Finite Group Databasebeta

← #6433759

Subgroups — #6433759

Order 512.

1876 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#35257C211yesmaximal, characteristic
3#30202C212yesmaximal, complemented
4#30199C212yesmaximal, complemented
5#30196C214yesmaximal, complemented
6#30193C214yesmaximal, complemented
7#30190C212yesmaximal, complemented
8#30187C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23744C211yesmaximal, characteristic, complemented
11#23633C212yesmaximal, complemented
12#23507C211yesmaximal, characteristic, complemented
13C2×#2043C211yesmaximal, characteristic, complemented
14#23457C212yesmaximal, complemented
15#20974C212yesmaximal
16#20974C212yesmaximal
17#20336C211yesmaximal, characteristic
18C22×#131C2212yes—
19C2×#544C2211yescharacteristic
20C2×#536C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#419C2212yes—
24C2×#419C2212yes—
25C22×#103C2212yes—
26C22×#103C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2851C2212yes—
31#2629C2212yes—
32#2628C2212yes—
33#2621C2214yes—
34#2547—21no—
35#2547—21no—
36#2546—21no—
37#2546—21no—
38#2486—22no—
39#2486—22no—
40#2454—21no—
41#2453—21no—
42#2452—21no—
43#2451—21no—
44#2450—22no—
45#2449—22no—
46#2317C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49#2310C2212yes—
50#2296C2211yescharacteristic
51C2×#389C2211yescharacteristic
52#2285C2211yescharacteristic
53#2101C2212yes—
54#2101C2212yes—
55#2101C2214yes—
56#2100C2211yescharacteristic
57#2100C2211yescharacteristic
58#2100C2212yes—
59#2099C2211yescharacteristic
60#2099C2211yescharacteristic
61#2099C2212yes—
62#2074C2214yes—
63#2074C2214yes—
64#2072C2214yes—
65#2072C2214yes—
66#2043C2212yes—
67#2043C2214yes—
68#2043C2212yes—
69#2043C2212yes—
70#2043C2212yes—
71#2043C2212yes—
72#2043C2212yes—
73#2026C2214yes—
74#2026C2214yes—
75#2024C2214yes—
76#2024C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1892C2214yes—
82#1892C2214yes—
83#1892C2214yes—
84#1892C2214yes—
85#1736C2212yes—
86#1736C2212yes—
87#1735C2212yes—
88#1735C2212yes—
89#1729C2212yes—
90#1729C2214yes—
91#1728C2211yescharacteristic
92#1728C2212yes—
93C2×#336C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8C2312yes—
97C2×#131—28no—
98C2×#131—44no—
99C2×#131—44no—
100C22×M16C2312yes—
101C22×M16C2314yes—
102C23×C8C2312yes—
103C23×C8C2314yes—
104#544—44no—
105#536—28no—
106#536—28no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—44no—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#124—22no—
118C2×#123—24no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2314yes—
123C4×#41—44no—
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8D411yescharacteristic
126C2×C4×D4D412yes—
127C2×C4×D4D412yes—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2312yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#419—44no—
143#419—44no—
144C2×#103—44no—
145C2×#103—44no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154#400—22no—
155#399—22no—
156#398—24no—
157#398—24no—
158#398—22no—
159#398—22no—
160#397—24no—
161#397—24no—
162#396—22no—
163#396—22no—
164#395—22no—
165#393—22no—
166#391—21no—
167#391—21no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2314yes—
173#390—21no—
174#390—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2311yescharacteristic
190#387—22no—
191#387—22no—
192#386—22no—
193#386—22no—
194#385—22no—
195#385—22no—
196#385—21no—
197#385—21no—
198#385—22no—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382C2312yes—
203#382C2312yes—
204#381—24no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—21no—
211#380—21no—
212#380C2312yes—
213#380C2312yes—
214#380C2312yes—
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C2×#95—21no—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2312yes—
239#336C2312yes—
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41#4118yes—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2414yes—
254C22×D4C2×D412yes—
255C22×D4C2×D411yescharacteristic
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265#131—432no—
266C2×M16—28no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—44no—
271C2×M16—44no—
272C22×C8—28no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8—24no—
276C22×C8—44no—
277C22×C8—44no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—48no—
288#125—44no—
289#125—44no—
290#124—48no—
291#123—416no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—48no—
297C22⋊Q8—84no—
298C4×Q8—42no—
299C4×Q8—42no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—44no—
303C4×D4—44no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117—24no—
308#117—22no—
309#117—22no—
310#117—44no—
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—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—82no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#103—44no—
377#103—44no—
378#103—44no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404#41—216no—
405#41—216no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—82no—
423C2×Q8—82no—
424C2×Q8#12414yes—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4#12418yes—
430C2×D4#12414yes—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4118yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478M16—416no—
479M16—48no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—416no—
490C2×C8—48no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C22⋊C4—24no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—24no—
548C42—42no—
549C42—42no—
550C42—44no—
551C42—42no—
552C42—42no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23C2×#12414yes—
564C23C2×#12412yes—
565C23#53412yes—
566C23#53712yes—
567C23C23×D411yescharacteristic
568Q8—28no—
569Q8—44no—
570Q8—44no—
571Q8—44no—
572Q8—44no—
573Q8—88no—
574Q8—164no—
575D4—216no—
576D4—28no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—88no—
630C2×C4—162no—
631C2×C4C2×#12414yes—
632C2×C4C2×#12412yes—
633C2×C4#53412yes—
634C2×C4#53712yes—
635C2×C4#53818yes—
636C2×C4#53918yes—
637C2×C4#57712yes—
638C2×C4#57812yes—
639C2×C4C22×#4112yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4C2×2−1+411yescharacteristic
643C8—42no—
644C8—42no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22#263914yes—
660C22C22×#12411yescharacteristic
661C22C2×#53411yescharacteristic
662C22C2×#53711yescharacteristic
663C22C2×#57711yescharacteristic
664C22C2×#57811yescharacteristic
665C22#358611yescharacteristic
666C22#358611yescharacteristic
667C4—24no—
668C4—22no—
669C4—22no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—84no—
675C4—82no—
676C4—81no—
677C4—81no—
678C4—82no—
679C4—82no—
680C4—81no—
681C4—82no—
682C4—81no—
683C4#263914yes—
684C4C2×#53812yes—
685C4C2×#53912yes—
686C2—24no—
687C2—44no—
688C2C2×#263911yescharacteristic
689C2#6062311yescharacteristic
690C2#6062711yescharacteristic
691C2#6075811yescharacteristic
692C2#6075911yescharacteristic
693C2#6092311yescharacteristic
694C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
3. #30202C2semidirect
4. #30199C2semidirect
5. #30196C2semidirect
6. #30193C2semidirect
7. #30190C2semidirect
8. #30187C2semidirect
9. #23747C2semidirect
10. #23744C2semidirect
11. #23633C2semidirect
12. #23507C2semidirect
13. C2×#2043C2semidirect
14. #23457C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#352572
#302022
#301992
#301962
#301932
#301902
#301872
#237472
#237442
#236332
#235072
C2×#20432
#234572
#209742
#203362