Finite Group Databasebeta

← #4929352

Subgroups — #4929352

Order 512.

3654 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic
3C2×#2432C211yesmaximal, characteristic
4C2×#2274C211yesmaximal, characteristic, complemented
5#15163C212yesmaximal, complemented
6#15128C212yesmaximal, complemented
7#15097C212yesmaximal, complemented
8#15070C212yesmaximal, complemented
9#14890C214yesmaximal, complemented
10#14761C214yesmaximal, complemented
11#14292C211yesmaximal, characteristic, complemented
12#14204C214yesmaximal, complemented
13#14112C214yesmaximal, complemented
14#13953C211yesmaximal, characteristic, complemented
15#13851C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#123C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#117C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23#2583—28no—
24#2432—28no—
25#2406—28no—
26C2×#398C2212yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#391C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2214yes—
35C2×#389C2214yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#385C2214yes—
43C2×#381C2211yescharacteristic
44#2274—28no—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55#1446C2212yes—
56#1446C2212yes—
57#1445C2212yes—
58#1444C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2214yes—
68#1438C2212yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2212yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2212yes—
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78#1434C2212yes—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—44no—
83C22×#41—44no—
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×#124—24no—
95C2×#124—44no—
96C2×#123—28no—
97C2×#123—44no—
98C2×#123—44no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×#117—28no—
103C2×#117—28no—
104C2×#117—28no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4D418yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#398—28no—
140#396—216no—
141#395—28no—
142#395—28no—
143#394—28no—
144#394—216no—
145#391—24no—
146#390—24no—
147#389—216no—
148#389—48no—
149#388—216no—
150#388—48no—
151#386—28no—
152#385—28no—
153#385—28no—
154#385—216no—
155#385—216no—
156#381—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—28no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163C4×C22⋊C4—28no—
164C4×C22⋊C4—44no—
165C4×C22⋊C4—44no—
166C4×C22⋊C4D418yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C2×#41—432no—
190C2×#41—432no—
191C2×#41—416no—
192C2×#41—416no—
193C2×#41—416no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—42no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2412yes—
251C4⋊Q8—416no—
252#124—416no—
253#123—432no—
254C22⋊Q8—416no—
255#117—264no—
256C2×C4⋊C4—232no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4#4118yescomplemented
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×D418yes—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42#4118yescomplemented
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C2×#4112yes—
381C24H(5,𝔽2)14yes—
382C24C2511yescharacteristic
383#41—4128no—
384C2×Q8—28no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—84no—
393C2×Q8—84no—
394C2×D4—216no—
395C2×D4—28no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—44no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C22×C4—216no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—216no—
421C22×C4—216no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—28no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4#121116yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4114yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C42−1+418yes—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—416no—
506C4⋊C4—416no—
507C4⋊C4—416no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—88no—
512C22⋊C4—216no—
513C22⋊C4—232no—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C42—216no—
524C42—216no—
525C42—44no—
526C42—44no—
527C42—84no—
528C42—82no—
529C42—82no—
530C42—84no—
531C42—84no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—28no—
547C23—28no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23C2×#12114yes—
571C23C23×D411yescharacteristic
572C23C23×D411yescharacteristic
573C23C22×#4111yescharacteristic
574C23C22×#4112yes—
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)12yes—
578C23C2×2−1+414yes—
579Q8—416no—
580Q8—88no—
581D4—416no—
582D4—432no—
583C2×C4—232no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—416no—
618C2×C4—432no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—88no—
656C2×C4C2×#12114yes—
657C2×C4C2×#12114yes—
658C2×C4#52518yes—
659C2×C4#53218yes—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—48no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—48no—
671C22—44no—
672C22—48no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—84no—
679C22C22×#12111yescharacteristic
680C22C2×#52512yes—
681C22C2×#53212yes—
682C22#344411yescharacteristic
683C22#344411yescharacteristic
684C22#348411yescharacteristic
685C22#348412yes—
686C22#348411yescharacteristic
687C22#348612yes—
688C22#349012yes—
689C22#349214yes—
690C22#349912yes—
691C22#350912yes—
692C22#351114yes—
693C22#352014yes—
694C22#352214yes—
695C4—28no—
696C4—24no—
697C4—24no—
698C4—28no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—44no—
703C4—48no—
704C4—44no—
705C4—82no—
706C4—82no—
707C4—84no—
708C2—44no—
709C2—84no—
710C2#3534811yescharacteristic
711C2#3543512yes—
712C2#3544012yes—
713C2#3566711yescharacteristic
714C2#3569312yes—
715C2#3569714yes—
716C2#3574012yes—
717C2#3662411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2583C2semidirect
4. C2×#2274C2semidirect
5. #15163C2semidirect
6. #15128C2semidirect
7. #15097C2semidirect
8. #15070C2semidirect
9. #14890C2semidirect
10. #14761C2semidirect
11. #14292C2semidirect
12. #14204C2semidirect
13. #14112C2semidirect
14. #13953C2semidirect
289. C2×C4⋊C4#41semidirect
342. C2×C42#41semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24062
C2×#24322
C2×#22742
#151632
#151282
#150972
#150702
#148902
#147612
#142922
#142042
#141122
#139532
#138512