Finite Group Databasebeta

← #6458299

Subgroups — #6458299

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35253C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#27959C214yesmaximal, complemented
4#27957C214yesmaximal, complemented
5#27956C212yesmaximal, complemented
6#27953C212yesmaximal, complemented
7#27952C212yesmaximal, complemented
8#27948C212yesmaximal, complemented
9#23662C212yesmaximal, complemented
10#23661C212yesmaximal, complemented
11#21326C212yesmaximal, complemented
12#20846C211yesmaximal, characteristic
13#20844C211yesmaximal, characteristic
14#20843C211yesmaximal, characteristic
15C2×#1853C211yesmaximal, characteristic
16#20802C212yesmaximal, complemented
17#20379C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#428C2212yes—
23C2×#428C2212yes—
24C2×#427C2212yes—
25C22×#106C2212yes—
26C2×#407C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C2×#405C2212yes—
29#2857C2212yes—
30#2851C2212yes—
31#2629C2212yes—
32#2628C2212yes—
33#2442—21no—
34#2441—21no—
35#2440—21no—
36#2439—21no—
37#2438—22no—
38#2437—22no—
39#2317—22no—
40#2317—21no—
41#2317—21no—
42#2317C2211yescharacteristic
43#2317C2214yes—
44#2317C2212yes—
45#2317C2212yes—
46#2317C2211yescharacteristic
47#2316C2211yescharacteristic
48C2×#389—21no—
49C2×#389—21no—
50C2×#385—22no—
51C2×#384C2212yes—
52C2×#382C2212yes—
53#2280C2211yescharacteristic
54C2×#380C2211yescharacteristic
55#2085C2214yescomplemented
56#2085C2214yescomplemented
57#2085C2214yescomplemented
58#2085C2214yescomplemented
59#2084C2214yescomplemented
60#2084C2214yescomplemented
61#2084C2214yescomplemented
62#2083C2214yescomplemented
63#1993C2214yescomplemented
64#1992C2214yescomplemented
65#1991C2214yescomplemented
66#1990C2214yescomplemented
67#1854C2212yes—
68#1854C2214yes—
69#1854C2212yes—
70#1853C2211yescharacteristic
71#1853C2212yes—
72#1853C2212yes—
73#1853C2212yes—
74#1853C2211yescharacteristic
75#1853C2212yes—
76#1853C2212yes—
77#1853C2214yes—
78#1852C2212yes—
79#1852C2212yes—
80#1852C2211yescharacteristic
81#1852C2211yescharacteristic
82#1852C2212yes—
83#1847C2214yescomplemented
84#1847C2214yescomplemented
85#1846C2214yescomplemented
86#1844C2214yescomplemented
87#1750C2214yes—
88#1749C2212yes—
89C4×C22⋊C8C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
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—44no—
110C2×#117D412yes—
111C2×#117D411yescharacteristic
112C2×#117D411yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—44no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4D412yes—
138C22×C42—21no—
139C22×C42C2312yes—
140#428—44no—
141#428—44no—
142#427—44no—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#106—44no—
148C2×#106C2312yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×#106C2312yes—
154#407Q818yescomplemented
155C2×C22⋊C8Q818yescomplemented
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2312yes—
161#405—28no—
162C2×#97C2314yes—
163C2×C4×C8C2314yes—
164#400—22no—
165#399—22no—
166#398—22no—
167#398—22no—
168#396—22no—
169#396—22no—
170#395—22no—
171#393—22no—
172#391—22no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389—24no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#387—22no—
189#387—22no—
190#385—22no—
191#385—22no—
192#385—24no—
193#385—24no—
194#385—24no—
195#385—24no—
196#385—22no—
197#385—22no—
198#384—21no—
199#384—22no—
200#384—22no—
201#384C2314yes—
202#384C2314yes—
203#384C2312yes—
204#382—21no—
205#382—22no—
206#382—22no—
207#382C2314yes—
208#382C2314yes—
209#382C2312yes—
210#381—21no—
211#380—22no—
212#380—21no—
213#380—21no—
214#380—22no—
215#380C2311yescharacteristic
216#380C2312yes—
217#380C2311yescharacteristic
218#380C2311yescharacteristic
219#380C2312yes—
220#380C2311yescharacteristic
221#379—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C2×#95—21no—
225C2×#95—22no—
226C2×#95—22no—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2314yes—
239C25C2×D411yescharacteristic
240C22×Q8—22no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—44no—
251C23×C4—82no—
252C23×C4—82no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×Q814yes—
259C23×C4C2×Q814yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16C2×Q818yes—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8C2×Q818yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C4⋊Q8—28no—
273C4⋊Q8—44no—
274#125—28no—
275#125—44no—
276C22⋊Q8—48no—
