Finite Group Databasebeta

← #6517048

Subgroups — #6517048

Order 512.

2374 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3216C211yesmaximal, characteristic, complemented
2C2×#2973C211yesmaximal, characteristic, complemented
3#35240C211yesmaximal, characteristic, complemented
4#34936C211yesmaximal, characteristic
5#28610C214yesmaximal, complemented
6#28492C214yesmaximal, complemented
7#25402C214yesmaximal, complemented
8#25398C214yesmaximal, complemented
9#24389C214yesmaximal, complemented
10#24383C214yesmaximal, complemented
11#21812C211yesmaximal, characteristic, complemented
12#21392C211yesmaximal, characteristic, complemented
13#21366C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C2×#525C2212yes—
17C2×#525C2212yes—
18C22×C4×D4C2211yescharacteristic
19#3216—28no—
20C2×#497C2212yescomplemented
21C2×#489C2212yescomplemented
22C2×#470C2214yescomplemented
23C2×#468C2214yescomplemented
24C2×#467C2214yescomplemented
25C2×#465C2214yescomplemented
26C2×#462C2212yescomplemented
27C2×#460C2212yescomplemented
28C2×#452C2214yes—
29#2973—28no—
30C2×#435C2214yes—
31C2×#435C2214yes—
32C2×#432C2211yescharacteristic
33C2×#432C2211yescharacteristic
34C2×#407C2212yes—
35C22×C22⋊C8C2212yes—
36#2848C2212yes—
37#2848C2212yes—
38#2633—22no—
39#2547C2212yes—
40#2542C2212yes—
41#2542C2212yes—
42#2541C2212yes—
43#2540C2214yes—
44#2539—21no—
45#2539—21no—
46#2536C2214yes—
47#2502—21no—
48#2501—21no—
49#2482—21no—
50#2482—21no—
51#2435—21no—
52#2433—21no—
53#2397—22no—
54#2326—22no—
55C4×#127—22no—
56#2283C2211yescharacteristic
57C4×#117C2211yescharacteristic
58#2036C2214yescomplemented
59#2035C2214yescomplemented
60#2034C2214yescomplemented
61#2033C2214yescomplemented
62#2001C2214yescomplemented
63#2001C2214yescomplemented
64#2000C2214yescomplemented
65#2000C2214yescomplemented
66#1987C2212yescomplemented
67#1983C2214yescomplemented
68#1981C2212yescomplemented
69#1961C2212yescomplemented
70#1960C2212yescomplemented
71#1949C2214yescomplemented
72#1895C2214yescomplemented
73#1894C2214yescomplemented
74#1893C2214yescomplemented
75#1892C2214yescomplemented
76#1873C2214yes—
77#1854C2214yes—
78#1788C2214yes—
79#1776C2214yescomplemented
80#1763C2214yescomplemented
81C22×#41C2312yes—
82C23×D4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×Q16C2314yes—
86C22×SD16C2314yes—
87C22×M16C2314yes—
88C23×C8C2314yes—
89#533—24no—
90#533—24no—
91#532—24no—
92#532—24no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127—22no—
97C2×#127C2311yescharacteristic
98C2×#125—21no—
99C2×#124—22no—
100C2×#124C2312yes—
101#525—48no—
102#525—48no—
103C2×#123—22no—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2314yes—
109C2×C22⋊Q8C2314yes—
110C2×C22⋊Q8C2314yes—
111C2×#121—21no—
112C2×#121—21no—
113C2×#121C2314yes—
114C2×#121C2314yes—
115C2×#120—22no—
116C2×#120—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2312yes—
119C2×C4×D4—24no—
120C2×C4×D4—24no—
121C2×C4×D4—24no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—24no—
125C2×C4×D4D412yes—
126C2×C4×D4D412yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117C2312yes—
131C22×C4⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42C2311yescharacteristic
135#497—28no—
136#489—28no—
137#470—48no—
138#468—48no—
139#467—48no—
140#465—48no—
141#462—28no—
142#460—28no—
143#452—216no—
144#435—216no—
145#435—216no—
146#432—24no—
147#432D418yescomplemented
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160#407—28no—
161C2×C22⋊C8—28no—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2312yes—
166C2×#97C2312yes—
167C2×#97C2312yes—
168#399—24no—
169#399—22no—
170#399—22no—
171#399—22no—
172#399—22no—
173#398—24no—
174#397—22no—
175#397—22no—
176#394—24no—
177#392—24no—
178#392—22no—
179#392—22no—
180#391—22no—
181#390—22no—
182#390—22no—
183#390—22no—
184#390—24no—
185#390—24no—
186#388—24no—
187#388—24no—
188#385—24no—
189#385—24no—
190#385—22no—
191#385—22no—
192#384—22no—
193#384—22no—
194#383—21no—
195#382C2312yes—
196#381—21no—
197#380C2312yes—
198#380C2314yes—
199#380C2314yes—
200#379—22no—
201#379—22no—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208#376—21no—
209#376—21no—
210C43—21no—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215C25C2×D411yescharacteristic
216C25C2×D411yescharacteristic
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—44no—
220C2×#41—44no—
221C22×Q8—21no—
222C22×Q8C2412yes—
223C22×Q8C2412yes—
224C22×D4—22no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—24no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—44no—
231C22×D4C2×D412yes—
232C22×D4C2×D412yes—
233C22×D4C2412yes—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C2×Q16—28no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×M16—28no—
