Finite Group Databasebeta

← #6684269

Subgroups — #6684269

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3387C211yesmaximal, characteristic, complemented
2C2×#3333C211yesmaximal, characteristic
3#39173C211yesmaximal, characteristic, complemented
4#36457C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#30789C214yesmaximal, complemented
7#30680C214yesmaximal, complemented
8#30278C211yesmaximal, characteristic, complemented
9#29827C211yesmaximal, characteristic, complemented
10#28713C211yesmaximal, characteristic, complemented
11#28645C211yesmaximal, characteristic, complemented
12#25969C214yesmaximal, complemented
13#25935C211yesmaximal, characteristic, complemented
14#25887C211yesmaximal, characteristic
15#25841C211yesmaximal, characteristic, complemented
16#25706C211yesmaximal, characteristic, complemented
17#24920C214yesmaximal, complemented
18#22214C211yesmaximal, characteristic, complemented
19#21119C211yesmaximal, characteristic, complemented
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#550C2211yescharacteristic
24#3387—28no—
25#3333—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#486C2211yescharacteristic
30C2×#486C2211yescharacteristic
31C2×#480C2212yes—
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2212yes—
37C2×#468C2212yes—
38C2×#460C2211yescharacteristic
39C2×#452C2212yes—
40C2×#446C2211yescharacteristic
41C2×#441C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#417C2212yes—
45#2862C2211yescharacteristic
46#2858—21no—
47#2858C2211yescharacteristic
48#2690—24no—
49#2642—21no—
50#2567C2211yescharacteristic
51#2565—21no—
52#2563—21no—
53#2563C2211yescharacteristic
54#2548—21no—
55#2548C2211yescharacteristic
56#2547—21no—
57#2547C2211yescharacteristic
58#2491—24no—
59#2491C2214yes—
60#2442C2214yes—
61#2421C2214yes—
62#2341C2214yes—
63#2326—21no—
64C4×#124—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2092C2214yes—
69#2092C2214yes—
70#2077C2214yes—
71#2067C2214yes—
72#2067C2214yes—
73#2057C2214yes—
74#2055C2214yes—
75#2036C2214yes—
76#2034C2214yes—
77#2025C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1982C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1936C2214yes—
86#1928C2212yes—
87#1924C2214yes—
88#1921C2212yes—
89#1880C2214yes—
90#1878C2212yes—
91#1863C2214yes—
92#1863C2212yes—
93#1852C2214yes—
94#1851C2214yes—
95#1842C2211yescharacteristic
96#1809C2214yes—
97#1805C2214yes—
98#1788C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41C2312yes—
104C22×Q16C2311yescharacteristic
105C22×SD16C2312yes—
106C22×M16C2312yes—
107Q82—24no—
108Q82—42no—
109#557—24no—
110#557—42no—
111#554—24no—
112#554—42no—
113#550—24no—
114#550—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×#127—21no—
123C2×#127C2311yescharacteristic
124C2×#125—21no—
125C2×#125—22no—
126C2×#125C2312yes—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124—24no—
130C2×#124—22no—
131C2×#124D411yescharacteristic
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8C2312yes—
134C2×#121—22no—
135C2×#121C2312yes—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×D4—21no—
145C2×C4×D4—21no—
146C2×C4×D4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148#501—28no—
149#500—42no—
150#498—24no—
151#486—24no—
152#486—42no—
153#480—28no—
154#479—28no—
155#477—24no—
156#477—24no—
157#475—24no—
158#475—28no—
159#468—28no—
160#460—24no—
161#452—28no—
162#446—24no—
163#441—28no—
164C4×Q16D418yes—
165#436—24no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2314yes—
169C2×#107C2312yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2314yes—
178#417—28no—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#103C2311yescharacteristic
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2314yes—
190C2×#102C2314yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2314yes—
193C2×#97C2312yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2311yescharacteristic
196#400—24no—
197#400—24no—
198#398—24no—
199#397—24no—
200#397—22no—
201#395—22no—
202#394—22no—
203#394—24no—
204#393—22no—
205#393—24no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391D411yescharacteristic
212#391C2312yes—
213#390—21no—
214#390—21no—
215#390—24no—
216#390—24no—
217#390—21no—
218#390—21no—
219#390D411yescharacteristic
220#390C2314yes—
221#390C2314yes—
222#389C2314yes—
223#389C2312yes—
224#388—24no—
225#388—24no—
226#388D414yes—
227#387—22no—
228#387—24no—
229#385—24no—
230#384—21no—
231#382—21no—
232#382D411yescharacteristic
233#382C2314yes—
234#380—24no—
235#380—24no—
236#380C2314yes—
237#380C2314yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—24no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
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—22no—
265C22×Q8—21no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2411yescharacteristic
268C22×Q8C2411yescharacteristic
269C22×D4—22no—
270C22×D4—22no—
271C22×D4—21no—
272C22×D4—21no—
273C22×D4C2×D411yescharacteristic
274C22×D4C2412yes—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4C2412yes—
