Finite Group Databasebeta

← #6508614

Subgroups — #6508614

Order 512.

3322 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3505C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3C22×#481C211yesmaximal, characteristic, complemented
4C2×#2487C211yesmaximal, characteristic, complemented
5#24095C214yesmaximal, complemented
6#24092C214yesmaximal, complemented
7#24067C214yesmaximal, complemented
8#24039C214yesmaximal, complemented
9#24036C214yesmaximal, complemented
10#24007C214yesmaximal, complemented
11#21759C211yesmaximal, characteristic, complemented
12#21758C211yesmaximal, characteristic, complemented
13#21738C211yesmaximal, characteristic, complemented
14#3505—28no—
15#3505—28no—
16C2×#553—22no—
17C2×#553C2212yes—
18C2×#552C2212yes—
19C2×#551—22no—
20C2×#548C2212yes—
21C2×#546C2212yes—
22C2×#539—21no—
23C2×#538—21no—
24C2×#534—21no—
25C2×#529C2212yes—
26C2×#525C2212yes—
27C22×C22⋊Q8C2211yescharacteristic
28C22×#121C2211yescharacteristic
29C2×#518—21no—
30C22×C4×D4C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C2×#484—28no—
33C2×#484C2214yescomplemented
34C2×#484C2214yescomplemented
35C2×#481—28no—
36C2×#481C2214yes—
37C2×#481C2214yescomplemented
38C22×#107C2212yescomplemented
39C22×#106C2212yescomplemented
40C22×#103C2212yescomplemented
41C22×#102C2212yescomplemented
42C22×C22⋊C8C2211yescharacteristic
43C22×C22⋊C8C2211yescharacteristic
44#2487—22no—
45#2487—22no—
46#2487C2214yes—
47#2487C2214yescomplemented
48C2×#400—21no—
49C2×#399—21no—
50C2×#389C2212yescomplemented
51C2×#386—21no—
52C2×#385—21no—
53C2×#380C2211yescharacteristic
54C2×#380C2212yescomplemented
55#2059C2218yescomplemented
56#2042C2218yescomplemented
57#2035C2218yescomplemented
58#2033C2218yescomplemented
59#1893C2214yescomplemented
60#1892C2214yescomplemented
61#1874C2214yes—
62#1872C2214yescomplemented
63#1869C2214yes—
64#1869C2214yescomplemented
65#1865C2218yescomplemented
66#1862C2218yescomplemented
67#1854C2218yescomplemented
68#1824C2212yescomplemented
69#1823C2212yescomplemented
70#1817C2212yescomplemented
71#1788C2218yescomplemented
72#1787C2212yescomplemented
73#1786C2212yescomplemented
74#1779C2212yescomplemented
75C22×#41C2312yes—
76C23×Q8C2311yescharacteristic
77C23×D4—21no—
78C23×D4D411yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8C2312yes—
84C23×C8C2312yes—
85#553—232no—
86#553—28no—
87#553—28no—
88#552—28no—
89#552—28no—
90#551—232no—
91#548—28no—
92#548—28no—
93#546—28no—
94#546—28no—
95#539—216no—
96#538—216no—
97#534—216no—
98#529—28no—
99#529—28no—
100C2×#126—22no—
101C2×#126—22no—
102C2×#126—22no—
103C2×#125—22no—
104C2×#125C2312yes—
105C2×#124—22no—
106C2×#124C2312yes—
107#525—28no—
108#525—28no—
109C2×#123—22no—
110C2×#123—22no—
111C2×#123—22no—
112C2×#123—22no—
113C2×#123—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8D418yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×#121—22no—
123C2×#121—22no—
124C2×#121—21no—
125C2×#121—21no—
126C2×#121—24no—
127C2×#121—22no—
128C2×#121D418yes—
129C2×#121C2312yes—
130C2×#121C2312yes—
131C2×#121C2312yes—
132C2×#120—22no—
133C2×#120—22no—
134#518—216no—
135C2×C4×Q8C2312yes—
136C2×C4×D4—24no—
137C2×C4×D4—24no—
138C2×C4×D4—22no—
139C2×C4×D4—22no—
140C2×C4×D4—22no—
141C2×C4×D4—22no—
142C2×C4×D4—22no—
143C2×C4×D4D412yes—
144C2×C4×D4D412yes—
145C2×C4×D4C2312yes—
146C2×C4×D4C2312yes—
147C2×C4×D4C2312yes—
148C2×#117—22no—
149C2×#117—22no—
150C2×#117C2312yes—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—24no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—21no—
156C22×C4⋊C4—21no—
157C22×C4⋊C4—22no—
158C22×C4⋊C4D411yescharacteristic
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2312yes—
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2311yescharacteristic
163C22×C4⋊C4C2311yescharacteristic
164C22×C4⋊C4C2312yes—
165C22×C22⋊C4—21no—
166C22×C22⋊C4—21no—
167C22×C22⋊C4—22no—
168C22×C22⋊C4—21no—
169C22×C22⋊C4—22no—
170C22×C22⋊C4D411yescharacteristic
171C22×C22⋊C4D411yescharacteristic
172C22×C42C2311yescharacteristic
