Finite Group Databasebeta

← #196401

Subgroups — #196401

Order 512.

1730 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1517C212yesmaximal, complemented
4#14682C211yesmaximal, characteristic, complemented
5#9114C214yesmaximal, complemented
6#9113C214yesmaximal, complemented
7#8026C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C2×#431—21no—
13C2×#431—28no—
14C2×#431C2212yes—
15C2×#431C2212yes—
16C22×C4⋊C8—21no—
17C22×C22⋊C8C2212yes—
18C22×C4×C8—21no—
19#2857C411yescharacteristic, complemented
20#2857C411yescharacteristic, complemented
21#2856C411yescharacteristic, complemented
22#2856C411yescharacteristic, complemented
23#2854—21no—
24#2854C2212yes—
25#2850—21no—
26#2850—21no—
27#2847—21no—
28#2847C2212yes—
29C4×#121—21no—
30C4×#121—21no—
31C4×#121C412yescomplemented
32C4×#121C412yescomplemented
33#2297—21no—
34#2285C2211yescharacteristic
35C2×#382C2212yes—
36C2×#380—21no—
37#1517—28no—
38C2×#327C2218yes—
39C4⋊#95—21no—
40#1434—21no—
41#1434C2212yes—
42C22⋊#95—21no—
43C22⋊#95C2212yes—
44#1294C2218yes—
45C22×#41—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50#532—24no—
51#532—24no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55#525—44no—
56#525—44no—
57C2×#123—22no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121C2×C412yes—
66C2×#121C2×C412yes—
67C2×#121C2312yes—
68C2×#120—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C2×#117—21no—
74C22×C4⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—24no—
77C22×C42—24no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#431—28no—
91#431—28no—
92#431—24no—
93#431—28no—
94#431D418yescomplemented
95C2×C4⋊C8—28no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—28no—
100C2×C22⋊C8C2314yes—
101C2×C4×C8—28no—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104#391—21no—
105#391—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390C2×C412yes—
111#390C2×C412yes—
112#389—21no—
113#389—21no—
114#389C2×C412yes—
115#389C2×C412yes—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#384C2×C411yescharacteristic
127#384C2×C411yescharacteristic
128#382—21no—
129#382—24no—
130#382—24no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380—44no—
135#380C2×C412yes—
136#380C2×C412yes—
137#380C2312yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2×C412yes—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95C2314yes—
147#327—216no—
148C25C2×D411yescharacteristic
149C2×#41—44no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—44no—
153C22×Q8—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22×C412yes—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4D814yes—
180C23×C4SD1614yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C22×C8—28no—
192C22×C8—24no—
193C22×C8—44no—
194C22×C8—44no—
195C4⋊Q8—24no—
196#127—24no—
197#124—42no—
198#124—42no—
199#123—44no—
200#123—44no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203#121—24no—
204#121—24no—
205#121—44no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211#121—44no—
212#120—44no—
213#120—44no—
214C4×D4—28no—
215C4×D4—48no—
216C4×D4—48no—
217C4×D4—42no—
218C4×D4—42no—
219#117—42no—
220#117—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D414yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4114yes—
283C4⋊C8—28no—
284C4⋊C8—28no—
285C22⋊C8—28no—
286C22⋊C8—24no—
287C22⋊C8C2×D418yes—
288C4×C8—28no—
289C4×C8—28no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—41no—
308C24—41no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24C2×D812yes—
312C24C2×SD1612yes—
313C24C22×D411yescharacteristic
314#41—88no—
315#41—88no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×D4—24no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—48no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4#10214yes—
403C22×C4#10214yes—
404C22×C4#10214yes—
405C22×C4#10214yes—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4#12114yes—
417C22×C4#12114yes—
418C22×C4C2×D812yes—
419C22×C4C2×D812yes—
420C22×C4C2×SD1612yes—
421C22×C4C2×SD1612yes—
422C22×C4#13514yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C2×#4112yes—
427C2×C8—28no—
428C2×C8—416no—
429C2×C8—44no—
430C2×C8—44no—
431C4⋊C4—24no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—88no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—88no—
456C42—22no—
457C42—22no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—44no—
471C42C4×D414yes—
472C42C4×D414yes—
473C42#12012yes—
474C42#12012yes—
475C42#12312yes—
476C42#12312yes—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—21no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23#41012yes—
509C23C2×#10212yes—
510C23C2×#10212yes—
511C23#42112yes—
512C23C22×C22⋊C411yescharacteristic
513C23C2×#12112yes—
514C23C22×D811yescharacteristic
515C23C22×SD1611yescharacteristic
516C23C2×#13512yes—
517Q8—84no—
518D4—84no—
519D4—84no—
520D4—88no—
521D4—84no—
522D4—84no—
523D4—88no—
524D4—84no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—88no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#38612yes—
586C2×C4#38612yes—
587C2×C4#38612yes—
588C2×C4#38612yes—
589C2×C4#41012yes—
590C2×C4C2×#10212yes—
591C2×C4C2×#10212yes—
592C2×C4C2×#10212yes—
593C2×C4C2×#10212yes—
594C2×C4#41612yes—
595C2×C4#41612yes—
596C2×C4#42112yes—
597C2×C4#46518yes—
598C2×C4#46618yes—
599C2×C4C2×C4×D412yes—
600C2×C4C2×#12011yescharacteristic
601C2×C4C2×#12311yescharacteristic
602C8—48no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—48no—
621C22—48no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22#151714yescomplemented
634C22C2×#38611yescharacteristic
635C22C2×#41011yescharacteristic
636C22C2×#41011yescharacteristic
637C22C22×#10211yescharacteristic
638C22C2×#41611yescharacteristic
639C22C2×#42111yescharacteristic
640C22C2×#42111yescharacteristic
641C22C2×#46512yes—
642C22C2×#46612yes—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—21no—
649C4—21no—
650C4—22no—
651C4—22no—
652C4—41no—
653C4—41no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—164no—
658C4#189314yes—
659C4#189314yes—
660C2—22no—
661C2—22no—
662C2—24no—
663C2—42no—
664C2—162no—
665C2C2×#151711yescharacteristic
666C2#1757111yescharacteristic
667C2#1757311yescharacteristic
668C2#1757411yescharacteristic
669C2#2090211yescharacteristic
670C2C2×#189311yescharacteristic
671C2#2098311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#431C2semidirect
3. C2×#1517C2semidirect
4. #14682C2semidirect
5. #9114C2semidirect
6. #9113C2semidirect
7. #8026C2semidirect
19. #2857C4semidirect
20. #2857C4semidirect
21. #2856C4semidirect
22. #2856C4semidirect
31. C4×#121C4semidirect
32. C4×#121C4semidirect
94. #431D4semidirect
633. C22#1517semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
#366242
C2×#15172
#146822
#91142
#91132
#80262