Finite Group Databasebeta

← #374157

Subgroups — #374157

Order 512.

1577 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
1#60124C211yesmaximal, characteristic, complemented
2#34810C211yesmaximal, characteristic
3#20979C212yesmaximal
4C2×#1740C211yesmaximal, characteristic, complemented
5#11212C212yesmaximal, complemented
6#11056C212yesmaximal, complemented
7#10959C212yesmaximal, complemented
8#10893C212yesmaximal, complemented
9#10733C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C22×#117C2212yes—
12C22×#117C2211yescharacteristic
13#2924—24no—
14#2924—24no—
15#2921—22no—
16#2920—22no—
17#2905C2212yes—
18C2×#421C2214yes—
19C2×#407C2212yes—
20#2873C2212yes—
21#2504C411yescharacteristic, complemented
22#2503C411yescharacteristic, complemented
23#2502C411yescharacteristic, complemented
24#2501C411yescharacteristic, complemented
25#2500C411yescharacteristic, complemented
26#2500C411yescharacteristic, complemented
27#2499C411yescharacteristic, complemented
28#2499C411yescharacteristic, complemented
29#2498—22no—
30#2412—22no—
31#2395—22no—
32C4×#123—22no—
33#2297C2211yescharacteristic
34#2297C2211yescharacteristic
35#2285C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#1897—24no—
39#1896—24no—
40#1740—22no—
41#1740C2212yes—
42#1740C2212yes—
43C2×#342—22no—
44#1733C2214yes—
45C2×#339C2214yes—
46C2×#339C2214yes—
47C22×#41C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16C2314yes—
50#525—22no—
51#525—22no—
52#525—22no—
53#525—22no—
54C2×#123—22no—
55C2×#123—21no—
56C2×C22⋊Q8C2×C411yescharacteristic
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×#121C2×C411yescharacteristic
59C2×#121C2×C411yescharacteristic
60C2×#120—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—28no—
70C2×#117—22no—
71C2×#117—44no—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D412yes—
75C2×#117D412yes—
76C2×#117C2314yes—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C22⋊C4—22no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D412yes—
86C22×C42—22no—
87C22×C42D412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#433—216no—
91#432—24no—
92#432—24no—
93#431—24no—
94#431—24no—
95#424—28no—
96#423—24no—
97#421—28no—
98#421—44no—
99C2×#104—28no—
100#407—22no—
101#407—22no—
102#407D414yes—
103#407D414yes—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106#405—28no—
107C4×M16—24no—
108#400—22no—
109#399—22no—
110#398—22no—
111#397—22no—
112#397—22no—
113#397—22no—
114#394—22no—
115#394—22no—
116#391C2×C411yescharacteristic, complemented
117#391C2×C411yescharacteristic, complemented
118#391C2×C411yescharacteristic, complemented
119#391C2×C411yescharacteristic, complemented
120#390C2×C411yescharacteristic, complemented
121#390C2×C411yescharacteristic, complemented
122#390C2×C411yescharacteristic, complemented
123#390C2×C411yescharacteristic, complemented
124#388—22no—
125#388—22no—
126#387—22no—
127#387—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#385—22no—
132#384C2×C411yescharacteristic, complemented
133#384C2×C411yescharacteristic, complemented
134#384C2×C411yescharacteristic, complemented
135#384C2×C411yescharacteristic, complemented
136#382C2×C411yescharacteristic, complemented
137#382C2×C411yescharacteristic, complemented
138#382C2×C411yescharacteristic, complemented
139#382C2×C411yescharacteristic, complemented
140#381—22no—
141#380—22no—
142#380—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4C2×C411yescharacteristic
150C4×C22⋊C4C2×C411yescharacteristic
151C4×C22⋊C4C2×C411yescharacteristic
152C4×C22⋊C4C2×C411yescharacteristic
153C2×#95—21no—
154#342—24no—
155#342—24no—
156#339—28no—
157#339—28no—
158#339—44no—
159C25C2×D411yescharacteristic
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—24no—
163C2×#41—22no—
164C2×#41—22no—
165C22×Q8C22×C411yescharacteristic
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22×C411yescharacteristic
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—42no—
177C23×C4C22⋊C412yes—
178C23×C4C22⋊C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4#4112yes—
187C23×C4#4112yes—
188C23×C4C2411yescharacteristic
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16C2×D418yes—
195C22×C8—24no—
196#123—48no—
197#123—42no—
198#123—42no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—22no—
207#120—42no—
208#120—42no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—42no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—42no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—22no—
219#117—24no—
220#117—24no—
221#117—24no—
222#117—24no—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—24no—
228#117—28no—
