Finite Group Databasebeta

← #777911

Subgroups — #777911

Order 512.

3730 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#2421C212yesmaximal, complemented
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#16558C212yesmaximal
5#16102C214yesmaximal, complemented
6#15928C214yesmaximal, complemented
7#15230C211yesmaximal, characteristic, complemented
8#15181C211yesmaximal, characteristic
9#15128C211yesmaximal, characteristic, complemented
10#15070C212yesmaximal
11#14884C212yesmaximal, complemented
12#14174C212yesmaximal, complemented
13#14129C212yesmaximal, complemented
14#13969C212yesmaximal, complemented
15#13931C211yesmaximal, characteristic, complemented
16#13879C212yesmaximal, complemented
17C4×#394C211yesmaximal, characteristic
18C2×#556—216no—
19C2×#556—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2214yes—
22C22×#124C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C4×Q8C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2211yescharacteristic
29C22×#117C2212yes—
30#2421—216no—
31C2×#400C2214yes—
32C2×#397C2214yes—
33C2×#395C2212yes—
34C2×#394C2211yescharacteristic
35Q8×C4⋊C4—28no—
36C2×#391C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#387C2214yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#382C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53#1446C2212yes—
54#1446C2212yes—
55#1446C2212yes—
56#1445C2214yes—
57#1444C2212yes—
58#1442C2211yescharacteristic
59#1442C2212yes—
60#1441C2212yes—
61#1441C2211yescharacteristic
62#1441C2214yes—
63#1441C2214yes—
64#1440C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2211yescharacteristic
73#1438C2212yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85#1434C2212yes—
86C4×#95C2211yescharacteristic
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4C2312yes—
91#556—2128no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—24no—
94C2×#126—216no—
95C2×#126—216no—
96C2×#126—216no—
97C2×#124—216no—
98C2×#124—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8D4116yes—
102C2×C4×Q8—24no—
103C2×C4×Q8—28no—
104C2×C4×Q8—24no—
105C2×C4×Q8—28no—
106C2×C4×Q8—216no—
107C2×C4×Q8—24no—
108C2×C4×Q8—28no—
109C2×C4×Q8—24no—
110C2×C4×Q8D418yes—
111C2×C4×D4—28no—
112C2×C4×D4—24no—
113C2×#117—28no—
114C2×#117—28no—
115C2×#117—28no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2312yes—
143C22×C42C2314yes—
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146C22×C42C2312yes—
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151#400—216no—
152#397—216no—
153#395—28no—
154#394—24no—
155#391—28no—
156#389—24no—
157#389—28no—
158#389—28no—
159#389—24no—
160#389—28no—
161#388—216no—
162#387—216no—
163#384—24no—
164#384—28no—
165#382—28no—
166#380—28no—
167#380—28no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—28no—
170C4×C22⋊C4—28no—
171C4×C22⋊C4—28no—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C25C2412yes—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—44no—
207C22×Q8C2×D414yes—
208C22×Q8C2×D418yes—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—42no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4C2×D418yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—232no—
238#126—2128no—
239#124—264no—
240C22⋊Q8—264no—
241C4×Q8—232no—
242C4×Q8—264no—
243C4×Q8—232no—
244C4×Q8—416no—
245C4×D4—232no—
246#117—264no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D4116yes—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4#4114yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4C2×D4116yes—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—28no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42C2×D414yes—
337C2×C42C2×D418yes—
338C2×C42#4114yes—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—22no—
355#95—28no—
356#95—24no—
357#95—22no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—42no—
363#95—42no—
364#95—44no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24C22×D414yes—
380C24C2511yescharacteristic
381C2×Q8—216no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—216no—
385C2×Q8—216no—
386C2×Q8—416no—
387C2×Q8—416no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8#120116yes—
391C2×D4—28no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—44no—
395C2×D4—44no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4C22×D414yes—
476C22×C4C22×D418yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D418yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C22×C42−1+414yes—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—28no—
499C4⋊C4—216no—
500C4⋊C4—232no—
501C4⋊C4—216no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—416no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C22⋊C4—232no—
514C22⋊C4—216no—
515C22⋊C4—232no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—48no—
521C22⋊C4—88no—
522C42—232no—
523C42—216no—
524C42—28no—
525C42—216no—
526C42—28no—
527C42—48no—
528C42—44no—
529C42—48no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23C23×D411yescharacteristic
566C23C23×D412yes—
567C23C22×#4111yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4111yescharacteristic
570C23C2×2−1+412yes—
571C23C2×2−1+411yescharacteristic
572C23C2×2−1+411yescharacteristic
573C23#58512yes—
574C23#58511yescharacteristic
575C23#58512yes—
576Q8—232no—
577Q8—432no—
578D4—416no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—432no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4C2×#12018yes—
665C2×C4C2×#12014yes—
666C2×C4#52514yes—
667C2×C4#53614yes—
668C2×C4#54718yes—
669C2×C4#54718yes—
670C2×C4#54814yes—
671C2×C4#55614yescomplemented
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22C22×#12011yescharacteristic
688C22C2×#52511yescharacteristic
689C22C2×#53611yescharacteristic
690C22C2×#54712yes—
691C22C2×#54712yes—
692C22C2×#54811yescharacteristic
693C22D4×#4112yes—
694C22C2×#55611yescharacteristic
695C22#348011yescharacteristic
696C22#348711yescharacteristic
697C22#348712yes—
698C22#348912yes—
699C22#348911yescharacteristic
700C22#349312yes—
701C22#349712yes—
702C22#350011yescharacteristic
703C22#350711yescharacteristic
704C22#351812yes—
705C22#352112yes—
706C22#352311yescharacteristic
707C22#355512yes—
708C22#355512yes—
709C22#356512yes—
710C4—24no—
711C4—28no—
712C4—24no—
713C4—42no—
714C4—48no—
715C4—44no—
716C4—44no—
717C4—42no—
718C4—42no—
719C4—44no—
720C4—82no—
721C4—82no—
722C4—81no—
723C4—82no—
724C4—84no—
725C4#240618yes—
726C2—84no—
727C2C2×#240611yescharacteristic
728C2#3575711yescharacteristic
729C2#3601911yescharacteristic
730C2#3642811yescharacteristic
731C2#3657911yescharacteristic
732C2#3917211yescharacteristic
733C2#4289111yescharacteristic
734C2#4526112yes—
735C2#4532612yes—
736C2#5911812yes—
737C2#5924412yes—
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×#2421C2semidirect
3. C2×Q8×C4⋊C4C2semidirect
5. #16102C2semidirect
6. #15928C2semidirect
7. #15230C2semidirect
9. #15128C2semidirect
11. #14884C2semidirect
12. #14174C2semidirect
13. #14129C2semidirect
14. #13969C2semidirect
15. #13931C2semidirect
16. #13879C2semidirect
671. C2×C4#556semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#24212
C2×Q8×C4⋊C42
#165582
#161022
#159282
#152302
#151812
#151282
#150702
#148842
#141742
#141292
#139692
#139312
#138792
C4×#3942