Finite Group Databasebeta

← #366777

Subgroups — #366777

Order 512.

1681 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
1#34810C211yesmaximal, characteristic
2C2×#1896C211yesmaximal, characteristic
3C2×#1871C211yesmaximal, characteristic
4#20837C211yesmaximal, characteristic, complemented
5C2×#1826C211yesmaximal, characteristic, complemented
6C22×#339C211yesmaximal, characteristic, complemented
7#13753C211yesmaximal, characteristic, complemented
8#11370C212yesmaximal, complemented
9#11053C212yesmaximal, complemented
10#10924C212yesmaximal, complemented
11#10840C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C2×#525C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#423C2212yes—
17C22×#104C2211yescharacteristic
18C22×#104C2212yes—
19C22×#104C2211yescharacteristic
20C2×#407C2211yescharacteristic
21#2504C411yescharacteristic, complemented
22#2503C411yescharacteristic, complemented
23#2502C411yescharacteristic, complemented
24#2501C411yescharacteristic, complemented
25#2500C412yescomplemented
26#2499C412yescomplemented
27#2498—22no—
28#2412—22no—
29#2395—22no—
30C4×#123—22no—
31#2297C2212yes—
32#2285C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×#376C2212yes—
36#1896—24no—
37#1871—24no—
38#1851—22no—
39#1850—22no—
40#1826—24no—
41C2×#339—24no—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2211yescharacteristic
48C2×#339C2212yes—
49#1728C2212yes—
50#1434—22no—
51C22⋊#95—22no—
52C4×#95C2212yes—
53C22×#41C2311yescharacteristic
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C22×M16—21no—
57C22×M16—22no—
58C22×M16—24no—
59C22×M16—22no—
60C22×M16—42no—
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2312yes—
64C23×C8—21no—
65#525—22no—
66#525—22no—
67#525—22no—
68#525—22no—
69C2×#123—21no—
70C2×#123—22no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×#120—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117—21no—
79C2×#117—21no—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117C2×C412yes—
85C2×#117C2312yes—
86C2×#117C2311yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4D412yes—
92C22×C42—24no—
93C22×C42—21no—
94C22×C42—21no—
95C22×C42—22no—
96C22×C42—42no—
97C22×C42D414yes—
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102#423—24no—
103#423—42no—
104C2×C4⋊C8—24no—
105C2×#104—24no—
106C2×#104—24no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—24no—
110C2×#104—42no—
111C2×#104—42no—
112#407—22no—
113#407—22no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116#400—22no—
117#399—22no—
118#398—22no—
119#397—24no—
120#397—22no—
121#394—24no—
122#391C2×C412yescomplemented
123#391C2×C412yescomplemented
124#390C2×C412yescomplemented
125#390C2×C412yescomplemented
126#388—24no—
127#387—24no—
128#386—24no—
129#386—22no—
130#385—22no—
131#384C2×C412yescomplemented
132#384C2×C412yescomplemented
133#382C2×C412yescomplemented
134#382C2×C412yescomplemented
135#381—22no—
136#380—24no—
137C4×C4⋊C4—24no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4C2×C412yes—
141C4×C22⋊C4C2×C411yescharacteristic
142C4×C22⋊C4C2×C411yescharacteristic
143C4×C22⋊C4D414yes—
144C4×C22⋊C4D412yes—
145C4×C22⋊C4D412yes—
146#376—24no—
147#376—42no—
148C2×#95—21no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95D412yes—
152#339—24no—
153#339—24no—
154#339—24no—
155#339—22no—
156#339—22no—
157#339—24no—
158#339—24no—
159#336—24no—
160C25C2×D411yescharacteristic
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C22×Q8C22×C411yescharacteristic
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D414yes—
196C23×C4#4112yes—
197C23×C4#4112yes—
198C23×C4C2411yescharacteristic
199C2×M16—22no—
200C2×M16—28no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—44no—
214C2×M16—48no—
215C2×M16—44no—
216C2×M16—42no—
217C2×M16#4114yes—
218C22×C8—22no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—44no—
222C22×C8—42no—
223#123—48no—
224#123—42no—
225#123—42no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#120—42no—
235#120—42no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240#117—24no—
241#117—24no—
242#117—22no—
243#117—22no—
244#117—21no—
245#117—21no—
246#117—22no—
247#117—22no—
248#117—21no—
249#117—21no—
250#117—42no—
251#117—42no—
252#117—42no—
253#117—42no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4C22⋊C412yes—
283C2×C22⋊C4C22⋊C412yes—
284C2×C22⋊C4C22⋊C414yes—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—28no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—21no—
304C2×C42—44no—
305C2×C42—48no—
306C2×C42—44no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C22×C412yes—
