Finite Group Databasebeta

← #6458880

Subgroups — #6458880

Order 512.

1900 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#35254C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#27874C214yesmaximal, complemented
4#27869C214yesmaximal, complemented
5#27863C212yesmaximal, complemented
6#27862C212yesmaximal, complemented
7#27857C212yesmaximal, complemented
8#27856C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23731C211yesmaximal, characteristic, complemented
11#23658C211yesmaximal, characteristic, complemented
12C2×#2071C211yesmaximal, characteristic, complemented
13#21063C212yesmaximal, complemented
14#21036C212yesmaximal, complemented
15#20975C212yesmaximal
16#20664C212yesmaximal, complemented
17#20336C211yesmaximal, characteristic, complemented
18C22×#131C2212yes—
19C2×#535C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#418C2212yes—
24C22×#102C2212yes—
25#2854C2212yes—
26#2854C2212yes—
27#2624C2212yes—
28#2624C2212yes—
29#2448—21no—
30#2448—21no—
31#2445—21no—
32#2445—21no—
33#2418—22no—
34#2418—22no—
35#2355—21no—
36#2355—21no—
37#2354—21no—
38#2354—21no—
39#2352—22no—
40#2352—22no—
41#2316C2211yescharacteristic
42#2314C2214yescomplemented
43#2311C2212yes—
44#2311C2211yescharacteristic
45#2309C2211yescharacteristic
46#2296C2211yescharacteristic
47C2×#390C2211yescharacteristic
48#2285C2211yescharacteristic
49#2285C2212yescomplemented
50#2285C2212yescomplemented
51C2×#384C2211yescharacteristic
52#2274C2212yes—
53#2101C2212yescomplemented
54#2101C2212yescomplemented
55#2100C2211yescharacteristic, complemented
56#2100C2211yescharacteristic, complemented
57#2099C2211yescharacteristic, complemented
58#2099C2211yescharacteristic, complemented
59#2094C2212yes—
60#2093C2212yes—
61#2092C2214yes—
62#2084C2212yescomplemented
63#2084C2212yescomplemented
64#2083C2212yescomplemented
65#2083C2212yescomplemented
66#2071C2212yes—
67#2071C2212yes—
68#2071C2214yes—
69#1926C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2214yescomplemented
74#1920C2214yescomplemented
75#1919C2214yescomplemented
76#1918C2214yescomplemented
77#1895C2214yes—
78#1895C2214yes—
79#1893C2214yes—
80#1893C2214yes—
81#1809C2214yescomplemented
82#1809C2214yescomplemented
83#1808C2214yescomplemented
84#1807C2214yescomplemented
85#1736C2212yes—
86#1735C2212yes—
87#1735C2212yes—
88C2×#340C2212yes—
89#1729C2212yes—
90#1729C2214yescomplemented
91#1728C2211yescharacteristic
92#1728C2212yescomplemented
93C2×#336C2211yescharacteristic
94C2×#336C2212yescomplemented
95C22×#41C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C2×#131—28no—
98C2×#131—28no—
99C2×#131—44no—
100C22×M16C2314yes—
101C22×M16C2312yes—
102C23×C8C2312yes—
103C23×C8C2314yes—
104#535—28no—
105C2×#127—21no—
106C2×#124—21no—
107C2×#123—22no—
108C2×#123—22no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#120—22no—
112C4×#41—28no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4D412yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×#117—22no—
120C2×#117—22no—
121C2×#117C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#428—28no—
135#428—44no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140#418—28no—
141C2×#102—44no—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146#399—22no—
147#399—22no—
148#397—22no—
149#397—22no—
150#394—22no—
151#394—22no—
152#394—22no—
153#394—22no—
154#391—22no—
155#391—21no—
156#390—22no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—21no—
167#389—21no—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#388—24no—
175#388—24no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—24no—
181#387—24no—
182#386—24no—
183#386—22no—
184#386—22no—
185#386—22no—
186#385—21no—
187#385—21no—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—22no—
195#382—22no—
196#382C2312yes—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2314yes—
203#380C2314yes—
204#380C2312yes—
205#380C2312yes—
206#379—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C22⋊C4—24no—
218#340C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C25—21no—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41#4118yes—
242C22×Q8C2×D411yescharacteristic
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—24no—
