Finite Group Databasebeta

← #615580

Subgroups — #615580

Order 512.

4446 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C212yesmaximal
2C22×#380C211yesmaximal, characteristic
3C22×C4×C22⋊C4C211yesmaximal, characteristic
4#15132C214yesmaximal, complemented
5#15061C214yesmaximal, complemented
6#15019C212yesmaximal, complemented
7#15014C212yesmaximal, complemented
8#14697C212yesmaximal, complemented
9#14685C212yesmaximal, complemented
10#14680C211yesmaximal, characteristic
11#14130C212yesmaximal, complemented
12#13992C214yesmaximal, complemented
13#13904C212yesmaximal, complemented
14#13869C212yesmaximal
15C25×C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2483—216no—
24C2×#397C2218yes—
25C2×#389C2214yescomplemented
26C2×#385C2214yes—
27C2×#385C2214yes—
28C2×#381C2214yes—
29C2×#381C2214yes—
30C2×#381C2214yes—
31C2×#380—28no—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#379C2212yes—
40C2×C4×C22⋊C4—28no—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2212yes—
47C22×#95C2211yescharacteristic
48#1445C2214yescomplemented
49#1445C2214yescomplemented
50#1444C2214yescomplemented
51#1443C2218yescomplemented
52#1442C2218yescomplemented
53#1442C2214yescomplemented
54#1440C2218yes—
55#1436C2218yescomplemented
56C4⋊#95C2214yescomplemented
57C4⋊#95C2214yescomplemented
58C4⋊#95C2214yescomplemented
59#1434C2218yescomplemented
60C22⋊#95C2214yes—
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C26C2311yescharacteristic
67C24×C4—28no—
68C24×C4—28no—
69C24×C4—44no—
70C24×C4D418yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2318yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4D418yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C42—24no—
111C22×C42—24no—
112C22×C42—28no—
113C22×C42—44no—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118#397—232no—
119#389—48no—
120#385—216no—
121#385—216no—
122#381—216no—
123#381—216no—
124#381—216no—
125#380—28no—
126#380—28no—
127#380—28no—
128#380—28no—
129#380—28no—
130#380—28no—
131#380—44no—
132#379—28no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—28no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95—28no—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C25—24no—
157C25—24no—
158C25—24no—
159C25—24no—
160C25—42no—
161C25—42no—
162C25C2×D414yes—
163C25C2×D414yes—
164C25#4114yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—232no—
183C23×C4—232no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—416no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—464no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4#4118yes—
223C23×C4#4118yes—
224C23×C4C2418yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C22⋊C4—232no—
252C2×C22⋊C4—232no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—432no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×D418yes—
279C2×C22⋊C4C2×D418yes—
280C2×C22⋊C4#4118yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—44no—
289C2×C42—416no—
290C2×C42—416no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—216no—
298#95—216no—
299#95—216no—
300#95—28no—
301#95—28no—
302#95—216no—
303#95—216no—
304#95—48no—
305#95—44no—
306#95—48no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—28no—
316C24—28no—
317C24—28no—
318C24—216no—
319C24—216no—
320C24—28no—
321C24—216no—
322C24—216no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—44no—
330C24—44no—
331C24—48no—
332C24—48no—
333C24—416no—
334C24—416no—
335C24—416no—
336C24—416no—
337C24#121116yes—
338C24C22×D412yes—
339C24C22×D412yes—
340C24C22×D412yes—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C2×#4112yes—
345C24C2×#4112yes—
346C24C2×#4112yes—
347C24C2×#4114yes—
348C24C2×#4114yes—
349C24C2×#4114yes—
350C242−1+414yes—
351C24C2511yescharacteristic
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—232no—
361C22×C4—232no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—216no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—416no—
417C22×C4—4128no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4#123116yes—
444C22×C4#123116yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—816no—
464C22⋊C4—216no—
465C22⋊C4—216no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—88no—
473C42—48no—
474C42—48no—
475C42—48no—
476C42—48no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—232no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—232no—
515C23—416no—
516C23—416no—
517C23—416no—
518C23—464no—
519C23—464no—
520C23—416no—
521C23—416no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C2×#12114yes—
527C23C2×#12114yes—
528C23C2×#12114yes—
529C23C2×#12314yes—
530C23C2×#12314yes—
531C23#52514yes—
532C23#52918yes—
533C23#53218yes—
534C23#53614yes—
535C23#54718yes—
536C23#55218yes—
537C23C23×D411yescharacteristic
538C23C23×D411yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4112yes—
541C23C22×#4112yes—
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+412yes—
545C23#58512yes—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—232no—
553C2×C4—232no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—416no—
566C2×C4—416no—
567C2×C4—416no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—416no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—416no—
584C2×C4—44no—
585C2×C4—416no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—816no—
604C2×C4—816no—
605C2×C4C2×#12314yes—
606C2×C4C2×#12314yes—
607C2×C4C2×#12314yes—
608C2×C4C2×#12314yes—
609C2×C4#53414yes—
610C2×C4#551116yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—216no—
615C22—216no—
616C22—216no—
617C22—216no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—464no—
631C22—42no—
632C22—48no—
633C22—48no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22#248618yes—
638C22#285418yes—
639C22C22×#12111yescharacteristic
640C22C22×#12311yescharacteristic
641C22C22×#12311yescharacteristic
642C22C2×#52511yescharacteristic
643C22C2×#52912yes—
644C22C2×#53212yes—
645C22C2×#53411yescharacteristic
646C22C2×#53611yescharacteristic
647C22#344911yescharacteristic
648C22C2×#54712yes—
649C22C2×#55114yes—
650C22C2×#55212yes—
651C22#347514yes—
652C22#349514yes—
653C22#349612yes—
654C22#350512yes—
655C22#352014yes—
656C4—24no—
657C4—24no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—84no—
664C4—88no—
665C4—84no—
666C4#248318yescomplemented
667C2—28no—
668C2—28no—
669C2—44no—
670C2C2×#248311yescharacteristic
671C2C2×#248611yescharacteristic
672C2#3480911yescharacteristic
673C2#3500011yescharacteristic
674C2#3508411yescharacteristic
675C2#3531111yescharacteristic
676C2#3616714yes—
677C2#3618314yes—
678C2C2×#285411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #15132C2semidirect
5. #15061C2semidirect
6. #15019C2semidirect
7. #15014C2semidirect
8. #14697C2semidirect
9. #14685C2semidirect
11. #14130C2semidirect
12. #13992C2semidirect
13. #13904C2semidirect
25. C2×#389C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
55. #1436C22semidirect
56. C4⋊#95C22semidirect
57. C4⋊#95C22semidirect
58. C4⋊#95C22semidirect
59. #1434C22semidirect
666. C4#2483semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C22×#3802
C22×C4×C22⋊C42
#151322
#150612
#150192
#150142
#146972
#146852
#146802
#141302
#139922
#139042
#138692