229#117—24no—
230#117—22no—
231#117—22no—
232#117—24no—
233#117—48no—
234#117—48no—
235#117—44no—
236#117#4118yes—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22⋊C412yes—
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4C22⋊C412yes—
279C2×C22⋊C4C22⋊C412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—48no—
294C2×C42—42no—
295C2×C42C22⋊C412yes—
296C2×C42C22⋊C412yes—
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C4⋊C8—28no—
304C4⋊C8—28no—
305#104—416no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310#97—28no—
311C4×C8—28no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—42no—
329C24—81no—
330C24C2×C22⋊C411yescharacteristic
331C24C2×C22⋊C411yescharacteristic
332C24#12112yes—
333C24#12112yes—
334C24C22×D411yescharacteristic
335#41—44no—
336#41—44no—
337#41—44no—
338#41—44no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4—82no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—82no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C412yes—
406C22×C4C2×C22⋊C412yes—
407C22×C4C4×D412yes—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4#12011yescharacteristic
412C22×C4#12011yescharacteristic
413C22×C4#12012yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12311yescharacteristic
417C22×C4#12311yescharacteristic
418C22×C4#12312yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4C2×#4114yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+412yes—
426M16—28no—
427M16—28no—
428M16—44no—
429M16—44no—
430C2×C8—28no—
431C2×C8—44no—
432C2×C8—44no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—28no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—28no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—42no—
462C22⋊C4—48no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—42no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—28no—
486C42—48no—
487C42—41no—
488C42—41no—
489C42—42no—
490C42—42no—
491C42—44no—
492C23—21no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—21no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—84no—
516C23—82no—
517C23—82no—
518C23#38612yes—
519C23#38611yescharacteristic
520C23#38611yescharacteristic
521C23C22×C22⋊C411yescharacteristic
522C23C2×#12112yes—
523C23#52512yes—
524C23#53511yescharacteristic
525C23#53611yescharacteristic
526Q8—44no—
527D4—44no—
528D4—82no—
529D4—84no—
530D4—82no—
531D4—82no—
532D4—82no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—48no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—84no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—816no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4#38611yescharacteristic
592C2×C4#38611yescharacteristic
593C2×C4#38612yes—
594C2×C4C2×C4×D412yes—
595C2×C4C4×#4112yes—
596C2×C4#52012yes—
597C2×C4C2×#12011yescharacteristic
598C2×C4C2×#12311yescharacteristic
599C2×C4#53212yes—
600C2×C4#53712yes—
601C2×C4#54614yes—
602C2×C4#54714yes—
603C2×C4#55114yes—
604C2×C4#55214yes—
605C8—48no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—21no—
610C22—22no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—21no—
615C22—22no—
616C22—21no—
617C22—22no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—81no—
626C22—81no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22#188412yes—
635C22C2×#38611yescharacteristic
636C22#229711yescharacteristic
637C22#231111yescharacteristic
638C22#241112yes—
639C22#241312yes—
640C4—21no—
641C4—22no—
642C4—21no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—88no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4#235914yes—
656C4#243012yes—
657C4#249812yes—
658C2—21no—
659C2—21no—
660C2—22no—
661C2—22no—
662C2—22no—
663C2—41no—
664C2—42no—
665C2—161no—
666C2#1413511yescharacteristic
667C2C2×#188411yescharacteristic
668C2#2094711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60124C2semidirect
4. C2×#1740C2semidirect
5. #11212C2semidirect
6. #11056C2semidirect
7. #10959C2semidirect
8. #10893C2semidirect
9. #10733C2semidirect
21. #2504C4semidirect
22. #2503C4semidirect
23. #2502C4semidirect
24. #2501C4semidirect
25. #2500C4semidirect
26. #2500C4semidirect
27. #2499C4semidirect
28. #2499C4semidirect
116. #391C2×C4semidirect
117. #391C2×C4semidirect
118. #391C2×C4semidirect
119. #391C2×C4semidirect
120. #390C2×C4semidirect
121. #390C2×C4semidirect
122. #390C2×C4semidirect
123. #390C2×C4semidirect
132. #384C2×C4semidirect
133. #384C2×C4semidirect
134. #384C2×C4semidirect
135. #384C2×C4semidirect
136. #382C2×C4semidirect
137. #382C2×C4semidirect
138. #382C2×C4semidirect
139. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#348102
#209792
C2×#17402
#112122
#110562
#109592
#108932
#107332