Finite Group Databasebeta

← #613465

Subgroups — #613465

Order 512.

3050 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#14935C212yesmaximal
4#14934C212yesmaximal
5#14863C214yesmaximal, complemented
6#14800C214yesmaximal, complemented
7#14702C212yesmaximal
8#14695C212yesmaximal
9#14685C211yesmaximal, characteristic
10#14130C211yesmaximal, characteristic
11#14053C212yesmaximal, complemented
12#14030C212yesmaximal, complemented
13#14006C212yesmaximal, complemented
14#13981C212yesmaximal, complemented
15#13907C211yesmaximal, characteristic
16#13881C211yesmaximal, characteristic
17C4×#381C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C23×C42C2212yes—
24#2440—28no—
25C2×#398C2212yes—
26C2×#395C2212yes—
27C2×#391C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#384C2211yescharacteristic
36C2×#382C2212yes—
37C2×#381C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4—28no—
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C22×#95C2212yes—
50#1446C2214yes—
51#1444C2212yes—
52#1443C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2212yes—
73#1434C2214yes—
74#1434C2214yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—44no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C42—24no—
122C22×C42—24no—
123C22×C42—28no—
124C22×C42—48no—
125C22×C42Q818yescomplemented
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2314yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133#398—28no—
134#395—28no—
135#391—48no—
136#389—24no—
137#389—24no—
138#389—28no—
139#389—28no—
140#387—48no—
141#385—28no—
142#385—28no—
143#384—24no—
144#382—28no—
145#381—24no—
146#381—24no—
147#381—44no—
148#380—24no—
149#380—24no—
150#380—44no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—216no—
153C4×C4⋊C4—28no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—24no—
156C2×#95—28no—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C25C2411yescharacteristic
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—28no—
192C22×Q8—42no—
193C22×Q8—44no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D418yes—
216C23×C4C2×D414yes—
217C23×C4C2×Q814yes—
218C23×C4C2×Q814yes—
219C23×C4C2×Q814yes—
220C23×C4#4114yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C4⋊Q8—416no—
238C22⋊Q8—432no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
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—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4#4118yes—
277C2×C4⋊C4#4118yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4#4114yes—
295C2×C22⋊C4#4114yes—
296C2×C42—232no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42—432no—
315C2×C42C2×Q818yes—
316C2×C42C2×Q818yes—
317#95—28no—
318#95—28no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
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—24no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24C22×D412yes—
353C24C22×Q812yes—
354C24C2×#4112yes—
355C24C2×#4112yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)12yes—
358C242−1+412yes—
359C24C2511yescharacteristic
360C2×Q8—48no—
361C2×Q8—416no—
362C2×Q8—48no—
363C2×Q8—84no—
364C2×Q8—88no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—216no—
421C22×C4—22no—
422C22×C4—22no—
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—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4C22⋊Q8116yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q812yes—
470C22×C4C22×Q814yes—
471C22×C4C22×Q812yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4118yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4118yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C42−1+414yes—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—216no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—88no—
497C4⋊C4—84no—
498C4⋊C4#123116yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C42—216no—
506C42—216no—
507C42—416no—
508C42—416no—
509C42—44no—
510C42—44no—
511C42—48no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23C2×C22⋊Q814yes—
529C23#53314yes—
530C23#54414yes—
531C23#54614yes—
532C23#54814yes—
533C23#55514yes—
534C23#55614yes—
535C23C23×D411yescharacteristic
536C23C23×Q811yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4112yes—
539C23C22×#4112yes—
540C23C22×#4111yescharacteristic
541C23C22×#4111yescharacteristic
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+411yescharacteristic
545C23C2×2−1+412yes—
546Q8—816no—
547C2×C4—232no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—232no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—432no—
581C2×C4—432no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—416no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4C2×C22⋊Q814yes—
637C2×C4C2×C22⋊Q814yes—
638C2×C4C2×#12318yes—
639C2×C4C2×#12314yes—
640C2×C4#52514yes—
641C2×C4#52918yes—
642C2×C4#53318yes—
643C2×C4#54418yes—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—28no—
648C22—24no—
649C22—28no—
650C22—24no—
651C22—28no—
652C22—28no—
653C22—24no—
654C22#243918yes—
655C22C22×C22⋊Q811yescharacteristic
656C22C22×#12311yescharacteristic
657C22C2×#52511yescharacteristic
658C22C2×#52912yes—
659C22C2×#53312yes—
660C22C2×#53311yescharacteristic
661C22C2×#54412yes—
662C22C2×#54411yescharacteristic
663C22C2×#54611yescharacteristic
664C22C2×#54811yescharacteristic
665C22C2×#55511yescharacteristic
666C22C2×#55611yescharacteristic
667C22#348112yes—
668C22#348112yes—
669C22#348212yes—
670C22#348212yes—
671C22#348411yescharacteristic
672C22#348611yescharacteristic
673C22#349212yes—
674C22#351912yes—
675C22#352012yes—
676C22#352212yes—
677C22#352312yes—
678C4—24no—
679C4—28no—
680C4—28no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—44no—
686C4—42no—
687C4—48no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—84no—
693C4#285518yes—
694C2—28no—
695C2C2×#243911yescharacteristic
696C2#3496612yes—
697C2#3522411yescharacteristic
698C2#3523912yes—
699C2#3623611yescharacteristic
700C2#3625712yes—
701C2#3642211yescharacteristic
702C2#3648211yescharacteristic
703C2#3648912yes—
704C2#3652811yescharacteristic
705C2C2×#285511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #14863C2semidirect
6. #14800C2semidirect
11. #14053C2semidirect
12. #14030C2semidirect
13. #14006C2semidirect
14. #13981C2semidirect
125. C22×C42Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C22×C4×C4⋊C42
#149352
#149342
#148632
#148002
#147022
#146952
#146852
#141302
#140532
#140302
#140062
#139812
#139072
#138812
C4×#3812