Finite Group Databasebeta

← #558165

Subgroups — #558165

Order 512.

3030 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#14702C212yesmaximal, complemented
3#14655C212yesmaximal, complemented
4#14599C212yesmaximal, complemented
5#14596C214yesmaximal, complemented
6#14515C212yesmaximal, complemented
7#14427C214yesmaximal
8#14142C212yesmaximal, complemented
9#14003C212yesmaximal, complemented
10#13997C212yesmaximal, complemented
11#13974C212yesmaximal, complemented
12#13922C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13831C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2409C4116yescomplemented
21C2×#395C2212yes—
22C2×#393C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2212yes—
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1446C2214yes—
42#1446C2214yes—
43#1445C2212yes—
44#1445C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1438C2212yes—
53#1437C2212yes—
54#1437C2214yes—
55#1437C2211yescharacteristic
56#1437C2212yes—
57#1437C2211yescharacteristic
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2211yescharacteristic
62#1436C2214yes—
63#1436C2211yescharacteristic
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C23×Q8—42no—
77C23×Q8—42no—
78C23×Q8—44no—
79C23×Q8—44no—
80C23×Q8C2×C418yescomplemented
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88C2×C22⋊Q8—44no—
89C2×C22⋊Q8—44no—
90C2×C22⋊Q8C2×C4116yes—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C2×C4×D4C2×C418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C42—44no—
110C22×C42—42no—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118#395C2×C4116yescomplemented
119#393C2×C4116yes—
120#391—44no—
121#391—44no—
122#391—44no—
123#391—44no—
124#391C2×C4116yescomplemented
125#389—44no—
126#388—216no—
127#386—28no—
128#386C2×C4116yes—
129#385C2×C4116yescomplemented
130#382—42no—
131#380—44no—
132#380—44no—
133#380C2×C418yescomplemented
134C4×C4⋊C4—42no—
135C4×C4⋊C4—44no—
136C4×C22⋊C4—44no—
137C4×C22⋊C4—42no—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—416no—
181C22×Q8—432no—
182C22×Q8—432no—
183C22×Q8—416no—
184C22×Q8—416no—
185C22×Q8C22×C418yes—
186C22×Q8C22×C418yes—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4C22×C414yes—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4C22×C414yes—
212C23×C4C22×C414yes—
213C23×C4C22×C418yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C22⋊Q8—432no—
228C4×D4—416no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C22×C4116yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C414yes—
271C2×C22⋊C4C22×C4116yes—
272C2×C22⋊C4#4118yes—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—416no—
287C2×C42—44no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C22×C414yes—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—28no—
298#95—44no—
299#95—42no—
300#95—41no—
301#95—41no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95C22×C418yes—
320#95C22×C4116yes—
321#95C22×C4116yes—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24C23×C412yes—
337C24C23×C412yes—
338C24C2×#4114yes—
339C242−1+414yes—
340C24C2511yescharacteristic
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—216no—
344C2×Q8—48no—
345C2×Q8—416no—
346C2×Q8—4128no—
347C2×Q8—48no—
348C2×Q8—416no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—48no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4C23×C412yes—
449C22×C4C23×C412yes—
450C22×C4C23×C418yes—
451C22×C4C23×C412yes—
452C22×C4C23×C418yes—
453C22×C4C23×C412yes—
454C22×C4C23×C412yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C42−1+414yes—
462C4⋊C4—216no—
463C4⋊C4—44no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C22⋊C4—216no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C42—28no—
478C42—44no—
479C42—44no—
480C42—48no—
481C42—48no—
482C42—88no—
483C42—84no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23C4×#4114yes—
525C23#51914yes—
526C23C24×C411yescharacteristic
527C23C22×#4112yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×2−1+412yes—
533C23C2×2−1+412yes—
534C23#58512yes—
535Q8—432no—
536Q8—432no—
537D4—416no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—432no—
563C2×C4—432no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4#51814yes—
619C2×C4#51818yes—
620C2×C4#56018yes—
621C2×C4#56014yes—
622C2×C4H(3,𝔽4)14yes—
623C22—28no—
624C22—28no—
625C22—24no—
626C22—24no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22C2×C4×#4111yescharacteristic
642C22C2×#51812yes—
643C22C2×#51811yescharacteristic
644C22C2×#51911yescharacteristic
645C22C4×2−1+412yes—
646C22#350512yes—
647C22#351112yes—
648C22#351412yes—
649C22#352012yes—
650C22#352012yes—
651C22#352012yes—
652C22C2×#56011yescharacteristic
653C22C2×#56012yes—
654C22C2×H(3,𝔽4)11yescharacteristic
655C22#353812yes—
656C22#354112yes—
657C22#354112yes—
658C22#355011yescharacteristic
659C22#355111yescharacteristic
660C22#355212yes—
661C22#356112yes—
662C4—48no—
663C4—42no—
664C4—42no—
665C4—48no—
666C4—44no—
667C4—44no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—84no—
672C4—82no—
673C4—84no—
674C4#237518yes—
675C2—44no—
676C2—44no—
677C2#3424511yescharacteristic
678C2#3432712yes—
679C2C2×#237511yescharacteristic
680C2#3441311yescharacteristic
681C2#3447512yes—
682C2#4566712yes—
683C2#4566912yes—
684C2#4567712yes—
685C2#4568712yes—
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. #14702C2semidirect
3. #14655C2semidirect
4. #14599C2semidirect
5. #14596C2semidirect
6. #14515C2semidirect
8. #14142C2semidirect
9. #14003C2semidirect
10. #13997C2semidirect
11. #13974C2semidirect
12. #13922C2semidirect
14. #13831C2semidirect
20. #2409C4semidirect
80. C23×Q8C2×C4semidirect
118. #395C2×C4semidirect
124. #391C2×C4semidirect
129. #385C2×C4semidirect
133. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#147022
#146552
#145992
#145962
#145152
#144272
#141422
#140032
#139972
#139742
#139222
#139052
#138312
C4×#3862