Finite Group Databasebeta

← #4435290

Subgroups — #4435290

Order 512.

3442 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2435C211yesmaximal, characteristic
2C2×#2667C211yesmaximal, characteristic, complemented
3#16886C212yesmaximal, complemented
4#16558C212yesmaximal, complemented
5#16371C214yesmaximal, complemented
6#15297C214yesmaximal, complemented
7#15284C214yesmaximal, complemented
8#15250C212yesmaximal, complemented
9#14999C214yesmaximal, complemented
10#14932C212yesmaximal, complemented
11#14660C211yesmaximal, characteristic, complemented
12#14335C211yesmaximal, characteristic
13#14292C211yesmaximal, characteristic, complemented
14#13902C211yesmaximal, characteristic
15#13807C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×#124C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19#2667—28no—
20#2435—28no—
21C2×#399C2212yes—
22C2×#398C2212yescomplemented
23C2×#398C2212yescomplemented
24C2×#397C2214yescomplemented
25C2×#396C2212yes—
26C2×#394C2212yes—
27C2×#394C2212yes—
28C2×#390C2212yes—
29C2×#390C2214yes—
30C2×#389C2212yescomplemented
31C2×#388C2214yescomplemented
32C2×#388C2214yescomplemented
33C2×#388C2214yescomplemented
34C2×#387C2212yescomplemented
35C2×#381C2211yescharacteristic
36C2×#381C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#379C2214yes—
39C2×#379C2212yes—
40C2×#379C2212yes—
41C2×#379C2214yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44#1446C2212yescomplemented
45#1446C2211yescharacteristic, complemented
46#1446C2211yescharacteristic, complemented
47#1445C2212yescomplemented
48#1445C2211yescharacteristic, complemented
49#1445C2211yescharacteristic, complemented
50#1441C2214yescomplemented
51#1440C2212yes—
52#1440C2212yes—
53#1440C2214yes—
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1438C2212yes—
66#1438C2212yes—
67#1438C2212yes—
68#1438C2212yes—
69#1438C2214yes—
70#1437C2214yes—
71#1437C2214yescomplemented
72#1436C2214yescomplemented
73#1434C2212yescomplemented
74#1434C2212yescomplemented
75C4×#95C2212yescomplemented
76C4×#95C2212yescomplemented
77C26C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4—24no—
80C23×D4—24no—
81C23×D4—24no—
82C23×D4—48no—
83C23×D4C2312yes—
84C23×D4C2312yes—
85C24×C4C2311yescharacteristic
86C2×#124—28no—
87C2×#124—48no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122#399—28no—
123#398—28no—
124#398—28no—
125#397—48no—
126#396—28no—
127#394—28no—
128#394—28no—
129#390—28no—
130#390—48no—
131#389—28no—
132#388—48no—
133#388—48no—
134#388—48no—
135#387—28no—
136#381—24no—
137#381—24no—
138#380—24no—
139#379—216no—
140#379—44no—
141#379—44no—
142#379—48no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4D418yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C25—24no—
169C25—28no—
170C25—24no—
171C25—24no—
172C25—44no—
173C25—44no—
174C25—44no—
175C25—42no—
176C25—42no—
177C25—42no—
178C25—42no—
179C25C2414yes—
180C25C2411yescharacteristic
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—44no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—432no—
196C22×D4—432no—
197C22×D4—432no—
198C22×D4—416no—
199C23×C4—44no—
200C23×C4C2×D414yes—
201C23×C4#4114yes—
202C23×C4#4114yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216#124—432no—
217C22⋊Q8—416no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
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⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D418yes—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42C2×D418yes—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—28no—
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#95—44no—
320#95—44no—
321#95C2×D414yes—
322#95C2×D414yes—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—216no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—416no—
344C24—416no—
345C24—48no—
346C24—416no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—84no—
351C24—84no—
352C24—84no—
353C24—84no—
354C24—84no—
355C24—84no—
356C24—88no—
357C24—88no—
358C24—88no—
359C24—88no—
360C24—88no—
361C24—88no—
362C24—88no—
363C24—88no—
364C24—88no—
365C24C22×D412yes—
366C24C2×#4112yes—
367C24C2×#4112yes—
368C24H(5,𝔽2)18yes—
369C24C2511yescharacteristic
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—84no—
373C2×Q8—84no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—4128no—
383C2×D4—48no—
384C2×D4—48no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
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—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4#124116yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)14yes—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—48no—
483C42—84no—
484C42—88no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—432no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—816no—
537C23—816no—
538C23—816no—
539C23—816no—
540C23—832no—
541C23—832no—
542C23—832no—
543C23—832no—
544C23—816no—
545C23—816no—
546C23C2×#12414yes—
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C22×#4111yescharacteristic
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)14yes—
552C23#58512yes—
553C23#58512yes—
554C23#58512yes—
555Q8—88no—
556D4—432no—
557D4—432no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
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×C4C2×#12414yes—
623C2×C4C2×#12414yes—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—44no—
628C22—48no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—864no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22C22×#12411yescharacteristic
649C22#344912yes—
650C22#346012yes—
651C22#348412yes—
652C22#349012yes—
653C22#349012yes—
654C22#349412yes—
655C22#349811yescharacteristic
656C22#350011yescharacteristic
657C22#351012yes—
658C22#351011yescharacteristic
659C22#351011yescharacteristic
660C22#353314yes—
661C22#353414yes—
662C22#354214yes—
663C22#356014yes—
664C4—24no—
665C4—24no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—82no—
670C4—82no—
671C4—84no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C2—44no—
678C2—88no—
679C2#3526411yescharacteristic
680C2#3533811yescharacteristic
681C2#3536711yescharacteristic
682C2#4050212yes—
683C2#4105414yes—
684C2#4195512yes—
685C2#5087812yes—
686C2#5104112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2667C2semidirect
3. #16886C2semidirect
4. #16558C2semidirect
5. #16371C2semidirect
6. #15297C2semidirect
7. #15284C2semidirect
8. #15250C2semidirect
9. #14999C2semidirect
10. #14932C2semidirect
11. #14660C2semidirect
13. #14292C2semidirect
15. #13807C2semidirect
22. C2×#398C22semidirect
23. C2×#398C22semidirect
24. C2×#397C22semidirect
30. C2×#389C22semidirect
31. C2×#388C22semidirect
32. C2×#388C22semidirect
33. C2×#388C22semidirect
34. C2×#387C22semidirect
44. #1446C22semidirect
45. #1446C22semidirect
46. #1446C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1441C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
71. #1437C22semidirect
72. #1436C22semidirect
73. #1434C22semidirect
74. #1434C22semidirect
75. C4×#95C22semidirect
76. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24352
C2×#26672
#168862
#165582
#163712
#152972
#152842
#152502
#149992
#149322
#146602
#143352
#142922
#139022
#138072