Finite Group Databasebeta

← #712179

Subgroups — #712179

Order 512.

2844 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2C2×#2624C211yesmaximal, characteristic
3C2×#2346C211yesmaximal, characteristic, complemented
4C2×#2316C211yesmaximal, characteristic, complemented
5#16564C212yesmaximal, complemented
6#16518C212yesmaximal, complemented
7#15302C214yesmaximal, complemented
8#15289C214yesmaximal, complemented
9#15225C212yesmaximal, complemented
10#15141C212yesmaximal, complemented
11#14910C212yesmaximal, complemented
12#14894C212yesmaximal, complemented
13#14141C212yesmaximal, complemented
14#14137C211yesmaximal, characteristic
15#14130C211yesmaximal, characteristic, complemented
16#14005C212yesmaximal, complemented
17#13853C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2624—28no—
23#2500—28no—
24C2×#399C2212yes—
25C2×#397C2212yes—
26C2×#396C2214yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#394C2212yes—
30#2346—28no—
31#2316—28no—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#388C2212yes—
45C2×#388C2214yes—
46C2×#388C2212yes—
47C2×#388C2214yes—
48C2×#386C2214yes—
49C2×#385C2212yes—
50C2×#384C2211yescharacteristic
51C2×#382C2212yes—
52C2×#382C2211yescharacteristic
53C2×#380C2214yes—
54C2×#380C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×#376C2212yes—
59#1445C2211yescharacteristic
60#1445C2211yescharacteristic
61#1445C2211yescharacteristic
62#1445C2211yescharacteristic
63#1445C2214yes—
64#1444C2212yes—
65#1444C2214yes—
66#1442C2212yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1440C2212yes—
72#1440C2212yes—
73#1440C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2212yes—
79#1439C2214yes—
80#1437C2214yes—
81#1436C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#121—24no—
92C2×#121—24no—
93C2×#121—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117D418yes—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2312yes—
138#399—28no—
139#397—28no—
140#396—48no—
141#394—216no—
142#394—28no—
143#394—28no—
144#390—28no—
145#390—24no—
146#390—24no—
147#390—28no—
148#389—28no—
149#389—28no—
150#389—28no—
151#389—28no—
152#389—28no—
153#389—28no—
154#389—28no—
155#389—48no—
156#388—216no—
157#388—28no—
158#388—28no—
159#388—48no—
160#386—48no—
161#385—44no—
162#384—24no—
163#382—24no—
164#382—28no—
165#380—216no—
166#380D418yes—
167C4×C4⋊C4—28no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—24no—
170#376—28no—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C25C2412yes—
197C25C2411yescharacteristic
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—44no—
207C22×D4—44no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4#4114yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233#121—416no—
234#117—232no—
235#117—416no—
236C2×C4⋊C4—232no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
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—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4#4118yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—41no—
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—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4C2×D418yes—
321C2×C22⋊C4C2×D418yes—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—28no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—44no—
337C2×C42C2×D418yes—
338#95—22no—
339#95—28no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—42no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363C24—22no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24C22×D412yes—
383C24C22×D412yes—
384C24C2×#4112yes—
385C24C2511yescharacteristic
386C2×D4—416no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—88no—
394C2×D4—84no—
395C2×D4—84no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—216no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
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—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4#121116yes—
480C22×C4C22×D414yes—
481C22×C4C22×D418yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C2×#4118yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4114yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C42−1+414yes—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—416no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—88no—
505C4⋊C4—88no—
506C4⋊C4—88no—
507C4⋊C4—84no—
508C4⋊C4—88no—
509C4⋊C4—84no—
510C22⋊C4—216no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—88no—
525C42—216no—
526C42—416no—
527C42—48no—
528C42—44no—
529C42—44no—
530C42—48no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23C2×#12114yes—
577C23C23×D411yescharacteristic
578C23C23×D411yescharacteristic
579C23C22×#4111yescharacteristic
580C23C22×#4112yes—
581C23C2×H(5,𝔽2)12yes—
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×2−1+412yes—
584C23C2×2−1+411yescharacteristic
585C23#58512yes—
586C23#58511yescharacteristic
587C23#58511yescharacteristic
588D4—816no—
589D4—88no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—232no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—816no—
675C2×C4C2×#12114yes—
676C2×C4C2×#12114yes—
677C2×C4#52514yes—
678C2×C4#53614yes—
679C2×C4#54718yes—
680C2×C4#55218yes—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—44no—
685C22—44no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—44no—
692C22—84no—
693C22—84no—
694C22—82no—
695C22—82no—
696C22—84no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—84no—
702C22—82no—
703C22C22×#12111yescharacteristic
704C22C2×#52511yescharacteristic
705C22C2×#53611yescharacteristic
706C22#346012yes—
707C22C2×#54712yes—
708C22C2×#55212yes—
709C22#348812yes—
710C22#348811yescharacteristic
711C22#348911yescharacteristic
712C22#349611yescharacteristic
713C22#349711yescharacteristic
714C22#349811yescharacteristic
715C22#350011yescharacteristic
716C22#351111yescharacteristic
717C22#351211yescharacteristic
718C22#351312yes—
719C22#351612yes—
720C22#351912yes—
721C22#352412yes—
722C22#354412yes—
723C22#354712yes—
724C22#354712yes—
725C22#356512yes—
726C4—24no—
727C4—48no—
728C4—42no—
729C4—42no—
730C4—48no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—81no—
738C4—81no—
739C4—82no—
740C4#241318yes—
741C2—44no—
742C2—162no—
743C2C2×#241311yescharacteristic
744C2#3525611yescharacteristic
745C2#3556411yescharacteristic
746C2#3598511yescharacteristic
747C2#3599511yescharacteristic
748C2#3826612yes—
749C2#3907912yes—
750C2#4296511yescharacteristic
751C2#4296811yescharacteristic
752C2#4488812yes—
753C2#4501612yes—
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
3. C2×#2346C2semidirect
4. C2×#2316C2semidirect
5. #16564C2semidirect
6. #16518C2semidirect
7. #15302C2semidirect
8. #15289C2semidirect
9. #15225C2semidirect
10. #15141C2semidirect
11. #14910C2semidirect
12. #14894C2semidirect
13. #14141C2semidirect
15. #14130C2semidirect
16. #14005C2semidirect
17. #13853C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#26242
C2×#23462
C2×#23162
#165642
#165182
#153022
#152892
#152252
#151412
#149102
#148942
#141412
#141372
#141302
#140052
#138532