Finite Group Databasebeta

← #6487689

Subgroups — #6487689

Order 512.

2071 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic, complemented
2C2×#3108C211yesmaximal, characteristic
3#35241C211yesmaximal, characteristic
4C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#32909C212yesmaximal
6#32906C212yesmaximal, complemented
7#29760C212yesmaximal
8#29756C212yesmaximal, complemented
9#24826C214yesmaximal, complemented
10#24818C212yesmaximal, complemented
11#24813C212yesmaximal, complemented
12#24741C214yesmaximal, complemented
13#24732C212yesmaximal, complemented
14#24729C212yesmaximal, complemented
15#21953C211yesmaximal, characteristic, complemented
16#21867C211yesmaximal, characteristic
17#21860C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#3113—28no—
23#3108—28no—
24C2×#483C2211yescharacteristic
25C2×#483C2212yes—
26C2×#480C2211yescharacteristic
27C2×#479C2212yes—
28C2×#475C2212yes—
29C2×#464C2212yes—
30C2×#464C2214yes—
31C2×#462C2212yes—
32C2×#462C2214yes—
33C2×#451C2211yescharacteristic
34C2×#451C2212yes—
35C2×#448C2211yescharacteristic
36C2×#441C2214yes—
37C2×C4×Q16C2212yes—
38C2×C4×SD16C2212yes—
39C2×#432C2211yescharacteristic
40C2×#431C2211yescharacteristic
41#2857C2212yes—
42#2850C2212yes—
43#2664—22no—
44#2568C2211yescharacteristic
45#2565C2212yes—
46#2547C2211yescharacteristic
47#2546C2211yescharacteristic
48#2545C2212yes—
49#2542C2211yescharacteristic
50#2498—22no—
51#2417—21no—
52#2417—21no—
53Q8×C4⋊C4—21no—
54Q8×C4⋊C4—21no—
55Q8×C4⋊C4—22no—
56Q8×C4⋊C4C2212yes—
57Q8×C4⋊C4C2212yes—
58Q8×C4⋊C4C2214yes—
59#2323—21no—
60#2323—21no—
61C2×#389—22no—
62C2×#389C2212yes—
63C2×#384—21no—
64C2×#384C2211yescharacteristic
65C2×C4×C4⋊C4—21no—
66C2×C4×C4⋊C4C2211yescharacteristic
67#2085C2214yes—
68#2084C2214yes—
69#2068C2214yes—
70#2066C2214yes—
71#2066C2214yes—
72#2065C2214yes—
73#2043C2214yes—
74#2042C2214yes—
75#1992C2212yes—
76#1990C2212yes—
77#1959C2212yes—
78#1956C2212yes—
79#1940C2214yes—
80#1939C2212yes—
81#1939C2212yes—
82#1880C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1878C2214yes—
86#1874C2212yes—
87#1872C2212yes—
88#1872C2212yes—
89#1872C2212yes—
90#1805C2214yes—
91#1803C2214yes—
92#1803C2212yes—
93#1803C2214yes—
94#1802C2212yes—
95#1797C2212yes—
96#1797C2212yes—
97C23×Q8C2311yescharacteristic
98C22×Q16C2312yes—
99C22×Q16C2314yes—
100C22×SD16C2312yes—
101#533—24no—
102#533—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106#529—24no—
107#529—24no—
108C2×#126—21no—
109C2×#125—21no—
110C2×#125C2311yescharacteristic
111C2×#124C2311yescharacteristic
112C2×#124C2312yes—
113C2×#123—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2311yescharacteristic
119C2×#121C2311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—22no—
124C2×C4×Q8—22no—
125C2×C4×Q8—24no—
126C2×C4×Q8—48no—
127C2×C4×Q8D418yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×Q8C2314yes—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×D4—21no—
135C2×C4×D4C2311yescharacteristic
136C2×#117—21no—
137C2×#117—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4D411yescharacteristic
144C22×C4⋊C4D411yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C42—21no—
148C22×C42—22no—
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151#483—24no—
152#483—28no—
153#480—24no—
154#479—28no—
155#475—28no—
156#464—28no—
157#464—48no—
158#462—28no—
159#462—48no—
160#451—24no—
161#451—28no—
162#448—24no—
163#441—216no—
164C4×Q16—28no—
165C4×SD16—28no—
166#432—24no—
167#431—24no—
168C2×#107C2312yes—
169C2×#106C2314yes—
