Finite Group Databasebeta

← #6458402

Subgroups — #6458402

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35253C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#27971C214yesmaximal, complemented
4#27970C214yesmaximal, complemented
5#27969C212yesmaximal, complemented
6#27965C212yesmaximal, complemented
7#27964C212yesmaximal, complemented
8#27961C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23748C212yesmaximal, complemented
11#21326C212yesmaximal, complemented
12#20763C212yesmaximal, complemented
13#20702C211yesmaximal, characteristic
14#20700C211yesmaximal, characteristic
15#20699C211yesmaximal, characteristic
16C2×#1824C211yesmaximal, characteristic
17#20379C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#405C2212yes—
25#2857C2212yes—
26#2851C2212yes—
27#2629C2212yes—
28#2628C2212yes—
29#2442—21no—
30#2441—21no—
31#2440—21no—
32#2439—21no—
33#2438—22no—
34#2437—22no—
35#2317—22no—
36#2317—21no—
37#2317—21no—
38#2317C2211yescharacteristic
39#2317C2212yes—
40#2317C2214yes—
41#2317C2212yes—
42#2317C2211yescharacteristic
43#2316C2211yescharacteristic
44C2×#389—21no—
45C2×#389—21no—
46C2×#385—22no—
47C2×#384C2212yes—
48C2×#382C2212yes—
49#2280C2211yescharacteristic
50C2×#380C2211yescharacteristic
51#2101C2214yescomplemented
52#2101C2214yescomplemented
53#2101C2214yescomplemented
54#2101C2214yescomplemented
55#2100C2214yescomplemented
56#2100C2214yescomplemented
57#2099C2214yescomplemented
58#2099C2214yescomplemented
59#1993C2214yescomplemented
60#1992C2214yescomplemented
61#1991C2214yescomplemented
62#1990C2214yescomplemented
63#1836C2214yescomplemented
64#1836C2214yescomplemented
65#1835C2214yescomplemented
66#1834C2214yescomplemented
67#1825C2212yes—
68#1825C2214yes—
69#1825C2212yes—
70#1824C2211yescharacteristic
71#1824C2212yes—
72#1824C2212yes—
73#1824C2212yes—
74#1824C2212yes—
75#1824C2211yescharacteristic
76#1824C2212yes—
77#1824C2214yes—
78#1823C2211yescharacteristic
79#1823C2212yes—
80#1823C2211yescharacteristic
81#1823C2212yes—
82#1823C2212yes—
83#1750C2214yes—
84#1749C2212yes—
85C4×C22⋊C8C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C24×C4—21no—
96C24×C4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×#125C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×#117—28no—
107C2×#117—21no—
108C2×#117—21no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117D412yes—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4D412yes—
140C22×C42—21no—
141C22×C42C2312yes—
142#407—24no—
143C2×C22⋊C8—24no—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149#405—28no—
150C2×#97C2314yes—
151C2×C4×C8C2314yes—
152#400—22no—
153#399—22no—
154#398—22no—
155#398—22no—
156#396—22no—
157#396—22no—
158#395—22no—
159#393—22no—
160#391—22no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—24no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—22no—
180#385—24no—
181#385—24no—
182#385—24no—
183#385—24no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—22no—
188#384—22no—
189#384C2314yes—
190#384C2314yes—
191#384C2312yes—
192#382—21no—
193#382—22no—
194#382—22no—
195#382C2314yes—
196#382C2314yes—
197#382C2312yes—
198#381—21no—
199#380—22no—
200#380—21no—
201#380—21no—
202#380—22no—
203#380C2311yescharacteristic
204#380C2311yescharacteristic
205#380C2311yescharacteristic
206#380C2312yes—
207#380C2311yescharacteristic
208#380C2312yes—
209#379—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C2×#95—21no—
213C2×#95—22no—
214C2×#95—22no—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237C25C2×D411yescharacteristic
238C22×Q8—22no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—44no—
249C23×C4—82no—
250C23×C4—82no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4#4114yes—
257C23×C4#4114yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C4⋊Q8—28no—
271C4⋊Q8—44no—
272#125—28no—