173#484—216no—
174#484—216no—
175#481—216no—
176#481—216no—
177C2×#107—28no—
178C2×#107C2318yes—
179C2×#106—28no—
180C2×#106C2318yes—
181C2×#103—28no—
182C2×#103C2318yes—
183C2×#103C2318yes—
184C2×#102—28no—
185C2×#102C2318yes—
186C2×#102C2318yes—
187C2×C22⋊C8—24no—
188C2×C22⋊C8—24no—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2318yes—
194#400—24no—
195#400—24no—
196#399—24no—
197#399—24no—
198#389—24no—
199#389C2318yes—
200#386—24no—
201#386—24no—
202#385—24no—
203#385—24no—
204#380—21no—
205#380—21no—
206#380—24no—
207#380C2312yes—
208#380C2312yes—
209#380C2318yes—
210C2×#95—21no—
211C2×#95—22no—
212C2×#95—22no—
213#340C2318yes—
214#340C2318yes—
215#336C2318yes—
216C25—21no—
217C25C2×D411yescharacteristic
218C25C2×D411yescharacteristic
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—48no—
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—44no—
225C22×Q8C2×D414yes—
226C22×Q8C2412yes—
227C22×D4—24no—
228C22×D4—28no—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—44no—
238C22×D4—44no—
239C22×D4—44no—
240C22×D4C2×D412yes—
241C22×D4C2×D412yes—
242C22×D4C2×D414yes—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—84no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D414yes—
267C23×C4C2×D414yes—
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C23×C4C2411yescharacteristic
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—44no—
275C22×C8—44no—
276C22×C8C2414yes—
277C22×C8C2414yes—
278#126—216no—
279#126—216no—
280#126—216no—
281#125—216no—
282#125—24no—
283#125—24no—
284#124—216no—
285#124—24no—
286#124—24no—
287#123—216no—
288#123—216no—
289#123—216no—
290#123—216no—
291#123—216no—
292C22⋊Q8—216no—
293C22⋊Q8—24no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8C2×D418yes—
300C22⋊Q8C2×D418yes—
301#121—216no—
302#121—216no—
303#121—216no—
304#121—24no—
305#121—28no—
306#121—28no—
307#121—24no—
308#121—24no—
309#121—24no—
310#121C2×D418yes—
311#121C2×D418yes—
312#120—216no—
313#120—216no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×D4—216no—
317C4×D4—216no—
318C4×D4—216no—
319C4×D4—216no—
320C4×D4—216no—
321C4×D4—24no—
322C4×D4—28no—
323C4×D4—28no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—24no—
329#117—216no—
330#117—216no—
331#117—24no—
332#117—24no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—28no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—84no—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2414yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C22⋊C4—216no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—28no—
387C2×C22⋊C4—44no—
388C2×C22⋊C4—84no—
389C2×C22⋊C4C2×D412yes—
390C2×C22⋊C4C2×D412yes—
391C2×C22⋊C4C2×D412yes—
392C2×C22⋊C4C2×D412yes—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42C2×D412yes—
398C2×C42C2412yes—
399#107—216no—
400#106—216no—
401#103—216no—
402#103—48no—
403#102—216no—
404#102—48no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409C22⋊C8—48no—
410#95—28no—
411#95—22no—
412#95—22no—
413#95—28no—
414C24—22no—
415C24—22no—
416C24—24no—
417C24—22no—
418C24—22no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—82no—
426C24—82no—
427C24#12014yes—
428C24#13514yes—
429C24#13714yes—
430C24C22×D412yes—
431C24C22×D411yescharacteristic
432C24C22×D411yescharacteristic
433C24C22×D411yescharacteristic
434#41—88no—
435#41—88no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—48no—
440C2×Q8—82no—
441C2×Q8—84no—
442C2×Q8C22×D414yes—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—24no—
446C2×D4—28no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—28no—
452C2×D4—28no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—28no—
456C2×D4—48no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—416no—
462C2×D4—432no—
463C2×D4—84no—
464C2×D4—82no—
465C2×D4—84no—
466C2×D4—84no—
467C2×D4C22×D414yes—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—28no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—82no—
513C22×C4—84no—
514C22×C4—84no—
