Finite Group Databasebeta

← #6815365

Subgroups — #6815365

Order 512.

1740 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×#3369C211yesmaximal, characteristic
2C2×#3359C211yesmaximal, characteristic
3#39619C211yesmaximal, characteristic, complemented
4#38954C211yesmaximal, characteristic
5#33255C211yesmaximal, characteristic, complemented
6#31246C214yesmaximal, complemented
7#31221C214yesmaximal
8#29831C211yesmaximal, characteristic, complemented
9#29828C211yesmaximal, characteristic
10#28646C211yesmaximal, characteristic, complemented
11#28638C211yesmaximal, characteristic, complemented
12#26838C214yesmaximal, complemented
13#26824C214yesmaximal
14#25837C211yesmaximal, characteristic, complemented
15#25834C211yesmaximal, characteristic
16#25634C211yesmaximal, characteristic, complemented
17#25632C211yesmaximal, characteristic
18#21714C211yesmaximal, characteristic, complemented
19#21279C211yesmaximal, characteristic
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3369—28no—
25#3359—28no—
26C2×#498C2211yescharacteristic
27C2×#494C2211yescharacteristic
28C2×#489C2212yes—
29C2×#487C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#475C2212yes—
33C2×#475C2212yes—
34C2×#475C2211yescharacteristic
35C2×#476C2211yescharacteristic
36C2×#464C2212yes—
37C2×#462C2211yescharacteristic
38C2×#461C2211yescharacteristic
39C2×#461C2212yes—
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#441C2212yes—
43C2×C4×SD16C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2713—24no—
48#2698—24no—
49#2660—21no—
50#2621—21no—
51#2568—21no—
52#2568C2211yescharacteristic
53#2568C2211yescharacteristic
54#2567C2211yescharacteristic
55#2565—21no—
56#2546C2211yescharacteristic
57#2499—21no—
58#2451C2214yes—
59#2451C2214yes—
60#2447C2214yes—
61#2446C2214yes—
62#2434—21no—
63#2327—21no—
64#2310—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2295C2211yescharacteristic
67#2293C2211yescharacteristic
68#2068C2214yes—
69#2068C2214yes—
70#2066C2214yes—
71#2066C2214yes—
72#2058C2214yes—
73#2058C2214yes—
74#2056C2214yes—
75#2055C2214yes—
76#2016C2214yes—
77#2015C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1981C2211yescharacteristic
84#1973C2212yes—
85#1971C2214yes—
86#1971C2214yes—
87#1944C2212yes—
88#1940C2214yes—
89#1936C2212yes—
90#1935C2212yes—
91#1935C2214yes—
92#1933C2214yes—
93#1878C2214yes—
94#1878C2214yes—
95#1865C2214yes—
96#1864C2214yes—
97#1842C2211yescharacteristic
98C4×#107C2211yescharacteristic
99#1771C2214yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C23×Q8—21no—
104C22×Q16C2312yes—
105C22×Q16C2314yes—
106C22×SD16C2312yes—
107C22×SD16C2311yescharacteristic
108Q82—24no—
109Q82—42no—
110#557—24no—
111#557—42no—
112#556—24no—
113#556—42no—
114D4×Q8—24no—
115D4×Q8—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8D411yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2312yes—
126C2×C4⋊Q8C2312yes—
127C2×#126—22no—
128C2×#125—21no—
129C2×#125—22no—
130C2×#124C2312yes—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—22no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4C2312yes—
143C2×C4×D4C2311yescharacteristic
144C2×#117—21no—
145C2×#117—21no—
146#498—42no—
147#494—42no—
148#489—28no—
149#487—28no—
150#487—24no—
151#487—24no—
152#476—24no—
153#475—24no—
154#475—28no—
155#475—28no—
156#464—28no—
157#462—24no—
158#461—28no—
159#461—24no—
160#446—24no—
161C8×Q8—24no—
162#441—28no—
163C4×SD16—28no—
164C4×SD16D418yes—
165C2×#107C2312yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×#106C2314yes—
169C2×#106C2311yescharacteristic
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2314yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2311yescharacteristic
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2311yescharacteristic
189C2×#97C2312yes—
190C2×#97C2312yes—
191C2×#97C2312yes—
192C2×C4×C8C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195#400—22no—
196#400—22no—
197#400—24no—
198#398—24no—
199#398—22no—
200#398—24no—
201#398—24no—
202#397—24no—
203#397—24no—
204#395—22no—
205#395—24no—
206#395—24no—
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#391—24no—
216#391D411yescharacteristic
217#391D411yescharacteristic
218#391C2314yes—
219#391C2312yes—
220#391C2314yes—
221#391C2314yes—
222#391C2314yes—
223#390C2312yes—
224#389D414yes—
225#389C2314yes—
226#389C2314yes—
227#389C2312yes—
228#389C2312yes—
229#388—22no—
230#387—24no—
231#387—24no—
232#387—24no—
