Finite Group Databasebeta

← #6686275

Subgroups — #6686275

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3365C211yesmaximal, characteristic, complemented
2C2×#3342C211yesmaximal, characteristic, complemented
3#39744C211yesmaximal, characteristic, complemented
4#36597C211yesmaximal, characteristic
5#33196C211yesmaximal, characteristic, complemented
6#30642C214yesmaximal, complemented
7#30602C214yesmaximal, complemented
8#30276C211yesmaximal, characteristic, complemented
9#29826C211yesmaximal, characteristic, complemented
10#28675C211yesmaximal, characteristic, complemented
11#28597C211yesmaximal, characteristic
12#26557C214yesmaximal, complemented
13#26347C214yesmaximal, complemented
14#25840C211yesmaximal, characteristic, complemented
15#25705C211yesmaximal, characteristic, complemented
16#25652C211yesmaximal, characteristic, complemented
17#25574C211yesmaximal, characteristic
18#22208C211yesmaximal, characteristic, complemented
19#21119C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#556C2211yescharacteristic
24#3365—28no—
25#3342—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#488C2212yes—
29C2×#486C2211yescharacteristic
30C2×#486C2211yescharacteristic
31C2×#485C2212yes—
32C2×#485C2212yes—
33C2×#475C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#478C2212yes—
37C2×#467C2212yes—
38C2×#464C2212yes—
39C2×#460C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2211yescharacteristic
42C2×C4×SD16C2212yes—
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2868C2211yescharacteristic
46#2858—21no—
47#2858C2211yescharacteristic
48#2721—24no—
49#2658—21no—
50#2554—24no—
51#2547C2211yescharacteristic
52#2546—21no—
53#2546C2211yescharacteristic
54#2546C2211yescharacteristic
55#2545—21no—
56#2545—21no—
57#2544—21no—
58#2541C2211yescharacteristic
59#2441C2214yes—
60#2426C2214yes—
61#2423C2214yes—
62#2363C2214yes—
63#2327—21no—
64C4×#124—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2093C2214yes—
69#2093C2214yes—
70#2086C2214yes—
71#2065C2214yes—
72#2059C2214yes—
73#2055C2214yes—
74#2042C2214yes—
75#2036C2214yes—
76#2033C2214yes—
77#2000C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1984C2211yescharacteristic
82#1982C2211yescharacteristic
83#1954C2211yescharacteristic
84#1948C2214yes—
85#1942C2214yes—
86#1928C2212yes—
87#1923C2214yes—
88#1921C2212yes—
89#1893C2214yes—
90#1878C2212yes—
91#1875C2214yes—
92#1864C2214yes—
93#1863C2212yes—
94#1863C2214yes—
95#1854C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1803C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2311yescharacteristic
104C22×SD16C2312yes—
105C23×C8C2312yes—
106#559—24no—
107#559—42no—
108#557—24no—
109#557—42no—
110#556—24no—
111#556—24no—
112#556—42no—
113#556—42no—
114C2×C4⋊Q8—21no—
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×#127—21no—
123C2×#127C2311yescharacteristic
124C2×#126—22no—
125C2×#126—22no—
126C2×#125—22no—
127C2×#125C2312yes—
128C2×#124—21no—
129C2×#124—21no—
130C2×#124—21no—
131C2×#124—21no—
132C2×#124—21no—
133C2×#124—21no—
134C2×#124D411yescharacteristic
135C2×#124C2312yes—
136C2×#123—22no—
137C2×C22⋊Q8C2312yes—
138C2×C22⋊Q8C2312yes—
139C2×#121C2314yes—
140C2×C4×Q8—21no—
141C2×C4×Q8—21no—
142C2×C4×Q8—21no—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×D4—21no—
149C2×C4×D4C2312yes—
150C2×C4×D4C2311yescharacteristic
151C2×#117C2312yes—
152C2×#117C2312yes—
153#500—24no—
154#500—42no—
155#488—28no—
156#486—24no—
157#486—42no—
158#485—28no—
159#485—28no—
160#478—28no—
161#475—24no—
162#475—24no—
163#475—24no—
164#467—28no—
165#464—28no—
166#460—24no—
167C8×Q8—24no—
168C4×Q16D418yescomplemented
169C4×SD16—28no—
170C8×D4—24no—
171#427—28no—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#107C2312yes—
175C2×#107C2314yes—
176C2×#107C2311yescharacteristic
177C2×#106C2312yes—
178C2×#106C2312yes—
179C2×#106C2312yes—
180C2×C4⋊C8C2312yes—
181C2×C4⋊C8C2311yescharacteristic
182C2×C4⋊C8C2311yescharacteristic
183C2×C4⋊C8C2314yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2312yes—
189C2×#103C2311yescharacteristic
190C2×#103C2311yescharacteristic
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#102C2312yes—
194C2×#102C2314yes—
195C2×#102C2314yes—
196C2×C22⋊C8C2312yes—
197C2×C22⋊C8C2314yes—
198C2×C4×C8C2311yescharacteristic
199C2×C4×C8C2312yes—
200C2×C4×C8C2311yescharacteristic
201#399—24no—
202#399—24no—
203#398—22no—
204#398—22no—
205#398—24no—
206#397—22no—
207#396—24no—
208#396—24no—
209#394—24no—
210#392—22no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391D411yescharacteristic
215#391C2312yes—
216#390—21no—
217#390—24no—
218#390—24no—
219#390—22no—
220#390D411yescharacteristic
221#390C2314yes—
222#389—24no—
223#389C2314yes—
224#389C2312yes—
225#389C2314yes—
226#388—24no—
227#388—24no—
228#388—24no—
229#388D414yes—
230#387—22no—
231#387—24no—
232#387—24no—
233#386—24no—
234#384—22no—
235#384C2314yes—
236#382D411yescharacteristic
237#380C2314yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2311yescharacteristic
244C4×C22⋊C4—24no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2411yescharacteristic
