Finite Group Databasebeta

← #2736595

Subgroups — #2736595

Order 512.

2872 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2814C212yesmaximal, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2422C211yesmaximal, characteristic
4#17108C214yesmaximal, complemented
5#17103C214yesmaximal, complemented
6#16606C212yesmaximal, complemented
7#16264C212yesmaximal, complemented
8#16251C212yesmaximal, complemented
9#15149C211yesmaximal, characteristic, complemented
10#15145C212yesmaximal, complemented
11#15143C211yesmaximal, characteristic, complemented
12#15138C212yesmaximal, complemented
13#15131C212yesmaximal, complemented
14#15130C212yesmaximal
15#15125C211yesmaximal, characteristic, complemented
16#13920C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#123C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×#121C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#2814—216no—
25#2542—28no—
26#2422—28no—
27C2×#399C2214yes—
28C2×#398C2214yes—
29C2×#398C2212yes—
30C2×#398C2214yes—
31C2×#397C2214yes—
32C2×#397C2212yes—
33C2×#397C2211yescharacteristic
34C2×#396C2214yes—
35C2×#395C2212yes—
36C2×#395C2212yes—
37C2×#394C2214yes—
38C2×#394C2212yes—
39C2×#394C2214yes—
40C2×#393C2212yes—
41C2×#390C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2212yes—
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#385C2212yes—
48C2×#384C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51C2×#380C2211yescharacteristic
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57#1446C2212yes—
58#1445C2211yescharacteristic
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1443C2212yes—
63#1442C2212yes—
64#1442C2211yescharacteristic
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2212yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2211yescharacteristic
78#1439C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—44no—
93C2×#123—28no—
94C2×#123—28no—
95C2×#123—44no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—48no—
100C2×C22⋊Q8—44no—
101C2×#121—24no—
102C2×#121—44no—
103C2×C4×D4—44no—
104C2×C4×D4D418yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#399—216no—
141#398—28no—
142#398—216no—
143#398—48no—
144#397—28no—
145#397—216no—
146#397D418yescomplemented
147#396—216no—
148#395—28no—
149#395—44no—
150#394—28no—
151#394—216no—
152#394—48no—
153#393—28no—
154#390—24no—
155#390—28no—
156#390—44no—
157#389—24no—
158#389—44no—
159#389—44no—
160#385—28no—
161#384—24no—
162#384—28no—
163#384—44no—
164#380—24no—
165#380—28no—
166#380—44no—
167C4×C4⋊C4—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—44no—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—44no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4C2×D414yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C2×D418yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—416no—
238#123—432no—
239C22⋊Q8—432no—
240C22⋊Q8—432no—
241#121—416no—
242C4×D4—416no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—41no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D414yes—
327C2×C22⋊C4C2×D418yes—
328C2×C22⋊C4C2×D418yes—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42C2×D414yes—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D414yes—
363#95C2×D418yes—
364C24—22no—
365C24—21no—
366C24—24no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—21no—
376C24—22no—
377C24—24no—
378C24—22no—
379C24—21no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—21no—
384C24C22×D414yes—
385C24C22×D412yes—
386C24C2511yescharacteristic
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—88no—
391C2×Q8—88no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—84no—
400C2×D4—88no—
401C2×D4—88no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—216no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C22×C42−1+414yes—
499C22×C42−1+414yes—
500C22×C42−1+412yes—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—416no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—84no—
514C4⋊C4—88no—
515C4⋊C4—88no—
516C4⋊C4—88no—
517C4⋊C4—88no—
518C22⋊C4—28no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C42—28no—
533C42—44no—
534C42—48no—
535C42—48no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—24no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—41no—
557C23—44no—
558C23—42no—
559C23—44no—
560C23—42no—
561C23—44no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—41no—
573C23—42no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23C23×D411yescharacteristic
583C23C23×D411yescharacteristic
584C23C2×H(5,𝔽2)12yes—
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)12yes—
588C23C2×2−1+412yes—
589C23C2×2−1+411yescharacteristic
590C23C2×2−1+411yescharacteristic
591C23C2×2−1+412yes—
592C23#58511yescharacteristic
593Q8—816no—
594D4—416no—
595D4—816no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—816no—
680C2×C4—88no—
681C2×C4#53414yes—
682C2×C4#53614yes—
683C2×C4#53714yes—
684C2×C4D4214yes—
685C2×C4#56218yes—
686C2×C4#56318yes—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—42no—
691C22—44no—
692C22—44no—
693C22—42no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—42no—
698C22—44no—
699C22—81no—
700C22—82no—
701C22—84no—
702C22—81no—
703C22—84no—
704C22—84no—
705C22—81no—
706C22—82no—
707C22—84no—
708C22—81no—
709C22C2×#53411yescharacteristic
710C22C2×#53611yescharacteristic
711C22C2×#53711yescharacteristic
712C22C2×D4211yescharacteristic
713C22#348411yescharacteristic
714C22#348911yescharacteristic
715C22#349611yescharacteristic
716C22#349812yes—
717C22#349911yescharacteristic
718C22#349911yescharacteristic
719C22#349912yes—
720C22C2×#56212yes—
721C22C2×#56312yes—
722C22#353612yes—
723C22#354312yes—
724C22#354412yes—
725C22#354711yescharacteristic
726C22#354712yes—
727C22#354811yescharacteristic
728C22#355212yes—
729C22#355212yes—
730C22#355212yes—
731C22#355412yes—
732C4—24no—
733C4—42no—
734C4—44no—
735C4—44no—
736C4—44no—
737C4—44no—
738C4—42no—
739C4—44no—
740C4—42no—
741C4—81no—
742C4—82no—
743C4—84no—
744C4—82no—
745C4—84no—
746C4#281418yescomplemented
747C2—84no—
748C2—82no—
749C2C2×#281411yescharacteristic
750C2#3933611yescharacteristic
751C2#3934011yescharacteristic
752C2#3936011yescharacteristic
753C2#4242811yescharacteristic
754C2#4248511yescharacteristic
755C2#4291711yescharacteristic
756C2#4548612yes—
757C2#4599312yes—
758C2#5295912yes—
759C2#5308212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2814C2semidirect
2. C2×#2542C2semidirect
4. #17108C2semidirect
5. #17103C2semidirect
6. #16606C2semidirect
7. #16264C2semidirect
8. #16251C2semidirect
9. #15149C2semidirect
10. #15145C2semidirect
11. #15143C2semidirect
12. #15138C2semidirect
13. #15131C2semidirect
15. #15125C2semidirect
16. #13920C2semidirect
17. C4×#397C2semidirect
146. #397D4semidirect
746. C4#2814semidirect

Maximal subgroups

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

subgroupindex
C2×#28142
C2×#25422
C2×#24222
#171082
#171032
#166062
#162642
#162512
#151492
#151452
#151432
#151382
#151312
#151302
#151252
#139202
C4×#3972