Finite Group Databasebeta

← #6683896

Subgroups — #6683896

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3380C211yesmaximal, characteristic
2C2×#3367C211yesmaximal, characteristic
3#39490C211yesmaximal, characteristic, complemented
4#36276C211yesmaximal, characteristic, complemented
5#33026C211yesmaximal, characteristic, complemented
6#30961C214yesmaximal, complemented
7#30823C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29865C211yesmaximal, characteristic, complemented
10#28682C211yesmaximal, characteristic, complemented
11#28639C211yesmaximal, characteristic, complemented
12#26386C214yesmaximal, complemented
13#26160C214yesmaximal
14#25934C211yesmaximal, characteristic, complemented
15#25835C211yesmaximal, characteristic, complemented
16#25834C211yesmaximal, characteristic, complemented
17#25646C211yesmaximal, characteristic, complemented
18#22240C211yesmaximal, characteristic, complemented
19#21118C211yesmaximal, characteristic
20C2×#556C2211yescharacteristic
21C2×#554C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3380—28no—
25#3367—28no—
26C2×#500C2211yescharacteristic
27C2×#499C2212yes—
28C2×#498C2211yescharacteristic
29C2×#489C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#477C2212yes—
33C2×#478C2212yes—
34C2×#478C2212yes—
35C2×#476C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#474C2211yescharacteristic
38C2×#463C2212yes—
39C2×#461C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×C4×D8C2212yes—
44C2×C4×D8C2211yescharacteristic
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2857—21no—
48#2689—24no—
49#2621—21no—
50#2568C2211yescharacteristic
51#2567C2211yescharacteristic
52#2567C2211yescharacteristic
53#2550C2214yes—
54#2549—24no—
55#2546C2211yescharacteristic
56#2503—21no—
57#2502—21no—
58#2499—21no—
59#2453C2214yes—
60#2447C2214yes—
61#2434—21no—
62#2349C2214yes—
63#2317—21no—
64C4×C4⋊Q8C2211yescharacteristic
65C2×#391—21no—
66#2293C2211yescharacteristic
67C42×D4C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2086C2214yes—
71#2085C2214yes—
72#2073C2214yes—
73#2066C2214yes—
74#2059C2214yes—
75#2058C2214yes—
76#2042C2214yes—
77#2026C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1981C2211yescharacteristic
84#1944C2214yes—
85#1933C2214yes—
86#1927C2214yes—
87#1926C2212yes—
88#1921C2212yes—
89#1917C2214yes—
90#1916C2214yes—
91#1879C2212yes—
92#1877C2214yes—
93#1877C2214yes—
94#1865C2214yes—
95#1864C2212yes—
96#1864C2214yes—
97#1864C2214yes—
98#1831C2211yescharacteristic
99C4×#106C2211yescharacteristic
100C4×#102C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C23×Q8—21no—
104C22×SD16C2312yes—
105C22×D8C2311yescharacteristic
106#556—24no—
107#556—42no—
108#554—24no—
109#554—24no—
110#554—42no—
111#554—42no—
112D4×Q8—24no—
113D4×Q8—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8D411yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×#126—22no—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×#124C2312yes—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2311yescharacteristic
134C2×C4×Q8C2311yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2312yes—
139C2×C4×D4C2311yescharacteristic
140C2×C4×D4C2311yescharacteristic
141C2×#117—21no—
142C2×#117—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C42—21no—
146#500—24no—
147#499—28no—
148#498—42no—
149#489—28no—
150#487—24no—
151#487—42no—
152#478—28no—
153#478—28no—
154#477—28no—
155#476—24no—
156#476—24no—
157#474—24no—
158#463—28no—
159#461—24no—
160#446—24no—
161C8×Q8—24no—
162#440—28no—
163C4×D8—28no—
164C4×D8D418yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2311yescharacteristic
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2314yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2314yes—
181C2×#103C2314yes—
182C2×#102C2312yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2311yescharacteristic
188C2×#102C2311yescharacteristic
189C2×#97C2312yes—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2312yes—
192C2×C4×C8C2311yescharacteristic
193#400—22no—
194#400—24no—
195#398—24no—
196#398—24no—
197#398—22no—
198#397—24no—
199#397—24no—
200#395—22no—
201#395—22no—
202#395—24no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391—21no—
210#391—22no—
211#391—24no—
212#391D411yescharacteristic
213#391D411yescharacteristic
214#391C2314yes—
215#391C2314yes—
216#390C2312yes—
217#389—24no—
218#389—24no—
219#389D414yes—
220#389C2314yes—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#388—22no—
225#387—24no—
226#385—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#381—24no—
237#380—24no—
238C4×C4⋊C4C2311yescharacteristic
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2314yes—
