Finite Group Databasebeta

← #4419443

Subgroups — #4419443

Order 512.

3500 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#2822C211yesmaximal, characteristic, complemented
3C2×#2584C211yesmaximal, characteristic, complemented
4C2×#2633C211yesmaximal, characteristic
5#16674C212yesmaximal, complemented
6#16673C212yesmaximal, complemented
7#16606C212yesmaximal, complemented
8#16601C212yesmaximal, complemented
9#15848C214yesmaximal, complemented
10#15271C214yesmaximal
11#14907C212yesmaximal, complemented
12#14906C212yesmaximal, complemented
13#14893C212yesmaximal, complemented
14#14786C212yesmaximal, complemented
15#14346C211yesmaximal, characteristic, complemented
16#14289C211yesmaximal, characteristic, complemented
17#13941C211yesmaximal, characteristic
18C2×#552—28no—
19C2×#552—28no—
20C2×#552—28no—
21C22×#126C2212yes—
22C22×#125C2211yescharacteristic
23C22×#125C2212yes—
24C22×#125C2211yescharacteristic
25C22×#123C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117C2212yes—
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#2822—28no—
34#2633—28no—
35#2584—28no—
36C2×#400C2212yes—
37C2×#400C2212yes—
38C2×#399C2211yescharacteristic
39C2×#399C2211yescharacteristic
40C2×#398C2212yes—
41C2×#398C2212yes—
42C2×#398C2212yes—
43C2×#398C2214yes—
44C2×#398C2212yes—
45C2×#397C2211yescharacteristic
46C2×#397C2211yescharacteristic
47C2×#395C2212yes—
48C2×#394C2212yes—
49C2×#390C2212yes—
50C2×#389C2212yes—
51C2×#389C2212yes—
52C2×#389C2214yes—
53C2×#385C2212yes—
54C2×#385C2212yes—
55C2×#385C2214yes—
56C2×#383C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#1446C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2212yes—
79#1439C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85#1434C2212yes—
86C22⋊#95C2214yes—
87C22⋊#95C2212yes—
88C22⋊#95C2212yes—
89C23×Q8C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94#552—2128no—
95C2×#126—28no—
96C2×#126—28no—
97C2×#126—28no—
98C2×#125—28no—
99C2×#125—28no—
100C2×#125—28no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125D418yes—
107C2×#123—28no—
108C2×#123—28no—
109C2×#123—28no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C4×Q8—24no—
117C2×C4×Q8—24no—
118C2×C4×Q8D418yes—
119C2×C4×D4—24no—
120C2×C4×D4—24no—
121C2×C4×D4D418yes—
122C2×#117—24no—
123C2×#117—24no—
124C2×#117—24no—
125C2×#117—28no—
126C2×#117—28no—
127C2×#117—28no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—24no—
130C22×C4⋊C4—24no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2314yes—
147C22×C4⋊C4C2314yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2314yes—
152C22×C22⋊C4C2312yes—
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2314yes—
157C22×C22⋊C4C2311yescharacteristic
158C22×C22⋊C4C2311yescharacteristic
159C22×C22⋊C4C2312yes—
160C22×C22⋊C4C2312yes—
161C22×C22⋊C4C2312yes—
162C22×C22⋊C4C2314yes—
163C22×C42C2312yes—
164C22×C42C2312yes—
165C22×C42C2312yes—
166C22×C42C2311yescharacteristic
167#400—28no—
168#400—28no—
169#399—24no—
170#399—24no—
171#398—28no—
172#398—28no—
173#398—216no—
174#398—28no—
175#398—28no—
176#397—24no—
177#397—24no—
178#395—28no—
179#394—28no—
180#390—44no—
181#389—216no—
182#389—28no—
183#389—28no—
184#385—28no—
185#385—28no—
186#385—48no—
187#383—24no—
188#382—24no—
189C4×C4⋊C4—24no—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2312yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C25C2411yescharacteristic
215C25C2411yescharacteristic
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—44no—
219C22×Q8C2×D414yes—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—41no—
225C22×D4—41no—
226C22×D4C2×D414yes—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2414yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257#126—264no—
258#125—264no—
259#125—232no—
260#125—416no—
261#123—264no—
262C22⋊Q8—232no—
263C22⋊Q8—232no—
264C4×Q8—232no—
265C4×D4—232no—
266#117—232no—
267#117—264no—
268C2×C4⋊C4—232no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D418yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D418yes—
