Finite Group Databasebeta

← #875059

Subgroups — #875059

Order 512.

4038 conjugacy classes of subgroups.

The subgroup lattice has 789 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2733C211yesmaximal, characteristic, complemented
2C2×#2436C212yesmaximal, complemented
3C22×#391C211yesmaximal, characteristic, complemented
4#17077C212yesmaximal, complemented
5#16652C212yesmaximal, complemented
6#15851C214yesmaximal, complemented
7#15317C214yesmaximal, complemented
8#14757C214yesmaximal, complemented
9#14719C212yesmaximal, complemented
10#14698C212yesmaximal, complemented
11#14492C212yesmaximal, complemented
12#14467C211yesmaximal, characteristic, complemented
13#13999C214yesmaximal, complemented
14C24×Q8C2211yescharacteristic
15C22×#124C2212yes—
16C22×#124C2212yes—
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#121C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2733—28no—
23#2436—216no—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#393C2214yes—
27C2×#393C2214yes—
28C2×#392C2214yes—
29C2×#391—28no—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2214yes—
36C2×#391C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#388C2214yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#380C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2214yes—
47#1444C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2212yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69C22⋊#95C2212yes—
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—42no—
73C23×Q8—42no—
74C23×Q8—42no—
75C23×Q8—42no—
76C23×Q8D418yescomplemented
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2314yes—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C23×D4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2312yes—
88C2×#124—28no—
89C2×#124—28no—
90C2×#124—28no—
91C2×#124—28no—
92C2×#124—44no—
93C2×#124—44no—
94C2×#121—28no—
95C2×#121—28no—
96C2×#121—44no—
97C2×#121—44no—
98C2×#121—44no—
99C2×#121—44no—
100C2×#121—44no—
101C2×#121—48no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4—42no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42—24no—
126C22×C42—42no—
127C22×C42—42no—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#394—28no—
134#394—44no—
135#393—216no—
136#393—216no—
137#392—216no—
138#391—28no—
139#391—28no—
140#391—24no—
141#391—24no—
142#391—24no—
143#391—216no—
144#391D418yescomplemented
145#390—28no—
146#390—44no—
147#388—216no—
148#386—216no—
149#386—216no—
150#386—216no—
151#386—48no—
152#380—44no—
153C4×C22⋊C4—28no—
154C2×#95—216no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25C2414yes—
173C25C2411yescharacteristic
174C22×Q8—232no—
175C22×Q8—24no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—416no—
182C22×Q8—41no—
183C22×Q8—41no—
184C22×Q8—42no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8—42no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8—416no—
193C22×Q8—416no—
194C22×Q8—416no—
195C22×Q8—416no—
196C22×Q8—416no—
197C22×Q8C2×D418yes—
198C22×Q8C2×D414yes—
199C22×Q8C2×D414yes—
200C22×Q8C2×D414yes—
201C22×Q8#4114yes—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4—28no—
205C22×D4—28no—
206C22×D4—28no—
207C22×D4—28no—
208C22×D4—28no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4—44no—
221C22×D4—42no—
222C22×D4—42no—
223C22×D4—42no—
224C22×D4—42no—
225C22×D4—42no—
226C22×D4—44no—
227C22×D4—42no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—28no—
244C23×C4—24no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4—44no—
251C23×C4C2×D414yes—
252C23×C4C2×D418yes—
253C23×C4#4118yes—
254C23×C4C2414yes—
255C23×C4C2414yes—
256C23×C4C2412yes—
257C23×C4C2414yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2412yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265#124—432no—
266#124—432no—
267#121—432no—
268#121—432no—
269#121—432no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—416no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C2×D414yes—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
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×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—41no—
340C2×C42—41no—
341C2×C42—42no—
342C2×C42—41no—
343C2×C42—41no—
344C2×C42—416no—
345C2×C42C2×D414yes—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—42no—
362#95C2×D4116yes—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24C22×D412yes—
385C24C2×#4114yes—
386C24H(5,𝔽2)12yes—
387C24H(5,𝔽2)12yes—
388C24H(5,𝔽2)14yes—
389C24C2511yescharacteristic
390C2×Q8—216no—
391C2×Q8—28no—
