Finite Group Databasebeta

← #6684242

Subgroups — #6684242

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3381C211yesmaximal, characteristic, complemented
2C2×#3302C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic, complemented
4#36164C211yesmaximal, characteristic
5#33195C211yesmaximal, characteristic, complemented
6#30782C214yesmaximal, complemented
7#30666C214yesmaximal, complemented
8#30280C211yesmaximal, characteristic, complemented
9#29830C211yesmaximal, characteristic, complemented
10#28706C211yesmaximal, characteristic, complemented
11#28639C211yesmaximal, characteristic, complemented
12#25964C214yesmaximal, complemented
13#25931C211yesmaximal, characteristic, complemented
14#25882C211yesmaximal, characteristic
15#25835C211yesmaximal, characteristic, complemented
16#25701C211yesmaximal, characteristic, complemented
17#24907C214yesmaximal, complemented
18#22208C211yesmaximal, characteristic, complemented
19#21117C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#547C2211yescharacteristic
24#3381—28no—
25#3302—28no—
26C2×#499C2212yes—
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#478C2212yes—
35C2×#474C2211yescharacteristic
36C2×#474C2211yescharacteristic
37C2×#467C2212yes—
38C2×#462C2211yescharacteristic
39C2×#452C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C4×SD16C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#416C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2850—21no—
48#2721—24no—
49#2658—21no—
50#2568C2211yescharacteristic
51#2563C2211yescharacteristic
52#2547C2211yescharacteristic
53#2546—21no—
54#2546C2211yescharacteristic
55#2545—21no—
56#2504—21no—
57#2503—21no—
58#2489C2214yes—
59#2488—24no—
60#2441C2214yes—
61#2423C2214yes—
62#2339C2214yes—
63#2327—21no—
64C4×#124C2211yescharacteristic
65#2295C2211yescharacteristic
66C2×#382—21no—
67C42×D4C2211yescharacteristic
68#2096C2214yes—
69#2096C2214yes—
70#2077C2214yes—
71#2068C2214yes—
72#2066C2214yes—
73#2059C2214yes—
74#2056C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2026C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1947C2214yes—
85#1933C2214yes—
86#1927C2212yes—
87#1923C2214yes—
88#1920C2212yes—
89#1878C2214yes—
90#1877C2212yes—
91#1864C2214yes—
92#1864C2212yes—
93#1854C2214yes—
94#1850C2214yes—
95#1842C2211yescharacteristic
96#1807C2214yes—
97#1802C2214yes—
98#1788C2214yes—
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41C2312yes—
104C22×Q16C2312yes—
105C22×SD16C2311yescharacteristic
106C23×C8C2312yes—
107#557—24no—
108#557—42no—
109#556—24no—
110#556—42no—
111#554—24no—
112#554—42no—
113#547—24no—
114#547—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×#126—22no—
124C2×#125—22no—
125C2×#125C2312yes—
126C2×#125C2312yes—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124D411yescharacteristic
130C2×#123—22no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×#121—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2311yescharacteristic
144C2×C4×D4C2311yescharacteristic
145C2×#117C2312yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C42—21no—
149#499—28no—
150#498—24no—
151#498—42no—
152#487—24no—
153#487—42no—
154#485—28no—
155#479—28no—
156#478—28no—
157#477—24no—
158#474—24no—
159#474—24no—
160#467—28no—
161#462—24no—
162#452—28no—
163C8×Q8—24no—
164C4×SD16—28no—
165C4×SD16D418yes—
166C8×D4—24no—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2314yes—
173C2×#106C2312yes—
174C2×#106C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2314yes—
179#416—28no—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2312yes—
187C2×#103C2311yescharacteristic
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2311yescharacteristic
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2314yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2312yes—
196C2×C4×C8C2311yescharacteristic
197#400—22no—
198#400—24no—
199#399—24no—
200#399—24no—
201#398—22no—
202#398—24no—
203#397—22no—
204#397—24no—
205#396—24no—
206#393—22no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391D411yescharacteristic
212#391C2312yes—
213#391C2314yes—
214#391C2314yes—
215#390—21no—
216#390D411yescharacteristic
217#389—24no—
218#389C2314yes—
219#389C2312yes—
220#388—24no—
221#388—24no—
222#388D414yes—
223#387—22no—
224#387—24no—
225#387—24no—
226#386—24no—
227#386—24no—
228#384—22no—
229#384C2314yes—
230#382—21no—
231#382—21no—
232#382—21no—
233#382—21no—
234#382—24no—
235#382D411yescharacteristic
236#381—24no—
237#380C2314yes—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C22⋊C4—22no—
244C4×C22⋊C4—21no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C2×#95—24no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C2×#41—24no—
259C2×#41—28no—
260C2×#41—42no—