170C2×#106C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2314yes—
174C2×C4⋊C8C2314yes—
175C2×#103C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2314yes—
180C2×#103C2314yes—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#102C2312yes—
184C2×#102C2314yes—
185C2×C22⋊C8C2311yescharacteristic
186C2×C22⋊C8C2311yescharacteristic
187C2×C22⋊C8C2312yes—
188C2×#97C2312yes—
189C2×C4×C8C2312yes—
190#400—22no—
191#399—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#397—22no—
196#396—22no—
197#396—22no—
198#391—21no—
199#391—21no—
200#391C2312yes—
201#391C2312yes—
202#391C2312yes—
203#390C2312yes—
204#389—21no—
205#389—21no—
206#389—22no—
207#389—22no—
208#389—22no—
209#389—22no—
210#389—24no—
211#389—24no—
212#389—24no—
213#389—24no—
214#389C2314yes—
215#389C2314yes—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#387—22no—
221#387—22no—
222#386—22no—
223#386—22no—
224#384—22no—
225#384—22no—
226#384—24no—
227#384—21no—
228#384—21no—
229#384C2312yes—
230#384C2312yes—
231#384C2312yes—
232#382C2312yes—
233#380—22no—
234#380—22no—
235#380—21no—
236#380—21no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—24no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D412yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2311yescharacteristic
246C2×#95—22no—
247C2×#95D412yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#336C2314yes—
253#336C2314yes—
254C22×Q8—24no—
255C22×Q8—24no—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—22no—
259C22×Q8C2×D414yes—
260C22×Q8C2411yescharacteristic
261C22×Q8C2412yes—
262C22×Q8C2412yes—
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D412yes—
273C23×C4C2411yescharacteristic
274C2×Q16—24no—
275C2×Q16—28no—
276C2×Q16—24no—
277C2×Q16—28no—
278C2×Q16—24no—
279C2×Q16—28no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2414yes—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—44no—
291#126—44no—
292#125—22no—
293#125—22no—
294#125—44no—
295#124—22no—
296#124—22no—
297#124—44no—
298#123—44no—
299C22⋊Q8—22no—
300C22⋊Q8—24no—
301C22⋊Q8—22no—
302C22⋊Q8—24no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306#121—22no—
307#121—22no—
308C4×Q8—28no—
309C4×Q8—28no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—28no—
313C4×Q8—28no—
314C4×Q8—22no—
315C4×Q8—24no—
316C4×Q8—22no—
317C4×Q8—24no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—48no—
321C4×Q8—48no—
322C4×Q8—44no—
323C4×Q8C2×D418yes—
324C4×Q8C2×D418yes—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—44no—
328#117—44no—
329#117—44no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—42no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4—48no—
362C2×C4⋊C4—82no—
363C2×C4⋊C4—84no—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D414yes—
369C2×C4⋊C4C2×D412yes—
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C42—28no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—21no—
390C2×C42—22no—
391C2×C42—24no—
392C2×C42—24no—
393C2×C42—44no—
394C2×C42—48no—
395C2×C42C2×D412yes—
396C2×C42C2×D412yes—
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D414yes—
400C2×C42#4114yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—24no—
406#106—24no—
407#106—28no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—44no—
411C4⋊C8—44no—
412#103—24no—
413#103—24no—
414#103—28no—
415#103—24no—
416#103—28no—
417#103—44no—
418#103—44no—
419#103—44no—
420#102—24no—
421#102—44no—
422C22⋊C8—22no—
423C22⋊C8—22no—
424C22⋊C8—24no—
425#97—24no—
426C4×C8—24no—
427#95—24no—
428#95—24no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95C2×D412yes—