392C2×Q8—216no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—48no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×Q8—4128no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8#121116yescomplemented
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—416no—
418C2×D4—416no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—416no—
423C2×D4—416no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—88no—
436C2×D4—84no—
437C2×D4—88no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—216no—
462C22×C4—216no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—216no—
480C22×C4—216no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—48no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4—48no—
518C22×C4—416no—
519C22×C4—48no—
520C22×C4—84no—
521C22×C4—88no—
522C22×C4C22×D414yes—
523C22×C4C22×D414yes—
524C22×C4C22×D412yes—
525C22×C4C22×D414yes—
526C22×C4C22×D414yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D414yes—
530C22×C4C2×#4114yes—
531C22×C4C2×#4114yes—
532C22×C4C2×#4112yes—
533C22×C4C2×#4114yes—
534C22×C4H(5,𝔽2)12yes—
535C22×C4H(5,𝔽2)12yes—
536C22×C42−1+414yes—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C22⋊C4—416no—
546C22⋊C4—416no—
547C22⋊C4—416no—
548C22⋊C4—416no—
549C22⋊C4—416no—
550C22⋊C4—416no—
551C22⋊C4—416no—
552C22⋊C4—416no—
553C22⋊C4—416no—
554C22⋊C4—88no—
555C22⋊C4—88no—
556C22⋊C4—88no—
557C42—44no—
558C42—48no—
559C42—44no—
560C42—48no—
561C42—48no—
562C23—22no—
563C23—22no—
564C23—24no—
565C23—24no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—24no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23#53414yes—
616C23#53818yes—
617C23#55018yes—
618C23#55514yes—
619C23H(3,𝔽4)14yes—
620C23C23×D411yescharacteristic
621C23C23×D411yescharacteristic
622C23C22×#4111yescharacteristic
623C23C22×#4112yes—
624C23C2×H(5,𝔽2)11yescharacteristic
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)11yescharacteristic
627C23C2×H(5,𝔽2)11yescharacteristic
628C23C2×H(5,𝔽2)12yes—
629C23C2×2−1+412yes—
630C23#58512yes—
631Q8—416no—
632Q8—432no—
633Q8—416no—
634D4—816no—
635D4—816no—
636D4—816no—
637D4—832no—
638C2×C4—232no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—24no—
667C2×C4—24no—
668C2×C4—24no—
669C2×C4—28no—
670C2×C4—24no—
671C2×C4—432no—
672C2×C4—432no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—42no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—42no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—816no—
711C2×C4C2×#12118yes—
712C2×C4C2×#12114yes—
713C2×C4#53414yes—
714C2×C4#54818yes—
715C22—24no—
716C22—28no—
717C22—24no—
718C22—28no—
719C22—28no—
720C22—24no—
721C22—24no—
722C22—28no—
723C22—24no—
724C22—24no—
725C22—84no—
726C22—84no—
727C22—84no—
728C22—84no—
729C22—84no—
730C22—84no—
731C22—84no—
732C22—84no—
733C22—84no—
734C22—84no—
735C22—84no—
736C22—84no—
737C22—84no—
738C22—84no—
739C22—84no—
740C22#273318yescomplemented
741C22C22×#12111yescharacteristic
742C22C2×#53411yescharacteristic
743C22C2×#53411yescharacteristic
744C22C2×#53812yes—
745C22C2×#54812yes—
746C22C2×#55012yes—
747C22C2×#55511yescharacteristic
748C22#348012yes—
749C22#348411yescharacteristic
750C22#348411yescharacteristic
751C22#348812yes—
752C22#349011yescharacteristic
753C22#349011yescharacteristic
754C22#349712yes—
755C22#350512yes—
756C22#351112yes—
757C22#352212yes—
758C22C2×H(3,𝔽4)11yescharacteristic
759C22#353512yes—
760C22#353612yes—
761C22#354411yescharacteristic
762C22#354511yescharacteristic
763C22#355812yes—
764C4—28no—
765C4—24no—
766C4—24no—
767C4—28no—
768C4—44no—
769C4—44no—
770C4—44no—
771C4—84no—
772C4—82no—
773C4—84no—
774C4—84no—
775C2—28no—
776C2—164no—
777C2#3498111yescharacteristic
778C2#3657612yes—
779C2#3719011yescharacteristic
780C2#3770312yes—
781C2#3848612yes—
782C2#3974312yes—
783C2#4597711yescharacteristic
784C2#4609911yescharacteristic
785C2C2×#273311yescharacteristic
786C2#5913511yescharacteristic
787C2#5988511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2733C2semidirect
2. C2×#2436C2semidirect
3. C22×#391C2semidirect
4. #17077C2semidirect
5. #16652C2semidirect
6. #15851C2semidirect
7. #15317C2semidirect
8. #14757C2semidirect
9. #14719C2semidirect
10. #14698C2semidirect
11. #14492C2semidirect
12. #14467C2semidirect
13. #13999C2semidirect
76. C23×Q8D4semidirect
144. #391D4semidirect
411. C2×Q8#121semidirect
740. C22#2733semidirect

Maximal subgroups

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

subgroupindex
C2×#27332
C2×#24362
C22×#3912
#170772
#166522
#158512
#153172
#147572
#147192
#146982
#144922
#144672
#139992