Finite Group Databasebeta

← #6477327

Subgroups — #6477327

Order 512.

3258 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#542C211yesmaximal, characteristic, complemented
2C2×#3193C211yesmaximal, characteristic, complemented
3C2×#3186C211yesmaximal, characteristic
4#35262C211yesmaximal, characteristic, complemented
5#33003C212yesmaximal, complemented
6#33002C212yesmaximal, complemented
7#29638C212yesmaximal, complemented
8#29623C212yesmaximal, complemented
9#25510C218yesmaximal, complemented
10#24073C214yesmaximal, complemented
11#23945C214yesmaximal, complemented
12#21767C211yesmaximal, characteristic, complemented
13#21762C211yesmaximal, characteristic, complemented
14#20685C211yesmaximal, characteristic, complemented
15C2×#542—28no—
16C2×#542—28no—
17C2×#542—22no—
18C2×#542—22no—
19C2×#542C2214yes—
20C2×#542C2214yescomplemented
21C22×C4⋊Q8C2211yescharacteristic, complemented
22C22×#126—21no—
23C22×#124—21no—
24C22×#124C2211yescharacteristic, complemented
25C22×#123—21no—
26C22×C22⋊Q8—21no—
27C22×C22⋊Q8C2211yescharacteristic
28C22×#121C2211yescharacteristic
29C22×#121C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic, complemented
31C22×C4×D4C2211yescharacteristic, complemented
32C22×#117C2211yescharacteristic
33#3193—28no—
34#3186—28no—
35C2×#484C2212yescomplemented
36C2×#480C2212yes—
37C2×#480C2212yes—
38C2×#480C2212yes—
39C2×#468C2214yescomplemented
40C2×#466C2214yes—
41C2×#448C2212yes—
42C2×#450C2212yescomplemented
43C2×#447C2212yes—
44C2×#447C2212yes—
45C2×#407C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2854C2212yes—
48#2854C2212yes—
49#2624C2214yescomplemented
50#2504—21no—
51#2501—21no—
52#2500—22no—
53#2482—22no—
54#2405—21no—
55#2405—21no—
56#2317C2212yescomplemented
57#2309C2212yescomplemented
58#2280C2211yescharacteristic
59#2057C2218yescomplemented
60#2042C2218yescomplemented
61#2023C2218yescomplemented
62#2012C2218yescomplemented
63#1895C2214yescomplemented
64#1892C2214yescomplemented
65#1880C2218yescomplemented
66#1879C2218yescomplemented
67#1875C2214yes—
68#1873C2214yescomplemented
69#1863C2218yescomplemented
70#1862C2218yescomplemented
71#1823C2212yescomplemented
72#1823C2212yescomplemented
73#1818C2214yescomplemented
74#1817C2212yescomplemented
75#1817C2212yescomplemented
76#1787C2212yescomplemented
77#1786C2212yescomplemented
78#1729C2214yescomplemented
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×D4—21no—
82C23×D4—21no—
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4—21no—
86C24×C4C2311yescharacteristic
87C22×SD16C2314yes—
88C22×D8C2314yes—
89C22×M16C2312yes—
90C23×C8C2312yes—
91#542—232no—
92#542—232no—
93#542—216no—
94#542—216no—
95#542—216no—
96#542—216no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8C2314yes—
101C2×#126—216no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#124—216no—
105C2×#124—24no—
106C2×#124—24no—
107C2×#124—24no—
108C2×#124—24no—
109C2×#124—22no—
110C2×#124C2314yes—
111C2×#123—216no—
112C2×#123—22no—
113C2×#123—22no—
114C2×#123—24no—
115C2×#123—24no—
116C2×C22⋊Q8—216no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8D418yes—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125C2×#121—24no—
126C2×#121—21no—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121—21no—
130C2×#121—24no—
131C2×#121—24no—
132C2×#121D418yes—
133C2×#121C2312yes—
134C2×#121C2312yes—
135C2×#121C2312yes—
136C2×#121C2312yes—
137C2×C4×Q8—24no—
138C2×C4×Q8—24no—
139C2×C4×Q8—22no—
140C2×C4×Q8C2314yes—
141C2×C4×D4—24no—
142C2×C4×D4—24no—
143C2×C4×D4—22no—
144C2×C4×D4C2314yes—
145C2×#117—24no—
146C2×#117—24no—
147C2×#117D412yes—
