Finite Group Databasebeta

← #6686263

Subgroups — #6686263

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3359C211yesmaximal, characteristic, complemented
2C2×#3340C211yesmaximal, characteristic
3#38958C211yesmaximal, characteristic
4#36508C211yesmaximal, characteristic, complemented
5#33202C211yesmaximal, characteristic, complemented
6#30636C214yesmaximal, complemented
7#30601C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29829C211yesmaximal, characteristic, complemented
10#28677C211yesmaximal, characteristic, complemented
11#28600C211yesmaximal, characteristic, complemented
12#26553C214yesmaximal, complemented
13#26337C214yesmaximal, complemented
14#25838C211yesmaximal, characteristic, complemented
15#25705C211yesmaximal, characteristic, complemented
16#25651C211yesmaximal, characteristic, complemented
17#25568C211yesmaximal, characteristic
18#22208C211yesmaximal, characteristic, complemented
19#21115C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#555C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3359—28no—
25#3340—28no—
26C2×#500C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#487C2212yes—
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#482C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#475C2212yes—
36C2×#474C2211yescharacteristic
37C2×#465C2212yes—
38C2×#461C2211yescharacteristic
39C2×#461C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C4×SD16C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2861—21no—
48#2700—24no—
49#2620—21no—
50#2567C2211yescharacteristic
51#2565—21no—
52#2547C2211yescharacteristic
53#2546C2211yescharacteristic
54#2545—21no—
55#2542C2211yescharacteristic
56#2519—24no—
57#2499—21no—
58#2439C2214yes—
59#2436—21no—
60#2429C2214yes—
61#2420C2214yes—
62#2362C2214yes—
63#2310—21no—
64C4×C4⋊Q8—21no—
65C4×#124C2211yescharacteristic
66#2293C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2088C2214yes—
71#2068C2214yes—
72#2057C2214yes—
73#2055C2214yes—
74#2043C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2000C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1941C2214yes—
86#1926C2212yes—
87#1923C2214yes—
88#1919C2212yes—
89#1892C2214yes—
90#1880C2212yes—
91#1873C2214yes—
92#1865C2212yes—
93#1863C2214yes—
94#1862C2214yes—
95#1854C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1805C2214yes—
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C23×Q8—21no—
104C22×SD16C2312yes—
105C22×SD16C2311yescharacteristic
106C23×C8C2312yes—
107#557—24no—
108#557—42no—
109#556—24no—
110#556—42no—
111#555—24no—
112#555—42no—
113D4×Q8—24no—
114D4×Q8—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×#126—22no—
123C2×#125—21no—
124C2×#125—22no—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124—22no—
128C2×#124D411yescharacteristic
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8C2312yes—
132C2×C22⋊Q8C2314yes—
133C2×#121—21no—
134C2×#121C2312yes—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C2×C4×D4C2311yescharacteristic
144C2×#117—21no—
145C2×#117—21no—
146C2×#117C2312yes—
147C2×#117C2312yes—
148#500—42no—
149#498—24no—
150#487—28no—
151#487—24no—
152#487—42no—
153#485—28no—
154#482—28no—
155#477—24no—
156#475—24no—
157#475—28no—
158#474—24no—
159#465—28no—
160#461—28no—
161#461—24no—
162C8×Q8—24no—
163C4×SD16—28no—
164C4×SD16D418yes—
165C8×D4—24no—
166#427—28no—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2314yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2314yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2312yes—
187C2×#103C2311yescharacteristic
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2311yescharacteristic
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2314yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195C2×C4×C8C2311yescharacteristic
196#399—22no—
197#398—22no—
198#398—24no—
199#397—22no—
200#397—24no—
201#396—24no—
202#396—24no—
203#395—22no—
204#394—24no—
205#393—24no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391—24no—
215#391D411yescharacteristic
216#391C2314yes—
217#390—21no—
218#390D411yescharacteristic
219#390C2312yes—
220#389—24no—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#388—24no—
225#388—24no—
226#388—24no—
227#388—22no—
228#388D414yes—
229#387—24no—
230#387—24no—
231#386—24no—
232#384—21no—
233#384—21no—
234#382—22no—
235#382D411yescharacteristic
236#382C2314yes—
237#381—24no—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—24no—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4C2314yes—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8—44no—
263C22×Q8—44no—
264C22×Q8—82no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2412yes—
267C22×Q8C2411yescharacteristic
268C22×D4—22no—
