Finite Group Databasebeta

← #1409291

Subgroups — #1409291

Order 512.

2874 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2491C211yesmaximal, characteristic
3C2×#2640C212yesmaximal, complemented
4#16558C212yesmaximal, complemented
5#16506C212yesmaximal, complemented
6#16375C214yesmaximal, complemented
7#16374C214yesmaximal, complemented
8#15150C211yesmaximal, characteristic, complemented
9#15147C211yesmaximal, characteristic, complemented
10#15144C212yesmaximal, complemented
11#15136C211yesmaximal, characteristic, complemented
12#15125C212yesmaximal, complemented
13#14894C212yesmaximal, complemented
14#14053C212yesmaximal, complemented
15#13998C212yesmaximal
16C4×#397C211yesmaximal, characteristic, complemented
17#13810C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21C22×#124C2212yes—
22C22×#121C2211yescharacteristic
23#2640—216no—
24#2551—28no—
25#2491—28no—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#397C2212yes—
30C2×#397C2212yes—
31C2×#397C2211yescharacteristic
32C2×#395C2214yes—
33C2×#394C2212yes—
34C2×#393C2214yes—
35C2×#393C2214yes—
36C2×#393C2212yes—
37C2×#393C2214yes—
38C2×#391C2214yes—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#390C2214yes—
42C2×#390C2212yes—
43C2×#390C2212yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#388C2212yes—
47C2×#384C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2212yes—
56#1446C2214yes—
57#1446C2214yes—
58#1445C2211yescharacteristic
59#1445C2212yes—
60#1445C2212yes—
61#1443C2212yes—
62#1442C2214yes—
63#1442C2211yescharacteristic
64#1441C2214yes—
65#1440C2212yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2211yescharacteristic
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1438C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82#1434C2212yes—
83#1434C2214yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×Q8C2314yes—
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—44no—
97C2×C4⋊Q8—44no—
98C2×C4⋊Q8—44no—
99C2×#124—28no—
100C2×#124—28no—
101C2×#124—44no—
102C2×#121—24no—
103C2×#121—28no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#397—28no—
138#397—28no—
139#397—28no—
140#397—44no—
141#397—44no—
142#397D418yescomplemented
143#395—216no—
144#394—28no—
145#393—216no—
146#393—216no—
147#393—28no—
148#393—48no—
149#391—216no—
150#390—24no—
151#390—216no—
152#390—24no—
153#390—44no—
154#390—44no—
155#389—28no—
156#389—28no—
157#388—28no—
158#384—24no—
159#380—24no—
160#380—44no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4D418yes—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—44no—
166#376—28no—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2311yescharacteristic
194C25C2412yes—
195C25C2411yescharacteristic
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—28no—
199C22×Q8—28no—
200C22×Q8—28no—
201C22×Q8—42no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—28no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—24no—
216C22×D4—42no—
217C22×D4—44no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C4⋊Q8—432no—
239C4⋊Q8—432no—
240C4⋊Q8—416no—
241#124—432no—
242#121—416no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4#4118yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
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—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×D418yes—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42C2×D418yes—
333C2×C42C2×D414yes—
334#95—28no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—22no—
348#95—24no—
349#95—44no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—42no—
357#95—44no—
358#95C2×D414yes—
359#95C2×D418yes—
360#95C2×D418yes—
361C24—21no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—24no—
380C24—22no—
381C24—24no—
382C24C22×D412yes—
383C24C2511yescharacteristic
384C2×Q8—48no—
385C2×Q8—416no—
386C2×Q8—416no—
387C2×Q8—416no—
388C2×Q8—48no—
389C2×Q8—84no—
390C2×Q8—88no—
391C2×Q8—88no—
392C2×Q8—88no—
393C2×Q8—88no—
394C2×Q8—88no—
395C2×Q8—88no—
396C2×D4—48no—
397C2×D4—416no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—84no—
405C2×D4—88no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
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—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C2×#4118yes—
493C22×C4C2×#4114yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C42−1+414yes—
499C22×C42−1+414yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—416no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—84no—
511C4⋊C4#124116yes—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—84no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C42—28no—
528C42—216no—
529C42—416no—
530C42—44no—
531C42—48no—
532C42—48no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23C23×D411yescharacteristic
580C23C23×D411yescharacteristic
581C23C22×#4112yes—
582C23C2×H(5,𝔽2)11yescharacteristic
583C23C2×H(5,𝔽2)11yescharacteristic
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×2−1+412yes—
587C23C2×2−1+412yes—
588C23#58512yes—
589C23#58511yescharacteristic
590Q8—816no—
591Q8—832no—
592D4—816no—
593D4—88no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
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—41no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—88no—
677C2×C4C2×#12418yes—
678C2×C4C2×#12414yes—
679C2×C4#53514yes—
680C2×C4#53614yes—
681C2×C4#54218yes—
682C2×C4#54218yes—
683C2×C4D4214yes—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—44no—
688C22—44no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—84no—
703C22—84no—
704C22—84no—
705C22—84no—
706C22C22×#12411yescharacteristic
707C22C2×#53511yescharacteristic
708C22C2×#53611yescharacteristic
709C22C2×#54212yes—
710C22C2×#54212yes—
711C22C2×D4211yescharacteristic
712C22D4×#4112yes—
713C22#348611yescharacteristic
714C22#348811yescharacteristic
715C22#349611yescharacteristic
716C22#349811yescharacteristic
717C22#349912yes—
718C22#350011yescharacteristic
719C22#351412yes—
720C22#351812yes—
721C22#352212yes—
722C22#354112yes—
723C22#354412yes—
724C22#354611yescharacteristic
725C22#354712yes—
726C22#354911yescharacteristic
727C22#356112yes—
728C22#356512yes—
729C4—28no—
730C4—24no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—44no—
735C4—44no—
736C4—42no—
737C4—44no—
738C4—82no—
739C4—81no—
740C4—84no—
741C4—82no—
742C4—84no—
743C4#264018yescomplemented
744C2—82no—
745C2—162no—
746C2C2×#264011yescharacteristic
747C2#3594811yescharacteristic
748C2D4×#12411yescharacteristic
749C2#3933711yescharacteristic
750C2#3934211yescharacteristic
751C2#3940312yes—
752C2#3940612yes—
753C2#4273112yes—
754C2#4273712yes—
755C2#4278811yescharacteristic
756C2#4296511yescharacteristic
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×#2551C2semidirect
3. C2×#2640C2semidirect
4. #16558C2semidirect
5. #16506C2semidirect
6. #16375C2semidirect
7. #16374C2semidirect
8. #15150C2semidirect
9. #15147C2semidirect
10. #15144C2semidirect
11. #15136C2semidirect
12. #15125C2semidirect
13. #14894C2semidirect
14. #14053C2semidirect
16. C4×#397C2semidirect
17. #13810C2semidirect
142. #397D4semidirect
743. C4#2640semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#24912
C2×#26402
#165582
#165062
#163752
#163742
#151502
#151472
#151442
#151362
#151252
#148942
#140532
#139982
C4×#3972
#138102