148C2×#117D412yes—
149C2×#117C2312yes—
150C2×#117C2312yes—
151C2×#117C2312yes—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4D411yescharacteristic
167C22×C22⋊C4D411yescharacteristic
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170C22×C42C2311yescharacteristic
171#484—28no—
172#480—28no—
173#480—28no—
174#480—28no—
175#468—216no—
176#466—216no—
177#450—28no—
178#448—28no—
179#447—28no—
180#447—28no—
181C2×#107C2314yes—
182C2×#107C2314yes—
183C2×#103C2314yes—
184C2×#103C2318yes—
185C2×#102C2314yes—
186C2×#102C2314yes—
187C2×#102C2314yes—
188C2×#102C2318yes—
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—
226C25—21no—
227C25C2×D411yescharacteristic
228C22×Q8—28no—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8—24no—
232C22×Q8—44no—
233C22×Q8—84no—
234C22×Q8C2×D414yes—
235C22×Q8C2412yes—
236C22×D4—28no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4—44no—
249C22×D4—84no—
250C22×D4—84no—
251C22×D4C2×D414yes—
252C22×D4C2412yes—
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—28no—
258C23×C4—24no—
259C23×C4—21no—
260C23×C4—44no—
261C23×C4—84no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D414yes—
270C23×C4C2×D414yes—
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C2×SD16—28no—
275C2×SD16—48no—
276C2×D8—28no—
277C2×D8—48no—
278C2×M16—24no—
279C2×M16—44no—
280C22×C8—24no—
281C22×C8—44no—
282C22×C8C2414yes—
283C22×C8C2414yes—
284C4⋊Q8—216no—
285C4⋊Q8—28no—
286C4⋊Q8—28no—
287#126—232no—
288#126—232no—
289#124—232no—
290#124—232no—
291#124—216no—
292#124—28no—
293#124—28no—
294#123—232no—
295#123—232no—
296#123—84no—
297#123—84no—
298C22⋊Q8—232no—
299C22⋊Q8—232no—
300C22⋊Q8—28no—
301C22⋊Q8—28no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8C2×D418yes—
305C22⋊Q8C2×D418yes—
306#121—28no—
307#121—28no—
308#121—24no—
309#121—24no—
310#121—24no—
311#121—24no—
312#121—24no—
313#121—24no—
314#121—44no—
315#121—44no—
316#121C2×D418yes—
317#121C2×D418yes—
318C4×Q8—216no—
319C4×Q8—28no—
320C4×Q8—28no—
321C4×D4—216no—
322C4×D4—28no—
323C4×D4—28no—
324#117—28no—
325#117—28no—
326#117—24no—
327#117—24no—
328#117—24no—
329#117—24no—
330#117—44no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—44no—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C42—28no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—24no—
393C2×C42—22no—
394C2×C42C2×D412yes—
395C2×C42C2×D412yes—
396C2×C42C2412yes—
397C2×C42C2412yes—
398#107—28no—
399#107—28no—
400#103—28no—
401#103—48no—
402#102—28no—
403#102—28no—
404#102—28no—
405#102—48no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409C22⋊C8—24no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414C24—24no—
415C24—24no—
416C24—22no—
417C24—22no—
418C24—42no—
419C24—42no—
420C24—44no—
421C24—44no—
422C24—44no—
423C24—44no—
424C24—42no—
425C24—82no—
426C24—82no—
427C24#12014yes—
428C24C22×D411yescharacteristic
429C24C22×D411yescharacteristic
430C24C22×D411yescharacteristic
431C2×Q8—216no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—28no—
435C2×Q8—48no—
436C2×Q8—44no—
437C2×Q8—48no—
438C2×Q8—88no—
439C2×Q8—816no—
440C2×Q8C22×D414yes—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—28no—
448C2×D4—28no—
449C2×D4—48no—
450C2×D4—44no—
451C2×D4—48no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—84no—
459C2×D4—84no—
460C2×D4—816no—
461C2×D4—84no—
462C2×D4—84no—
463C2×D4—82no—
464C2×D4—82no—
465C2×D4—816no—
466C2×D4C22×D414yes—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—216no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4—88no—
497C22×C4—82no—
498C22×C4—816no—
499C22×C4#12014yes—
500C22×C4#12014yes—