242C4×C22⋊C4—21no—
243C4×C22⋊C4—21no—
244C4×C22⋊C4—22no—
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—21no—
262C22×Q8—44no—
263C22×Q8—44no—
264C22×Q8—82no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2412yes—
268C22×D4C2411yescharacteristic
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4—22no—
276C2×SD16—24no—
277C2×SD16—28no—
278C2×D8—24no—
279C2×D8C2×D414yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—42no—
295C4⋊Q8—42no—
296C4⋊Q8—82no—
297#126—48no—
298#125—24no—
299#125—24no—
300#125—42no—
301#125—42no—
302#124—24no—
303#124—42no—
304C22⋊Q8—48no—
305C22⋊Q8—48no—
306C22⋊Q8—48no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—24no—
316C4×Q8—42no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—42no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4C2×D414yes—
328C4×D4C2×D414yes—
329#117—44no—
330#117—44no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
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—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
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×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—82no—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—24no—
385#107—24no—
386#107C2×D414yes—
387#106—24no—
388#106—24no—
389#106—24no—
390#106—44no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—28no—
399#103—44no—
400#102—24no—
401#102—24no—
402#102—24no—
403#102—24no—
404#102—24no—
405#102C2×D414yes—
406#102C2×D414yes—
407#97—24no—
408C4×C8—24no—
409C4×C8—22no—
410C4×C8C2×D414yes—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—21no—
415#95—22no—
416#95—22no—
417#95—21no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95C2×D414yes—
425#95C2×D414yes—
426C24—21no—
427C24—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
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—42no—
444C2×Q8—42no—
445C2×Q8—82no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—816no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4C22×D412yes—
455C2×D4C22×D412yes—
456C2×D42−1+414yes—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—28no—
461C22×C4—24no—
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—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2511yescharacteristic
490SD16—48no—
491D8—28no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8C22×D412yes—
502C2×C8C22×D412yes—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—44no—
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—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4C22×D412yes—
536C4⋊C4C22×D412yes—
537C4⋊C42−1+414yes—
538C22⋊C4—28no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—42no—
557C42—44no—
558C42—42no—
559C42—42no—
560C42—82no—
561C42C22×D412yes—
562C42H(5,𝔽2)12yes—
563C23—22no—
564C23—24no—
565C23—22no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23C23×D411yescharacteristic
574Q8—44no—
575Q8—44no—
576Q8—42no—
577Q8—42no—
578Q8—44no—
579Q8—88no—
580Q8—84no—
581Q8—162no—
582Q8—162no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—24no—
587D4—42no—
588D4#53614yes—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
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—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4—164no—
632C2×C4—164no—
633C2×C4D4212yes—
634C2×C4D4212yes—
635C2×C4#57512yes—
636C2×C4#57712yes—
637C2×C4C23×D411yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+412yes—
640C2×C4C2×2−1+412yes—
641C8—22no—
642C8—22no—
643C8—42no—
644C8—42no—
645C8—42no—
646C22—24no—
647C22—42no—
648C22—44no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—82no—
655C22—82no—
656C22C2×D4211yescharacteristic
657C22#348612yes—
658C22C2×#57511yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358412yes—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#326714yes—
677C4C2×#53611yescharacteristic
678C4#349911yescharacteristic
679C4#355212yes—
680C2—44no—
681C2—82no—
682C2#3767511yescharacteristic
683C2C2×#326711yescharacteristic
684C2#6109011yescharacteristic
685C2#6183912yes—
686C2#6186412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
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. #39490C2semidirect
4. #36276C2semidirect
5. #33026C2semidirect
6. #30961C2semidirect
7. #30823C2semidirect
8. #30281C2semidirect
9. #29865C2semidirect
10. #28682C2semidirect
11. #28639C2semidirect
12. #26386C2semidirect
14. #25934C2semidirect
15. #25835C2semidirect
16. #25834C2semidirect
17. #25646C2semidirect
18. #22240C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33802
C2×#33672
#394902
#362762
#330262
#309612
#308232
#302812
#298652
#286822
#286392
#263862
#261602
#259342
#258352
#258342
#256462
#222402
#211182