Finite Group Databasebeta

← #4432453

Subgroups — #4432453

Order 512.

2830 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2819C211yesmaximal, characteristic
2C2×#2592C211yesmaximal, characteristic, complemented
3C2×#2632C211yesmaximal, characteristic, complemented
4C2×#2351C211yesmaximal, characteristic, complemented
5#16566C212yesmaximal, complemented
6#16549C214yesmaximal, complemented
7#16516C212yesmaximal, complemented
8#15633C214yesmaximal, complemented
9#15617C214yesmaximal, complemented
10#14883C214yesmaximal, complemented
11#14866C212yesmaximal, complemented
12#14815C212yesmaximal, complemented
13#14296C211yesmaximal, characteristic, complemented
14#14288C211yesmaximal, characteristic, complemented
15#14153C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20#2819—28no—
21#2632—28no—
22#2592—28no—
23C2×#399C2212yes—
24C2×#397C2212yes—
25C2×#396C2214yes—
26C2×#395C2212yes—
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33C2×#393C2212yes—
34C2×#393C2212yes—
35C2×#392C2212yes—
36C2×#392C2212yes—
37#2351—28no—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#390C2214yes—
43C2×#389C2214yes—
44C2×#389C2212yes—
45C2×#388C2214yes—
46C2×#388C2212yes—
47C2×#388C2212yes—
48C2×#383C2211yescharacteristic
49C2×#382C2212yes—
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#1446C2211yescharacteristic
55#1446C2211yescharacteristic
56#1444C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1438C2214yes—
71#1438C2211yescharacteristic
72#1438C2211yescharacteristic
73#1437C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78#1434C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2314yes—
85C2×#124—24no—
86C2×#124—24no—
87C2×#124—24no—
88C2×#124—24no—
89C2×#124—24no—
90C2×#124—24no—
91C2×C4×Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—48no—
95C2×C4×Q8D418yescomplemented
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#399—28no—
128#397—28no—
129#396—48no—
130#395—28no—
131#394—216no—
132#394—28no—
133#394—48no—
134#394—48no—
135#393—28no—
136#393—28no—
137#393—28no—
138#393—28no—
139#392—28no—
140#392—28no—
141#391—28no—
142#391—44no—
143#390—28no—
144#390—48no—
145#390—44no—
146#389—28no—
147#389—216no—
148#388—216no—
149#388—28no—
150#388—28no—
151#383—24no—
152#382—28no—
153#382D418yes—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2412yes—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—44no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8C2×D414yes—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—28no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224#124—416no—
225#124—416no—
226C4×Q8—232no—
227C4×Q8—432no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D418yes—
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—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
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⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42C2×D418yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42#4114yes—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—28no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95C2×D418yes—
335#95C2×D418yes—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—216no—
360C2×Q8—28no—
361C2×Q8—216no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—84no—
367C2×Q8—82no—
368C2×Q8—84no—
369C2×Q8—82no—
370C2×Q8—84no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—84no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—88no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
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—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4C22×D418yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C42−1+414yes—
457C4⋊C4—216no—
458C4⋊C4—28no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—88no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C42—216no—
484C42—416no—
485C42—48no—
486C42—84no—
487C42#121116yes—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—42no—
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—
532C23C23×D411yescharacteristic
533C23C23×D411yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×H(5,𝔽2)12yes—
537C23C2×2−1+412yes—
538C23#58512yes—
539C23#58514yes—
540Q8—232no—
541Q8—88no—
542Q8—88no—
543D4—88no—
544D4—88no—
545D4—832no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
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—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4C2×#12114yes—
613C2×C4C2×#12114yes—
614C2×C4C2×#12114yes—
615C2×C4#53718yes—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—82no—
635C22—82no—
636C22C22×#12111yescharacteristic
637C22C2×#53712yes—
638C22#346012yes—
639C22#348411yescharacteristic
640C22#348611yescharacteristic
641C22#348712yes—
642C22#348812yes—
643C22#349212yes—
644C22#349811yescharacteristic
645C22#350012yes—
646C22#350011yescharacteristic
647C22#351012yes—
648C22#354014yes—
649C22#355414yes—
650C22#355814yes—
651C22#356514yes—
652C4—24no—
653C4—24no—
654C4—28no—
655C4—24no—
656C4—44no—
657C4—48no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4—82no—
665C2—162no—
666C2—162no—
667C2#3522311yescharacteristic
668C2#3525611yescharacteristic
669C2#3567711yescharacteristic
670C2#4111514yes—
671C2#4288112yes—
672C2#4288212yes—
673C2#5069312yes—
674C2#5069612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2819C2
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×#2592C2semidirect
3. C2×#2632C2semidirect
4. C2×#2351C2semidirect
5. #16566C2semidirect
6. #16549C2semidirect
7. #16516C2semidirect
8. #15633C2semidirect
9. #15617C2semidirect
10. #14883C2semidirect
11. #14866C2semidirect
12. #14815C2semidirect
13. #14296C2semidirect
14. #14288C2semidirect
15. #14153C2semidirect
95. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28192
C2×#25922
C2×#26322
C2×#23512
#165662
#165492
#165162
#156332
#156172
#148832
#148662
#148152
#142962
#142882
#141532