Finite Group Databasebeta

← #641738

Subgroups — #641738

Order 512.

5574 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2848C211yesmaximal, characteristic
3C2×#2659C212yesmaximal, complemented
4#16606C214yesmaximal, complemented
5#16365C212yesmaximal, complemented
6#16254C212yesmaximal, complemented
7#14874C212yesmaximal, complemented
8#14821C212yesmaximal, complemented
9#14786C212yesmaximal, complemented
10#14758C214yesmaximal, complemented
11#14720C212yesmaximal, complemented
12#14281C212yesmaximal, complemented
13#13877C214yesmaximal
14C2×C22⋊#95C211yesmaximal, characteristic
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28C23×C22⋊C4C2212yes—
29C23×C22⋊C4C2212yes—
30#2848—28no—
31#2659—216no—
32C2×#398C2214yescomplemented
33C2×#398C2214yescomplemented
34C2×#395C2214yescomplemented
35C2×#395C2214yescomplemented
36C2×#395C2214yescomplemented
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2218yes—
46C2×#379C2212yes—
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1446C2214yescomplemented
50#1446C2214yescomplemented
51#1445C2214yescomplemented
52#1444C2218yescomplemented
53#1444C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2218yescomplemented
58#1439C2218yes—
59#1439C2218yes—
60C22⋊#95C2214yes—
61C22⋊#95C2212yes—
62C22⋊#95C2214yes—
63C22⋊#95C2218yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C26C2311yescharacteristic
68C23×Q8—28no—
69C23×Q8—24no—
70C23×Q8—44no—
71C23×Q8—44no—
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C24×C4—28no—
76C24×C4—44no—
77C24×C4—44no—
78C24×C4D418yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2314yes—
87C24×C4C2311yescharacteristic
88C2×C22⋊Q8—2128no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—216no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—216no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8—44no—
104C2×C22⋊Q8D4116yes—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—216no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2318yes—
117C22×C4⋊C4C2318yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4—28no—
122C22×C22⋊C4—216no—
123C22×C22⋊C4—44no—
124C22×C22⋊C4—44no—
125C22×C22⋊C4Q8116yescomplemented
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136#398—216no—
137#398—216no—
138#395—216no—
139#395—216no—
140#395—48no—
141#385—28no—
142#385—28no—
143#385—28no—
144#385—48no—
145#385—48no—
146#380—24no—
147#380—24no—
148#380—24no—
149#380—232no—
150#379—28no—
151C2×#95—28no—
152C2×#95—44no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C25—24no—
167C25—22no—
168C25—24no—
169C25—24no—
170C25—22no—
171C25—42no—
172C25—42no—
173C25C2×D414yes—
174C25C2×Q818yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—232no—
179C22×Q8—28no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—28no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—464no—
192C22×Q8—416no—
193C22×Q8—416no—
194C22×Q8C2×D418yes—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—232no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—416no—
216C23×C4—416no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—464no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D418yes—
238C23×C4C2×Q8116yescomplemented
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2414yes—
245C23×C4C2411yescharacteristic
246C23×C4C2418yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C22⋊Q8—264no—
251C22⋊Q8—264no—
252C22⋊Q8—264no—
253C22⋊Q8—264no—
254C22⋊Q8—416no—
255C22⋊Q8—416no—
256C22⋊Q8—416no—
257C2×C4⋊C4—232no—
258C2×C4⋊C4—264no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—216no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4C2×D4116yes—
287C2×C4⋊C4C2×D418yes—
288C2×C22⋊C4—264no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—432no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4C2×D4116yes—
313C2×C22⋊C4C2×Q8116yes—
314C2×C22⋊C4C2×Q8116yescomplemented
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—216no—
321#95—216no—
322#95—28no—
323#95—44no—
324#95—48no—
325#95—48no—
326#95—48no—
327#95—48no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—28no—
338C24—28no—
339C24—28no—
340C24—28no—
341C24—216no—
342C24—216no—
