Finite Group Databasebeta

← #6508811

Subgroups — #6508811

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic, complemented
2C2×#2973C211yesmaximal, characteristic
3#35010C211yesmaximal, characteristic
4C2×#2485C211yesmaximal, characteristic
5#28102C214yesmaximal, complemented
6#27668C214yesmaximal, complemented
7#25247C214yesmaximal, complemented
8#25241C214yesmaximal, complemented
9#24609C212yesmaximal, complemented
10#24606C212yesmaximal, complemented
11#24552C212yesmaximal, complemented
12#24551C212yesmaximal, complemented
13#21942C211yesmaximal, characteristic, complemented
14#21765C211yesmaximal, characteristic, complemented
15#20702C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3095—28no—
20C2×#484C2212yes—
21C2×#485C2214yescomplemented
22C2×#483C2212yes—
23#2973—28no—
24C2×#435C2214yes—
25C2×#435C2214yes—
26C2×#432C2212yes—
27C2×#428C2212yescomplemented
28C2×#419C2212yescomplemented
29C2×#407C2211yescharacteristic
30C2×#407C2211yescharacteristic
31C22×C22⋊C8C2212yes—
32#2855C2212yes—
33#2848C2212yes—
34#2627—21no—
35#2626—21no—
36#2579—22no—
37#2564—22no—
38#2487C2212yes—
39#2485—22no—
40#2485—22no—
41#2485C2214yes—
42#2485C2214yes—
43#2417C2212yes—
44#2417C2212yes—
45#2409C2212yes—
46C2×#397—21no—
47C2×#394—21no—
48#2315—21no—
49#2308—21no—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#387—21no—
54C2×#380C2211yescharacteristic
55C2×#380C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58#2088C2214yescomplemented
59#2086C2214yescomplemented
60#2077C2214yescomplemented
61#2076C2214yescomplemented
62#2068C2214yescomplemented
63#2066C2214yescomplemented
64#2036C2214yescomplemented
65#2035C2214yescomplemented
66#1944C2214yescomplemented
67#1943C2214yescomplemented
68#1935C2214yescomplemented
69#1935C2214yescomplemented
70#1894C2214yescomplemented
71#1892C2214yescomplemented
72#1892C2214yescomplemented
73#1872C2212yes—
74#1872C2212yes—
75#1853C2212yes—
76#1853C2212yescomplemented
77#1852C2212yes—
78#1852C2212yescomplemented
79#1825C2214yescomplemented
80#1824C2212yes—
81#1823C2212yes—
82#1823C2212yescomplemented
83#1809C2214yescomplemented
84#1808C2214yescomplemented
85#1807C2214yescomplemented
86#1787C2212yescomplemented
87#1736C2212yescomplemented
88#1736C2212yescomplemented
89C23×Q8C2311yescharacteristic
90C23×D4D411yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C22×M16C2312yes—
94C22×M16C2312yes—
95C23×C8C2314yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4D412yes—
107C2×C4×D4D412yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117—21no—
111C2×#117—21no—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#485—216no—
138#484—28no—
139#483—28no—
140#435—216no—
141#435—216no—
142#432—28no—
143#428—28no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2314yes—
149#419—28no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154#407—24no—
155#407D418yes—
156C2×C22⋊C8—28no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×#97C2314yes—
163#400—22no—
164#399—22no—
165#398—22no—
166#398—22no—
167#397—24no—
168#397—24no—
169#396—22no—
170#396—22no—
171#395—22no—
172#394—24no—
173#394—24no—
174#393—22no—
175#391C2312yes—
176#391C2312yes—
177#389—22no—
178#389—22no—
179#389C2314yes—
180#389C2314yes—
181#389C2312yes—
182#389C2312yes—
183#388—24no—
184#388—24no—
185#387—24no—
186#387—24no—
187#387—22no—
188#387—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#381—22no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380C2312yes—
209#380C2314yes—
210#380C2312yes—
211#380C2314yes—
212#380C2312yes—
213#380C2312yes—
214#380C2312yes—
215#380C2312yes—
216#379—22no—
217C2×#95—21no—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95—21no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231C25C2×D411yescharacteristic
232C25C2×D411yescharacteristic
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—42no—
237C22×Q8—42no—
238C22×Q8C2412yes—
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—42no—
242C22×D4C2×D412yes—
243C22×D4C2×D412yes—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4—42no—
257C23×C4—42no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D414yes—
266C23×C4#4114yes—
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×M16—24no—
272C2×M16—42no—
273C2×M16—42no—
274C2×M16—42no—
275C2×M16—42no—
276C2×M16C2×D418yes—
277C22×C8—28no—
278C22×C8—44no—
279C22×C8—44no—
280C22×C8C2414yes—
281C22×C8C2414yes—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296#117—24no—
297#117—24no—
298#117—24no—
299#117—24no—
300#117—44no—
301#117—44no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—44no—
360C2×C22⋊C4—82no—
