Finite Group Databasebeta

← #533607

Subgroups — #533607

Order 512.

2752 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2#14609C212yesmaximal, complemented
3#14476C212yesmaximal, complemented
4#14243C214yesmaximal, complemented
5#14232C212yesmaximal, complemented
6#14082C214yesmaximal
7#14053C212yesmaximal, complemented
8#13994C212yesmaximal, complemented
9#13973C212yesmaximal, complemented
10#13871C211yesmaximal, characteristic
11#13810C212yesmaximal, complemented
12#13805C212yesmaximal, complemented
13#13801C212yesmaximal, complemented
14C4×#380C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2563C4116yescomplemented
21C2×#393C2212yes—
22C2×#393C2212yes—
23C2×#391C2214yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#386C2212yes—
28C2×#382C2212yes—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×#376C2214yes—
41#1446C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1440C2214yes—
46#1440C2212yes—
47#1439C2214yes—
48#1438C2212yes—
49#1437C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1437C2211yescharacteristic
53#1437C2212yes—
54#1437C2211yescharacteristic
55#1436C2214yes—
56#1436C2212yes—
57#1436C2211yescharacteristic
58#1436C2211yescharacteristic
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64#1434C2214yes—
65#1434C2214yes—
66#1434C2214yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C4×#95C2214yes—
74C4×#95C2214yes—
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8C2×C418yescomplemented
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C2×C4×D4C2×C418yes—
87C2×#117—28no—
88C2×#117—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42—28no—
107C22×C42—42no—
108C22×C42C2314yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2314yes—
118#393C2×C4116yes—
119#393C2×C4116yescomplemented
120#391—48no—
121#390C2×C4116yes—
122#389—44no—
123#388—48no—
124#386—28no—
125#382C2×C4116yescomplemented
126#380—28no—
127#380—42no—
128#380C2×C4116yes—
129#380C2×C4116yescomplemented
130C4×C4⋊C4—28no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—44no—
133C4×C4⋊C4C2×C418yescomplemented
134C4×C22⋊C4—28no—
135C4×C22⋊C4—48no—
136C4×C22⋊C4—42no—
137#376—216no—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—44no—
170C22×Q8—44no—
171C22×Q8C22×C418yes—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4C22×C414yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C22×C414yes—
193C23×C4C22×C414yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C4⋊Q8—416no—
208C4×D4—416no—
209#117—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C418yes—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C418yes—
230C2×C4⋊C4#4118yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4C22×C414yes—
256C2×C22⋊C4C22×C414yes—
257C2×C22⋊C4C22×C4116yes—
258C2×C22⋊C4C22×C4116yes—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—28no—
267C2×C42—24no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—28no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—416no—
277C2×C42—42no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42C22×C414yes—
284C2×C42C22×C418yes—
285C2×C42#4118yes—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—28no—
296#95—24no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—41no—
302#95—41no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95C22×C418yes—
310#95C22×C4116yes—
311#95C22×C4116yes—
312#95#4118yes—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24C23×C412yes—
329C24C23×C412yes—
330C242−1+414yes—
331C24C2511yescharacteristic
332C2×Q8—48no—
333C2×Q8—416no—
334C2×Q8—88no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4—48no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—216no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4C23×C412yes—
423C22×C4C23×C412yes—
424C22×C4C23×C418yes—
425C22×C4C23×C412yes—
426C22×C4C23×C418yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—216no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—84no—
457C42—216no—
458C42—28no—
459C42—216no—
460C42—416no—
461C42—44no—
462C42—44no—
463C42—48no—
464C42—416no—
465C42—48no—
466C42—88no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23#51914yes—
509C23C24×C411yescharacteristic
510C23C22×#4112yes—
511C23C22×#4112yes—
512C23C22×#4112yes—
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×2−1+412yes—
517C23#58512yes—
518Q8—816no—
519D4—416no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—432no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4C4×#4114yes—
598C2×C4C4×#4118yes—
599C2×C4#51814yes—
600C2×C4#55114yes—
601C2×C4#55114yes—
602C2×C4#55518yes—
603C2×C4#55518yes—
604C22—24no—
605C22—28no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—44no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22C2×C4×#4112yes—
623C22C2×C4×#4111yescharacteristic
624C22C2×#51811yescharacteristic
625C22C2×#51911yescharacteristic
626C22C4×H(5,𝔽2)12yes—
627C22C2×#55111yescharacteristic
628C22C2×#55111yescharacteristic
629C22C2×#55512yes—
630C22C2×#55512yes—
631C22#347512yes—
632C22#347512yes—
633C22#348312yes—
634C22#349212yes—
635C22#349512yes—
636C22#351612yes—
637C22#352012yes—
638C22#352012yes—
639C22#352012yes—
640C22#354212yes—
641C22#354911yescharacteristic
642C22#355111yescharacteristic
643C4—28no—
644C4—24no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—44no—
649C4—48no—
650C4—42no—
651C4—48no—
652C4—48no—
653C4—82no—
654C4—84no—
655C4#237218yes—
656C2—44no—
657C2—82no—
658C2C2×#237211yescharacteristic
659C2C4×#55512yes—
660C2#3424511yescharacteristic
661C2#3429912yes—
662C2#3441511yescharacteristic
663C2#5920312yes—
664C2#5923312yes—
665C2#5923712yes—
666C2#5938012yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14609C2semidirect
3. #14476C2semidirect
4. #14243C2semidirect
5. #14232C2semidirect
7. #14053C2semidirect
8. #13994C2semidirect
9. #13973C2semidirect
11. #13810C2semidirect
12. #13805C2semidirect
13. #13801C2semidirect
14. C4×#380C2semidirect
20. #2563C4semidirect
83. C2×C4⋊Q8C2×C4semidirect
119. #393C2×C4semidirect
125. #382C2×C4semidirect
129. #380C2×C4semidirect
133. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#146092
#144762
#142432
#142322
#140822
#140532
#139942
#139732
#138712
#138102
#138052
#138012
C4×#3802
C4×#3862