
60C22×Q16Q814yescomplemented
61C23×C8C2312yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8D414yes—
81C2×C4×Q8D414yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4D412yes—
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C42—21no—
95C22×C42—21no—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98#441—232no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#106—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—42no—
116C2×#103—42no—
117C2×#97—22no—
118C2×#97—22no—
119C2×#97D414yes—
120#400—28no—
121#398—216no—
122#395—28no—
123#391—28no—
124#391—24no—
125#391—24no—
126#391—24no—
127#391—24no—
128#389—28no—
129#389—24no—
130#389—24no—
131#389—24no—
132#389—24no—
133#387—216no—
134#384—22no—
135#384D414yes—
136#382—22no—
137#382D414yes—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—21no—
146#340—24no—
147#340—24no—
148#340—24no—
149#340—42no—
150#336—24no—
151#336—24no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—42no—
179C22×Q8C2×D412yes—
180C22×Q8C2×D412yes—
181C22×Q8C2×Q812yes—
182C22×Q8C2×Q812yes—
183C22×Q8#4112yes—
184C22×Q8#4112yes—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2411yescharacteristic
196C2×Q16—28no—
197C2×Q16—28no—
198C2×Q16—28no—
199C2×Q16—28no—
200C2×Q16—28no—
201C2×Q16—28no—
202C2×Q16—28no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8C2×D412yes—
221C22×C8C2×D412yes—
222C22×C8C2×Q812yes—
223C22×C8#4112yes—
224C4×Q8—28no—
225C4×Q8—28no—
226C4×Q8—28no—
227C4×Q8—28no—
228C4×Q8—44no—
229C4×Q8—44no—
230C4×Q8—44no—
231C4×Q8—44no—
232C4×Q8—44no—
233C4×Q8—44no—
234C4×Q8—44no—
235C4×Q8—44no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—84no—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42#4112yes—
299#106—28no—
300C4⋊C8—44no—
301#103—28no—
302#103—28no—
303#103—48no—
304#103—48no—
305#97—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314C24C22×D411yescharacteristic
315C2×Q8—28no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
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—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—44no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—44no—
374C2×Q8C22⋊Q814yes—
375C2×Q8C22⋊Q814yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
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—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
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—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4#12012yes—
428C22×C4#12012yes—
429C22×C4#12012yes—
430C22×C4#12012yes—
431C22×C4#12112yes—
432C22×C4#12112yes—
433C22×C4#13714yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×Q811yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C42−1+411yescharacteristic
446C22×C42−1+411yescharacteristic
447C22×C42−1+411yescharacteristic
448Q16—232no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8C22⋊Q814yes—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—48no—
479C4⋊C4—416no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—84no—
493C4⋊C4—88no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—48no—
499C42—48no—
500C42#12014yes—
501C42#12114yes—
502C42#12114yes—
503C42#12714yes—
504C23C2×#12011yescharacteristic
505C23C2×#12011yescharacteristic
506C23C2×#12111yescharacteristic
507C23#54811yescharacteristic
508C23#54811yescharacteristic
509C23D4×Q811yescharacteristic
510C23#55011yescharacteristic
511C23C2×#13712yes—
512C23#57512yes—
513C23#57712yes—
514C23#57812yes—
515Q8—24no—
516Q8—24no—
517Q8—24no—
518Q8—24no—
519Q8—416no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—82no—
578C2×C4—84no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—84no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4#39214yes—
589C2×C4#39214yes—
590C2×C4C2×#12011yescharacteristic
591C2×C4C2×#12111yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4C2×C22⋊Q811yescharacteristic
595C2×C4C2×C22⋊Q811yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52511yescharacteristic
598C2×C4#52511yescharacteristic
599C2×C4C2×#12711yescharacteristic
600C2×C4#53611yescharacteristic
601C2×C4#53611yescharacteristic
602C2×C4#53611yescharacteristic
603C2×C4#53711yescharacteristic
604C2×C4#53711yescharacteristic
605C2×C4#53711yescharacteristic
606C2×C4#54611yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54711yescharacteristic
609C2×C4#54711yescharacteristic
610C2×C4#54811yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4D4×Q811yescharacteristic
613C2×C4D4×Q811yescharacteristic
614C2×C4#55211yescharacteristic
615C2×C4#55211yescharacteristic
616C2×C4#55411yescharacteristic
617C2×C4#55611yescharacteristic
618C8—24no—
619C8—42no—
620C8—82no—
621C22C2×#39211yescharacteristic
622C22#240711yescharacteristic
623C22#240711yescharacteristic
624C22#240911yescharacteristic
625C22#240911yescharacteristic
626C22#253611yescharacteristic
627C22#253811yescharacteristic
628C22#300411yescharacteristic
629C22#300511yescharacteristic
630C22#300712yes—
631C22#300911yescharacteristic
632C22#301111yescharacteristic
633C22#301312yes—
634C22#305612yes—
635C22#306312yes—
636C22#336912yes—
637C22#337112yes—
638C22#337412yes—
639C22#337712yes—
640C22#338712yes—
641C22#339012yes—
642C22#339812yes—
643C22#340112yes—
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—84no—
653C4—84no—
654C4#241511yescharacteristic
655C4#241511yescharacteristic
656C4#241711yescharacteristic
657C4#241711yescharacteristic
658C4#254011yescharacteristic
659C4#254211yescharacteristic
660C4#261611yescharacteristic
661C4#284811yescharacteristic
662C2#1517711yescharacteristic
663C2#2339211yescharacteristic
664C2#2339411yescharacteristic
665C2#2420211yescharacteristic
666C2#2420311yescharacteristic
667C2#2420712yes—
668C2#2434411yescharacteristic
669C2#2434511yescharacteristic
670C2#2434812yes—
671C2#2819812yes—
672C2#2827812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
60. C22×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4412
C2×#25822
C2×#25812
C2×#19852
C2×#18052
C2×#18032
#113272
#110672
#110242
#108432