269C22×D4C2×D411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—22no—
273C23×C4C2412yes—
274C2×SD16—24no—
275C2×SD16—42no—
276C2×SD16—44no—
277C2×SD16C2×D414yes—
278C22×C8—24no—
279C22×C8—44no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C4⋊Q8—24no—
286C4⋊Q8—24no—
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—42no—
296C4⋊Q8—42no—
297C4⋊Q8—44no—
298#126—48no—
299#125—44no—
300#125—48no—
301#124—24no—
302#124—44no—
303#124—48no—
304#124—82no—
305C22⋊Q8—24no—
306C22⋊Q8—28no—
307C22⋊Q8—48no—
308C22⋊Q8—44no—
309C22⋊Q8—42no—
310C22⋊Q8—44no—
311#121—24no—
312#121—42no—
313#121—44no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—42no—
320C4×Q8—42no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8C2×D414yes—
324C4×D4—24no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—42no—
329C4×D4C2×D414yes—
330#117—24no—
331#117—24no—
332#117—42no—
333#117—42no—
334#117—44no—
335#117—82no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D414yes—
374C2×C22⋊C4C2×D414yes—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D412yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389#107—24no—
390#107—24no—
391#107—44no—
392#106—24no—
393#106—24no—
394#106—24no—
395#106—24no—
396#106C2×D414yes—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8—44no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—42no—
407#103—44no—
408#103—44no—
409#103C2×D414yes—
410#102—24no—
411#102—24no—
412#102C2×D414yes—
413C22⋊C8—24no—
414C22⋊C8—44no—
415C4×C8—24no—
416C4×C8—22no—
417C4×C8C2×D414yes—
418#95—22no—
419#95—22no—
420#95—21no—
421#95—24no—
422#95—21no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95C2×D414yes—
430#95C2×D414yes—
431C24—21no—
432C24C22×D412yes—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—44no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—82no—
452C2×Q8—82no—
453C2×Q8—816no—
454C2×Q8#13514yes—
455C2×Q8C22×D412yes—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—44no—
463C2×D4—84no—
464C2×D4C22×D412yes—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—82no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2511yescharacteristic
492SD16—44no—
493SD16—48no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—48no—
504C2×C8C22×D412yes—
505C2×C8C22×D412yes—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
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—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—48no—
534C4⋊C4—84no—
535C4⋊C4—82no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D412yes—
538C4⋊C42−1+414yes—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—28no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—84no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—42no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42—44no—
564C42—44no—
565C42—42no—
566C42—42no—
567C42C22×D412yes—
568C42H(5,𝔽2)12yes—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—42no—
576C23—42no—
577C23—41no—
578C23—41no—
579C23C23×D411yescharacteristic
580Q8—28no—
581Q8—22no—
582Q8—22no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—44no—
587Q8—88no—
588Q8—84no—
589Q8—84no—
590Q8—162no—
591D4—22no—
592D4—22no—
593D4—44no—
594D4—42no—
595D4—164no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—88no—
637C2×C4D4212yes—
638C2×C4D4212yes—
639C2×C4C2×#13514yes—
640C2×C4C2×#13512yes—
641C2×C4#57712yes—
642C2×C4C23×D411yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×2−1+412yes—
645C2×C4#58512yes—
646C8—22no—
647C8—22no—
648C8—42no—
649C8—42no—
650C8—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22—82no—
660C22—82no—
661C22C2×D4211yescharacteristic
662C22#348912yes—
663C22C22×#13511yescharacteristic
664C22C2×#57711yescharacteristic
665C22#358612yes—
666C4—24no—
667C4—22no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4#328214yes—
682C4#344911yescharacteristic
683C4#349911yescharacteristic
684C4#355412yes—
685C2—82no—
686C2—82no—
687C2#3768011yescharacteristic
688C2C2×#328211yescharacteristic
689C2#6108311yescharacteristic
690C2#6210512yes—
691C2#6219112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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×#3359C2semidirect
4. #36508C2semidirect
5. #33202C2semidirect
6. #30636C2semidirect
7. #30601C2semidirect
8. #30281C2semidirect
9. #29829C2semidirect
10. #28677C2semidirect
11. #28600C2semidirect
12. #26553C2semidirect
13. #26337C2semidirect
14. #25838C2semidirect
15. #25705C2semidirect
16. #25651C2semidirect
18. #22208C2semidirect
19. #21115C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33592
C2×#33402
#389582
#365082
#332022
#306362
#306012
#302812
#298292
#286772
#286002
#265532
#263372
#258382
#257052
#256512
#255682
#222082
#211152