277C22⋊Q8—84no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—28no—
282#117—28no—
283#117—24no—
284#117—24no—
285#117—24no—
286#117—24no—
287#117—42no—
288#117—42no—
289#117—44no—
290#117—44no—
291#117#4118yes—
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—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—84no—
326C2×C4⋊C4—84no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—28no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—82no—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—28no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—44no—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369#107—42no—
370#107—42no—
371#107—44no—
372#107—44no—
373#106—44no—
374#106—44no—
375#106—42no—
376#106—42no—
377#106—44no—
378#106—44no—
379C22⋊C8C2×Q814yes—
380C22⋊C8C2×Q814yes—
381C22⋊C8C2×Q818yes—
382C22⋊C8C2×Q818yes—
383C22⋊C8C2×Q818yes—
384#97—28no—
385C4×C8—28no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—41no—
407C24—41no—
408C24C4⋊Q814yes—
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411C24C22×D411yescharacteristic
412C2×Q8—44no—
413C2×Q8—82no—
414C2×Q8—82no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—816no—
464C22×C4C4⋊Q814yes—
465C22×C4C4⋊Q814yes—
466C22×C4C4⋊Q814yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q812yes—
477C22×C4C22×Q812yes—
478C22×C4C22×Q812yes—
479C22×C4C22×Q812yes—
480C22×C4C2×#4114yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4C2511yescharacteristic
483M16—216no—
484C2×C8—216no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8C22×Q814yes—
492C2×C8C22×Q814yes—
493C2×C8C22×Q818yes—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4#12314yes—
531C4⋊C4#12314yes—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C22⋊C4—82no—
545C22⋊C4#12318yes—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—44no—
550C42—42no—
551C42—44no—
552C42—42no—
553C42C2×#4118yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—28no—
565C23—21no—
566C23—21no—
567C23—28no—
568C23—21no—
569C23—21no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23C2×#12312yes—
576C23C2×C4⋊Q812yes—
577C23C2×C4⋊Q812yes—
578C23C2×C4⋊Q812yes—
579C23#53512yes—
580C23#57712yes—
581C23#57712yes—
582C23C23×D411yescharacteristic
583Q8—164no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4—162no—
640C2×C4—162no—
641C2×C4C2×#12314yes—
642C2×C4C2×#12314yes—
643C2×C4C2×#12312yes—
644C2×C4C2×C4⋊Q812yes—
645C2×C4C2×C4⋊Q812yes—
646C2×C4C2×C4⋊Q812yes—
647C2×C4#53512yes—
648C2×C4#55418yes—
649C2×C4#55418yes—
650C2×C4C23×Q811yescharacteristic
651C2×C4C23×Q811yescharacteristic
652C2×C4C22×#4112yes—
653C2×C4C2×H(5,𝔽2)12yes—
654C8—24no—
655C8—24no—
656C8—28no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—416no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22#250214yes—
678C22#316314yes—
679C22C22×#12311yescharacteristic
680C22C22×C4⋊Q811yescharacteristic
681C22C2×#53511yescharacteristic
682C22C2×#57711yescharacteristic
683C22C2×#57711yescharacteristic
684C22#358511yescharacteristic
685C22#358611yescharacteristic
686C4—24no—
687C4—24no—
688C4—22no—
689C4—22no—
690C4—42no—
691C4—42no—
692C4—81no—
693C4—82no—
694C4—81no—
695C4—81no—
696C4—82no—
697C4—84no—
698C4—81no—
699C4—82no—
700C4—82no—
701C4#250214yes—
702C4C2×#55412yes—
703C4C2×#55412yes—
704C2—24no—
705C2—24no—
706C2—42no—
707C2C2×#250211yescharacteristic
708C2#6052211yescharacteristic
709C2#6052511yescharacteristic
710C2C2×#316311yescharacteristic
711C2#6066411yescharacteristic
712C2#6081211yescharacteristic
713C2#6081311yescharacteristic
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. #27959C2semidirect
4. #27957C2semidirect
5. #27956C2semidirect
6. #27953C2semidirect
7. #27952C2semidirect
8. #27948C2semidirect
9. #23662C2semidirect
10. #23661C2semidirect
11. #21326C2semidirect
16. #20802C2semidirect
55. #2085C22semidirect
56. #2085C22semidirect
57. #2085C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2084C22semidirect
61. #2084C22semidirect
62. #2083C22semidirect
63. #1993C22semidirect
64. #1992C22semidirect
65. #1991C22semidirect
66. #1990C22semidirect
83. #1847C22semidirect
84. #1847C22semidirect
85. #1846C22semidirect
86. #1844C22semidirect
154. #407Q8semidirect
155. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#352532
C2×#23172
#279592
#279572
#279562
#279532
#279522
#279482
#236622
#236612
#213262
#208462
#208442
#208432
C2×#18532
#208022
#203792