Finite Group Databasebeta

← #6683898

Subgroups — #6683898

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×SD16C211yesmaximal, characteristic
2C2×#3372C211yesmaximal, characteristic
3#39619C211yesmaximal, characteristic, complemented
4#36276C211yesmaximal, characteristic
5#33026C211yesmaximal, characteristic
6#30956C214yesmaximal, complemented
7#30828C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29867C211yesmaximal, characteristic, complemented
10#28684C211yesmaximal, characteristic, complemented
11#28638C211yesmaximal, characteristic, complemented
12#26396C214yesmaximal
13#26170C214yesmaximal
14#25929C211yesmaximal, characteristic
15#25841C211yesmaximal, characteristic, complemented
16#25839C211yesmaximal, characteristic
17#25647C211yesmaximal, characteristic, complemented
18#22240C211yesmaximal, characteristic, complemented
19#21114C211yesmaximal, characteristic
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24Q8×SD16—28no—
25#3372—28no—
26C2×#500C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#498C2212yes—
29C2×#491C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#475C2212yes—
36C2×#476C2211yescharacteristic
37C2×#476C2212yes—
38C2×#461C2211yescharacteristic
39C2×#461C2212yes—
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×C4×SD16C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2857—21no—
48#2713—24no—
49#2660—21no—
50#2568—21no—
51#2568C2211yescharacteristic
52#2567C2211yescharacteristic
53#2567C2211yescharacteristic
54#2565—21no—
55#2550C2214yes—
56#2549—24no—
57#2546C2211yescharacteristic
58#2502—21no—
59#2451C2214yes—
60#2446C2214yes—
61#2434—21no—
62#2349C2214yes—
63#2327—21no—
64C4×C4⋊Q8C2211yescharacteristic
65#2295C2211yescharacteristic
66C2×#391—21no—
67C42×D4C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2088C2214yes—
71#2085C2214yes—
72#2074C2214yes—
73#2068C2214yes—
74#2059C2214yes—
75#2057C2214yes—
76#2043C2214yes—
77#2024C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1981C2211yescharacteristic
84#1943C2214yes—
85#1933C2214yes—
86#1928C2212yes—
87#1927C2214yes—
88#1919C2212yes—
89#1917C2214yes—
90#1916C2214yes—
91#1877C2214yes—
92#1877C2214yes—
93#1877C2212yes—
94#1865C2214yes—
95#1863C2214yes—
96#1862C2212yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C23×Q8—21no—
104C22×SD16C2312yes—
105C22×SD16C2311yescharacteristic
106Q82—24no—
107Q82—42no—
108#557—24no—
109#557—42no—
110#554—24no—
111#554—42no—
112D4×Q8—24no—
113D4×Q8—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8D411yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2312yes—
125C2×#125—21no—
126C2×#125—22no—
127C2×#125C2312yes—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C22×C4⋊C4—21no—
144C22×C42—21no—
145#500—42no—
146#498—28no—
147#498—24no—
148#491—28no—
149#487—24no—
150#487—42no—
151#479—28no—
152#477—24no—
153#476—24no—
154#476—28no—
155#475—24no—
156#475—28no—
157#461—28no—
158#461—24no—
159#446—24no—
160C8×Q8—24no—
161#440—28no—
162C4×SD16—28no—
163C4×SD16D418yes—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#107C2314yes—
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2311yescharacteristic
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2314yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2311yescharacteristic
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2311yescharacteristic
188C2×#97C2312yes—
189C2×C4×C8C2311yescharacteristic
190C2×C4×C8C2312yes—
191C2×C4×C8C2311yescharacteristic
192#400—22no—
193#400—24no—
194#398—24no—
195#398—24no—
196#398—22no—
197#398—24no—
198#395—22no—
199#395—22no—
200#395—24no—
201#395—24no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391—22no—
210#391—24no—
211#391—24no—
212#391D411yescharacteristic
213#391D411yescharacteristic
214#391C2314yes—
215#391C2312yes—
216#391C2314yes—
217#389—24no—
218#389D414yes—
219#389C2314yes—
220#389C2314yes—
221#389C2312yes—
222#389C2314yes—
223#387—24no—
224#387—24no—
225#387—22no—
226#387—24no—
227#385—24no—
228#385—24no—
229#384—21no—
230#384—21no—
231#384D411yescharacteristic
232#382—21no—
233#382—21no—
234#382—21no—
235#382C2314yes—
236#380—24no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4C2311yescharacteristic
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2314yes—
