Finite Group Databasebeta

← #6461279

Subgroups — #6461279

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic, complemented
2#34269C211yesmaximal, characteristic
3#29175C214yesmaximal, complemented
4#29174C214yesmaximal, complemented
5#29167C214yesmaximal, complemented
6#29161C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23551C211yesmaximal, characteristic, complemented
10#23454C211yesmaximal, characteristic, complemented
11#21038C211yesmaximal, characteristic, complemented
12#21037C211yesmaximal, characteristic, complemented
13#20975C211yesmaximal, characteristic
14#20975C211yesmaximal, characteristic
15#20879C211yesmaximal, characteristic, complemented
16#20879C211yesmaximal, characteristic, complemented
17#20663C211yesmaximal, characteristic, complemented
18#20662C211yesmaximal, characteristic, complemented
19#20336C211yesmaximal, characteristic, complemented
20C22×#131C2211yescharacteristic
21C22×#131C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×#533C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×C4×#41C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#427C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34#2851C2214yescomplemented
35#2628C2214yescomplemented
36#2581—22no—
37#2580—22no—
38#2564—24no—
39#2367—22no—
40#2366—22no—
41#2358—24no—
42#2324C2214yes—
43#2316C2211yescharacteristic
44#2316C2211yescharacteristic
45#2311C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48C4×#121C2214yes—
49#2296C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#382C2211yescharacteristic
52#2274C2211yescharacteristic
53#2101C2214yescomplemented
54#2101C2214yescomplemented
55#2092C2212yes—
56#2092C2212yes—
57#2092C2212yes—
58#2092C2212yes—
59#2059C2212yescomplemented
60#2059C2212yescomplemented
61#2058C2212yescomplemented
62#2057C2212yescomplemented
63#2025C2214yes—
64#2023C2214yes—
65#1921C2214yescomplemented
66#1920C2214yescomplemented
67#1919C2214yescomplemented
68#1918C2214yescomplemented
69#1895C2214yes—
70#1895C2214yes—
71#1893C2214yes—
72#1893C2214yes—
73#1864C2214yes—
74#1864C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1809C2214yescomplemented
78#1809C2214yescomplemented
79#1808C2214yescomplemented
80#1807C2214yescomplemented
81#1736C2211yescharacteristic
82#1736C2211yescharacteristic
83#1736C2211yescharacteristic
84#1736C2211yescharacteristic
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1735C2211yescharacteristic
88#1735C2211yescharacteristic
89#1729C2211yescharacteristic
90#1729C2211yescharacteristic
91#1729C2212yescomplemented
92#1729C2212yescomplemented
93#1728C2211yescharacteristic
94#1728C2212yescomplemented
95C2×#336C2211yescharacteristic
96C2×#336C2212yescomplemented
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C2×#131—24no—
100C2×#131—24no—
101C2×#131—24no—
102C2×#131—24no—
103C2×#131—42no—
104C2×#131—42no—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C22×M16C2312yes—
108C22×M16C2312yes—
109C23×C8C2312yes—
110C23×C8C2312yes—
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#535—24no—
114#535Q818yescomplemented
115#533—42no—
116#533—42no—
117C2×C4⋊Q8C2312yes—
118C2×#127—21no—
119C2×#125C2312yes—
120C2×#124—21no—
121C2×C22⋊Q8—22no—
122C2×#121C2314yes—
123C2×#120—22no—
124#518—42no—
125#518—42no—
126C4×#41—42no—
127C4×#41—42no—
128C2×C4×Q8—22no—
129C2×C4×Q8D412yes—
130C2×C4×D4—24no—
131C2×C4×D4—22no—
132C2×C4×D4D414yes—
133C2×C4×D4D412yes—
134C2×C4×D4C2314yes—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C22⋊C4—22no—
145C22×C22⋊C4—22no—
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148#427—42no—
149#427—42no—
150C2×#107—24no—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#106—24no—
156C2×#106C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160#418—42no—
161#418—42no—
162C2×#102—24no—
163C2×#102—24no—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2314yes—
170#400—24no—
171#399—24no—
172#395—24no—
173#393—24no—
174#391—21no—
175#391—21no—
176#391—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—24no—
181#390—22no—
182#390C2312yes—
183#390C2312yes—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2314yes—
189#389C2314yes—
190#388—24no—
191#388—24no—
192#387—24no—
193#387—24no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—24no—
198#386—24no—
199#385—24no—
200#385—24no—
201#385—22no—
202#385—24no—
203#382—22no—
204#382C2312yes—
205#382C2312yes—
206#380—22no—
207#380—22no—
208#380C2314yes—
209#380C2314yes—
210#380C2314yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4C2314yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—24no—
216C2×#95—22no—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
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×Q8—22no—
245C22×Q8C2×D411yescharacteristic
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2×D412yes—
254C22×D4C2×D411yescharacteristic
255C22×D4C2×Q814yes—
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—24no—
261C23×C4C2×D412yes—
262C23×C4C2×D414yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265#131—416no—
