Finite Group Databasebeta

← #6816570

Subgroups — #6816570

Order 512.

1730 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic, complemented
2C2×#3340C211yesmaximal, characteristic
3#39489C211yesmaximal, characteristic
4#39172C211yesmaximal, characteristic, complemented
5#33314C211yesmaximal, characteristic, complemented
6#31101C214yesmaximal, complemented
7#31100C214yesmaximal, complemented
8#29869C211yesmaximal, characteristic, complemented
9#29865C211yesmaximal, characteristic, complemented
10#28705C211yesmaximal, characteristic, complemented
11#28684C211yesmaximal, characteristic, complemented
12#26069C214yesmaximal, complemented
13#25841C211yesmaximal, characteristic, complemented
14#25835C211yesmaximal, characteristic, complemented
15#25704C211yesmaximal, characteristic
16#25701C211yesmaximal, characteristic, complemented
17#25041C214yesmaximal, complemented
18#21633C211yesmaximal, characteristic, complemented
19#21279C211yesmaximal, characteristic, complemented
20C2×Q82C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#555C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3385—28no—
25#3340—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#485C2212yes—
32C2×#482C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#477C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#474C2211yescharacteristic
38C2×#474C2212yes—
39C2×#465C2212yes—
40C2×#446C2211yescharacteristic
41C2×#440C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#427C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2716—24no—
48#2674—24no—
49#2640—21no—
50#2620—21no—
51#2568—21no—
52#2567C2211yescharacteristic
53#2563C2211yescharacteristic
54#2563C2211yescharacteristic
55#2546C2211yescharacteristic
56#2545—21no—
57#2501—21no—
58#2499—21no—
59#2441C2214yes—
60#2439C2214yes—
61#2422C2214yes—
62#2421C2214yes—
63#2327—21no—
64#2317—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67#2293C2211yescharacteristic
68#2088C2214yes—
69#2085C2214yes—
70#2083C2214yes—
71#2076C2214yes—
72#2073C2214yes—
73#2072C2214yes—
74#2059C2214yes—
75#2057C2214yes—
76#2035C2214yes—
77#2033C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1987C2211yescharacteristic
82#1984C2211yescharacteristic
83#1973C2212yes—
84#1954C2211yescharacteristic
85#1944C2212yes—
86#1943C2214yes—
87#1936C2212yes—
88#1935C2212yes—
89#1924C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1854C2214yes—
93#1854C2214yes—
94#1847C2214yes—
95#1847C2214yes—
96#1831C2211yescharacteristic
97#1808C2214yes—
98C4×#107C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#102C2211yescharacteristic
101#1752C2214yes—
102C4×#97C2211yescharacteristic
103C22×SD16C2311yescharacteristic
104C23×C8C2312yes—
105Q82—24no—
106Q82—42no—
107#556—24no—
108#556—42no—
109#555—24no—
110#555—42no—
111#554—24no—
112#554—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×#126—22no—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×#124—22no—
126C2×#124D411yescharacteristic
127C2×#123—24no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8C2312yes—
131C2×#121—21no—
132C2×#121C2312yes—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×D4C2312yes—
139C2×C4×D4C2311yescharacteristic
140C2×C4×D4C2311yescharacteristic
141C2×#117—21no—
142C2×#117—21no—
143C2×#117C2312yes—
144C2×#117C2312yes—
145C22×C4⋊C4—21no—
146#501—28no—
147#500—24no—
148#500—42no—
149#498—24no—
150#498—42no—
151#485—28no—
152#482—28no—
153#479—28no—
154#477—24no—
155#477—24no—
156#476—24no—
157#474—24no—
158#474—28no—
159#465—28no—
160#446—24no—
161#440—28no—
162C4×SD16D418yes—
163C8×D4—24no—
164#427—28no—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2314yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2314yes—
176C2×#106C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2311yescharacteristic
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2311yescharacteristic
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2314yes—
190C2×#97C2312yes—
191C2×#97C2312yes—
192C2×#97C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195#400—24no—
196#399—24no—
197#398—22no—
198#397—24no—
199#397—24no—
200#396—24no—
201#396—24no—
202#395—22no—
203#394—24no—
204#393—22no—
205#393—22no—
206#393—24no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—22no—
212#391—24no—
213#391D411yescharacteristic
214#391C2312yes—
215#390—21no—
216#390—21no—
217#390D411yescharacteristic
218#390C2312yes—
219#389—24no—
220#389C2314yes—
221#389C2314yes—
222#389C2312yes—
223#389C2312yes—
224#388—24no—
225#388—22no—
226#388D414yes—
227#387—22no—
228#387—24no—
229#387—24no—
230#387—24no—