515C22×C4—82no—
516C22×C4—816no—
517C22×C4#12014yes—
518C22×C4#12014yes—
519C22×C4#12014yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C2511yescharacteristic
533C2×C8—24no—
534C2×C8—24no—
535C2×C8—28no—
536C2×C8—28no—
537C2×C8—48no—
538C2×C8—48no—
539C4⋊C4—28no—
540C4⋊C4—28no—
541C4⋊C4—28no—
542C4⋊C4—28no—
543C4⋊C4—28no—
544C4⋊C4—28no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—28no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—88no—
559C4⋊C4C22×D414yes—
560C4⋊C4C22×D414yes—
561C22⋊C4—28no—
562C22⋊C4—28no—
563C22⋊C4—28no—
564C22⋊C4—28no—
565C22⋊C4—28no—
566C22⋊C4—28no—
567C22⋊C4—24no—
568C22⋊C4—28no—
569C22⋊C4—28no—
570C22⋊C4—24no—
571C22⋊C4—28no—
572C22⋊C4—28no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—88no—
578C42—28no—
579C42—28no—
580C42—24no—
581C42—22no—
582C42—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—24no—
590C23—22no—
591C23—24no—
592C23—24no—
593C23—22no—
594C23—22no—
595C23—44no—
596C23—44no—
597C23—48no—
598C23—48no—
599C23—48no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—42no—
604C23—42no—
605C23—48no—
606C23—82no—
607C23—84no—
608C23—82no—
609C23—84no—
610C23—84no—
611C23—164no—
612C23—164no—
613C23#452116yes—
614C23C2×#12012yes—
615C23C2×#12012yes—
616C23C2×#12012yes—
617C23C2×#13512yes—
618C23C2×#13512yes—
619C23C2×#13512yes—
620C23C2×#13712yes—
621C23C2×#13712yes—
622C23C2×#13712yes—
623C23C23×D411yescharacteristic
624Q8—28no—
625Q8—44no—
626Q8—84no—
627D4—28no—
628D4—48no—
629D4—48no—
630D4—416no—
631D4—416no—
632D4—44no—
633D4—84no—
634D4—88no—
635D4—88no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—48no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—84no—
673C2×C4—82no—
674C2×C4—82no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—88no—
678C2×C4—88no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4—88no—
682C2×C4—88no—
683C2×C4C2×#12012yes—
684C2×C4C2×#12012yes—
685C2×C4C2×#12012yes—
686C2×C4C23×D411yescharacteristic
687C2×C4C23×D411yescharacteristic
688C2×C4#58514yes—
689C8—44no—
690C8—44no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—416no—
704C22—44no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—42no—
712C22—88no—
713C22—84no—
714C22—84no—
715C22—168no—
716C22C2×#45214yes—
717C22C2×#45214yes—
718C22C2×#45214yes—
719C22C22×#12011yescharacteristic
720C22#348512yes—
721C22C22×#13511yescharacteristic
722C22C22×#13711yescharacteristic
723C22#358512yes—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—42no—
728C4—44no—
729C4—42no—
730C4—44no—
731C4—42no—
732C4—44no—
733C4—84no—
734C4—84no—
735C4#348512yes—
736C4#348512yes—
737C2—24no—
738C2—24no—
739C2—24no—
740C2—82no—
741C2—84no—
742C2#3544611yescharacteristic
743C2C22×#45211yescharacteristic
744C2#6036911yescharacteristic
745C2#6169912yes—
746C2#6170912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3505C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#484C2semidirect
3. C22×#481C2semidirect
4. C2×#2487C2semidirect
5. #24095C2semidirect
6. #24092C2semidirect
7. #24067C2semidirect
8. #24039C2semidirect
9. #24036C2semidirect
10. #24007C2semidirect
11. #21759C2semidirect
12. #21758C2semidirect
13. #21738C2semidirect
33. C2×#484C22semidirect
34. C2×#484C22semidirect
37. C2×#481C22semidirect
38. C22×#107C22semidirect
39. C22×#106C22semidirect
40. C22×#103C22semidirect
41. C22×#102C22semidirect
47. #2487C22semidirect
50. C2×#389C22semidirect
54. C2×#380C22semidirect
55. #2059C22semidirect
56. #2042C22semidirect
57. #2035C22semidirect
58. #2033C22semidirect
59. #1893C22semidirect
60. #1892C22semidirect
62. #1872C22semidirect
64. #1869C22semidirect
65. #1865C22semidirect
66. #1862C22semidirect
67. #1854C22semidirect
68. #1824C22semidirect
69. #1823C22semidirect
70. #1817C22semidirect
71. #1788C22semidirect
72. #1787C22semidirect
73. #1786C22semidirect
74. #1779C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35052
C22×#4842
C22×#4812
C2×#24872
#240952
#240922
#240672
#240392
#240362
#240072
#217592
#217582
#217382