Finite Group Databasebeta

← #196410

Subgroups — #196410

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1517C212yesmaximal, complemented
4#14682C211yesmaximal, characteristic, complemented
5#9118C214yesmaximal, complemented
6#9117C214yesmaximal, complemented
7#8026C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432C2212yes—
14C2×#431—21no—
15C2×#431C2212yes—
16C2×#423—21no—
17C2×#407C2212yes—
18C2×C4×M16—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
49C22×M16—22no—
50#532—24no—
51#532—24no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55#525—48no—
56C2×#123—22no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121C2×C412yes—
65C2×#121C2×C412yes—
66C2×#121C2312yes—
67C2×#120—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×#117—21no—
73C22×C4⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—24no—
76C22×C42—24no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—28no—
90#432—28no—
91#431—28no—
92#431—24no—
93#431D418yescomplemented
94#423—28no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97#407—28no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8C2314yes—
100C4×M16—28no—
101C2×#97—22no—
102C2×C4×C8—22no—
103#391—21no—
104#391—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390C2×C412yes—
110#390C2×C412yes—
111#389—21no—
112#389—21no—
113#389C2×C412yes—
114#389C2×C412yes—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#384C2×C411yescharacteristic
126#384C2×C411yescharacteristic
127#382—21no—
128#382—24no—
129#382—24no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—44no—
134#380C2×C412yes—
135#380C2×C412yes—
136#380C2312yes—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2×C412yes—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95C2314yes—
146#327—216no—
147C25C2×D411yescharacteristic
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—42no—
151C2×#41—42no—
152C22×Q8—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C22×C412yes—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2411yescharacteristic
190C2×M16—28no—
191C2×M16—44no—
192C2×M16—44no—
193C22×C8—24no—
194C4⋊Q8—24no—
195#127—24no—
196#124—42no—
197#124—42no—
198#123—48no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201#121—24no—
202#121—24no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—44no—
210#120—48no—
211C4×D4—28no—
212C4×D4—48no—
213C4×D4—48no—
214C4×D4—42no—
215C4×D4—42no—
216#117—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
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—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D414yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4114yes—
279C4⋊C8—28no—
280C4⋊C8—28no—
281C22⋊C8—28no—
282C22⋊C8—24no—
283C22⋊C8C2×D418yes—
284#97—28no—
285C4×C8—28no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—41no—
306C24—41no—
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309C24C22×D411yescharacteristic
310#41—88no—
311#41—88no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×D4—24no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
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—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—48no—
393C22×C4—48no—
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—44no—
403C22×C4—44no—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
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×C4#12114yes—
415C22×C4#12114yes—
416C22×C4AGL(1,ℤ/8)12yes—
417C22×C4AGL(1,ℤ/8)14yes—
418C22×C4#13712yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C2×#4112yes—
423M16—416no—
424C2×C8—28no—
425C2×C8—44no—
426C2×C8—44no—
427C4⋊C4—24no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—88no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—88no—
451C42—22no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—44no—
466C42C4×D414yes—
467C42C4×D414yes—
468C42#12012yes—
469C42#12012yes—
470C42#12312yes—
471C42#12312yes—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—24no—
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—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23#41012yes—
506C23#42112yes—
507C23C22×C22⋊C411yescharacteristic
508C23C2×#12112yes—
509C23C2×AGL(1,ℤ/8)12yes—
510C23C2×AGL(1,ℤ/8)11yescharacteristic
511C23C2×#13711yescharacteristic
512Q8—84no—
513D4—84no—
514D4—84no—
515D4—88no—
516D4—88no—
517D4—84no—
518D4—84no—
519D4—84no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
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—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
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—88no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4#38612yes—
583C2×C4#38612yes—
584C2×C4#38612yes—
585C2×C4#38612yes—
586C2×C4#41012yes—
587C2×C4#41712yes—
588C2×C4#41712yes—
589C2×C4#41812yes—
590C2×C4#41812yes—
591C2×C4#42112yes—
592C2×C4#46818yes—
593C2×C4#46918yes—
594C2×C4C2×C4×D412yes—
595C2×C4C2×#12011yescharacteristic
596C2×C4C2×#12311yescharacteristic
597C8—48no—
598C22—22no—
599C22—22no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—48no—
613C22—48no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22C2×#38611yescharacteristic
630C22C2×#41011yescharacteristic
631C22C2×#41011yescharacteristic
632C22C2×#41811yescharacteristic
633C22C2×#41711yescharacteristic
634C22C2×#42111yescharacteristic
635C22C2×#42111yescharacteristic
636C22C2×#46812yes—
637C22C2×#46912yes—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—21no—
642C4—21no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—164no—
653C4#189514yes—
654C4#189514yes—
655C2—22no—
656C2—22no—
657C2—42no—
658C2—42no—
659C2—162no—
660C2#1758311yescharacteristic
661C2#1758511yescharacteristic
662C2#1758811yescharacteristic
663C2#1758911yescharacteristic
664C2#2090211yescharacteristic
665C2C2×#189511yescharacteristic
666C2#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. C2×#2920C2semidirect
3. C2×#1517C2semidirect
4. #14682C2semidirect
5. #9118C2semidirect
6. #9117C2semidirect
7. #8026C2semidirect
19. #2857C4semidirect
20. #2857C4semidirect
21. #2856C4semidirect
22. #2856C4semidirect
31. C4×#121C4semidirect
32. C4×#121C4semidirect
93. #431D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366242
C2×#15172
#146822
#91182
#91172
#80262