Finite Group Databasebeta

← #6477325

Subgroups — #6477325

Order 512.

3042 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic, complemented
2C2×#3193C211yesmaximal, characteristic
3C2×#3187C211yesmaximal, characteristic
4#35262C211yesmaximal, characteristic
5#33005C212yesmaximal, complemented
6#33003C212yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#29624C212yesmaximal, complemented
9#25511C218yesmaximal, complemented
10#24071C214yesmaximal, complemented
11#23942C214yesmaximal, complemented
12#21768C211yesmaximal, characteristic, complemented
13#21762C211yesmaximal, characteristic, complemented
14#20685C211yesmaximal, characteristic, complemented
15C2×#541—28no—
16C2×#541—28no—
17C2×#541—22no—
18C2×#541—22no—
19C2×#541C2214yes—
20C2×#541C2214yescomplemented
21C22×C4⋊Q8C2211yescharacteristic, complemented
22C22×#126—21no—
23C22×#126—21no—
24C22×#125—21no—
25C22×#125—21no—
26C22×#125C2211yescharacteristic, complemented
27C22×C22⋊Q8C2211yescharacteristic
28C22×C22⋊Q8C2211yescharacteristic
29C22×#121C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic, complemented
31C22×C4×Q8C2211yescharacteristic, complemented
32C22×#117C2211yescharacteristic
33#3193—28no—
34#3187—28no—
35C2×#484C2212yes—
36C2×#481C2212yes—
37C2×#483C2212yes—
38C2×#480C2212yes—
39C2×#468C2214yes—
40C2×#465C2214yes—
41C2×#448C2212yes—
42C2×#448C2212yes—
43C2×#450C2212yes—
44C2×#450C2212yes—
45C2×#407C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2854C2212yes—
48#2854C2212yes—
49#2624C2214yes—
50#2504—21no—
51#2501—21no—
52#2500—22no—
53#2482—22no—
54#2405—21no—
55#2405—21no—
56#2317C2212yes—
57#2309C2212yes—
58#2280C2211yescharacteristic
59#2059C2218yescomplemented
60#2043C2218yescomplemented
61#2025C2218yescomplemented
62#2013C2218yescomplemented
63#1894C2214yescomplemented
64#1892C2214yescomplemented
65#1880C2218yescomplemented
66#1880C2218yescomplemented
67#1873C2214yes—
68#1873C2214yes—
69#1865C2218yescomplemented
70#1863C2218yescomplemented
71#1824C2212yescomplemented
72#1823C2212yescomplemented
73#1818C2214yes—
74#1817C2212yescomplemented
75#1817C2212yescomplemented
76#1787C2212yescomplemented
77#1787C2212yescomplemented
78#1729C2214yescomplemented
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C22×M16C2312yes—
87C23×C8C2312yes—
88#541—232no—
89#541—232no—
90#541—216no—
91#541—216no—
92#541—216no—
93#541—216no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8C2314yes—
98C2×#126—216no—
99C2×#126—216no—
100C2×#126—24no—
101C2×#126—24no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#125—216no—
105C2×#125—216no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125—22no—
113C2×#125C2314yes—
114C2×#123—22no—
115C2×#123—22no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8D418yes—
123C2×C22⋊Q8D418yes—
124C2×C22⋊Q8C2312yes—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2312yes—
128C2×#121—21no—
129C2×#121—21no—
130C2×#121—24no—
131C2×#121—24no—
132C2×#121C2312yes—
133C2×#121C2312yes—
134C2×C4×Q8—24no—
135C2×C4×Q8—24no—
136C2×C4×Q8—24no—
137C2×C4×Q8—24no—
138C2×C4×Q8—22no—
139C2×C4×Q8—22no—
140C2×C4×Q8C2314yes—
141C2×C4×Q8C2314yes—
142C2×#117—24no—
143C2×#117—24no—
144C2×#117D412yes—
145C2×#117D412yes—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C2×#117C2312yes—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4—21no—
156C22×C4⋊C4—21no—
157C22×C4⋊C4D411yescharacteristic
158C22×C4⋊C4D411yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2311yescharacteristic
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4D411yescharacteristic
166C22×C22⋊C4D411yescharacteristic
167C22×C42—21no—
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170C22×C42C2311yescharacteristic
171#484—28no—
172#483—28no—
173#481—28no—
174#480—28no—
175#468—216no—
176#465—216no—
177#450—28no—
178#450—28no—
179#448—28no—
180#448—28no—
181C2×#107C2314yes—
182C2×#106C2314yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2318yes—
186C2×#103C2318yes—
187C2×#102C2314yes—
188C2×#102C2314yes—
189#407—24no—
190C2×C22⋊C8—24no—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2312yes—
195#399—24no—
196#397—24no—
197#394—24no—
198#394—24no—
199#390—22no—
200#390—22no—
201#390C2314yes—
202#390C2314yes—
203#389C2314yes—
204#389C2314yes—
205#388—24no—
206#388—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384—22no—
212#382—22no—
213#381—21no—
214#380C2312yes—
215#380C2312yes—
216#379—21no—
217#379—22no—
218#379—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8C2×D414yes—
234C22×Q8C2×D414yes—
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—84no—
245C22×D4C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D414yes—
260C23×C4C2×D414yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—48no—
267C2×SD16—48no—
268C2×M16—24no—
269C2×M16—44no—
270C22×C8—24no—