231#385—24no—
232#384—21no—
233#384—21no—
234#384—21no—
235#384C2314yes—
236#382—21no—
237#382—21no—
238#382D411yescharacteristic
239#382C2314yes—
240#380C2314yes—
241#380C2314yes—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4C2314yes—
247#376—21no—
248C43C2311yescharacteristic
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—21no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×D4—22no—
266C22×D4C2×D411yescharacteristic
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4—22no—
270C23×C4C2412yes—
271C2×SD16—24no—
272C2×SD16C2×D414yes—
273C22×C8—24no—
274C22×C8—28no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C4⋊Q8—24no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—42no—
290C4⋊Q8—44no—
291C4⋊Q8—48no—
292C4⋊Q8—44no—
293C4⋊Q8—84no—
294#126—48no—
295#125—24no—
296#125—42no—
297#124—42no—
298#124—44no—
299#124—48no—
300#123—88no—
301C22⋊Q8—24no—
302C22⋊Q8—48no—
303C22⋊Q8—42no—
304C22⋊Q8—44no—
305#121—24no—
306#121—42no—
307#121—44no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—24no—
312C4×Q8—42no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8C2×D414yes—
316C4×D4—24no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—42no—
321C4×D4C2×D414yes—
322#117—24no—
323#117—24no—
324#117—44no—
325#117—42no—
326#117—42no—
327#117—44no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D414yes—
368C2×C22⋊C4C2×D414yes—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D412yes—
378C2×C42C2412yes—
379C2×C42C2412yes—
380C2×C42C2411yescharacteristic
381#107—24no—
382#107—24no—
383#107—24no—
384#107—24no—
385#107—24no—
386#107—24no—
387#107—44no—
388#106—24no—
389#106—24no—
390#106—24no—
391#106—44no—
392#106C2×D414yes—
393C4⋊C8—24no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103C2×D414yes—
400#102—24no—
401#102—24no—
402#102—24no—
403#102C2×D414yes—
404C22⋊C8—24no—
405C22⋊C8—28no—
406#97—24no—
407#97—24no—
408#97—24no—
409C4×C8—22no—
410C4×C8C2×D414yes—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—21no—
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—
426C24C22×D412yes—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—82no—
441C2×Q8—82no—
442C2×Q8—82no—
443C2×Q8—84no—
444C2×Q8C22×D412yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—44no—
452C2×D4—84no—
453C2×D4C22×D412yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—84no—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2511yescharacteristic
480SD16—28no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—48no—
489C2×C8C22×D412yes—
490C2×C8C22×D412yes—
491C2×C82−1+414yes—
492C2×C82−1+414yes—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—82no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4C22×D412yes—
527C4⋊C4C22×D412yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—42no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—42no—
553C42—42no—
554C42C22×D412yes—
555C42H(5,𝔽2)12yes—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23—42no—
563C23—42no—
564C23—41no—
565C23—41no—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—44no—
570Q8—42no—
571Q8—84no—
572Q8—162no—
573Q8—162no—
574Q8—162no—
575Q8—164no—
576D4—22no—
577D4—22no—
578D4—44no—
579D4—42no—
580D4—164no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—164no—
622C2×C4D4212yes—
623C2×C4D4212yes—
624C2×C4#57512yes—
625C2×C4#57812yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+412yes—
629C2×C4C2×2−1+412yes—
630C8—22no—
631C8—24no—
632C8—42no—
633C8—42no—
634C8#53614yes—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—82no—
644C22—82no—
645C22C2×D4211yescharacteristic
646C22#349912yes—
647C22C2×#57511yescharacteristic
648C22C2×#57811yescharacteristic
649C22#358512yes—
650C4—22no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4#326914yes—
666C4C2×#53611yescharacteristic
667C4#349911yescharacteristic
668C4#355212yes—
669C2—82no—
670C2—82no—
671C2#3936011yescharacteristic
672C2C2×#326911yescharacteristic
673C2#6108211yescharacteristic
674C2#6250812yes—
675C2#6251912yes—
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×#3385C2semidirect
4. #39172C2semidirect
5. #33314C2semidirect
6. #31101C2semidirect
7. #31100C2semidirect
8. #29869C2semidirect
9. #29865C2semidirect
10. #28705C2semidirect
11. #28684C2semidirect
12. #26069C2semidirect
13. #25841C2semidirect
14. #25835C2semidirect
16. #25701C2semidirect
17. #25041C2semidirect
18. #21633C2semidirect
19. #21279C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33852
C2×#33402
#394892
#391722
#333142
#311012
#311002
#298692
#298652
#287052
#286842
#260692
#258412
#258352
#257042
#257012
#250412
#216332
#212792