310C2×C42C22×C412yes—
311C2×C42C2×D414yes—
312C2×C42C2×D412yes—
313C2×C42C2×D414yes—
314C2×C42C2×D412yes—
315C2×C42#4114yes—
316C4⋊C8—48no—
317#104—48no—
318#104—44no—
319#104—44no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—48no—
328#95—48no—
329#95C22⋊C414yes—
330#95C22⋊C412yes—
331#95C22⋊C412yes—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—21no—
336C24—42no—
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24#12112yes—
340C24#12112yes—
341C24C22×D411yescharacteristic
342#41—41no—
343#41—41no—
344#41—42no—
345#41—41no—
346#41—41no—
347#41—42no—
348#41—42no—
349#41—42no—
350#41—42no—
351#41—42no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—82no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—432no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—82no—
422C22×C4C2×C22⋊C414yes—
423C22×C4C2×C22⋊C414yes—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C4×D412yes—
427C22×C4C4×D412yes—
428C22×C4C4×D412yes—
429C22×C4C4×D412yes—
430C22×C4#12112yes—
431C22×C4#12112yes—
432C22×C4#12412yes—
433C22×C4#12412yes—
434C22×C4#12712yes—
435C22×C4#12712yes—
436C22×C4C23×C411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+412yes—
446M16—24no—
447M16—24no—
448M16—48no—
449M16—42no—
450M16—42no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—42no—
456M16—42no—
457C2×C8—24no—
458C2×C8—48no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—42no—
464C2×C8—42no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—84no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—44no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—84no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—48no—
507C42—42no—
508C42—41no—
509C42—42no—
510C42—42no—
511C42—41no—
512C42—42no—
513C42—42no—
514C42—44no—
515C42#12314yescomplemented
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—42no—
529C23—42no—
530C23—48no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23#39012yes—
537C23#39012yes—
538C23#42114yes—
539C23C22×C22⋊C411yescharacteristic
540C23C2×#12111yescharacteristic
541C23C2×#12111yescharacteristic
542C23#54612yes—
543C23#54712yes—
544Q8—41no—
545Q8—42no—
546Q8—41no—
547D4—41no—
548D4—42no—
549D4—41no—
550D4—84no—
551D4—81no—
552D4—81no—
553D4—82no—
554D4—84no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—432no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—82no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4#39012yes—
610C2×C4#39012yes—
611C2×C4C2×C4×D411yescharacteristic
612C2×C4C2×C4×D411yescharacteristic
613C2×C4C4×#4112yes—
614C2×C4#52012yes—
615C2×C4C2×#12311yescharacteristic
616C2×C4#52511yescharacteristic
617C2×C4C2×#12411yescharacteristic
618C2×C4#52912yes—
619C2×C4C2×#12711yescharacteristic
620C2×C4#53411yescharacteristic
621C2×C4#53611yescharacteristic
622C2×C4#53912yes—
623C2×C4#54612yes—
624C2×C4#54612yes—
625C2×C4#54712yes—
626C2×C4#54712yes—
627C2×C4#55112yes—
628C2×C4#55212yes—
629C8—44no—
630C8—81no—
631C8—81no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—44no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—84no—
652C22#196314yes—
653C22C2×#39011yescharacteristic
654C22#233912yes—
655C22#241211yescharacteristic
656C22#241311yescharacteristic
657C22#242512yes—
658C22C2×#42112yes—
659C22#290112yes—
660C4—22no—
661C4—22no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—84no—
667C4—81no—
668C4—82no—
669C4—81no—
670C4—82no—
671C4—81no—
672C4C4×#12311yescharacteristic
673C4#232211yescharacteristic
674C4#235912yes—
675C4#248812yes—
676C4#250111yescharacteristic
677C4#253911yescharacteristic
678C2—21no—
679C2—22no—
680C2—21no—
681C2—24no—
682C2—42no—
683C2—161no—
684C2#1399611yescharacteristic
685C2C2×#196311yescharacteristic
686C2#2121011yescharacteristic
687C2#2250614yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20837C2semidirect
5. C2×#1826C2semidirect
6. C22×#339C2semidirect
7. #13753C2semidirect
8. #11370C2semidirect
9. #11053C2semidirect
10. #10924C2semidirect
11. #10840C2semidirect
21. #2504C4semidirect
22. #2503C4semidirect
23. #2502C4semidirect
24. #2501C4semidirect
25. #2500C4semidirect
26. #2499C4semidirect
122. #391C2×C4semidirect
123. #391C2×C4semidirect
124. #390C2×C4semidirect
125. #390C2×C4semidirect
131. #384C2×C4semidirect
132. #384C2×C4semidirect
133. #382C2×C4semidirect
134. #382C2×C4semidirect
515. C42#123semidirect

Maximal subgroups

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

subgroupindex
#348102
C2×#18962
C2×#18712
#208372
C2×#18262
C22×#3392
#137532
#113702
#110532
#109242
#108402