263C22×Q8C2411yescharacteristic
264C22×D4—24no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4C2×D411yescharacteristic
268C22×D4C2412yes—
269C23×C4—22no—
270C23×C4C2412yes—
271C2×Q16—42no—
272C2×Q16C2×D414yes—
273C2×SD16—24no—
274C2×SD16—44no—
275C22×C8—24no—
276C22×C8—44no—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2411yescharacteristic
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—82no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293#127—22no—
294#127—22no—
295#127—82no—
296#126—48no—
297#126—48no—
298#125—24no—
299#125—42no—
300#125—48no—
301#124—24no—
302#124—24no—
303#124—42no—
304#124—44no—
305#124—44no—
306#124—44no—
307#124—44no—
308#124—82no—
309#124—82no—
310#123—48no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—42no—
314C22⋊Q8—42no—
315#121—28no—
316#121—44no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—42no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8C2×D414yes—
329C4×Q8C2×D414yes—
330C4×D4—24no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—42no—
334C4×D4—44no—
335#117—24no—
336#117—24no—
337#117—42no—
338#117—42no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D414yes—
378C2×C22⋊C4C2×D414yes—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—22no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D412yes—
390C2×C42C2×D411yescharacteristic
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394#107—24no—
395#107—24no—
396#107—24no—
397#107—44no—
398#107C2×D414yes—
399#106—24no—
400#106—24no—
401#106—24no—
402C4⋊C8—24no—
403C4⋊C8—22no—
404C4⋊C8—22no—
405C4⋊C8—44no—
406#103—24no—
407#103—24no—
408#103—24no—
409#103—24no—
410#103—42no—
411#103C2×D414yes—
412#103C2×D414yes—
413#102—24no—
414#102—24no—
415#102—24no—
416#102—44no—
417#102—44no—
418C22⋊C8—24no—
419C22⋊C8—44no—
420C4×C8—24no—
421C4×C8—22no—
422C4×C8C2×D414yes—
423#95—22no—
424#95—22no—
425#95—24no—
426#95—22no—
427#95—21no—
428#95—22no—
429#95—21no—
430#95—24no—
431#95—24no—
432#95C2×D414yes—
433#95C2×D414yes—
434C24—22no—
435C24C22×D412yes—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8C22×D412yes—
455C2×Q8C22×D412yes—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—48no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—82no—
469C2×D4—84no—
470C2×D4—84no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—84no—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2511yescharacteristic
494Q16—44no—
495SD16—48no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—48no—
506C2×C8C22×D412yes—
507C2×C8C22×D412yes—
508C4⋊C4—24no—
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—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—48no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4C22×D412yes—
539C4⋊C4C22×D412yes—
540C4⋊C42−1+414yes—
541C22⋊C4—24no—
542C22⋊C4—28no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—48no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—42no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42—44no—
567C42—42no—
568C42—42no—
569C42—82no—
570C42#13514yes—
571C42C22×D412yes—
572C42H(5,𝔽2)12yes—
573C23—24no—
574C23—22no—
575C23—24no—
576C23—22no—
577C23—22no—
578C23—44no—
579C23—42no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23C23×D411yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8—22no—
588Q8—44no—
589Q8—42no—
590Q8—42no—
591Q8—84no—
592Q8—84no—
593Q8—84no—
594D4—42no—
595D4—42no—
596D4—44no—
597D4—44no—
598D4—84no—
599D4—162no—
600D4—168no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—164no—
641C2×C4D4212yes—
642C2×C4D4212yes—
643C2×C4C2×#13514yes—
644C2×C4C2×#13512yes—
645C2×C4#57612yes—
646C2×C4C23×D411yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+412yes—
649C2×C4#58512yes—
650C8—22no—
651C8—22no—
652C8—42no—
653C8—42no—
654C8—42no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—42no—
659C22—44no—
660C22—82no—
661C22—82no—
662C22—84no—
663C22—82no—
664C22—84no—
665C22C2×D4211yescharacteristic
666C22#348912yes—
667C22C22×#13511yescharacteristic
668C22C2×#57611yescharacteristic
669C22#358612yes—
670C4—24no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#328714yes—
686C4#344911yescharacteristic
687C4#349911yescharacteristic
688C4#355412yes—
689C2—82no—
690C2—162no—
691C2#3768011yescharacteristic
692C2C2×#328711yescharacteristic
693C2#6108311yescharacteristic
694C2#6210112yes—
695C2#6218012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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×#3365C2semidirect
2. C2×#3342C2semidirect
3. #39744C2semidirect
5. #33196C2semidirect
6. #30642C2semidirect
7. #30602C2semidirect
8. #30276C2semidirect
9. #29826C2semidirect
10. #28675C2semidirect
12. #26557C2semidirect
13. #26347C2semidirect
14. #25840C2semidirect
15. #25705C2semidirect
16. #25652C2semidirect
18. #22208C2semidirect
19. #21119C2semidirect
168. C4×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33652
C2×#33422
#397442
#365972
#331962
#306422
#306022
#302762
#298262
#286752
#285972
#265572
#263472
#258402
#257052
#256522
#255742
#222082
#211192