271C22×C8—44no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—216no—
275C4⋊Q8—28no—
276C4⋊Q8—28no—
277#126—232no—
278#126—232no—
279#126—232no—
280#126—232no—
281#125—232no—
282#125—232no—
283#125—232no—
284#125—232no—
285#125—216no—
286#125—28no—
287#125—28no—
288#123—84no—
289#123—84no—
290C22⋊Q8—28no—
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8C2×D418yes—
299C22⋊Q8C2×D418yes—
300C22⋊Q8C2×D418yes—
301C22⋊Q8C2×D418yes—
302#121—24no—
303#121—24no—
304#121—24no—
305#121—24no—
306#121—44no—
307#121—44no—
308C4×Q8—216no—
309C4×Q8—216no—
310C4×Q8—28no—
311C4×Q8—28no—
312C4×Q8—28no—
313C4×Q8—28no—
314#117—28no—
315#117—28no—
316#117—24no—
317#117—24no—
318#117—24no—
319#117—24no—
320#117—44no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—44no—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C42—28no—
383C2×C42—28no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—24no—
389C2×C42—24no—
390C2×C42—22no—
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42C2412yes—
394C2×C42C2412yes—
395#107—28no—
396#106—28no—
397#103—28no—
398#103—28no—
399#103—48no—
400#103—48no—
401#102—28no—
402#102—28no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411C24—24no—
412C24—22no—
413C24—42no—
414C24—42no—
415C24—44no—
416C24—42no—
417C24—82no—
418C24—82no—
419C24#12014yes—
420C24C22×D411yescharacteristic
421C24C22×D411yescharacteristic
422C24C22×D411yescharacteristic
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—28no—
428C2×Q8—28no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—48no—
434C2×Q8—48no—
435C2×Q8C22×D414yes—
436C2×Q8C22×D414yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—28no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—84no—
447C2×D4—84no—
448C2×D4—82no—
449C2×D4—82no—
450C2×D4—816no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—82no—
483C22×C4#12014yes—
484C22×C4#12014yes—
485C22×C4#12014yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C2511yescharacteristic
500SD16—88no—
501SD16—88no—
502M16—84no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—28no—
506C2×C8—28no—
507C2×C8—84no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—216no—
513C4⋊C4—216no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—28no—
529C4⋊C4—42no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4C22×D414yes—
533C4⋊C4C22×D414yes—
534C22⋊C4—28no—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—28no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—82no—
551C42—216no—
552C42—216no—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—44no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—28no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23—88no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—168no—
581C23C2×#12012yes—
582C23C2×#12012yes—
583C23C2×#12012yes—
584C23#57714yes—
585C23C23×D411yescharacteristic
586Q8—28no—
587Q8—28no—
588Q8—48no—
589Q8—48no—
590Q8—44no—
591Q8—44no—
592D4—44no—
593D4—44no—
594D4—84no—
595D4—88no—
596D4—88no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4C2×#12012yes—
641C2×C4C2×#12012yes—
642C2×C4C2×#12012yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C23×D411yescharacteristic
645C2×C4#58514yes—
646C8—44no—
647C8—44no—
648C22—22no—
649C22—28no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—44no—
658C22—44no—
659C22—88no—
660C22—82no—
661C22—84no—
662C22—84no—
663C22—82no—
664C22—82no—
665C22—168no—
666C22#301014yes—
667C22C22×#12011yescharacteristic
668C22#344912yes—
669C22C2×#57712yes—
670C22#358612yes—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—44no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—84no—
683C4—82no—
684C4#348512yes—
685C4#348512yes—
686C2—24no—
687C2—42no—
688C2—42no—
689C2—162no—
690C2#3508811yescharacteristic
691C2C2×#301011yescharacteristic
692C2#6036811yescharacteristic
693C2#6098214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#541C2semidirect
5. #33005C2semidirect
6. #33003C2semidirect
7. #29640C2semidirect
8. #29624C2semidirect
9. #25511C2semidirect
10. #24071C2semidirect
11. #23942C2semidirect
12. #21768C2semidirect
13. #21762C2semidirect
14. #20685C2semidirect
20. C2×#541C22semidirect
21. C22×C4⋊Q8C22semidirect
26. C22×#125C22semidirect
30. C22×C4×Q8C22semidirect
31. C22×C4×Q8C22semidirect
59. #2059C22semidirect
60. #2043C22semidirect
61. #2025C22semidirect
62. #2013C22semidirect
63. #1894C22semidirect
64. #1892C22semidirect
65. #1880C22semidirect
66. #1880C22semidirect
69. #1865C22semidirect
70. #1863C22semidirect
71. #1824C22semidirect
72. #1823C22semidirect
74. #1817C22semidirect
75. #1817C22semidirect
76. #1787C22semidirect
77. #1787C22semidirect
78. #1729C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#31932
C2×#31872
#352622
#330052
#330032
#296402
#296242
#255112
#240712
#239422
#217682
#217622
#206852