233#387—22no—
234#387—24no—
235#385—24no—
236#384—21no—
237#384D411yescharacteristic
238#382—21no—
239#382—21no—
240#382—21no—
241#382—21no—
242#382—22no—
243#381—24no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2314yes—
248C4×C22⋊C4—21no—
249#376—21no—
250C43C2311yescharacteristic
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8—44no—
263C22×Q8—44no—
264C22×Q8—82no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2412yes—
268C22×Q8C2412yes—
269C22×Q8C2411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—22no—
273C2×Q16—24no—
274C2×Q16—28no—
275C2×Q16—44no—
276C2×Q16—44no—
277C2×Q16—44no—
278C2×SD16—24no—
279C2×SD16—42no—
280C2×SD16—44no—
281C2×SD16C2×D414yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2411yescharacteristic
285C22×C8C2411yescharacteristic
286C22×C8C2412yes—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—42no—
300C4⋊Q8—42no—
301C4⋊Q8—44no—
302C4⋊Q8—42no—
303C4⋊Q8—48no—
304C4⋊Q8—82no—
305#126—48no—
306#125—44no—
307#125—48no—
308#124—24no—
309#124—42no—
310C22⋊Q8—48no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8C2×D414yes—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—42no—
330C4×D4C2×D414yes—
331#117—44no—
332#117—82no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
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×C42C2412yes—
381C2×C42C2411yescharacteristic
382#107—42no—
383#106—24no—
384#106—24no—
385#106—44no—
386#106C2×D414yes—
387C4⋊C8—24no—
388C4⋊C8—24no—
389C4⋊C8—22no—
390C4⋊C8—22no—
391C4⋊C8—44no—
392#103—24no—
393#103—24no—
394#103—24no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—44no—
401#103—44no—
402#103C2×D414yes—
403#102—24no—
404#102—24no—
405#102C2×D414yes—
406#97—24no—
407#97—24no—
408#97—42no—
409C4×C8—24no—
410C4×C8—22no—
411C4×C8C2×D414yes—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—21no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95C2×D414yes—
429#95C2×D414yes—
430C24—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—44no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×Q8—84no—
457C2×Q8—816no—
458C2×Q8#13714yes—
459C2×Q8C22×D412yes—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4C22×D412yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—82no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488Q16—48no—
489Q16—88no—
490SD16—44no—
491SD16—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8C22×D412yes—
501C2×C8C22×D412yes—
502C2×C82−1+414yes—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—28no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—82no—
536C4⋊C4—84no—
537C4⋊C4C22×D412yes—
538C4⋊C4C22×D412yes—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—82no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—42no—
555C42—44no—
556C42—44no—
557C42—44no—
558C42—42no—
559C42—42no—
560C42C22×D412yes—
561C42H(5,𝔽2)12yes—
562C23—22no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23C23×D411yescharacteristic
568Q8—28no—
569Q8—22no—
570Q8—22no—
571Q8—44no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—44no—
578Q8—44no—
579Q8—88no—
580Q8—84no—
581Q8—162no—
582Q8—164no—
583D4—22no—
584D4—22no—
585D4—42no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—88no—
626C2×C4D4212yes—
627C2×C4D4212yes—
628C2×C4C2×#13714yes—
629C2×C4C2×#13712yes—
630C2×C4#57712yes—
631C2×C4C23×D411yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+412yes—
634C2×C4#58512yes—
635C8—22no—
636C8—22no—
637C8—24no—
638C8—42no—
639C8—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—82no—
644C22—82no—
645C22C2×D4211yescharacteristic
646C22#349712yes—
647C22C22×#13711yescharacteristic
648C22C2×#57711yescharacteristic
649C22#358612yes—
650C4—24no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4#326614yes—
667C4#344911yescharacteristic
668C4#349911yescharacteristic
669C4#356112yes—
670C2—82no—
671C2#3935611yescharacteristic
672C2C2×#326611yescharacteristic
673C2#6108311yescharacteristic
674C2#6239812yes—
675C2#6240712yes—
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
5. #33255C2semidirect
6. #31246C2semidirect
8. #29831C2semidirect
10. #28646C2semidirect
11. #28638C2semidirect
12. #26838C2semidirect
14. #25837C2semidirect
16. #25634C2semidirect
18. #21714C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33692
C2×#33592
#396192
#389542
#332552
#312462
#312212
#298312
#298282
#286462
#286382
#268382
#268242
#258372
#258342
#256342
#256322
#217142
#212792