278C2×Q16—24no—
279C2×Q16C2×D414yes—
280C2×SD16—24no—
281C2×SD16—44no—
282C2×M16—24no—
283C2×M16—44no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2411yescharacteristic
288C22×C8C2411yescharacteristic
289C4⋊Q8—24no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—42no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8—48no—
300#127—22no—
301#127—22no—
302#127—44no—
303#125—24no—
304#125—42no—
305#125—44no—
306#125—48no—
307#124—42no—
308#124—44no—
309#124—44no—
310#124—84no—
311#124—88no—
312C22⋊Q8—24no—
313C22⋊Q8—48no—
314C22⋊Q8—42no—
315#121—24no—
316#121—48no—
317#121—42no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—24no—
325C4×Q8—42no—
326C4×Q8—42no—
327C4×Q8—42no—
328C4×Q8—44no—
329C4×Q8—44no—
330C4×Q8C2×D414yes—
331C4×Q8C2×D414yes—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—44no—
335C4×D4—44no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—42no—
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—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
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—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388#107—24no—
389#107—24no—
390#107—42no—
391#107—44no—
392#107C2×D414yes—
393#106—24no—
394#106—24no—
395#106—24no—
396C4⋊C8—24no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—44no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—42no—
405#103C2×D414yes—
406#103C2×D414yes—
407#102—24no—
408#102—24no—
409#102—24no—
410#102—44no—
411#102—44no—
412C22⋊C8—24no—
413C22⋊C8—44no—
414#97—24no—
415C4×C8—22no—
416C4×C8C2×D414yes—
417#95—22no—
418#95—22no—
419#95—24no—
420#95—22no—
421#95—21no—
422#95—22no—
423#95—21no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95C2×D414yes—
429#95C2×D414yes—
430C24—22no—
431C24C22×D412yes—
432#41—48no—
433#41—84no—
434#41—88no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—82no—
450C2×Q8—84no—
451C2×Q8—82no—
452C2×Q8C22×D412yes—
453C2×Q8C22×D412yes—
454C2×Q82−1+414yes—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—44no—
463C2×D4—48no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—82no—
470C2×D4—84no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—84no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2511yescharacteristic
496Q16—28no—
497SD16—84no—
498M16—84no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8C22×D412yes—
509C2×C8C22×D412yes—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
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—22no—
526C4⋊C4—22no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—82no—
539C4⋊C4—84no—
540C4⋊C4C22×D412yes—
541C4⋊C4C22×D412yes—
542C22⋊C4—24no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—42no—
562C42—44no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42—42no—
567C42—42no—
568C42C22×D412yes—
569C42H(5,𝔽2)12yes—
570C23—24no—
571C23—22no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—44no—
576C23—42no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23C23×D411yescharacteristic
581Q8—22no—
582Q8—24no—
583Q8—22no—
584Q8—22no—
585Q8—44no—
586Q8—42no—
587Q8—84no—
588Q8—162no—
589Q8—162no—
590Q8—164no—
591Q8#53614yes—
592D4—42no—
593D4—42no—
594D4—44no—
595D4—44no—
596D4—88no—
597D4—84no—
598D4—162no—
599D4—164no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
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—22no—
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—44no—
625C2×C4—42no—
626C2×C4—44no—
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—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—88no—
639C2×C4—164no—
640C2×C4D4212yes—
641C2×C4D4212yes—
642C2×C4#57512yes—
643C2×C4#57812yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×2−1+412yes—
647C2×C4#58512yes—
648C8—22no—
649C8—22no—
650C8—42no—
651C8—42no—
652C8—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—84no—
662C22—84no—
663C22C2×D4211yescharacteristic
664C22#348912yes—
665C22C2×#57511yescharacteristic
666C22C2×#57811yescharacteristic
667C22#358512yes—
668C4—24no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#327714yes—
684C4C2×#53611yescharacteristic
685C4#350011yescharacteristic
686C4#356112yes—
687C2—82no—
688C2—162no—
689C2#3767611yescharacteristic
690C2C2×#327711yescharacteristic
691C2#6108211yescharacteristic
692C2#6212312yes—
693C2#6219312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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×#3387C2semidirect
3. #39173C2semidirect
5. #33202C2semidirect
6. #30789C2semidirect
7. #30680C2semidirect
8. #30278C2semidirect
9. #29827C2semidirect
10. #28713C2semidirect
11. #28645C2semidirect
12. #25969C2semidirect
13. #25935C2semidirect
15. #25841C2semidirect
16. #25706C2semidirect
17. #24920C2semidirect
18. #22214C2semidirect
19. #21119C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33872
C2×#33332
#391732
#364572
#332022
#307892
#306802
#302782
#298272
#287132
#286452
#259692
#259352
#258872
#258412
#257062
#249202
#222142
#211192