Finite Group Databasebeta

← #734989

Subgroups — #734989

Order 512.

3262 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×#2580C211yesmaximal, characteristic, complemented
3C2×#2398C211yesmaximal, characteristic, complemented
4C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#16549C214yesmaximal, complemented
6#15679C214yesmaximal, complemented
7#15316C214yesmaximal, complemented
8#15131C212yesmaximal, complemented
9#15130C212yesmaximal, complemented
10#14883C214yesmaximal, complemented
11#14276C211yesmaximal, characteristic, complemented
12#13998C212yesmaximal, complemented
13#13973C212yesmaximal, complemented
14#13905C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic, complemented
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2580—28no—
23#2537—28no—
24#2398—28no—
25C2×#400C2212yes—
26C2×#399C2212yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#394C2214yes—
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#393C2212yes—
34C2×#393C2212yes—
35Q8×C4⋊C4—28no—
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2211yescharacteristic
39C2×#390C2214yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#385C2214yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57#1445C2212yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1438C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2211yescharacteristic
77#1434C2214yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2314yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×#121—28no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—48no—
89C2×#121—44no—
90C2×C4×Q8—28no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—48no—
94C2×C4×Q8D418yescomplemented
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#400—28no—
136#399—28no—
137#397—28no—
138#396—48no—
139#394—28no—
140#394—216no—
141#394—48no—
142#394—48no—
143#393—28no—
144#393—28no—
145#390—24no—
146#390—24no—
147#390—28no—
148#390—48no—
149#389—216no—
150#389—28no—
151#389—28no—
152#389—44no—
153#385—216no—
154#384—28no—
155#384D418yes—
156#382—24no—
157#380—28no—
158#380—28no—
159#380—28no—
160#380—44no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C22⋊C4—28no—
164C4×C22⋊C4—28no—
165#376—24no—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2412yes—
186C25C2412yes—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8C2×D414yes—
192C22×D4—24no—
193C22×D4—28no—
194C22×D4—28no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—28no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4#4114yes—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236#121—432no—
237#121—432no—
238C4×Q8—232no—
239C4×Q8—432no—
240C4×D4—416no—
241C4×D4—416no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
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⋊C4C2×D418yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4#4114yes—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42C2×D418yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42#4114yes—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—24no—
349#95—28no—
350#95—24no—
351#95—24no—
352#95—28no—
353#95—28no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24C2×#4114yes—
385C24H(5,𝔽2)14yes—
386C24C2511yescharacteristic
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—216no—
390C2×Q8—28no—
391C2×Q8—216no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—416no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4C22×D418yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C42−1+414yes—
503C22×C42−1+412yes—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—28no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—88no—
516C4⋊C4—88no—
517C4⋊C4—88no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—416no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C42—216no—
533C42—416no—
534C42—48no—
535C42—44no—
536C42—44no—
537C42#121116yes—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—42no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—44no—
585C23—42no—
586C23—44no—
587C23—42no—
588C23—42no—
589C23—44no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—42no—
594C23—44no—
595C23—42no—
596C23C23×D411yescharacteristic
597C23C23×D411yescharacteristic
598C23C22×#4111yescharacteristic
599C23C22×#4111yescharacteristic
600C23C22×#4111yescharacteristic
601C23C2×H(5,𝔽2)12yes—
602C23C2×H(5,𝔽2)11yescharacteristic
603C23C2×H(5,𝔽2)12yes—
604C23C2×2−1+412yes—
605C23C2×2−1+411yescharacteristic
606C23#58512yes—
607Q8—232no—
608D4—416no—
609D4—416no—
610D4—832no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—28no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—816no—
683C2×C4—816no—
684C2×C4C2×#12114yes—
685C2×C4C2×#12114yes—
686C2×C4C2×#12114yes—
687C2×C4#52914yes—
688C2×C4#53718yes—
689C2×C4#54214yes—
690C2×C4#54614yes—
691C2×C4#54814yes—
692C2×C4#54814yes—
693C2×C4#55014yes—
694C22—44no—
695C22—44no—
696C22—48no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—48no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—82no—
707C22—84no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—84no—
719C22—82no—
720C22C22×#12111yescharacteristic
721C22C2×#52911yescharacteristic
722C22C2×#53712yes—
723C22C2×#54211yescharacteristic
724C22C2×#54611yescharacteristic
725C22C2×#54811yescharacteristic
726C22C2×#54811yescharacteristic
727C22C2×#55011yescharacteristic
728C22#348411yescharacteristic
729C22#348411yescharacteristic
730C22#348611yescharacteristic
731C22#348711yescharacteristic
732C22#348812yes—
733C22#349212yes—
734C22#349412yes—
735C22#351412yes—
736C22#351512yes—
737C22#351912yes—
738C22#352212yes—
739C22#354312yes—
740C22#354412yes—
741C22#354712yes—
742C22#355412yes—
743C4—24no—
744C4—28no—
745C4—24no—
746C4—48no—
747C4—44no—
748C4—44no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—84no—
754C4—82no—
755C4#242618yes—
756C2—84no—
757C2—84no—
758C2C2×#242611yescharacteristic
759C2#3522311yescharacteristic
760C2#3556311yescharacteristic
761C2#3594712yes—
762C2#3912612yes—
763C2#3913712yes—
764C2#4251512yes—
765C2#4493611yescharacteristic
766C2#4493711yescharacteristic
767C2#4493811yescharacteristic
768C2#4533511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#2580C2semidirect
3. C2×#2398C2semidirect
4. C2×Q8×C4⋊C4C2semidirect
5. #16549C2semidirect
6. #15679C2semidirect
7. #15316C2semidirect
8. #15131C2semidirect
9. #15130C2semidirect
10. #14883C2semidirect
11. #14276C2semidirect
12. #13998C2semidirect
13. #13973C2semidirect
14. #13905C2semidirect
15. #13785C2semidirect
94. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25802
C2×#23982
C2×Q8×C4⋊C42
#165492
#156792
#153162
#151312
#151302
#148832
#142762
#139982
#139732
#139052
#137852