247C22×D4—24no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261#131—432no—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—28no—
265C2×M16—28no—
266C2×M16—44no—
267C2×M16#4118yes—
268C22×C8—28no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—28no—
273C22×C8—44no—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2414yes—
277#127—44no—
278#124—44no—
279#123—84no—
280#123—84no—
281#121—28no—
282#121—28no—
283#120—48no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—24no—
290#117—28no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—22no—
296#117—22no—
297#117—84no—
298#117—84no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—82no—
350C2×C22⋊C4—82no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—28no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—28no—
369#107—44no—
370#106—28no—
371#106—44no—
372#102—28no—
373#102—28no—
374#102—44no—
375#102—44no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390C24—22no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—81no—
395C24—81no—
396C24—81no—
397C24—81no—
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400#41—216no—
401#41—216no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8#12314yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4#12318yes—
425C2×D4#12314yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4118yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469M16—216no—
470M16—416no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—416no—
480C2×C8—48no—
481C2×C8C2×#4118yes—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—24no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—22no—
542C42—42no—
543C42—44no—
544C42—42no—
545C42—84no—
546C42—84no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—162no—
568C23—162no—
569C23—162no—
570C23—162no—
571C23C2×#12314yes—
572C23C2×#12312yes—
573C23#53212yes—
574C23#53712yes—
575C23C23×D411yescharacteristic
576Q8—28no—
577D4—216no—
578D4—28no—
579D4—44no—
580D4—44no—
581D4—84no—
582D4—164no—
583D4—164no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—162no—
633C2×C4—164no—
634C2×C4C2×#12314yes—
635C2×C4C2×#12312yes—
636C2×C4#53212yes—
637C2×C4#53712yes—
638C2×C4#55118yes—
639C2×C4#55218yes—
640C2×C4#57612yes—
641C2×C4#57712yes—
642C2×C4C22×#4112yes—
643C2×C4C22×#4112yes—
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C8—28no—
647C8—42no—
648C8—42no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—168no—
670C22#249814yes—
671C22C22×#12311yescharacteristic
672C22C2×#53211yescharacteristic
673C22C2×#53711yescharacteristic
674C22C2×#57611yescharacteristic
675C22C2×#57711yescharacteristic
676C22#358511yescharacteristic
677C22#358611yescharacteristic
678C4—22no—
679C4—22no—
680C4—24no—
681C4—42no—
682C4—41no—
683C4—42no—
684C4—41no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—84no—
692C4—81no—
693C4#249814yes—
694C4C2×#55112yes—
695C4C2×#55212yes—
696C2—24no—
697C2—44no—
698C2—162no—
699C2C2×#249811yescharacteristic
700C2#6052111yescharacteristic
701C2#6052511yescharacteristic
702C2#6072211yescharacteristic
703C2#6072711yescharacteristic
704C2#6092211yescharacteristic
705C2#6092311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #27874C2semidirect
4. #27869C2semidirect
5. #27863C2semidirect
6. #27862C2semidirect
7. #27857C2semidirect
8. #27856C2semidirect
9. #23744C2semidirect
10. #23731C2semidirect
11. #23658C2semidirect
12. C2×#2071C2semidirect
13. #21063C2semidirect
14. #21036C2semidirect
16. #20664C2semidirect
17. #20336C2semidirect
42. #2314C22semidirect
49. #2285C22semidirect
50. #2285C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
58. #2099C22semidirect
62. #2084C22semidirect
63. #2084C22semidirect
64. #2083C22semidirect
65. #2083C22semidirect
73. #1921C22semidirect
74. #1920C22semidirect
75. #1919C22semidirect
76. #1918C22semidirect
81. #1809C22semidirect
82. #1809C22semidirect
83. #1808C22semidirect
84. #1807C22semidirect
90. #1729C22semidirect
92. #1728C22semidirect
94. C2×#336C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#341882
#278742
#278692
#278632
#278622
#278572
#278562
#237442
#237312
#236582
C2×#20712
#210632
#210362
#209752
#206642
#203362