Finite Group Databasebeta

← #6631341

Subgroups — #6631341

Order 512.

1617 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#35280C211yesmaximal, characteristic, complemented
2#35255C211yesmaximal, characteristic
3#32756C214yesmaximal, complemented
4#32755C214yesmaximal, complemented
5#32743C212yesmaximal, complemented
6#32742C212yesmaximal, complemented
7#32741C212yesmaximal, complemented
8#32740C212yesmaximal, complemented
9#23671C212yesmaximal, complemented
10C2×#2084C211yesmaximal, characteristic, complemented
11#23640C212yesmaximal, complemented
12#23630C211yesmaximal, characteristic, complemented
13#23610C212yesmaximal, complemented
14#23554C211yesmaximal, characteristic, complemented
15#23485C212yesmaximal
16#23451C211yesmaximal, characteristic, complemented
17#20630C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C22×#106C2212yes—
23C2×#424C2212yes—
24C2×#417C2212yes—
25C2×#416C2212yes—
26C2×#408C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2855C2212yes—
30#2854C2212yes—
31#2629C2214yes—
32#2625C2212yes—
33#2624C2212yes—
34#2582—21no—
35#2582—21no—
36#2538—21no—
37#2538—21no—
38#2444—22no—
39#2443—22no—
40#2423—21no—
41#2422—21no—
42#2421—21no—
43#2420—21no—
44#2417—22no—
45#2417—22no—
46#2317C2211yescharacteristic
47#2316C2212yes—
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2212yes—
51#2297C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2088C2214yes—
55#2088C2214yes—
56#2087C2214yes—
57#2086C2214yes—
58#2084C2212yes—
59#2084C2212yes—
60#2084C2214yes—
61#2077C2214yes—
62#2077C2214yes—
63#2077C2214yes—
64#2077C2214yes—
65#2072C2212yes—
66#2072C2212yes—
67#2071C2212yes—
68#2071C2212yes—
69#2066C2214yes—
70#2066C2214yes—
71#2065C2214yes—
72#2065C2214yes—
73#2059C2214yes—
74#2058C2212yes—
75#2057C2212yes—
76#2036C2214yes—
77#2035C2214yes—
78#2034C2214yes—
79#2033C2214yes—
80#2026C2212yes—
81#2025C2212yes—
82#2024C2211yescharacteristic
83#2024C2211yescharacteristic
84#2023C2211yescharacteristic
85#2023C2211yescharacteristic
86#1798C2214yes—
87#1797C2212yes—
88#1797C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#537—44no—
100#533—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121C2312yes—
114C4×#41—44no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×D4D412yes—
120C2×C4×D4D412yes—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#428—44no—
136C2×#107C2314yes—
137C2×#106—44no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141#424—44no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144#417—44no—
145#416—44no—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152#408—44no—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155#400—22no—
156#399—22no—
157#398—24no—
158#398—24no—
159#398—22no—
160#397—22no—
161#397—22no—
162#397—22no—
163#396—24no—
164#396—24no—
165#394—22no—
166#394—22no—
167#391—22no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391C2312yes—
172#391C2311yescharacteristic
173#391C2311yescharacteristic
174#390—21no—
175#390—22no—
176#390C2312yes—
177#390C2311yescharacteristic
178#390C2311yescharacteristic
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2314yes—
188#389C2314yes—
189#389C2312yes—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#387—22no—
194#387—22no—
195#386—24no—
196#386—22no—
197#386—22no—
198#386—21no—
199#386—21no—
200#386—22no—
201#385—22no—
202#385—22no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384C2312yes—
207#384C2312yes—
208#382—21no—
209#381—22no—
210#380—22no—
211#380—22no—
212#380—21no—
213#380—21no—
214#380—24no—
215#380C2312yes—
216#380C2312yes—
217#380C2314yes—
218#380C2314yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C2×#95—21no—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237C2×#41—24no—
238C2×#41—22no—
239C2×#41—24no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—22no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4—22no—
251C22×D4C2×D412yes—
252C22×D4C2×D411yescharacteristic
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C4⋊Q8—42no—
274C4⋊Q8—42no—
275#125—42no—
276#125—42no—
277#123—84no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—84no—
283C22⋊Q8—84no—
284#121—44no—
285#121—48no—
286#121—44no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4—44no—
295#117—24no—
296#117—24no—
297#117—24no—
298#117—24no—
299#117—44no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—44no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365#107—44no—
366#106—44no—
367#106—44no—
368#106—44no—
369C4⋊C8—44no—
370C4⋊C8—44no—
371#103—42no—
372#103—42no—
373#103—44no—
374#102—42no—
375#102—42no—
376#102—44no—
377C22⋊C8—44no—
378C22⋊C8—44no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395C24—21no—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398#41—48no—
399#41—48no—
400#41—48no—
401#41—48no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—82no—
430C2×D4—82no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4H(5,𝔽2)14yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+414yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480M16—48no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—48no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—82no—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—81no—
534C22⋊C4—81no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—82no—
541C42—22no—
542C42—22no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—42no—
548C42—44no—
549C42—42no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23#53412yes—
569C23#53612yes—
570C23#53712yes—
571C23C23×D411yescharacteristic
572Q8—44no—
573Q8—42no—
574Q8—42no—
575Q8—84no—
576Q8—84no—
577D4—48no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—84no—
582D4—164no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—162no—
635C2×C4—164no—
636C2×C4#53412yes—
637C2×C4#53612yes—
638C2×C4#53712yes—
639C2×C4#56218yes—
640C2×C4#56318yes—
641C2×C4#57712yes—
642C2×C4#57712yes—
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)12yes—
645C2×C4C2×2−1+412yes—
646C2×C4C2×2−1+411yescharacteristic
647C8—42no—
648C8—42no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22#281414yes—
671C22C2×#53411yescharacteristic
672C22C2×#53611yescharacteristic
673C22C2×#53711yescharacteristic
674C22C2×#57711yescharacteristic
675C22C2×#57711yescharacteristic
676C22#358511yescharacteristic
677C22#358611yescharacteristic
678C4—22no—
679C4—22no—
680C4—42no—
681C4—41no—
682C4—42no—
683C4—41no—
684C4—42no—
685C4—81no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—81no—
690C4—81no—
691C4—82no—
692C4—84no—
693C4#281414yes—
694C4C2×#56212yes—
695C4C2×#56312yes—
696C2—24no—
697C2—82no—
698C2—161no—
699C2C2×#281411yescharacteristic
700C2#6076311yescharacteristic
701C2#6076511yescharacteristic
702C2#6088111yescharacteristic
703C2#6088711yescharacteristic
704C2#6092311yescharacteristic
705C2#6092411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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. #35280C2semidirect
3. #32756C2semidirect
4. #32755C2semidirect
5. #32743C2semidirect
6. #32742C2semidirect
7. #32741C2semidirect
8. #32740C2semidirect
9. #23671C2semidirect
10. C2×#2084C2semidirect
11. #23640C2semidirect
12. #23630C2semidirect
13. #23610C2semidirect
14. #23554C2semidirect
16. #23451C2semidirect
17. #20630C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#352552
#327562
#327552
#327432
#327422
#327412
#327402
#236712
C2×#20842
#236402
#236302
#236102
#235542
#234852
#234512
#206302