Finite Group Databasebeta

← #556906

Subgroups — #556906

Order 512.

2886 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×C4×#121C211yesmaximal, characteristic
2#14670C212yesmaximal, complemented
3#14667C212yesmaximal, complemented
4#14355C212yesmaximal, complemented
5#14351C212yesmaximal, complemented
6#14275C212yesmaximal, complemented
7#14271C212yesmaximal
8#14268C214yesmaximal, complemented
9#14140C214yesmaximal, complemented
10#13929C212yesmaximal, complemented
11#13912C212yesmaximal, complemented
12#13899C212yesmaximal, complemented
13#13894C212yesmaximal, complemented
14C4×#380C211yesmaximal, characteristic, complemented
15C2×#1434C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C4×#121C4116yescomplemented
21C2×#390C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388C2214yes—
24C2×#388C2212yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#385C2214yes—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41#1445C2214yes—
42#1445C2214yes—
43#1444C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68#1434C2212yes—
69#1434C2212yes—
70#1434C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×#121—28no—
85C2×#121C2×C418yes—
86C2×C4×D4—44no—
87C2×C4×D4—48no—
88C2×C4×D4C2×C4116yes—
89C2×C4×D4C2×C418yescomplemented
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C42—44no—
109C22×C42C2×C418yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116#390C2×C418yescomplemented
117#389C2×C418yescomplemented
118#388—48no—
119#388C2×C4116yescomplemented
120#386—48no—
121#386—48no—
122#386C2×C4116yes—
123#385—48no—
124#381—44no—
125#380—28no—
126#380—44no—
127#380C2×C418yes—
128C4×C4⋊C4—44no—
129C4×C4⋊C4C2×C418yescomplemented
130C4×C22⋊C4—44no—
131C4×C22⋊C4C2×C4116yescomplemented
132C2×#95—28no—
133C2×#95—42no—
134C2×#95—42no—
135C2×#95—44no—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C25C2412yes—
164C25C2411yescharacteristic
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—42no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4C22×C418yes—
174C22×D4C22×C414yes—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—42no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C22×C414yes—
205C23×C4C22×C414yes—
206C23×C4C22×C418yes—
207C23×C4C22×C414yes—
208C23×C4#4118yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#121—232no—
223C4×D4—416no—
224C4×D4—432no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C414yes—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C418yes—
270C2×C22⋊C4C22×C418yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—28no—
274C2×C42—416no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42C22×C414yes—
281C2×C42C22×C414yes—
282C2×C42C22×C418yes—
283C2×C42C22×C414yes—
284C2×C42C22×C414yes—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—42no—
294#95—42no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95C22×C418yes—
310#95C22×C418yes—
311#95C22×C418yes—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—22no—
328C24C23×C414yes—
329C24C23×C412yes—
330C24C2×#4114yes—
331C24H(5,𝔽2)14yes—
332C24H(5,𝔽2)12yes—
333C242−1+412yes—
334C24C2511yescharacteristic
335C2×D4—216no—
336C2×D4—216no—
337C2×D4—216no—
338C2×D4—28no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—416no—
344C2×D4—416no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4#117116yes—
438C22×C4C23×C412yes—
439C22×C4C23×C414yes—
440C22×C4C23×C414yes—
441C22×C4C23×C414yes—
442C22×C4C23×C412yes—
443C22×C4C23×C412yes—
444C22×C4C23×C412yes—
445C22×C4C23×C412yes—
446C22×C4C23×C412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4H(5,𝔽2)14yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C4⋊C4—28no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C22⋊C4—216no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C42—28no—
471C42—28no—
472C42—44no—
473C42—44no—
474C42—48no—
475C42—88no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23C2×#11714yes—
518C23#51814yes—
519C23#52014yes—
520C23#54018yes—
521C23#54218yes—
522C23C24×C411yescharacteristic
523C23C22×#4112yes—
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23C2×2−1+411yescharacteristic
530C23#58511yescharacteristic
531C23#58511yescharacteristic
532C23#58512yes—
533D4—416no—
534D4—432no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—416no—
567C2×C4—416no—
568C2×C4—416no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—816no—
608C2×C4—816no—
609C2×C4—816no—
610C2×C4C2×#11714yes—
611C2×C4C2×#11714yes—
612C2×C4#51814yes—
613C2×C4#52014yes—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—28no—
620C22—24no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22#231918yes—
636C22C22×#11711yescharacteristic
637C22C2×#51811yescharacteristic
638C22C2×#51811yescharacteristic
639C22C2×#52011yescharacteristic
640C22C2×#52011yescharacteristic
641C22#342811yescharacteristic
642C22#342811yescharacteristic
643C22C2×#54012yes—
644C22C2×#54212yes—
645C22#345912yes—
646C22#345912yes—
647C22#346012yes—
648C22#346112yes—
649C22#353412yes—
650C22#353512yes—
651C22#353612yes—
652C22#353712yes—
653C22#355811yescharacteristic
654C22#355911yescharacteristic
655C22#356012yes—
656C22#356012yes—
657C22#356111yescharacteristic
658C22#356311yescharacteristic
659C4—24no—
660C4—24no—
661C4—24no—
662C4—24no—
663C4—44no—
664C4—42no—
665C4—44no—
666C4—42no—
667C4—44no—
668C4—84no—
669C4—84no—
670C4—84no—
671C4—84no—
672C2—28no—
673C2—84no—
674C2C2×#231911yescharacteristic
675C2#3405711yescharacteristic
676C2#3406011yescharacteristic
677C2#3437911yescharacteristic
678C2#3438211yescharacteristic
679C2#3438611yescharacteristic
680C2#3438811yescharacteristic
681C2#3745912yes—
682C2#3746212yes—
683C2#3746412yes—
684C2#3746512yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14670C2semidirect
3. #14667C2semidirect
4. #14355C2semidirect
5. #14351C2semidirect
6. #14275C2semidirect
8. #14268C2semidirect
9. #14140C2semidirect
10. #13929C2semidirect
11. #13912C2semidirect
12. #13899C2semidirect
13. #13894C2semidirect
14. C4×#380C2semidirect
15. C2×#1434C2semidirect
20. C4×#121C4semidirect
89. C2×C4×D4C2×C4semidirect
116. #390C2×C4semidirect
117. #389C2×C4semidirect
119. #388C2×C4semidirect
129. C4×C4⋊C4C2×C4semidirect
131. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#146702
#146672
#143552
#143512
#142752
#142712
#142682
#141402
#139292
#139122
#138992
#138942
C4×#3802
C2×#14342