Finite Group Databasebeta

← #196695

Subgroups — #196695

Order 512.

1704 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×#423C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1557C212yesmaximal, complemented
5#9141C212yesmaximal, complemented
6#9137C212yesmaximal, complemented
7#9133C212yesmaximal, complemented
8#9129C212yesmaximal, complemented
9#8042C212yesmaximal, complemented
10C23×M16—21no—
11C2×#532C2211yescharacteristic
12C2×#525—21no—
13C23×C42C2211yescharacteristic
14C2×#432C2212yes—
15C2×#423—21no—
16C2×#423—21no—
17C2×#423—28no—
18C2×#423C2212yes—
19C2×#423C2212yes—
20C22×C4⋊C8—21no—
21C22×C4⋊C8C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2856C2211yescharacteristic
26#2854—21no—
27#2854C412yes—
28#2850—21no—
29#2850—21no—
30#2847—21no—
31#2847C412yes—
32C4×#121—21no—
33C4×#121—21no—
34C4×#121C412yes—
35C4×#121C412yes—
36#2297—21no—
37#2285C2211yescharacteristic
38#1817—21no—
39#1797C2212yes—
40#1797C2212yes—
41#1779—21no—
42#1749—21no—
43#1749—21no—
44#1557—28no—
45C2×#332C2214yescomplemented
46C2×#331C2214yescomplemented
47#1296C2214yescomplemented
48#1296C2214yescomplemented
49C22×#41—21no—
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C22×M16—28no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—44no—
57C22×M16—44no—
58C23×C8—22no—
59#532—24no—
60#532—24no—
61C2×C4⋊Q8C2311yescharacteristic
62C2×#127C2311yescharacteristic
63C2×#124—21no—
64#525—48no—
65C2×#123—22no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121C2×C412yes—
74C2×#121C2×C412yes—
75C2×#121C2×C412yes—
76C2×#120—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—21no—
80C2×C4×D4C2×C412yes—
81C2×#117—21no—
82C22×C42—24no—
83C22×C42—24no—
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#432—24no—
97#432—24no—
98#423—28no—
99#423—28no—
100#423—24no—
101#423—28no—
102#423D418yescomplemented
103C2×C4⋊C8—28no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—28no—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2314yes—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×#97—22no—
114#391—21no—
115#391—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#390C2×C412yes—
121#390C2×C412yes—
122#389—21no—
123#389—21no—
124#389C2×C412yes—
125#389C2×C412yes—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#386—22no—
135#386—22no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#382—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380C2×C412yes—
143#380C2×C412yes—
144#380C2×C412yes—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2×C412yes—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149#336—22no—
150#336—22no—
151#332—28no—
152#331—28no—
153C25C22⋊C411yescharacteristic
154C2×#41—44no—
155C2×#41—44no—
156C2×#41—44no—
157C22×Q8—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22×C412yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—432no—
198C2×M16—44no—
199C2×M16—44no—
200C2×M16—44no—
201C2×M16—44no—
202C22×C8—28no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8—44no—
206C22×C8—44no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209#127—22no—
210#127—22no—
211#124—44no—
212#123—48no—
213C22⋊Q8—44no—
214C22⋊Q8—44no—
215#121—28no—
216#121—44no—
217#121—44no—
218#121—42no—
219#121—42no—
220#121—42no—
221#121—42no—
222#121—44no—
223#121—42no—
224#121—42no—
225#121—44no—
226#120—48no—
227C4×D4—28no—
228C4×D4—48no—
229C4×D4—48no—
230C4×D4—42no—
231C4×D4—42no—
232#117—44no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
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×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D414yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42#4114yes—
288C2×C42C2411yescharacteristic
289C4⋊C8—28no—
290C4⋊C8—28no—
291C4⋊C8—28no—
292C4⋊C8—24no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295C4⋊C8C2×D418yes—
296C22⋊C8—44no—
297C22⋊C8—44no—
298C22⋊C8—44no—
299#97—44no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—41no—
313C24—41no—
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317#41—816no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×D4—24no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
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—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
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×C4C2×C22⋊C411yescharacteristic
417C22×C4C4×D414yes—
418C22×C4#12012yes—
419C22×C4#12114yes—
420C22×C4#12312yes—
421C22×C4AGL(1,ℤ/8)12yes—
422C22×C4AGL(1,ℤ/8)12yes—
423C22×C4AGL(1,ℤ/8)14yes—
424C22×C4#13712yes—
425C22×C4#13712yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429M16—416no—
430M16—416no—
431C2×C8—28no—
432C2×C8—416no—
433C2×C8—44no—
434C2×C8—44no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—44no—
438C4⋊C4—24no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—21no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—44no—
477C42C4×D414yes—
478C42#12012yes—
479C42#12114yes—
480C42#12312yes—
481C42C22×D412yes—
482C42C2×#4112yes—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—28no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23#38612yes—
514C23#41312yes—
515C23#41712yes—
516C23#41712yes—
517C23#41812yes—
518C23C22×C22⋊C411yescharacteristic
519Q8—84no—
520D4—84no—
521D4—88no—
522D4—88no—
523D4—88no—
524D4—81no—
525D4—81no—
526D4—81no—
527D4—81no—
528D4—84no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
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—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—88no—
588C2×C4#38612yes—
589C2×C4#38612yes—
590C2×C4#38612yes—
591C2×C4#41312yes—
592C2×C4#41712yes—
593C2×C4#41712yes—
594C2×C4#41812yes—
595C2×C4C2×C4×D412yes—
596C2×C4C2×#12011yescharacteristic
597C2×C4C2×#12112yes—
598C2×C4C2×#12311yescharacteristic
599C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
600C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×#13711yescharacteristic
603C2×C4C2×#13711yescharacteristic
604C2×C4#57512yes—
605C8—48no—
606C8—84no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—416no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22#155714yescomplemented
636C22C2×#38611yescharacteristic
637C22C2×#41311yescharacteristic
638C22C2×#41311yescharacteristic
639C22C2×#41811yescharacteristic
640C22C2×#41711yescharacteristic
641C22C2×#41711yescharacteristic
642C22#289611yescharacteristic
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—41no—
652C4—41no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#300311yescharacteristic
657C4#300411yescharacteristic
658C4#305514yes—
659C4#309211yescharacteristic
660C4#309211yescharacteristic
661C2—24no—
662C2—42no—
663C2—42no—
664C2—162no—
665C2C2×#155711yescharacteristic
666C2#1773711yescharacteristic
667C2#1774511yescharacteristic
668C2#1775311yescharacteristic
669C2#2090911yescharacteristic
670C2#2097311yescharacteristic
671C2#2098911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#423C2semidirect
3. #20715C2semidirect
4. C2×#1557C2semidirect
5. #9141C2semidirect
6. #9137C2semidirect
7. #9133C2semidirect
8. #9129C2semidirect
9. #8042C2semidirect
45. C2×#332C22semidirect
46. C2×#331C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect
102. #423D4semidirect
635. C22#1557semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
#366242
#207152
C2×#15572
#91412
#91372
#91332
#91292
#80422