Finite Group Databasebeta

← #6477696

Subgroups — #6477696

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2C22×#433C211yesmaximal, characteristic, complemented
3#35311C211yesmaximal, characteristic, complemented
4#35262C211yesmaximal, characteristic
5#33134C212yesmaximal, complemented
6#33128C212yesmaximal, complemented
7#29775C212yesmaximal, complemented
8#29774C212yesmaximal, complemented
9#24618C214yesmaximal, complemented
10#24616C214yesmaximal, complemented
11#24518C214yesmaximal, complemented
12#24516C214yesmaximal, complemented
13#21939C211yesmaximal, characteristic, complemented
14#21762C211yesmaximal, characteristic, complemented
15#21761C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3187—28no—
20C2×#481C2212yes—
21C2×#483C2212yes—
22C2×#465C2212yes—
23C2×#465C2212yes—
24C2×#448C2212yes—
25C2×#450C2212yes—
26C2×#433—28no—
27C2×#433C2214yes—
28C2×#433C2214yescomplemented
29C22×C4⋊C8C2211yescharacteristic, complemented
30C22×C4⋊C8C2211yescharacteristic, complemented
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2211yescharacteristic, complemented
34C22×C4×C8C2211yescharacteristic, complemented
35#2855C2212yescomplemented
36#2855C2212yescomplemented
37#2854C2212yes—
38#2854C2212yes—
39#2625C2214yes—
40#2624C2214yes—
41#2504—21no—
42#2503—21no—
43#2502—21no—
44#2501—21no—
45#2500—22no—
46#2499—22no—
47#2484—22no—
48#2483—21no—
49#2483—21no—
50#2482—22no—
51#2405—21no—
52#2405—21no—
53#2317C2212yes—
54#2317C2212yescomplemented
55#2310C2212yescomplemented
56#2309C2212yes—
57#2280C2211yescharacteristic
58#2280C2211yescharacteristic
59#2088C2214yescomplemented
60#2088C2214yescomplemented
61#2068C2214yescomplemented
62#2067C2214yescomplemented
63#2066C2214yescomplemented
64#2065C2214yescomplemented
65#2055C2214yescomplemented
66#2055C2214yescomplemented
67#1973C2214yescomplemented
68#1968C2214yescomplemented
69#1943C2214yescomplemented
70#1941C2214yescomplemented
71#1936C2214yescomplemented
72#1935C2214yescomplemented
73#1934C2214yescomplemented
74#1933C2214yescomplemented
75#1893C2212yes—
76#1893C2212yes—
77#1892C2212yescomplemented
78#1892C2212yescomplemented
79#1875C2214yes—
80#1873C2214yes—
81#1854C2214yes—
82#1854C2214yescomplemented
83#1824C2212yes—
84#1823C2212yescomplemented
85#1808C2212yescomplemented
86#1807C2212yescomplemented
87#1787C2212yescomplemented
88#1786C2212yes—
89C23×Q8C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C22×SD16C2312yes—
94C22×SD16C2312yes—
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×#121—21no—
108C2×#121—21no—
109C2×#121—24no—
110C2×#121—24no—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×#117—24no—
114C2×#117—24no—
115C2×#117D412yes—
116C2×#117D412yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#483—28no—
133#481—28no—
134#465—28no—
135#465—28no—
136#450—28no—
137#448—28no—
138#433—216no—
139#433—216no—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×#106C2314yes—
144C2×C4⋊C8—24no—
145C2×C4⋊C8—24no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8—24no—
157C2×C22⋊C8D418yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×#97—24no—
163C2×#97C2314yes—
164C2×C4×C8—24no—
165C2×C4×C8C2314yes—
166#400—24no—
167#399—24no—
168#398—24no—
169#397—24no—
170#397—24no—
171#397—24no—
172#394—24no—
173#394—24no—
174#391—22no—
175#391—22no—
176#391C2314yes—
177#391C2314yes—
178#390—22no—
179#390—22no—
180#390C2314yes—
181#390C2314yes—
182#389C2314yes—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#388—24no—
187#388—24no—
188#387—24no—
189#387—24no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#384—22no—
199#384—22no—
200#382—22no—
201#382—22no—
202#381—22no—
203#381—22no—
204#381—21no—
205#381—21no—
206#381—22no—
207#381—22no—
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211#380C2312yes—
212#379—21no—
213#379—21no—
214#379—22no—
215#379—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226C25—21no—
227C25C2×D411yescharacteristic
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8C2412yes—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—42no—
240C22×D4—42no—
241C22×D4—44no—
242C22×D4—84no—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D414yes—
257C23×C4#4114yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—44no—