273#125—44no—
274C22⋊Q8—48no—
275C22⋊Q8—84no—
276#117—24no—
277#117—24no—
278#117—24no—
279#117—28no—
280#117—28no—
281#117—24no—
282#117—24no—
283#117—24no—
284#117—24no—
285#117—24no—
286#117—24no—
287#117—44no—
288#117—44no—
289#117#4118yes—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—84no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—28no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—82no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—28no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—44no—
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2412yes—
370C2×C42C2412yes—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376#97—28no—
377C4×C8—28no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—41no—
399C24—41no—
400C24#12414yes—
401C24C22×D411yescharacteristic
402C24C22×D411yescharacteristic
403C24C22×D411yescharacteristic
404C2×Q8—44no—
405C2×Q8—82no—
406C2×Q8—82no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—816no—
456C22×C4#12414yes—
457C22×C4#12414yes—
458C22×C4#12414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C42−1+414yes—
474C22×C4C2511yescharacteristic
475M16—48no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—48no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4#12314yes—
523C4⋊C4#12314yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—81no—
535C22⋊C4—81no—
536C22⋊C4—82no—
537C22⋊C4#12318yes—
538C42—24no—
539C42—24no—
540C42—24no—
541C42—42no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42C2×#4118yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—28no—
557C23—21no—
558C23—21no—
559C23—28no—
560C23—21no—
561C23—21no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23C2×#12312yes—
568C23C2×#12412yes—
569C23C2×#12412yes—
570C23C2×#12412yes—
571C23#53612yes—
572C23#57712yes—
573C23#57812yes—
574C23C23×D411yescharacteristic
575Q8—164no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—88no—
628C2×C4—88no—
629C2×C4—88no—
630C2×C4—84no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4C2×#12314yes—
634C2×C4C2×#12314yes—
635C2×C4C2×#12312yes—
636C2×C4C2×#12412yes—
637C2×C4C2×#12412yes—
638C2×C4C2×#12412yes—
639C2×C4#53612yes—
640C2×C4#55218yes—
641C2×C4#55218yes—
642C2×C4C22×#4112yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×2−1+412yes—
646C8—42no—
647C8—42no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—416no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22#250314yes—
670C22#313714yes—
671C22C22×#12311yescharacteristic
672C22C22×#12411yescharacteristic
673C22C2×#53611yescharacteristic
674C22C2×#57711yescharacteristic
675C22C2×#57811yescharacteristic
676C22#358511yescharacteristic
677C22#358511yescharacteristic
678C4—24no—
679C4—24no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4—81no—
689C4—82no—
690C4—81no—
691C4—82no—
692C4—84no—
693C4#250314yes—
694C4C2×#55212yes—
695C4C2×#55212yes—
696C2—24no—
697C2—24no—
698C2—42no—
699C2C2×#250311yescharacteristic
700C2#6052511yescharacteristic
701C2#6052611yescharacteristic
702C2C2×#313711yescharacteristic
703C2#6061911yescharacteristic
704C2#6087311yescharacteristic
705C2#6087411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27971C2semidirect
4. #27970C2semidirect
5. #27969C2semidirect
6. #27965C2semidirect
7. #27964C2semidirect
8. #27961C2semidirect
9. #23749C2semidirect
10. #23748C2semidirect
11. #21326C2semidirect
12. #20763C2semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
58. #2099C22semidirect
59. #1993C22semidirect
60. #1992C22semidirect
61. #1991C22semidirect
62. #1990C22semidirect
63. #1836C22semidirect
64. #1836C22semidirect
65. #1835C22semidirect
66. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#352532
C2×#23172
#279712
#279702
#279692
#279652
#279642
#279612
#237492
#237482
#213262
#207632
#207022
#207002
#206992
C2×#18242
#203792