343C24—232no—
344C24—22no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—44no—
352C24—44no—
353C24—416no—
354C24—416no—
355C24—48no—
356C24—48no—
357C24—416no—
358C24—48no—
359C24—48no—
360C24C4⋊Q8116yes—
361C24C22×D412yes—
362C24C22×D412yes—
363C24C22×D414yes—
364C24C22×D412yes—
365C24C22×Q814yes—
366C24C22×Q814yes—
367C24C22×Q814yes—
368C24H(5,𝔽2)14yes—
369C24H(5,𝔽2)12yes—
370C242−1+414yes—
371C242−1+412yes—
372C24C2511yescharacteristic
373C2×Q8—216no—
374C2×Q8—216no—
375C2×Q8—416no—
376C2×Q8—432no—
377C2×Q8—4128no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—416no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—48no—
386C2×Q8—88no—
387C22×C4—232no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—232no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—432no—
460C22×C4—4128no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4#120116yes—
471C22×C4C22×D418yes—
472C22×C4C22×D418yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×Q8116yes—
480C22×C4H(5,𝔽2)14yes—
481C4⋊C4—216no—
482C4⋊C4—232no—
483C4⋊C4—216no—
484C4⋊C4—232no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—816no—
492C4⋊C4—816no—
493C22⋊C4—232no—
494C22⋊C4—232no—
495C22⋊C4—232no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—232no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—232no—
538C23—28no—
539C23—28no—
540C23—416no—
541C23—416no—
542C23—464no—
543C23—464no—
544C23—416no—
545C23—416no—
546C23—416no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23C2×#12014yes—
554C23C2×C4⋊Q814yes—
555C23C2×C4⋊Q814yes—
556C23C2×C4⋊Q814yes—
557C23#53414yes—
558C23#53614yes—
559C23#53714yes—
560C23#54318yes—
561C23#54418yes—
562C23D4×Q8116yes—
563C23C23×D411yescharacteristic
564C23C23×D412yes—
565C23C23×Q812yes—
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×2−1+412yes—
570C23C2×2−1+411yescharacteristic
571C23#58512yes—
572Q8—432no—
573Q8—432no—
574Q8—88no—
575C2×C4—232no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—216no—
591C2×C4—216no—
592C2×C4—216no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—216no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—416no—
601C2×C4—432no—
602C2×C4—432no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—832no—
634C2×C4C2×#12014yes—
635C2×C4C2×#12014yes—
636C2×C4#53414yes—
637C22—24no—
638C22—216no—
639C22—216no—
640C22—216no—
641C22—216no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—464no—
657C22—48no—
658C22—48no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22#243418yes—
664C22#265918yescomplemented
665C22C22×#12011yescharacteristic
666C22C22×C4⋊Q811yescharacteristic
667C22C2×#53411yescharacteristic
668C22C2×#53411yescharacteristic
669C22C2×#53611yescharacteristic
670C22C2×#53711yescharacteristic
671C22#344911yescharacteristic
672C22C2×#54312yes—
673C22C2×#54412yes—
674C22C2×D4×Q814yes—
675C22#348414yes—
676C22#348812yes—
677C22#348912yes—
678C22#350114yes—
679C22#353514yes—
680C22#353612yes—
681C22#353812yes—
682C4—24no—
683C4—24no—
684C4—24no—
685C4—416no—
686C4—42no—
687C4—48no—
688C4—48no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C2—28no—
694C2—28no—
695C2—44no—
696C2C2×#265911yescharacteristic
697C2C2×#243411yescharacteristic
698C2#3498011yescharacteristic
699C2#3501111yescharacteristic
700C2#3525111yescharacteristic
701C2#3744211yescharacteristic
702C2#3744711yescharacteristic
703C2#3768414yes—
704C2#3768514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2659C2semidirect
4. #16606C2semidirect
5. #16365C2semidirect
6. #16254C2semidirect
7. #14874C2semidirect
8. #14821C2semidirect
9. #14786C2semidirect
10. #14758C2semidirect
11. #14720C2semidirect
12. #14281C2semidirect
32. C2×#398C22semidirect
33. C2×#398C22semidirect
34. C2×#395C22semidirect
35. C2×#395C22semidirect
36. C2×#395C22semidirect
49. #1446C22semidirect
50. #1446C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
125. C22×C22⋊C4Q8semidirect
238. C23×C4C2×Q8semidirect
314. C2×C22⋊C4C2×Q8semidirect
664. C22#2659semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#28482
C2×#26592
#166062
#163652
#162542
#148742
#148212
#147862
#147582
#147202
#142812
#138772
C2×C22⋊#952