Finite Group Databasebeta

← #615504

Subgroups — #615504

Order 512.

4449 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic
3C2×#2280C211yesmaximal, characteristic, complemented
4C22×C4×C22⋊C4C211yesmaximal, characteristic, complemented
5#15002C214yesmaximal, complemented
6#14997C214yesmaximal, complemented
7#14727C212yesmaximal, complemented
8#14718C212yesmaximal
9#14706C211yesmaximal, characteristic, complemented
10#14705C211yesmaximal, characteristic, complemented
11#14684C211yesmaximal, characteristic, complemented
12#14202C214yesmaximal, complemented
13#14199C214yesmaximal, complemented
14#14135C212yesmaximal, complemented
15#13962C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#120C2211yescharacteristic
20C22×#117C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2853—28no—
25#2849—28no—
26C2×#397C2214yes—
27C2×#394C2214yes—
28C2×#388C2214yescomplemented
29C2×#387C2214yescomplemented
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38#2280—28no—
39C2×#381C2211yescharacteristic
40C2×#381C2211yescharacteristic
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#380C2214yes—
44C2×#379C2211yescharacteristic
45C2×#379C2212yes—
46C2×C4×C22⋊C4—28no—
47C2×C4×C22⋊C4C2214yes—
48C2×C4×C22⋊C4C2214yes—
49C2×C4×C22⋊C4C2214yes—
50C22×#95C2212yes—
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yescomplemented
60#1438C2214yes—
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67C4⋊#95C2214yescomplemented
68#1434C2214yescomplemented
69#1434C2214yescomplemented
70#1434C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C26C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—28no—
81C24×C4—44no—
82C24×C4D418yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—24no—
90C2×#123—24no—
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—28no—
94C2×#120—24no—
95C2×#120—44no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—216no—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42—24no—
125C22×C42—24no—
126C22×C42—24no—
127C22×C42—24no—
128C22×C42—44no—
129C22×C42C2314yes—
130C22×C42C2314yes—
131C22×C42C2314yes—
132#397—216no—
133#394—216no—
134#388—216no—
135#387—216no—
136#386—28no—
137#386—28no—
138#386—28no—
139#386—44no—
140#385—28no—
141#385—28no—
142#385—28no—
143#385—44no—
144#381—24no—
145#381—24no—
146#381—24no—
147#381D418yescomplemented
148#380—216no—
149#379—24no—
150#379—28no—
151C4×C22⋊C4—216no—
152C4×C22⋊C4—216no—
153C4×C22⋊C4—216no—
154C2×#95—28no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25—28no—
176C25—24no—
177C25—22no—
178C25—22no—
179C25—24no—
180C25—44no—
181C25C2×D414yes—
182C25C2411yescharacteristic
183C25C2412yes—
184C25C2411yescharacteristic
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—44no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—232no—
203C23×C4—232no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—416no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—432no—
226C23×C4—432no—
227C23×C4C2×D418yes—
228C23×C4C2×D418yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4#4114yes—
233C23×C4#4118yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246#123—416no—
247#123—416no—
248#120—416no—
249#117—432no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C22⋊C4—264no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4#41116yes—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—416no—
322C2×C42—416no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95C2×D4116yes—
343C24—24no—
344C24—24no—
345C24—232no—
346C24—28no—
347C24—216no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—28no—
358C24—28no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—24no—
364C24—22no—
365C24—24no—
366C24—48no—
367C24—416no—
368C24—416no—
369C24—416no—
370C24—416no—
371C24—416no—
372C24—48no—
373C24—48no—
374C24C22×D414yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C2×#4118yes—
378C24C2×#4112yes—
379C24C2×#4114yes—
380C24H(5,𝔽2)14yes—
381C24C2511yescharacteristic
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—88no—
391C2×D4—88no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—4128no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4#121116yes—
482C22×C4#123116yes—
483C22×C4C22×D418yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C2×#4118yes—
489C22×C4C2×#4118yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—88no—
499C22⋊C4—232no—
500C22⋊C4—232no—
501C22⋊C4—216no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—88no—
516C42—416no—
517C42—416no—
518C42—416no—
519C23—24no—
520C23—28no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—232no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—432no—
541C23—432no—
542C23—464no—
543C23—416no—
544C23—464no—
545C23—416no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—44no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23C2×#12114yes—
570C23C2×#12314yes—
571C23#52518yes—
572C23#53414yes—
573C23#54618yes—
574C23#55118yes—
575C23C23×D411yescharacteristic
576C23C23×D411yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4112yes—
579C23C22×#4114yes—
580C23C2×H(5,𝔽2)12yes—
581C23#58514yes—
582D4—816no—
583D4—88no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—232no—
588C2×C4—232no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—432no—
610C2×C4—432no—
611C2×C4—432no—
612C2×C4—432no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4C2×#12118yes—
654C2×C4C2×#12314yes—
655C2×C4C2×#12314yes—
656C22—24no—
657C22—216no—
658C22—24no—
659C22—216no—
660C22—28no—
661C22—28no—
662C22—28no—
663C22—28no—
664C22—28no—
665C22—464no—
666C22—48no—
667C22—48no—
668C22—44no—
669C22—44no—
670C22—48no—
671C22—48no—
672C22—44no—
673C22—44no—
674C22—81no—
675C22—84no—
676C22—81no—
677C22—81no—
678C22—84no—
679C22—82no—
680C22—82no—
681C22#241218yes—
682C22C22×#12111yescharacteristic
683C22C22×#12311yescharacteristic
684C22C2×#52512yes—
685C22#344414yes—
686C22C2×#53411yescharacteristic
687C22#344912yes—
688C22C2×#54612yes—
689C22D4×#4114yes—
690C22C2×#55112yes—
691C22#347514yes—
692C22#349314yes—
693C22#351314yes—
694C22#351314yes—
695C4—28no—
696C4—24no—
697C4—24no—
698C4—48no—
699C4—48no—
700C4—48no—
701C4—81no—
702C4—84no—
703C4—81no—
704C4—81no—
705C4—82no—
706C4—82no—
707C2—28no—
708C2—48no—
709C2—161no—
710C2C2×#241211yescharacteristic
711C2#3473712yes—
712C2#3506212yes—
713C2#3515711yescharacteristic
714C2#3613814yes—
715C2#3619814yes—
716C2#3662611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2853C2semidirect
3. C2×#2280C2semidirect
4. C22×C4×C22⋊C4C2semidirect
5. #15002C2semidirect
6. #14997C2semidirect
7. #14727C2semidirect
9. #14706C2semidirect
10. #14705C2semidirect
11. #14684C2semidirect
12. #14202C2semidirect
13. #14199C2semidirect
14. #14135C2semidirect
15. #13962C2semidirect
28. C2×#388C22semidirect
29. C2×#387C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
59. #1439C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. #1434C22semidirect
69. #1434C22semidirect
70. #1434C22semidirect
147. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×#28492
C2×#22802
C22×C4×C22⋊C42
#150022
#149972
#147272
#147182
#147062
#147052
#146842
#142022
#141992
#141352
#139622