438#95C2×D412yes—
439#95C2×D412yes—
440#95C2×D412yes—
441C24—21no—
442C24C22×D411yescharacteristic
443C2×Q8—24no—
444C2×Q8—216no—
445C2×Q8—24no—
446C2×Q8—216no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—24no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8C22×D414yes—
463C2×Q8H(5,𝔽2)14yes—
464C2×Q82−1+414yes—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4#12114yes—
505C22×C4#12114yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C2×#4112yes—
517C22×C4C2511yescharacteristic
518Q16—48no—
519Q16—416no—
520SD16—48no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—24no—
530C4⋊C4—28no—
531C4⋊C4—28no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—42no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—84no—
566C4⋊C4—84no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4#13714yes—
570C4⋊C4C22×D414yes—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—42no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—42no—
577C22⋊C4—42no—
578C22⋊C4—44no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C42—28no—
582C42—28no—
583C42—24no—
584C42—24no—
585C42—42no—
586C42—42no—
587C42—48no—
588C42—48no—
589C42—42no—
590C42—44no—
591C42#12114yes—
592C42#12114yes—
593C42C22×D412yes—
594C42C22×D412yes—
595C42C2×#4112yes—
596C42C2×#4112yes—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23C2×#12112yes—
609C23#57712yes—
610C23#57812yes—
611C23C23×D411yescharacteristic
612Q8—216no—
613Q8—24no—
614Q8—24no—
615Q8—24no—
616Q8—42no—
617Q8—44no—
618Q8—42no—
619Q8—44no—
620Q8—84no—
621Q8#53718yes—
622D4—42no—
623D4—42no—
624D4—84no—
625C2×C4—216no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—21no—
630C2×C4—24no—
631C2×C4—21no—
632C2×C4—28no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—28no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—41no—
672C2×C4—44no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—84no—
676C2×C4—88no—
677C2×C4—84no—
678C2×C4C2×#12112yes—
679C2×C4C2×#12112yes—
680C2×C4C2×#12112yes—
681C2×C4C2×#12112yes—
682C2×C4C2×#12112yes—
683C2×C4C2×#13714yes—
684C2×C4C2×#13712yes—
685C2×C4#57512yes—
686C2×C4C23×D411yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)12yes—
689C2×C4C2×2−1+412yes—
690C8—42no—
691C8—42no—
692C8—44no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22#304614yes—
707C22C22×#12111yescharacteristic
708C22#348411yescharacteristic
709C22#348611yescharacteristic
710C22C22×#13711yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57711yescharacteristic
713C22C2×#57811yescharacteristic
714C4—22no—
715C4—22no—
716C4—28no—
717C4—22no—
718C4—24no—
719C4—22no—
720C4—44no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—81no—
725C4—81no—
726C4—82no—
727C4—84no—
728C4—81no—
729C4—82no—
730C4#303714yes—
731C4C2×#53712yes—
732C4#349212yes—
733C2—24no—
734C2—161no—
735C2#3522311yescharacteristic
736C2C2×#303711yescharacteristic
737C2C2×#304611yescharacteristic
738C2#6159611yescharacteristic
739C2#6160211yescharacteristic
740C2#6183811yescharacteristic
741C2#6184211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3113C2semidirect
4. C2×Q8×C4⋊C4C2semidirect
6. #32906C2semidirect
8. #29756C2semidirect
9. #24826C2semidirect
10. #24818C2semidirect
11. #24813C2semidirect
12. #24741C2semidirect
13. #24732C2semidirect
14. #24729C2semidirect
15. #21953C2semidirect
17. #21860C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#31082
#352412
C2×Q8×C4⋊C42
#329092
#329062
#297602
#297562
#248262
#248182
#248132
#247412
#247322
#247292
#219532
#218672
#218602