242C4×C22⋊C4—21no—
243C4×C22⋊C4—22no—
244#376—21no—
245C2×#95—24no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—21no—
263C22×Q8—44no—
264C22×Q8—44no—
265C22×Q8—82no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2×D411yescharacteristic
268C22×Q8C2412yes—
269C22×Q8C2411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—28no—
278C2×SD16C2×D414yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C22×C8C2411yescharacteristic
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—42no—
297C4⋊Q8—42no—
298C4⋊Q8—44no—
299C4⋊Q8—42no—
300C4⋊Q8—48no—
301C4⋊Q8—82no—
302#125—24no—
303#125—42no—
304#125—44no—
305#125—48no—
306C22⋊Q8—48no—
307C22⋊Q8—48no—
308C22⋊Q8—44no—
309C22⋊Q8—44no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8C2×D414yes—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—42no—
328C4×D4—42no—
329C4×D4C2×D414yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
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×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—82no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D412yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380#107—24no—
381#107—24no—
382#107—44no—
383#106—24no—
384#106—24no—
385#106—24no—
386#106—24no—
387#106C2×D414yes—
388C4⋊C8—24no—
389C4⋊C8—24no—
390C4⋊C8—22no—
391C4⋊C8—22no—
392#103—24no—
393#103—24no—
394#103—24no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—28no—
399#103—44no—
400#103C2×D414yes—
401#102—24no—
402#102—24no—
403#102C2×D414yes—
404#97—24no—
405C4×C8—24no—
406C4×C8—22no—
407C4×C8C2×D414yes—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—22no—
412#95—22no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95C2×D414yes—
424#95C2×D414yes—
425C24—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×Q8—84no—
450C2×Q8—816no—
451C2×Q8#13714yes—
452C2×Q8C22×D412yes—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4C22×D412yes—
456C2×D42−1+414yes—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—28no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—84no—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2511yescharacteristic
486SD16—28no—
487SD16—48no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8C22×D412yes—
498C2×C8C22×D412yes—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—28no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—82no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4C22×D412yes—
535C4⋊C4C22×D412yes—
536C4⋊C42−1+414yes—
537C22⋊C4—28no—
538C22⋊C4—24no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—42no—
551C42—44no—
552C42—44no—
553C42—42no—
554C42—42no—
555C42—82no—
556C42C22×D412yes—
557C42H(5,𝔽2)12yes—
558C23—22no—
559C23—24no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—22no—
566Q8—22no—
567Q8—44no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—44no—
572Q8—88no—
573Q8—84no—
574Q8—162no—
575Q8—162no—
576Q8—164no—
577D4—22no—
578D4—24no—
579D4#53614yes—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—88no—
620C2×C4—88no—
621C2×C4—164no—
622C2×C4D4212yes—
623C2×C4D4212yes—
624C2×C4C2×#13714yes—
625C2×C4C2×#13712yes—
626C2×C4#57712yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+412yes—
630C2×C4C2×2−1+412yes—
631C8—22no—
632C8—22no—
633C8—42no—
634C8—42no—
635C8—42no—
636C22—24no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—44no—
641C22C2×D4211yescharacteristic
642C22#348612yes—
643C22C22×#13711yescharacteristic
644C22C2×#57711yescharacteristic
645C22#358412yes—
646C4—24no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4#326614yes—
663C4C2×#53611yescharacteristic
664C4#349911yescharacteristic
665C4#355212yes—
666C2—44no—
667C2#3767511yescharacteristic
668C2C2×#326611yescharacteristic
669C2#6108811yescharacteristic
670C2#6184612yes—
671C2#6186412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #39619C2semidirect
6. #30956C2semidirect
7. #30828C2semidirect
8. #30281C2semidirect
9. #29867C2semidirect
10. #28684C2semidirect
11. #28638C2semidirect
15. #25841C2semidirect
17. #25647C2semidirect
18. #22240C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×SD162
C2×#33722
#396192
#362762
#330262
#309562
#308282
#302812
#298672
#286842
#286382
#263962
#261702
#259292
#258412
#258392
#256472
#222402
#211142