Finite Group Databasebeta

← #392817

Subgroups — #392817

Order 512.

2028 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#97C211yesmaximal, characteristic, complemented
2C2×#1936C212yesmaximal, complemented
3C2×#1893C211yesmaximal, characteristic
4C2×#1893C211yesmaximal, characteristic
5#14698C211yesmaximal, characteristic
6#14697C211yesmaximal, characteristic, complemented
7#10927C212yesmaximal, complemented
8#10881C212yesmaximal, complemented
9#10821C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#97—28no—
20C22×#97—24no—
21C22×#97C2212yes—
22C22×#97C2212yes—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#386—21no—
26C2×#386—21no—
27C2×#385—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#1936—28no—
32#1893—24no—
33#1893C418yescomplemented
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1437—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C23×D4—21no—
47C23×D4C2311yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8C2×C414yes—
60C23×C8D414yes—
61C23×C8C2312yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121C2×C414yes—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—24no—
85C22×C42—24no—
86C22×C42D412yes—
87C22×C42C2312yes—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102C2×C418yescomplemented
95C2×#97—232no—
96C2×#97—24no—
97C2×#97—24no—
98C2×#97—24no—
99C2×#97—24no—
100C2×#97—24no—
101C2×#97D418yescomplemented
102#390—24no—
103#390—24no—
104#389—24no—
105#389—24no—
106#386—44no—
107#386—44no—
108#385—44no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380C2×C414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123#340—22no—
124#340—22no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340C2×C418yescomplemented
130C25—21no—
131C25C2×D411yescharacteristic
132C22×D4—24no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C22×C412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C22×C8—216no—
179C22×C8—216no—
180C22×C8—216no—
181C22×C8—28no—
182C22×C8—28no—
183C22×C8—28no—
184C22×C8—28no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—216no—
196C22×C8—28no—
197C22×C8—28no—
198C22×C8C22×C414yes—
199C22×C8C2×D414yes—
200C22×C8C2×D414yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22×C8#4114yes—
204C22×C8#4114yes—
205#121—28no—
206#121—28no—
207#121—44no—
208#121—44no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C414yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C42—28no—
257C2×C42—216no—
258C2×C42—28no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42C2×D414yes—
267C2×C42#4114yes—
268#102—216no—
269#102—44no—
270#102—44no—
271#97—216no—
272#97—216no—
273#95—24no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—44no—
280#95—42no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—42no—
287C24—41no—
288C24—41no—
289C24—42no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—42no—
295C24—42no—
296C24C4×D412yes—
297C24#12112yes—
298C24#12412yes—
299C24AGL(1,ℤ/8)12yes—
300C24AGL(1,ℤ/8)12yes—
301C24AGL(1,ℤ/8)12yes—
302C24#13712yes—
303C24C22×D411yescharacteristic
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—24no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—84no—
326C2×D4—84no—
327C2×D4—84no—
328C2×D4—84no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—28no—
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—24no—
360C22×C4—22no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4#11714yes—
399C22×C4C4×D414yes—
400C22×C4C4×D412yes—
401C22×C4#12114yes—
402C22×C4#12114yes—
403C22×C4#12112yes—
404C22×C4#12314yes—
405C22×C4#12412yes—
406C22×C4#12614yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4111yescharacteristic
417C2×C8—232no—
418C2×C8—232no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8C4×D418yes—
434C2×C8#12014yes—
435C2×C8#12314yes—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—88no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—216no—
462C42—24no—
463C42—24no—
464C42—24no—
465C42—24no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—81no—
488C23—82no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23#38814yes—
506C23#44014yes—
507C23#44214yes—
508C23#46814yes—
509C23#46914yes—
510C23#48914yes—
511C23#49014yes—
512C23C2×C4×D411yescharacteristic
513C23C2×#12112yes—
514C23C2×#12111yescharacteristic
515C23C2×#12411yescharacteristic
516C23#53212yes—
517C23C2×AGL(1,ℤ/8)11yescharacteristic
518C23C2×AGL(1,ℤ/8)11yescharacteristic
519C23C2×AGL(1,ℤ/8)11yescharacteristic
520C23C2×#13711yescharacteristic
521C23#57512yes—
522C23#57512yes—
523D4—44no—
524D4—42no—
525D4—42no—
526D4—816no—
527D4—81no—
528D4—82no—
529D4—81no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#38814yes—
588C2×C4C2×#11711yescharacteristic
589C2×C4C2×C4×D412yes—
590C2×C4C2×C4×D411yescharacteristic
591C2×C4C4×#4112yes—
592C2×C4C2×#12011yescharacteristic
593C2×C4C2×#12311yescharacteristic
594C2×C4C2×#12311yescharacteristic
595C2×C4#52512yes—
596C2×C4#52512yes—
597C2×C4#52512yes—
598C2×C4#52512yes—
599C2×C4#52511yescharacteristic
600C2×C4#52511yescharacteristic
601C2×C4C2×#12611yescharacteristic
602C2×C4#52912yes—
603C2×C4#52912yes—
604C2×C4#52912yes—
605C2×C4#52912yes—
606C8—24no—
607C8—24no—
608C8—28no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—162no—
625C22—162no—
626C22—161no—
627C22—161no—
628C22—161no—
629C22—161no—
630C22#193618yescomplemented
631C22C2×#38811yescharacteristic
632C22C4×#12112yes—
633C22#284711yescharacteristic
634C22#285012yes—
635C22#285411yescharacteristic
636C22C2×#44011yescharacteristic
637C22C2×#44211yescharacteristic
638C22#294312yes—
639C22C2×#46811yescharacteristic
640C22C2×#46911yescharacteristic
641C22#305411yescharacteristic
642C22#305411yescharacteristic
643C22#305512yes—
644C22#305512yes—
645C22#305512yes—
646C22#305612yes—
647C22C2×#48911yescharacteristic
648C22C2×#49011yescharacteristic
649C22#313512yes—
650C22#318012yes—
651C22#318012yes—
652C22#318012yes—
653C22#318112yes—
654C4—24no—
655C4—44no—
656C4—44no—
657C4—84no—
658C4—82no—
659C4—162no—
660C4—161no—
661C4—161no—
662C4#229012yes—
663C4C4×#12011yescharacteristic
664C4C4×#12311yescharacteristic
665C4#284911yescharacteristic
666C4#285212yes—
667C4#285311yescharacteristic
668C2—28no—
669C2—162no—
670C2#1469611yescharacteristic
671C2C2×#193611yescharacteristic
672C2#2108611yescharacteristic
673C2#2152612yes—
674C2#2152812yes—
675C2#3293511yescharacteristic
676C2#3293611yescharacteristic
677C2#3305112yes—
678C2#3305212yes—
679C2#3313411yescharacteristic
680C2#3313511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#97C2semidirect
2. C2×#1936C2semidirect
6. #14697C2semidirect
7. #10927C2semidirect
8. #10881C2semidirect
9. #10821C2semidirect
10. #10721C2semidirect
11. #10721C2semidirect
33. #1893C4semidirect
94. C2×#102C2×C4semidirect
101. C2×#97D4semidirect
129. #340C2×C4semidirect
630. C22#1936semidirect

Maximal subgroups

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

subgroupindex
C23×#972
C2×#19362
C2×#18932
#146982
#146972
#109272
#108812
#108212
#107212