501C22×C4#12014yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C2511yescharacteristic
516SD16—88no—
517D8—88no—
518M16—84no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—28no—
522C2×C8—28no—
523C2×C8—84no—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—28no—
539C4⋊C4—42no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4C22×D414yes—
543C4⋊C4C22×D414yes—
544C22⋊C4—216no—
545C22⋊C4—216no—
546C22⋊C4—216no—
547C22⋊C4—28no—
548C22⋊C4—28no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—28no—
554C22⋊C4—28no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C22⋊C4—84no—
563C22⋊C4—82no—
564C42—216no—
565C42—24no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—44no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—28no—
581C23—24no—
582C23—44no—
583C23—44no—
584C23—48no—
585C23—48no—
586C23—44no—
587C23—44no—
588C23—48no—
589C23—44no—
590C23—44no—
591C23—42no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—88no—
596C23—816no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23—168no—
601C23C2×#12012yes—
602C23C2×#12012yes—
603C23C2×#12012yes—
604C23#57614yes—
605C23C23×D411yescharacteristic
606Q8—28no—
607Q8—48no—
608Q8—44no—
609Q8—816no—
610D4—28no—
611D4—48no—
612D4—44no—
613D4—44no—
614D4—44no—
615D4—84no—
616D4—88no—
617D4—88no—
618D4—88no—
619D4—88no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—48no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—48no—
650C2×C4—48no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—816no—
660C2×C4—84no—
661C2×C4C2×#12012yes—
662C2×C4C2×#12012yes—
663C2×C4C2×#12012yes—
664C2×C4C23×D411yescharacteristic
665C2×C4C23×D411yescharacteristic
666C2×C4#58514yes—
667C8—44no—
668C8—44no—
669C22—22no—
670C22—28no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—88no—
684C22—816no—
685C22—88no—
686C22—82no—
687C22—84no—
688C22—84no—
689C22—82no—
690C22—82no—
691C22—168no—
692C22#300814yes—
693C22C22×#12011yescharacteristic
694C22#344912yes—
695C22C2×#57612yes—
696C22#358612yes—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—44no—
705C4—42no—
706C4—42no—
707C4—84no—
708C4—82no—
709C4#348512yes—
710C4#348512yes—
711C2—24no—
712C2—42no—
713C2—42no—
714C2—84no—
715C2—162no—
716C2#3508811yescharacteristic
717C2C2×#300811yescharacteristic
718C2#6036811yescharacteristic
719C2#6097614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3186C2
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×#542C2semidirect
2. C2×#3193C2semidirect
4. #35262C2semidirect
5. #33003C2semidirect
6. #33002C2semidirect
7. #29638C2semidirect
8. #29623C2semidirect
9. #25510C2semidirect
10. #24073C2semidirect
11. #23945C2semidirect
12. #21767C2semidirect
13. #21762C2semidirect
14. #20685C2semidirect
20. C2×#542C22semidirect
21. C22×C4⋊Q8C22semidirect
24. C22×#124C22semidirect
30. C22×C4×Q8C22semidirect
31. C22×C4×D4C22semidirect
35. C2×#484C22semidirect
39. C2×#468C22semidirect
42. C2×#450C22semidirect
49. #2624C22semidirect
56. #2317C22semidirect
57. #2309C22semidirect
59. #2057C22semidirect
60. #2042C22semidirect
61. #2023C22semidirect
62. #2012C22semidirect
63. #1895C22semidirect
64. #1892C22semidirect
65. #1880C22semidirect
66. #1879C22semidirect
68. #1873C22semidirect
69. #1863C22semidirect
70. #1862C22semidirect
71. #1823C22semidirect
72. #1823C22semidirect
73. #1818C22semidirect
74. #1817C22semidirect
75. #1817C22semidirect
76. #1787C22semidirect
77. #1786C22semidirect
78. #1729C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5422
C2×#31932
C2×#31862
#352622
#330032
#330022
#296382
#296232
#255102
#240732
#239452
#217672
#217622
#206852