Finite Group Databasebeta

← #6508848

Subgroups — #6508848

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic, complemented
2C2×#2976C211yesmaximal, characteristic, complemented
3#37437C211yesmaximal, characteristic
4#35227C211yesmaximal, characteristic, complemented
5#28104C212yesmaximal, complemented
6#28097C212yesmaximal, complemented
7#27663C212yesmaximal, complemented
8#27658C212yesmaximal, complemented
9#25260C214yesmaximal, complemented
10#25246C214yesmaximal, complemented
11#24618C214yesmaximal, complemented
12#24568C214yesmaximal, complemented
13#21939C211yesmaximal, characteristic, complemented
14#21767C211yesmaximal, characteristic, complemented
15#20700C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#481—28no—
20C2×#481C2214yes—
21C2×#481C2214yes—
22#2976—28no—
23C2×#436C2212yescomplemented
24C2×#435C2212yes—
25C2×#435C2212yescomplemented
26C2×C8×D4C2212yes—
27C2×#433C2212yescomplemented
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C2×#408C2212yescomplemented
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2849—22no—
36#2816—21no—
37#2815—21no—
38#2661—22no—
39#2624C2212yes—
40#2623—22no—
41#2615C2212yes—
42#2613—22no—
43#2561—21no—
44#2539—21no—
45#2504—21no—
46#2503—21no—
47#2502—21no—
48#2501—21no—
49#2487C2212yes—
50#2487C2212yes—
51#2486C2212yes—
52#2486C2214yescomplemented
53#2414C2214yes—
54#2408C2212yes—
55#2317C2212yescomplemented
56#2311C2212yescomplemented
57#2280C2211yescharacteristic
58#2280C2211yescharacteristic
59#2088C2214yescomplemented
60#2088C2214yescomplemented
61#2077C2214yescomplemented
62#2077C2214yescomplemented
63#2067C2214yescomplemented
64#2065C2214yescomplemented
65#2034C2214yescomplemented
66#2033C2214yescomplemented
67#1943C2214yescomplemented
68#1941C2214yescomplemented
69#1936C2214yescomplemented
70#1934C2214yescomplemented
71#1895C2212yes—
72#1895C2212yes—
73#1895C2214yescomplemented
74#1893C2214yescomplemented
75#1875C2214yes—
76#1854C2214yescomplemented
77#1854C2214yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1823C2214yescomplemented
82#1809C2214yescomplemented
83#1808C2212yes—
84#1807C2212yes—
85#1807C2214yescomplemented
86#1786C2212yes—
87#1736C2212yescomplemented
88#1736C2212yescomplemented
89C22×#41C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4—21no—
93C24×C4C2311yescharacteristic
94C22×M16C2312yes—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98C23×C8C2311yescharacteristic
99C23×C8C2311yescharacteristic
100#525—24no—
101#525—24no—
102C2×#123—24no—
103C2×#123—22no—
104C2×#123D412yes—
105C2×C22⋊Q8D411yescharacteristic
106C2×C22⋊Q8D411yescharacteristic
107C2×#121—21no—
108C2×#121—21no—
109C2×#121—24no—
110C2×#121—24no—
111C2×#121D411yescharacteristic
112C2×#121D411yescharacteristic
113C2×#121C2312yes—
114C2×#121C2312yes—
115C2×#121C2314yes—
116C2×#120—22no—
117C2×#120D412yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—24no—
124C22×C4⋊C4—24no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134#481—216no—
135#481—216no—
136#436—28no—
137#435—28no—
138#435—28no—
139C8×D4—28no—
140#433—28no—
141C2×#107C2314yes—
142C2×#106—24no—
143C2×#106—24no—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#102—24no—
150C2×#102—24no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155#408—28no—
156#407—24no—
157C2×C22⋊C8D418yescomplemented
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166#400—22no—
167#400—22no—
168#399—22no—
169#399—22no—
170#399—22no—
171#398—22no—
172#398—24no—
173#397—22no—
174#397—24no—
175#397—22no—
176#394—22no—
177#394—24no—
178#394—24no—
179#393—22no—
180#392—24no—
181#392—24no—
182#392—22no—
183#392—22no—
184#391—22no—
185#390—22no—
186#390—24no—
187#390C2314yes—
188#389C2314yes—
189#389C2314yes—
190#389C2314yes—
191#388—24no—
192#386—24no—
193#386—22no—
194#386—22no—
195#385—24no—
196#385—22no—
197#385—22no—
198#384—22no—
199#382—22no—
200#381—21no—
201#381—21no—
202#381—22no—
203#381—24no—
204#380C2314yes—
205#380C2314yes—
206#380C2312yes—
207#380C2312yes—
208#380C2314yes—
209#380C2312yes—
210#380C2312yes—
211#379—21no—
212#379—21no—
213#379—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226C25—21no—
227C25C2×D411yescharacteristic
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—42no—
231C2×#41—42no—
232C22×Q8C2×D411yescharacteristic
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—42no—
240C22×D4—42no—
241C22×D4—44no—
242C22×D4—84no—
243C22×D4C2×D412yes—
244C22×D4C2×D412yes—
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4—44no—
256C23×C4—84no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D414yes—
