Finite Group Databasebeta

← #719157

Subgroups — #719157

Order 512.

3142 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C212yesmaximal
2C2×#2399C211yesmaximal, characteristic, complemented
3C2×#2349C211yesmaximal, characteristic
4#16305C214yesmaximal, complemented
5#15614C214yesmaximal, complemented
6#15305C214yesmaximal, complemented
7#15202C212yesmaximal, complemented
8#15133C212yesmaximal, complemented
9#14758C214yesmaximal, complemented
10#14276C211yesmaximal, characteristic, complemented
11#14018C212yesmaximal
12#13954C211yesmaximal, characteristic, complemented
13#13910C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic
15#13785C211yesmaximal, characteristic
16C22×C22⋊Q8C2214yes—
17C22×C4×Q8C2212yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2538—216no—
21#2399—28no—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#398C2214yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29#2349—28no—
30C2×#390C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#384C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2214yes—
45C2×#380C2214yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×#376C2211yescharacteristic
51C22×#95C2212yes—
52#1446C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—48no—
84C2×C22⋊Q8—48no—
85C2×C4×Q8—28no—
86C2×C4×Q8—28no—
87C2×C4×Q8—44no—
88C2×C4×D4—28no—
89C2×C4×D4—24no—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—44no—
93C2×C4×D4Q818yescomplemented
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#398—28no—
127#398—28no—
128#398—48no—
129#396—216no—
130#396—216no—
131#395—28no—
132#395—28no—
133#390—28no—
134#390D418yes—
135#390Q818yescomplemented
136#389—28no—
137#389—28no—
138#389—28no—
139#389—48no—
140#389—44no—
141#385—216no—
142#385—48no—
143#384—28no—
144#384—24no—
145#382—24no—
146#380—216no—
147#380—28no—
148#380—48no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4D418yes—
153#376—24no—
154C2×#95—28no—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C25C2412yes—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4C2×D414yes—
191C22×D4C2×Q814yes—
192C22×D4C2×Q814yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×Q818yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C22⋊Q8—464no—
220C4×Q8—432no—
221C4×D4—232no—
222C4×D4—432no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2×Q814yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
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⋊C4C2×D4116yes—
297C2×C22⋊C4C2×Q818yes—
298C2×C22⋊C4C2×Q8116yes—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—44no—
310C2×C42C2×D418yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×Q814yes—
314C2×C42C2×Q814yes—
315C2×C42#4114yes—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—22no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24C22×D414yes—
348C24C22×Q814yes—
349C24C22×Q814yes—
350C24C2511yescharacteristic
351C2×Q8—216no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4C4⋊Q8116yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—216no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q812yes—
456C22×C4C22×Q812yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+414yes—
464C22×C42−1+412yes—
465C4⋊C4—28no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—216no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—88no—
491C42—216no—
492C42—28no—
493C42—28no—
494C42—48no—
495C42—84no—
496C42C22⋊Q8116yescomplemented
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23C2×C4⋊Q818yes—
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×Q811yescharacteristic
523C23C23×Q811yescharacteristic
524C23C22×#4111yescharacteristic
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×2−1+412yes—
529C23C2×2−1+411yescharacteristic
530C23#58512yes—
531Q8—432no—
532D4—232no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—816no—
612C2×C4—832no—
613C2×C4C2×C22⋊Q814yes—
614C2×C4C2×C22⋊Q814yes—
615C2×C4C2×C22⋊Q814yes—
616C2×C4C2×C4⋊Q814yes—
617C2×C4#53718yes—
618C2×C4#54414yes—
619C2×C4D4214yes—
620C2×C4#54714yes—
621C2×C4D4×Q814yes—
622C2×C4#55414yes—
623C22—216no—
624C22—28no—
625C22—48no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—48no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22C22×C22⋊Q811yescharacteristic
635C22C22×C4⋊Q811yescharacteristic
636C22C2×#53712yes—
637C22C2×#54411yescharacteristic
638C22C2×D4211yescharacteristic
639C22C2×#54711yescharacteristic
640C22C2×D4×Q811yescharacteristic
641C22C2×#55411yescharacteristic
642C22#348412yes—
643C22#348812yes—
644C22#348912yes—
645C22#350112yes—
646C22#350112yes—
647C22#350312yes—
648C22#350312yes—
649C22#350811yescharacteristic
650C22#350811yescharacteristic
651C22#350911yescharacteristic
652C22#351011yescharacteristic
653C22#353612yes—
654C22#354212yes—
655C22#354812yes—
656C22#355512yes—
657C4—28no—
658C4—24no—
659C4—24no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—82no—
668C4—84no—
669C4—84no—
670C4#244018yes—
671C2—48no—
672C2C2×#244011yescharacteristic
673C2#3528011yescharacteristic
674C2#3551511yescharacteristic
675C2#3592812yes—
676C2#3764812yes—
677C2#3959812yes—
678C2#4249412yes—
679C2#4493111yescharacteristic
680C2#4494711yescharacteristic
681C2#4542911yescharacteristic
682C2#4543411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2399C2semidirect
4. #16305C2semidirect
5. #15614C2semidirect
6. #15305C2semidirect
7. #15202C2semidirect
8. #15133C2semidirect
9. #14758C2semidirect
10. #14276C2semidirect
12. #13954C2semidirect
13. #13910C2semidirect
93. C2×C4×D4Q8semidirect
135. #390Q8semidirect
496. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#23992
C2×#23492
#163052
#156142
#153052
#152022
#151332
#147582
#142762
#140182
#139542
#139102
#139062
#137852