328C2×C4⋊C4C2×D418yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—28no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—28no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4C2×D414yes—
359C2×C22⋊C4C2×D414yes—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—42no—
367C2×C42—42no—
368C2×C42C2×D418yes—
369C2×C42C2×D414yes—
370C2×C42C2×D414yes—
371C2×C42C2×D414yes—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—28no—
382#95—28no—
383#95—28no—
384#95—28no—
385#95—28no—
386#95—28no—
387#95—28no—
388#95—44no—
389#95—44no—
390#95—44no—
391#95—44no—
392#95—44no—
393#95—44no—
394#95—44no—
395#95—44no—
396C24—21no—
397C24—21no—
398C24—24no—
399C24—24no—
400C24—21no—
401C24—21no—
402C24—21no—
403C24—21no—
404C24—21no—
405C24—24no—
406C24—21no—
407C24—24no—
408C24—21no—
409C24—21no—
410C24C22×D412yes—
411C24C22×D412yes—
412C24H(5,𝔽2)12yes—
413C242−1+412yes—
414C24C2511yescharacteristic
415C2×Q8—28no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—48no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—216no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D414yes—
511C22×C4C22×D414yes—
512C22×C4C22×D414yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C42−1+418yes—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—216no—
528C4⋊C4—28no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—216no—
532C4⋊C4—28no—
533C4⋊C4—216no—
534C4⋊C4—216no—
535C4⋊C4—416no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—88no—
544C4⋊C4—88no—
545C22⋊C4—216no—
546C22⋊C4—216no—
547C22⋊C4—216no—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—416no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—88no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—216no—
557C42—216no—
558C42—84no—
559C42#120116yes—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—24no—
568C23—28no—
569C23—28no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—24no—
574C23—24no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—42no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23#53714yes—
594C23C23×D411yescharacteristic
595C23C23×D411yescharacteristic
596C23C23×D411yescharacteristic
597C23C2×H(5,𝔽2)11yescharacteristic
598C23C2×2−1+411yescharacteristic
599C23C2×2−1+414yes—
600C23#58514yes—
601C23#58511yescharacteristic
602C23#58511yescharacteristic
603Q8—416no—
604D4—416no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—416no—
636C2×C4—416no—
637C2×C4—432no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4C2×#12014yes—
689C2×C4C2×#12014yes—
690C2×C4C2×#12014yes—
691C22—28no—
692C22—24no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—42no—
701C22—42no—
702C22—44no—
703C22—44no—
704C22—42no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22C22×#12011yescharacteristic
710C22C2×#53711yescharacteristic
711C22#344912yes—
712C22#344911yescharacteristic
713C22#348512yes—
714C22#348612yes—
715C22#348712yes—
716C22#348811yescharacteristic
717C22#348811yescharacteristic
718C22#348911yescharacteristic
719C22#348911yescharacteristic
720C22#349912yes—
721C22#349912yes—
722C22#353514yes—
723C22#355314yes—
724C22#355714yes—
725C22#356314yes—
726C4—24no—
727C4—24no—
728C4—24no—
729C4—48no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—44no—
734C4—44no—
735C4—44no—
736C4—82no—
737C4—84no—
738C4—82no—
739C4—82no—
740C4—82no—
741C2—44no—
742C2—82no—
743C2#3514911yescharacteristic
744C2#3525111yescharacteristic
745C2#3566611yescharacteristic
746C2#4137312yes—
747C2#4138312yes—
748C2#4199112yes—
749C2#4217112yes—
750C2#5089512yes—
751C2#5106612yes—
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×#2822C2semidirect
3. C2×#2584C2semidirect
5. #16674C2semidirect
6. #16673C2semidirect
7. #16606C2semidirect
8. #16601C2semidirect
9. #15848C2semidirect
11. #14907C2semidirect
12. #14906C2semidirect
13. #14893C2semidirect
14. #14786C2semidirect
15. #14346C2semidirect
16. #14289C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#28222
C2×#25842
C2×#26332
#166742
#166732
#166062
#166012
#158482
#152712
#149072
#149062
#148932
#147862
#143462
#142892
#139412