Finite Group Databasebeta

← #6458258

Subgroups — #6458258

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34269C211yesmaximal, characteristic
2#34184C211yesmaximal, characteristic, complemented
3#27688C214yesmaximal, complemented
4#27687C214yesmaximal, complemented
5#27684C214yesmaximal, complemented
6#27679C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23744C211yesmaximal, characteristic, complemented
9#23731C211yesmaximal, characteristic, complemented
10C2×#2092C211yesmaximal, characteristic, complemented
11#21174C211yesmaximal, characteristic, complemented
12#21173C211yesmaximal, characteristic, complemented
13#20914C211yesmaximal, characteristic
14#20914C211yesmaximal, characteristic
15#20848C211yesmaximal, characteristic, complemented
16#20847C211yesmaximal, characteristic, complemented
17#20650C211yesmaximal, characteristic, complemented
18#20650C211yesmaximal, characteristic, complemented
19#20630C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×C4×#41C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#424C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#408C2211yescharacteristic
37#2367—22no—
38#2366—22no—
39#2358—24no—
40#2351—22no—
41#2349—24no—
42#2349—22no—
43#2324C2214yes—
44#2317C2211yescharacteristic
45#2317C2211yescharacteristic
46#2317C2212yescomplemented
47#2317C2212yescomplemented
48#2311C2211yescharacteristic
49C4×#121C2214yes—
50#2296C2211yescharacteristic
51C2×#390C2211yescharacteristic
52#2285C2211yescharacteristic
53#2285C2212yescomplemented
54C2×#382C2211yescharacteristic
55C2×#382C2212yescomplemented
56#2274C2211yescharacteristic
57#2101C2212yescomplemented
58#2101C2214yescomplemented
59#2101C2212yescomplemented
60#2101C2214yescomplemented
61#2100C2212yescomplemented
62#2099C2212yescomplemented
63#2094C2212yes—
64#2093C2212yes—
65#2092C2212yes—
66#2092C2214yes—
67#2092C2212yes—
68#2092C2214yes—
69#1949C2214yescomplemented
70#1949C2214yescomplemented
71#1948C2214yescomplemented
72#1947C2214yescomplemented
73#1875C2214yes—
74#1875C2214yes—
75#1874C2214yes—
76#1874C2214yes—
77#1854C2214yescomplemented
78#1854C2214yescomplemented
79#1853C2214yescomplemented
80#1852C2214yescomplemented
81#1804C2214yes—
82#1804C2214yes—
83#1802C2214yes—
84#1802C2214yes—
85#1798C2214yescomplemented
86#1797C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#535—24no—
104#535Q818yes—
105C2×#127—21no—
106C2×#124—21no—
107C2×#121C2314yes—
108C2×#120—22no—
109#520—42no—
110#520—42no—
111#520—42no—
112#520—42no—
113#518—24no—
114#518—24no—
115C4×#41—24no—
116C4×#41—24no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8D412yes—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—24no—
125C2×C4×D4—22no—
126C2×C4×D4D414yes—
127C2×C4×D4D412yes—
128C2×C4×D4C2314yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2312yes—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C22⋊C4—22no—
143C22×C22⋊C4—22no—
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146#428—42no—
147#428—42no—
148C2×#107—42no—
149C2×#107C2314yes—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#106—42no—
153C2×#106C2314yes—
154C2×#106C2312yes—
155C2×#106C2312yes—
156#424—24no—
157#424—24no—
158C2×C4⋊C8C2314yes—
159C2×C4⋊C8C2314yes—
160#418—42no—
161#418—42no—
162C2×#102—42no—
163C2×#102—42no—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170#408—24no—
171#408—24no—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2314yes—
174#391—21no—
175#390—21no—
176#390—24no—
177#390—22no—
178#390C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389—24no—
183#389—24no—
184#389C2312yes—
185#389C2312yes—
186#388—24no—
187#388—24no—
188#386—22no—
189#386—24no—
190#386—24no—
191#385—22no—
192#385—24no—
193#385—22no—
194#385—22no—
195#385—24no—
196#385—24no—
197#384C2312yes—
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—22no—
202#382—24no—
203#382C2312yes—
204#382C2312yes—
205#382C2312yes—
206#382C2312yes—
207#382C2311yescharacteristic
208#382C2311yescharacteristic
209#380—22no—
210#380—24no—
211#380C2314yes—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—24no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4C2314yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—24no—
219C2×#95—22no—
220C2×#95—24no—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41C2×Q814yes—
243C2×#41#4114yes—
244C22×Q8C2×D411yescharacteristic
245C22×D4—24no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C22×D4C2×Q814yes—
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—24no—
262C23×C4C2×D412yes—
263C23×C4C2×D414yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282#127—28no—
283#124—28no—
284#121—28no—
285#121C2×Q8116yes—
286#120—216no—
287C4×Q8—28no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×D4—28no—
291C4×D4—216no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—48no—
