Finite Group Databasebeta

← #196362

Subgroups — #196362

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1502C212yesmaximal, complemented
4C4×#380C211yesmaximal, characteristic, complemented
5#9114C214yesmaximal, complemented
6#9113C214yesmaximal, complemented
7#8014C212yesmaximal, 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×#380—21no—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4—21no—
38#1502—28no—
39C2×#327C2218yes—
40C4⋊#95—21no—
41#1434C2212yes—
42C22⋊#95C2212yes—
43C4×#95—21no—
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×C42—22no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#431—28no—
92#431—28no—
93#431—24no—
94#431—28no—
95#431D418yescomplemented
96C2×C4⋊C8—28no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—28no—
101C2×C22⋊C8C2314yes—
102C2×C4×C8—28no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105#391—21no—
106#391—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#390C2×C412yes—
112#390C2×C412yes—
113#389—21no—
114#389—21no—
115#389C2×C412yes—
116#389C2×C412yes—
117#388—22no—
118#388—22no—
119#388—22no—
120#388—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#384C2×C411yescharacteristic
128#384C2×C411yescharacteristic
129#382—21no—
130#380—28no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380C2×C412yes—
135#380C2×C412yes—
136#380C2312yes—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4C2×C412yes—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—44no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95C2314yes—
148#327—216no—
149C25C2×D411yescharacteristic
150C2×#41—44no—
151C2×#41—42no—
152C2×#41—42no—
153C2×#41—44no—
154C22×Q8—21no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22×C412yes—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
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×C4D814yes—
181C23×C4SD1614yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C22×C8—28no—
193C22×C8—24no—
194C22×C8—44no—
195C22×C8—44no—
196C4⋊Q8—24no—
197#127—24no—
198#124—42no—
199#124—42no—
200#123—44no—
201#123—44no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204#121—24no—
205#121—24no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211#121—44no—
212#121—44no—
213#120—44no—
214#120—44no—
215C4×D4—28no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—42no—
219C4×D4—42no—
220#117—42no—
221#117—42no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C22⋊C4—28no—
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—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
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×C42—22no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42—44no—
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×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D414yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4114yes—
287C4⋊C8—28no—
288C4⋊C8—28no—
289C22⋊C8—28no—
290C22⋊C8—24no—
291C22⋊C8C2×D418yes—
292C4×C8—28no—
293C4×C8—28no—
294#95—28no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—44no—
304#95—44no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C2×D812yes—
315C24C2×SD1612yes—
316C24C22×D411yescharacteristic
317#41—88no—
318#41—88no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×D4—24no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
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—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—416no—
393C22×C4—48no—
394C22×C4—48no—
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×C4#10214yes—
405C22×C4#10214yes—
406C22×C4#10214yes—
407C22×C4#10214yes—
408C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4#12114yes—
420C22×C4#12114yes—
421C22×C4C2×D812yes—
422C22×C4C2×D812yes—
423C22×C4C2×SD1612yes—
424C22×C4C2×SD1612yes—
425C22×C4#13514yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C2×#4112yes—
430C2×C8—28no—
431C2×C8—416no—
432C2×C8—44no—
433C2×C8—44no—
434C4⋊C4—24no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—416no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—416no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C42—22no—
460C42—22no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—44no—
474C42—88no—
475C42C4×D414yes—
476C42C4×D414yes—
477C42#12012yes—
478C42#12012yes—
479C42#12312yes—
480C42#12312yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—21no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23#41012yes—
513C23C2×#10212yes—
514C23C2×#10212yes—
515C23C2×#10412yes—
516C23C22×C22⋊C411yescharacteristic
517C23C2×#12112yes—
518C23C22×D811yescharacteristic
519C23C22×SD1611yescharacteristic
520C23C2×#13512yes—
521Q8—84no—
522D4—84no—
523D4—88no—
524D4—88no—
525D4—84no—
526D4—84no—
527D4—84no—
528D4—84no—
529C2×C4—24no—
530C2×C4—24no—
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—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—88no—
587C2×C4#38612yes—
588C2×C4#38612yes—
589C2×C4#38612yes—
590C2×C4#38612yes—
591C2×C4#41012yes—
592C2×C4C2×#10212yes—
593C2×C4C2×#10212yes—
594C2×C4C2×#10212yes—
595C2×C4C2×#10212yes—
596C2×C4#41612yes—
597C2×C4#41612yes—
598C2×C4C2×#10412yes—
599C2×C4C2×#10412yes—
600C2×C4#42112yes—
601C2×C4#46518yes—
602C2×C4#46618yes—
603C2×C4C2×C4×D412yes—
604C2×C4C2×#12011yescharacteristic
605C2×C4C2×#12311yescharacteristic
606C8—48no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—48no—
625C22—48no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22#150214yescomplemented
638C22C2×#38611yescharacteristic
639C22C2×#41011yescharacteristic
640C22C2×#41011yescharacteristic
641C22C22×#10211yescharacteristic
642C22C2×#41611yescharacteristic
643C22C22×#10411yescharacteristic
644C22C2×#42111yescharacteristic
645C22C2×#46512yes—
646C22C2×#46612yes—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—41no—
656C4—41no—
657C4—88no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4#187114yes—
662C4#189314yes—
663C4#189314yes—
664C2—22no—
665C2—24no—
666C2—22no—
667C2—42no—
668C2—162no—
669C2C2×#150211yescharacteristic
670C2#1753311yescharacteristic
671C2#1757411yescharacteristic
672C2#1757611yescharacteristic
673C2C2×#187111yescharacteristic
674C2C2×#189311yescharacteristic
675C2#2098311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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×#1502C2semidirect
4. C4×#380C2semidirect
5. #9114C2semidirect
6. #9113C2semidirect
7. #8014C2semidirect
19. #2857C4semidirect
20. #2857C4semidirect
21. #2856C4semidirect
22. #2856C4semidirect
31. C4×#121C4semidirect
32. C4×#121C4semidirect
95. #431D4semidirect
637. C22#1502semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
#366242
C2×#15022
C4×#3802
#91142
#91132
#80142