261C23×C4#4114yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—24no—
269C2×M16—44no—
270C2×M16—42no—
271C2×M16—42no—
272C22×C8—24no—
273C22×C8—42no—
274C22×C8—42no—
275C22×C8—44no—
276C22×C8C2×D414yes—
277C22×C8C2×D414yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282#123—28no—
283#123—84no—
284#123—88no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287#121—24no—
288#121—24no—
289#121—24no—
290#121—24no—
291#121—24no—
292#121—24no—
293#121—44no—
294#121—44no—
295#121—44no—
296#121—44no—
297#120—28no—
298#120—84no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—24no—
303#117—22no—
304#117—22no—
305#117—42no—
306#117—42no—
307#117—42no—
308#117—42no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2414yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42C2412yes—
362#107—44no—
363#106—28no—
364#106—28no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368#102—28no—
369#102—28no—
370#102—44no—
371#102—44no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8C2×D418yes—
377C22⋊C8C2×D418yes—
378#97—24no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390C24—24no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—44no—
396C24—82no—
397C24—82no—
398C24#12114yes—
399C24C22×D412yes—
400C24C22×D411yescharacteristic
401C24C22×D411yescharacteristic
402C24C22×D411yescharacteristic
403#41—82no—
404#41—82no—
405#41—84no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—84no—
427C2×D4—82no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—816no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—84no—
465C22×C4—816no—
466C22×C4#12114yes—
467C22×C4#12114yes—
468C22×C4#12114yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2511yescharacteristic
482M16—84no—
483M16—82no—
484M16—82no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8—84no—
494C2×C8C22×D414yes—
495C2×C8C22×D414yes—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C42—22no—
538C42—22no—
539C42—44no—
540C42—82no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—28no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—44no—
556C23—44no—
557C23—416no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—168no—
567C23C2×#12112yes—
568C23C2×#12112yes—
569C23C2×#12112yes—
570C23#57714yes—
571C23C23×D411yescharacteristic
572Q8—82no—
573D4—48no—
574D4—42no—
575D4—42no—
576D4—44no—
577D4—82no—
578D4—84no—
579D4—84no—
580D4—84no—
581D4—88no—
582D4—88no—
583D4—168no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12112yes—
629C2×C4C23×D411yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4#58512yes—
632C2×C4#58512yes—
633C8—24no—
634C8—24no—
635C8—42no—
636C8—42no—
637C22—22no—
638C22—28no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—416no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—42no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—84no—
658C22—82no—
659C22—168no—
660C22#305714yes—
661C22C22×#12111yescharacteristic
662C22#348512yes—
663C22C2×#57712yes—
664C22#358512yes—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—84no—
675C4—84no—
676C4#348511yescharacteristic
677C4#348511yescharacteristic
678C4#349112yes—
679C2—24no—
680C2—42no—
681C2—42no—
682C2—82no—
683C2—162no—
684C2#3544711yescharacteristic
685C2C2×#305711yescharacteristic
686C2#6045411yescharacteristic
687C2#6174614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37437C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#481C2semidirect
2. C2×#2976C2semidirect
4. #35227C2semidirect
5. #28104C2semidirect
6. #28097C2semidirect
7. #27663C2semidirect
8. #27658C2semidirect
9. #25260C2semidirect
10. #25246C2semidirect
11. #24618C2semidirect
12. #24568C2semidirect
13. #21939C2semidirect
14. #21767C2semidirect
15. #20700C2semidirect
23. C2×#436C22semidirect
25. C2×#435C22semidirect
27. C2×#433C22semidirect
32. C2×#408C22semidirect
52. #2486C22semidirect
55. #2317C22semidirect
56. #2311C22semidirect
59. #2088C22semidirect
60. #2088C22semidirect
61. #2077C22semidirect
62. #2077C22semidirect
63. #2067C22semidirect
64. #2065C22semidirect
65. #2034C22semidirect
66. #2033C22semidirect
67. #1943C22semidirect
68. #1941C22semidirect
69. #1936C22semidirect
70. #1934C22semidirect
73. #1895C22semidirect
74. #1893C22semidirect
76. #1854C22semidirect
77. #1854C22semidirect
81. #1823C22semidirect
82. #1809C22semidirect
85. #1807C22semidirect
87. #1736C22semidirect
88. #1736C22semidirect
157. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C2×#29762
#374372
#352272
#281042
#280972
#276632
#276582
#252602
#252462
#246182
#245682
#219392
#217672
#207002