264C2×SD16—44no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—42no—
268C22×C8—42no—
269C22×C8—42no—
270C22×C8—42no—
271C22×C8C2×D414yes—
272C22×C8C2×D414yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277#123—84no—
278#123—84no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285#121—24no—
286#121—24no—
287#121—24no—
288#121—24no—
289#121—44no—
290#121—44no—
291#117—28no—
292#117—28no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—42no—
298#117—42no—
299#117—42no—
300#117—42no—
301C2×C4⋊C4—24no—
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—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2414yes—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356#107—44no—
357#106—24no—
358#106—24no—
359#106—44no—
360C4⋊C8—28no—
361C4⋊C8—28no—
362#103—24no—
363#103—24no—
364#103—44no—
365#103—44no—
366#102—24no—
367#102—24no—
368#102—44no—
369#102—44no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8C2×D418yes—
373C22⋊C8C2×D418yes—
374#97—28no—
375C4×C8—28no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388C24—24no—
389C24—22no—
390C24—42no—
391C24—42no—
392C24—44no—
393C24—42no—
394C24—82no—
395C24—82no—
396C24#12114yes—
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—84no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—84no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—816no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4#12114yes—
457C22×C4#12114yes—
458C22×C4#12114yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473SD16—84no—
474SD16—84no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—44no—
482C2×C8—44no—
483C2×C8—44no—
484C2×C8—44no—
485C2×C8C22×D414yes—
486C2×C8C22×D414yes—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—44no—
532C42—44no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—28no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—88no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—168no—
554C23C2×#12112yes—
555C23C2×#12112yes—
556C23C2×#12112yes—
557C23#57714yes—
558C23C23×D411yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—44no—
562Q8—84no—
563D4—42no—
564D4—42no—
565D4—44no—
566D4—84no—
567D4—88no—
568D4—88no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#12112yes—
615C2×C4C23×D411yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4#58512yes—
618C2×C4#58512yes—
619C8—24no—
620C8—24no—
621C8—42no—
622C8—42no—
623C22—22no—
624C22—22no—
625C22—28no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—44no—
633C22—44no—
634C22—88no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22—168no—
641C22#305714yes—
642C22C22×#12111yescharacteristic
643C22#344912yes—
644C22C2×#57712yes—
645C22#358412yes—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—84no—
657C4—84no—
658C4#348511yescharacteristic
659C4#348511yescharacteristic
660C4#349112yes—
661C2—24no—
662C2—42no—
663C2—42no—
664C2—162no—
665C2#3508911yescharacteristic
666C2C2×#305711yescharacteristic
667C2#6045311yescharacteristic
668C2#6100114yes—
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
2. C22×#433C2semidirect
3. #35311C2semidirect
5. #33134C2semidirect
6. #33128C2semidirect
7. #29775C2semidirect
8. #29774C2semidirect
9. #24618C2semidirect
10. #24616C2semidirect
11. #24518C2semidirect
12. #24516C2semidirect
13. #21939C2semidirect
14. #21762C2semidirect
28. C2×#433C22semidirect
29. C22×C4⋊C8C22semidirect
30. C22×C4⋊C8C22semidirect
33. C22×#97C22semidirect
34. C22×C4×C8C22semidirect
35. #2855C22semidirect
36. #2855C22semidirect
54. #2317C22semidirect
55. #2310C22semidirect
59. #2088C22semidirect
60. #2088C22semidirect
61. #2068C22semidirect
62. #2067C22semidirect
63. #2066C22semidirect
64. #2065C22semidirect
65. #2055C22semidirect
66. #2055C22semidirect
67. #1973C22semidirect
68. #1968C22semidirect
69. #1943C22semidirect
70. #1941C22semidirect
71. #1936C22semidirect
72. #1935C22semidirect
73. #1934C22semidirect
74. #1933C22semidirect
77. #1892C22semidirect
78. #1892C22semidirect
82. #1854C22semidirect
84. #1823C22semidirect
85. #1808C22semidirect
86. #1807C22semidirect
87. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
C22×#4332
#353112
#352622
#331342
#331282
#297752
#297742
#246182
#246162
#245182
#245162
#219392
#217622
#217612