295C4×D4—48no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—88no—
300#117—22no—
301#117—24no—
302#117—24no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—42no—
310#117—42no—
311#117—42no—
312#117—42no—
313#117C2×Q814yes—
314#117#4114yes—
315#117#4114yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
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—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4—84no—
361C2×C22⋊C4C2×D414yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—24no—
374C2×C42C2×D414yes—
375C2×C42C2×D412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2412yes—
379#107—44no—
380#107—42no—
381#107—42no—
382#106—44no—
383#106—42no—
384#106—42no—
385C4⋊C8—28no—
386C4⋊C8—28no—
387#102—44no—
388#102—44no—
389#102—42no—
390#102—42no—
391#102—42no—
392#102—42no—
393C22⋊C8—28no—
394C22⋊C8—28no—
395#95—24no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405C24—22no—
406C24—24no—
407C24—42no—
408C24—42no—
409C24—82no—
410C24—82no—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41—28no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8C22⋊Q814yes—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—216no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4C22⋊Q818yes—
440C2×D4C22⋊Q814yes—
441C2×D4C22×Q814yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—48no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—84no—
472C22×C4—82no—
473C22×C4—84no—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×Q814yes—
480C22×C4C22×Q812yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4C2511yescharacteristic
488M16—44no—
489M16—44no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—44no—
503C2×C8—44no—
504C4⋊C4—28no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4#12318yes—
534C4⋊C4C22×Q814yes—
535C22⋊C4—24no—
536C22⋊C4—216no—
537C22⋊C4—28no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4#12318yes—
548C42—24no—
549C42—28no—
550C42—24no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—44no—
556C42—44no—
557C42—42no—
558C42—42no—
559C42—42no—
560C42—84no—
561C42C2×#4114yes—
562C42C2×#4114yes—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—48no—
575C23—88no—
576C23—164no—
577C23—162no—
578C23—162no—
579C23C2×C22⋊Q814yes—
580C23C2×C22⋊Q812yes—
581C23C2×#12312yes—
582C23#53512yes—
583C23C23×D411yescharacteristic
584Q8—28no—
585D4—216no—
586D4—24no—
587D4—24no—
588D4—28no—
589D4—42no—
590D4—42no—
591D4—48no—
592D4—816no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4—168no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4C2×C22⋊Q814yes—
636C2×C4C2×C22⋊Q812yes—
637C2×C4C2×#12318yes—
638C2×C4C2×#12312yes—
639C2×C4#53512yes—
640C2×C4#55314yes—
641C2×C4#55314yes—
642C2×C4#55414yes—
643C2×C4#55414yes—
644C2×C4C23×Q811yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C8—42no—
651C8—42no—
652C8—42no—
653C8—42no—
654C22—24no—
655C22—24no—
656C22—22no—
657C22—22no—
658C22—28no—
659C22—24no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—48no—
664C22—44no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—168no—
669C22#248714yes—
670C22C22×C22⋊Q811yescharacteristic
671C22C22×#12311yescharacteristic
672C22C2×#53511yescharacteristic
673C22#358412yes—
674C22#358512yes—
675C4—28no—
676C4—24no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4—81no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4#248714yes—
689C4C2×#55311yescharacteristic
690C4C2×#55311yescharacteristic
691C4C2×#55411yescharacteristic
692C4C2×#55411yescharacteristic
693C2—24no—
694C2—44no—
695C2—84no—
696C2C2×#248711yescharacteristic
697C2#6049612yes—
698C2#6052312yes—
699C2#6081911yescharacteristic
700C2#6082511yescharacteristic
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. #34184C2semidirect
3. #27688C2semidirect
4. #27687C2semidirect
5. #27684C2semidirect
6. #27679C2semidirect
7. #23746C2semidirect
8. #23744C2semidirect
9. #23731C2semidirect
10. C2×#2092C2semidirect
11. #21174C2semidirect
12. #21173C2semidirect
15. #20848C2semidirect
16. #20847C2semidirect
17. #20650C2semidirect
18. #20650C2semidirect
19. #20630C2semidirect
46. #2317C22semidirect
47. #2317C22semidirect
53. #2285C22semidirect
55. C2×#382C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2101C22semidirect
60. #2101C22semidirect
61. #2100C22semidirect
62. #2099C22semidirect
69. #1949C22semidirect
70. #1949C22semidirect
71. #1948C22semidirect
72. #1947C22semidirect
77. #1854C22semidirect
78. #1854C22semidirect
79. #1853C22semidirect
80. #1852C22semidirect
85. #1798C22semidirect
86. #1797C22semidirect

Maximal subgroups

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

subgroupindex
#342692
#341842
#276882
#276872
#276842
#276792
#237462
#237442
#237312
C2×#20922
#211742
#211732
#209142
#208482
#208472
#206502
#206302