361C2×C22⋊C4—82no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—44no—
371C2×C42C2×D412yes—
372C2×C42C2412yes—
373#107—28no—
374#107—44no—
375#106—28no—
376#106—44no—
377C4⋊C8—28no—
378#103—28no—
379#103—28no—
380#103—44no—
381#103—44no—
382C22⋊C8—24no—
383C22⋊C8—24no—
384C22⋊C8—28no—
385C22⋊C8C2×D418yes—
386C22⋊C8C2×D418yes—
387#97—28no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24—41no—
412C24—41no—
413C24—41no—
414C24—41no—
415C24—41no—
416C24#12114yes—
417C24#13714yes—
418C24C22×D411yescharacteristic
419C24C22×D411yescharacteristic
420C24C22×D412yes—
421C24C22×D411yescharacteristic
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—48no—
427C2×Q8—42no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×Q8—84no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—416no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—84no—
493C22×C4—84no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4#12114yes—
503C22×C4#12114yes—
504C22×C4#12114yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4C2511yescharacteristic
518M16—48no—
519M16—84no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—88no—
526C2×C8C22×D418yes—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—48no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—81no—
550C4⋊C4—81no—
551C4⋊C4—84no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—48no—
564C22⋊C4—44no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—84no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C42—24no—
573C42—22no—
574C42—22no—
575C42—44no—
576C42—44no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—22no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—21no—
586C23—21no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—22no—
599C23—28no—
600C23—28no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—82no—
606C23—82no—
607C23—82no—
608C23—82no—
609C23—82no—
610C23—82no—
611C23—82no—
612C23—82no—
613C23C2×#12112yes—
614C23C2×#12112yes—
615C23C2×#12112yes—
616C23C2×#13714yes—
617C23C2×#13712yes—
618C23#57512yes—
619C23C23×D411yescharacteristic
620Q8—44no—
621Q8—44no—
622Q8—84no—
623D4—48no—
624D4—48no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—84no—
677C2×C4—82no—
678C2×C4—82no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4—84no—
685C2×C4—162no—
686C2×C4—162no—
687C2×C4C2×#12112yes—
688C2×C4C2×#12112yes—
689C2×C4C2×#12112yes—
690C2×C4C23×D411yescharacteristic
691C2×C4C22×#4111yescharacteristic
692C2×C4#58512yes—
693C2×C4#58512yes—
694C8—28no—
695C8—44no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—24no—
703C22—24no—
704C22—22no—
705C22—22no—
706C22—24no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—42no—
717C22—41no—
718C22—41no—
719C22—42no—
720C22—41no—
721C22—42no—
722C22—42no—
723C22—88no—
724C22—88no—
725C22#305614yes—
726C22C22×#12111yescharacteristic
727C22#348511yescharacteristic
728C22#348511yescharacteristic
729C22C22×#13711yescharacteristic
730C22C2×#57511yescharacteristic
731C22#358511yescharacteristic
732C22#358611yescharacteristic
733C4—22no—
734C4—22no—
735C4—22no—
736C4—42no—
737C4—42no—
738C4—42no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4#348512yes—
748C4#349112yes—
749C2—24no—
750C2—44no—
751C2—42no—
752C2—42no—
753C2#3544711yescharacteristic
754C2C2×#305611yescharacteristic
755C2#6045611yescharacteristic
756C2#6170511yescharacteristic
757C2#6170811yescharacteristic
758C2#6171011yescharacteristic
759C2#6171211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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
1. C2×#3095C2semidirect
5. #28102C2semidirect
6. #27668C2semidirect
7. #25247C2semidirect
8. #25241C2semidirect
9. #24609C2semidirect
10. #24606C2semidirect
11. #24552C2semidirect
12. #24551C2semidirect
13. #21942C2semidirect
14. #21765C2semidirect
15. #20702C2semidirect
21. C2×#485C22semidirect
27. C2×#428C22semidirect
28. C2×#419C22semidirect
58. #2088C22semidirect
59. #2086C22semidirect
60. #2077C22semidirect
61. #2076C22semidirect
62. #2068C22semidirect
63. #2066C22semidirect
64. #2036C22semidirect
65. #2035C22semidirect
66. #1944C22semidirect
67. #1943C22semidirect
68. #1935C22semidirect
69. #1935C22semidirect
70. #1894C22semidirect
71. #1892C22semidirect
72. #1892C22semidirect
76. #1853C22semidirect
78. #1852C22semidirect
79. #1825C22semidirect
82. #1823C22semidirect
83. #1809C22semidirect
84. #1808C22semidirect
85. #1807C22semidirect
86. #1787C22semidirect
87. #1736C22semidirect
88. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C2×#29732
#350102
C2×#24852
#281022
#276682
#252472
#252412
#246092
#246062
#245522
#245512
#219422
#217652
#207022