266#131—416no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—24no—
276C2×M16—24no—
277C2×M16—42no—
278C2×M16—42no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8—24no—
287C22×C8—42no—
288C22×C8—42no—
289C22×C8#4114yes—
290C22×C8#4114yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C4⋊Q8—42no—
298C4⋊Q8—42no—
299#127—28no—
300#125—42no—
301#125—42no—
302#124—28no—
303C22⋊Q8—48no—
304#121—28no—
305#121C2×Q8116yes—
306#120—216no—
307C4×Q8—44no—
308C4×Q8—48no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—48no—
313C4×D4—48no—
314C4×D4—88no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—42no—
325#117—42no—
326#117C2×Q814yes—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—48no—
364C2×C22⋊C4—84no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D414yes—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—24no—
374C2×C42—42no—
375C2×C42—42no—
376C2×C42C2×D412yes—
377C2×C42C2×D414yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—24no—
382#107—24no—
383#107—42no—
384#107—42no—
385#106—24no—
386#106—24no—
387#106—42no—
388#106—42no—
389#102—24no—
390#102—24no—
391#102—24no—
392#102—24no—
393#102—44no—
394#102—44no—
395#95—24no—
396#95—24no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
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×Q8—44no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8C22⋊Q814yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—216no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4C22⋊Q818yes—
443C2×D4C22⋊Q814yes—
444C2×D4C22×Q814yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—48no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×Q814yes—
477C22×C4C22×Q812yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4C2511yescharacteristic
485M16—48no—
486M16—48no—
487M16—44no—
488M16—44no—
489C2×C8—28no—
490C2×C8—28no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—48no—
506C2×C8—48no—
507C2×C8C2×#4112yes—
508C2×C8C2×#4112yes—
509C2×C8C2×#4112yes—
510C2×C8C2×#4112yes—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—84no—
534C4⋊C4—82no—
535C4⋊C4—84no—
536C4⋊C4C22×Q814yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—216no—
540C22⋊C4—44no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—24no—
551C42—24no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—22no—
557C42—22no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—84no—
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—162no—
577C23—162no—
578C23—164no—
579C23C2×C22⋊Q814yes—
580C23C2×C22⋊Q812yes—
581C23#53212yes—
582C23#53412yes—
583C23C23×D411yescharacteristic
584Q8—28no—
585Q8—88no—
586D4—216no—
587D4—28no—
588D4—24no—
589D4—24no—
590D4—48no—
591D4—42no—
592D4—42no—
593D4—816no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—84no—
628C2×C4—88no—
629C2×C4—168no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4C2×C22⋊Q814yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4#53212yes—
636C2×C4#53412yes—
637C2×C4#55114yes—
638C2×C4#55114yes—
639C2×C4#55414yes—
640C2×C4#55414yes—
641C2×C4C23×Q811yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C8—22no—
648C8—22no—
649C8—22no—
650C8—22no—
651C8—42no—
652C8—42no—
653C22—22no—
654C22—22no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—28no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—82no—
665C22—82no—
666C22—84no—
667C22—168no—
668C22#256414yes—
669C22C22×C22⋊Q811yescharacteristic
670C22C2×#53211yescharacteristic
671C22C2×#53411yescharacteristic
672C22#358412yes—
673C22#358512yes—
674C4—24no—
675C4—44no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#256414yes—
686C4C2×#55111yescharacteristic
687C4C2×#55111yescharacteristic
688C4C2×#55411yescharacteristic
689C4C2×#55411yescharacteristic
690C2—24no—
691C2—44no—
692C2—84no—
693C2C2×#256411yescharacteristic
694C2#6049612yes—
695C2#6071311yescharacteristic
696C2#6071911yescharacteristic
697C2#6076712yes—
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
1. #35334C2semidirect
3. #29175C2semidirect
4. #29174C2semidirect
5. #29167C2semidirect
6. #29161C2semidirect
7. #23746C2semidirect
8. C2×#2092C2semidirect
9. #23551C2semidirect
10. #23454C2semidirect
11. #21038C2semidirect
12. #21037C2semidirect
15. #20879C2semidirect
16. #20879C2semidirect
17. #20663C2semidirect
18. #20662C2semidirect
19. #20336C2semidirect
34. #2851C22semidirect
35. #2628C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
59. #2059C22semidirect
60. #2059C22semidirect
61. #2058C22semidirect
62. #2057C22semidirect
65. #1921C22semidirect
66. #1920C22semidirect
67. #1919C22semidirect
68. #1918C22semidirect
77. #1809C22semidirect
78. #1809C22semidirect
79. #1808C22semidirect
80. #1807C22semidirect
91. #1729C22semidirect
92. #1729C22semidirect
94. #1728C22semidirect
96. C2×#336C22semidirect
114. #535Q8semidirect

Maximal subgroups

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

subgroupindex
#353342
#342692
#291752
#291742
#291672
#291612
#237462
C2×#20922
#235512
#234542
#210382
#210372
#209752
#208792
#206632
#206622
#203362