Finite Group Databasebeta

← #1394103

Subgroups — #1394103

Order 512.

2710 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2628C212yesmaximal
3C2×#2344C211yesmaximal, characteristic
4#16603C212yesmaximal, complemented
5#16565C212yesmaximal, complemented
6#16329C214yesmaximal, complemented
7#16300C214yesmaximal
8#15158C211yesmaximal, characteristic, complemented
9#15143C211yesmaximal, characteristic, complemented
10#15139C212yesmaximal, complemented
11#15135C212yesmaximal
12#15125C211yesmaximal, characteristic
13#14887C212yesmaximal
14#14005C212yesmaximal, complemented
15#13918C211yesmaximal, characteristic, complemented
16#13907C212yesmaximal
17C4×#393C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23#2628—216no—
24#2546—28no—
25C2×#400C2214yes—
26C2×#399C2214yes—
27C2×#398C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#393C2212yes—
32C2×#393C2211yescharacteristic
33#2344—28no—
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#384C2212yes—
45C2×#382C2212yes—
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2214yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54#1446C2212yes—
55#1446C2211yescharacteristic
56#1445C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1444C2214yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2211yescharacteristic
63#1442C2212yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2211yescharacteristic
75#1439C2212yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C2×#117—28no—
98C2×#117—28no—
99C2×#117—44no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4Q818yescomplemented
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#400—216no—
134#399—216no—
135#398—216no—
136#395—28no—
137#395—28no—
138#395—44no—
139#393—28no—
140#393D418yes—
141#391—28no—
142#389—24no—
143#389—28no—
144#389—28no—
145#389—28no—
146#389—24no—
147#389—216no—
148#389—44no—
149#387—28no—
150#387—28no—
151#384—44no—
152#382—28no—
153#381—28no—
154#380—24no—
155#380—216no—
156#380—44no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—44no—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C2×#95C2312yes—
192C25C2411yescharacteristic
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8C2×D414yes—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4C2×Q814yes—
214C23×C4C2×Q818yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C4⋊Q8—432no—
232C4⋊Q8—416no—
233C4×Q8—416no—
234#117—432no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
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—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
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⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×Q818yes—
283C2×C4⋊C4C2×Q814yes—
284C2×C4⋊C4C2×Q814yes—
285C2×C4⋊C4#4114yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42C2×D418yes—
319C2×C42C2×D414yes—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—42no—
339#95—44no—
340#95—42no—
341#95—42no—
342#95—44no—
343#95—42no—
344#95—42no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95C2×D418yes—
349#95C2×D414yes—
350#95C2×D418yes—
351C24—21no—
352C24—21no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—21no—
359C24C22×D412yes—
360C24C22×Q812yes—
361C242−1+412yes—
362C24C2511yescharacteristic
363C2×Q8—28no—
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—88no—
372C2×Q8—88no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—216no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—216no—
421C22×C4—28no—
422C22×C4—216no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q814yes—
473C22×C4C22×Q814yes—
474C22×C4C22×Q812yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C42−1+414yes—
480C22×C42−1+414yes—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C4⋊C4C22⋊Q8116yes—
499C22⋊C4—28no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—88no—
505C42—28no—
506C42—216no—
507C42—48no—
508C42—48no—
509C42—44no—
510C42—48no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23D4×Q814yes—
534C23C23×D411yescharacteristic
535C23C23×D411yescharacteristic
536C23C23×Q811yescharacteristic
537C23C22×#4111yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×2−1+412yes—
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+412yes—
543C23#58511yescharacteristic
544C23#58512yes—
545Q8—416no—
546Q8—816no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—432no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—416no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4C2×C22⋊Q818yes—
639C2×C4C2×C22⋊Q814yes—
640C2×C4#53514yes—
641C2×C4#53614yes—
642C2×C4#54218yes—
643C2×C4#54418yes—
644C2×C4D4214yes—
645C22—28no—
646C22—24no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—44no—
656C22C22×C22⋊Q811yescharacteristic
657C22C2×#53511yescharacteristic
658C22C2×#53611yescharacteristic
659C22C2×#54212yes—
660C22C2×#54412yes—
661C22C2×D4211yescharacteristic
662C22C2×D4×Q811yescharacteristic
663C22D4×#4111yescharacteristic
664C22#348711yescharacteristic
665C22#348911yescharacteristic
666C22#349612yes—
667C22#349911yescharacteristic
668C22#349912yes—
669C22#350111yescharacteristic
670C22#350212yes—
671C22#351612yes—
672C22#352011yescharacteristic
673C22#353812yes—
674C22#354212yes—
675C22#354712yes—
676C22#355212yes—
677C22#356312yes—
678C22#356512yes—
679C4—24no—
680C4—28no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—44no—
689C4—81no—
690C4—82no—
691C4—84no—
692C4—84no—
693C4—82no—
694C4#262818yescomplemented
695C2—44no—
696C2C2×#262811yescharacteristic
697C2#3584511yescharacteristic
698C2D4×C22⋊Q811yescharacteristic
699C2#3933911yescharacteristic
700C2#3936011yescharacteristic
701C2#3936812yes—
702C2#3939812yes—
703C2#4248812yes—
704C2#4263712yes—
705C2#4288811yescharacteristic
706C2#4299411yescharacteristic
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
1. C2×#2546C2semidirect
4. #16603C2semidirect
5. #16565C2semidirect
6. #16329C2semidirect
8. #15158C2semidirect
9. #15143C2semidirect
10. #15139C2semidirect
14. #14005C2semidirect
15. #13918C2semidirect
102. C22×C4⋊C4Q8semidirect
694. C4#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#26282
C2×#23442
#166032
#165652
#163292
#163002
#151582
#151432
#151392
#151352
#151252
#148872
#140052
#139182
#139072
C4×#3932