261C2×#41—44no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2412yes—
267C22×Q8C2411yescharacteristic
268C22×D4—22no—
269C22×D4C2×D411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4C2412yes—
276C2×Q16—24no—
277C2×Q16—44no—
278C2×SD16—24no—
279C2×SD16C2×D414yes—
280C22×C8—24no—
281C22×C8—44no—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—44no—
296C4⋊Q8—44no—
297C4⋊Q8—82no—
298C4⋊Q8—82no—
299#126—48no—
300#125—24no—
301#125—24no—
302#125—42no—
303#125—42no—
304#125—48no—
305#124—24no—
306#124—44no—
307#124—82no—
308#123—48no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—48no—
312C22⋊Q8—42no—
313C22⋊Q8—42no—
314C22⋊Q8—44no—
315#121—44no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—24no—
320C4×Q8—42no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8C2×D414yes—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—42no—
330C4×D4—42no—
331C4×D4C2×D414yes—
332#117—24no—
333#117—42no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D414yes—
373C2×C22⋊C4C2×D414yes—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—44no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D412yes—
383C2×C42C2×D411yescharacteristic
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387#107—24no—
388#107—24no—
389#107—24no—
390#106—24no—
391#106—24no—
392#106—42no—
393#106—44no—
394#106C2×D414yes—
395C4⋊C8—24no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398C4⋊C8—44no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—44no—
404#103—42no—
405#103—44no—
406#103C2×D414yes—
407#102—24no—
408#102—24no—
409#102—24no—
410#102C2×D414yes—
411C22⋊C8—24no—
412C22⋊C8—44no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8C2×D414yes—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—21no—
422#95—22no—
423#95—22no—
424#95—21no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95C2×D414yes—
430#95C2×D414yes—
431C24—21no—
432C24C22×D412yes—
433#41—48no—
434#41—84no—
435#41—88no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—82no—
454C2×Q8C22×D412yes—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—44no—
461C2×D4—48no—
462C2×D4C22×D412yes—
463C2×D42−1+414yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—82no—
486C22×C4—84no—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C2511yescharacteristic
495Q16—84no—
496SD16—28no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—84no—
507C2×C8C22×D412yes—
508C2×C8C22×D412yes—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
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—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—48no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4C22×D412yes—
542C4⋊C4C22×D412yes—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—28no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—42no—
561C42—44no—
562C42—44no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42#13514yes—
567C42C22×D412yes—
568C42H(5,𝔽2)12yes—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—22no—
573C23—24no—
574C23—22no—
575C23—22no—
576C23—42no—
577C23—41no—
578C23—41no—
579C23C23×D411yescharacteristic
580Q8—22no—
581Q8—22no—
582Q8—44no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—44no—
587Q8—84no—
588Q8—84no—
589Q8—162no—
590D4—22no—
591D4—24no—
592D4—44no—
593D4—88no—
594D4#53614yes—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4—164no—
638C2×C4D4212yes—
639C2×C4D4212yes—
640C2×C4C2×#13514yes—
641C2×C4C2×#13512yes—
642C2×C4#57712yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+412yes—
646C2×C4#58512yes—
647C8—22no—
648C8—22no—
649C8—42no—
650C8—42no—
651C8—42no—
652C22—24no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—42no—
661C22—44no—
662C22C2×D4211yescharacteristic
663C22#348912yes—
664C22C22×#13511yescharacteristic
665C22C2×#57711yescharacteristic
666C22#358612yes—
667C4—24no—
668C4—22no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#328414yes—
683C4C2×#53611yescharacteristic
684C4#350011yescharacteristic
685C4#356112yes—
686C2—44no—
687C2—82no—
688C2#3767611yescharacteristic
689C2C2×#328411yescharacteristic
690C2#6108311yescharacteristic
691C2#6209712yes—
692C2#6218212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3381C2semidirect
3. #39744C2semidirect
5. #33195C2semidirect
6. #30782C2semidirect
7. #30666C2semidirect
8. #30280C2semidirect
9. #29830C2semidirect
10. #28706C2semidirect
11. #28639C2semidirect
12. #25964C2semidirect
13. #25931C2semidirect
15. #25835C2semidirect
16. #25701C2semidirect
17. #24907C2semidirect
18. #22208C2semidirect
19. #21117C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33812
C2×#33022
#397442
#361642
#331952
#307822
#306662
#302802
#298302
#287062
#286392
#259642
#259312
#258822
#258352
#257012
#249072
#222082
#211172