Finite Group Databasebeta

← #5025888

Subgroups — #5025888

Order 512.

4552 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#560C211yesmaximal, characteristic
2C2×#2412C212yesmaximal, complemented
3C2×#2622C211yesmaximal, characteristic, complemented
4#15982C212yesmaximal, complemented
5#15982C212yesmaximal, complemented
6#15949C212yesmaximal, complemented
7#15911C214yesmaximal, complemented
8#15312C214yesmaximal, complemented
9#15277C214yesmaximal, complemented
10#15274C214yesmaximal, complemented
11#15261C212yesmaximal, complemented
12#14255C212yesmaximal, complemented
13#14134C211yesmaximal, characteristic, complemented
14C2×#560—28no—
15C2×#560—216no—
16C22×#127C2211yescharacteristic
17C22×#126C2212yes—
18C22×#123C2212yes—
19C22×#123C2211yescharacteristic
20C22×#123C2212yes—
21C22×#121C2212yes—
22C22×#121C2212yes—
23C22×#121C2212yes—
24C22×#120C2211yescharacteristic
25C22×#120C2212yes—
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28#2622—28no—
29#2412—216no—
30C2×#397C2214yes—
31C2×#397C2214yes—
32C2×#394C2214yes—
33C2×#394C2214yes—
34C2×#394C2214yescomplemented
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2214yescomplemented
43C2×#381C2212yes—
44C2×#380C2214yes—
45C2×#379C2211yescharacteristic
46#1444C2214yescomplemented
47#1444C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1440C2211yescharacteristic
54#1440C2211yescharacteristic
55#1440C2212yes—
56#1440C2212yes—
57#1440C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1438C2212yes—
61#1438C2212yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2214yescomplemented
65#1438C2214yescomplemented
66#1436C2214yescomplemented
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2214yescomplemented
70C22⋊#95C2214yescomplemented
71C26C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2312yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2314yes—
83#560—2128no—
84C2×#127—28no—
85C2×#127—24no—
86C2×#126—28no—
87C2×#126—28no—
88C2×#126—28no—
89C2×#123—28no—
90C2×#123—24no—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—28no—
94C2×#123—28no—
95C2×#123—48no—
96C2×#121—28no—
97C2×#121—28no—
98C2×#121—28no—
99C2×#121—28no—
100C2×#121—28no—
101C2×#121—28no—
102C2×#121—28no—
103C2×#121—28no—
104C2×#121—28no—
105C2×#120—28no—
106C2×#120—28no—
107C2×#120—28no—
108C2×#120D418yes—
109C2×#120D4116yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—28no—
118C22×C22⋊C4—28no—
119C22×C22⋊C4—48no—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C42C2311yescharacteristic
140#397—216no—
141#397—216no—
142#394—216no—
143#394—216no—
144#394—48no—
145#386—28no—
146#386—28no—
147#386—216no—
148#386—48no—
149#386—48no—
150#385—28no—
151#385—28no—
152#385—48no—
153#381—28no—
154#380—216no—
155#379—24no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25—44no—
172C25—44no—
173C25—44no—
174C25C2×D418yes—
175C25C2×D414yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C25C2412yes—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4C2×D418yes—
207C22×D4C2×D418yes—
208C22×D4C2×D418yes—
209C22×D4C2×D418yes—
210C22×D4C2×D414yes—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2414yes—
240#127—232no—
241#126—264no—
242#123—232no—
243#123—264no—
244#123—432no—
245#121—264no—
246#121—264no—
247#121—264no—
248#120—232no—
249#120—264no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—432no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—416no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×D418yes—
318C2×C22⋊C4C2×D418yes—
319C2×C22⋊C4C2×D418yes—
320C2×C22⋊C4C2×D418yes—
321C2×C22⋊C4C2×D414yes—
322C2×C42—22no—
323C2×C42—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—416no—
364C24—416no—
365C24—416no—
366C24—88no—
367C24—88no—
368C24—88no—
369C24—88no—
370C24#120116yes—
371C24C22×D414yes—
372C24C22×D412yes—
373C24C22×D414yes—
374C24C22×D414yes—
375C24C22×D414yes—
376C24C22×D414yes—
377C24C22×D412yes—
378C24C2511yescharacteristic
379C2×D4—216no—
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—216no—
385C2×D4—28no—
386C2×D4—48no—
387C2×D4—416no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—416no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4—84no—
401C2×D4—84no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—88no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
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—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—216no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4H(5,𝔽2)18yes—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—416no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C22⋊C4—216no—
488C22⋊C4—28no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—28no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C42—28no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—432no—
534C23—416no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—832no—
572C23—832no—
573C23—832no—
574C23C2×#12018yes—
575C23C2×#12014yes—
576C23C23×D412yes—
577C23C23×D411yescharacteristic
578C23C2×H(5,𝔽2)14yes—
579C23#58514yes—
580C23#58514yes—
581D4—816no—
582D4—816no—
583D4—816no—
584D4—816no—
585D4—88no—
586D4—816no—
587D4—88no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
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—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—816no—
640C2×C4—88no—
641C2×C4—832no—
642C2×C4—816no—
643C22—28no—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—28no—
648C22—48no—
649C22—48no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—48no—
654C22—48no—
655C22—48no—
656C22—48no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—48no—
661C22—42no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22—832no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22C22×#12011yescharacteristic
673C22#348412yes—
674C22#348512yes—
675C22#348514yes—
676C22#348512yes—
677C22#349614yes—
678C22#349614yes—
679C22#354214yes—
680C22#356014yes—
681C22#356214yes—
682C22#356614yes—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—48no—
687C4—44no—
688C4—42no—
689C4—84no—
690C4—84no—
691C4—84no—
692C2—44no—
693C2—48no—
694C2—84no—
695C2—82no—
696C2#3534712yes—
697C2#3544611yescharacteristic
698C2#4785014yes—
699C2#4793014yes—
700C2#4881014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#560C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2412C2semidirect
3. C2×#2622C2semidirect
4. #15982C2semidirect
5. #15982C2semidirect
6. #15949C2semidirect
7. #15911C2semidirect
8. #15312C2semidirect
9. #15277C2semidirect
10. #15274C2semidirect
11. #15261C2semidirect
12. #14255C2semidirect
13. #14134C2semidirect
34. C2×#394C22semidirect
42. C2×#385C22semidirect
46. #1444C22semidirect
47. #1444C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
64. #1438C22semidirect
65. #1438C22semidirect
66. #1436C22semidirect
69. C22⋊#95C22semidirect
70. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5602
C2×#24122
C2×#26222
#159822
#159492
#159112
#153122
#152772
#152742
#152612
#142552
#141342