257C2×M16—44no—
258C2×M16—44no—
259C22×C8—28no—
260C22×C8—44no—
261C22×C8—44no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270#127—22no—
271#127—22no—
272#127—48no—
273#125—44no—
274#124—24no—
275#124—24no—
276#124—48no—
277#123—48no—
278#123—48no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290#121—28no—
291#121—28no—
292#121—44no—
293#121—44no—
294#121—44no—
295#121—44no—
296#120—48no—
297#120—48no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—44no—
301C4×D4—28no—
302C4×D4—28no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—416no—
308C4×D4—416no—
309C4×D4—44no—
310C4×D4—44no—
311#117—28no—
312#117—44no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2414yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D414yes—
357C2×C42#4114yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—28no—
361#106—28no—
362C4⋊C8—24no—
363C4⋊C8C2×D418yes—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—44no—
371#102—44no—
372C22⋊C8—24no—
373C22⋊C8—28no—
374C22⋊C8—28no—
375C22⋊C8C2×D418yes—
376#97—24no—
377#97C2×D418yes—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386C24—22no—
387C24—24no—
388C24—22no—
389C24—22no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397C24C22×D412yes—
398#41—88no—
399#41—88no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—44no—
423C2×D4—416no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—82no—
432C2×D4—84no—
433C2×D4—84no—
434C22×C4—24no—
435C22×C4—24no—
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—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4#12114yes—
466C22×C4#12114yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C2×#4112yes—
476C22×C4C2511yescharacteristic
477Q16—88no—
478SD16—88no—
479M16—88no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—88no—
486C2×C8C22×D418yes—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4H(5,𝔽2)18yes—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—28no—
524C42—44no—
525C42—44no—
526C42—44no—
527C42—82no—
528C42#12114yes—
529C42#12114yes—
530C42C22×D412yes—
531C42C22×D412yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C23—28no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23—88no—
558C23—88no—
559C23C2×#12112yes—
560C23#57514yes—
561C23C23×D411yescharacteristic
562Q8—42no—
563Q8—42no—
564Q8—44no—
565Q8—44no—
566Q8—44no—
567Q8—84no—
568D4—48no—
569D4—48no—
570D4—44no—
571D4—44no—
572D4—84no—
573D4—84no—
574D4—816no—
575D4—816no—
576D4—88no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—28no—
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—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4C2×#12112yes—
615C2×C4C2×#12112yes—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#12112yes—
619C2×C4C23×D411yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)14yes—
622C8—28no—
623C8—44no—
624C22—22no—
625C22—24no—
626C22—22no—
627C22—28no—
628C22—22no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—88no—
639C22—88no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22—82no—
645C22C22×#12111yescharacteristic
646C22#348412yes—
647C22C2×#57512yes—
648C22#358612yes—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—48no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—84no—
659C4—82no—
660C4—82no—
661C4#349812yes—
662C4#350812yes—
663C2—44no—
664C2—42no—
665C2—82no—
666C2—162no—
667C2#3552811yescharacteristic
668C2#6044411yescharacteristic
669C2#6045511yescharacteristic
670C2#6158014yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34936C2
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×#3216C2semidirect
2. C2×#2973C2semidirect
3. #35240C2semidirect
5. #28610C2semidirect
6. #28492C2semidirect
7. #25402C2semidirect
8. #25398C2semidirect
9. #24389C2semidirect
10. #24383C2semidirect
11. #21812C2semidirect
12. #21392C2semidirect
13. #21366C2semidirect
20. C2×#497C22semidirect
21. C2×#489C22semidirect
22. C2×#470C22semidirect
23. C2×#468C22semidirect
24. C2×#467C22semidirect
25. C2×#465C22semidirect
26. C2×#462C22semidirect
27. C2×#460C22semidirect
58. #2036C22semidirect
59. #2035C22semidirect
60. #2034C22semidirect
61. #2033C22semidirect
62. #2001C22semidirect
63. #2001C22semidirect
64. #2000C22semidirect
65. #2000C22semidirect
66. #1987C22semidirect
67. #1983C22semidirect
68. #1981C22semidirect
69. #1961C22semidirect
70. #1960C22semidirect
71. #1949C22semidirect
72. #1895C22semidirect
73. #1894C22semidirect
74. #1893C22semidirect
75. #1892C22semidirect
79. #1776C22semidirect
80. #1763C22semidirect
147. #432D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32162
C2×#29732
#352402
#349362
#286102
#284922
#254022
#253982
#243892
#243832
#218122
#213922
#213662