Finite Group Databasebeta

← #6454235

Subgroups — #6454235

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2C22×#440C211yesmaximal, characteristic
3#35248C211yesmaximal, characteristic, complemented
4#34190C211yesmaximal, characteristic
5#24834C212yesmaximal, complemented
6#24748C212yesmaximal, complemented
7#24697C212yesmaximal, complemented
8#24646C212yesmaximal, complemented
9#24577C212yesmaximal, complemented
10#24527C212yesmaximal
11#24480C212yesmaximal, complemented
12#24358C212yesmaximal
13#23689C214yesmaximal, complemented
14#21514C214yesmaximal, complemented
15#21407C211yesmaximal, characteristic, complemented
16#21353C211yesmaximal, characteristic, complemented
17#20735C211yesmaximal, characteristic, complemented
18C23×SD16C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#3101—28no—
23C2×#483C2212yes—
24C2×#476C2211yescharacteristic
25C2×#476C2211yescharacteristic
26C2×#461C2214yes—
27C2×#461C2211yescharacteristic
28C2×#461C2211yescharacteristic
29C2×#450C2212yes—
30C2×#440—28no—
31C2×#440C2212yes—
32C2×#440C2212yes—
33C2×#440C2212yes—
34C2×#440C2212yes—
35C2×C4×SD16C2212yes—
36C2×C4×SD16C2212yes—
37C2×#431C2211yescharacteristic
38C22×#107C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×#97C2211yescharacteristic
42#2855—22no—
43#2634—22no—
44#2581—22no—
45#2568C2212yescomplemented
46#2546C2212yescomplemented
47#2503C2212yes—
48#2484—21no—
49#2484—21no—
50#2434C2212yes—
51#2357C2212yes—
52#2353—22no—
53#2349C2212yes—
54#2344C2212yes—
55#2342—22no—
56D4×C22⋊C4C2212yes—
57#2327C2214yescomplemented
58#2319C2212yes—
59#2318—22no—
60#2283C2211yescharacteristic
61#2283C2211yescharacteristic
62#2280—21no—
63C2×#381—21no—
64#2275C2212yes—
65#2085C2214yescomplemented
66#2073C2214yes—
67#2068C2214yescomplemented
68#2066C2214yescomplemented
69#2055C2214yes—
70#2055C2214yes—
71#2043C2214yescomplemented
72#2013C2214yes—
73#1993C2212yescomplemented
74#1981C2214yescomplemented
75#1960C2212yescomplemented
76#1949C2214yescomplemented
77#1940C2212yescomplemented
78#1940C2212yescomplemented
79#1928C2214yescomplemented
80#1927C2214yescomplemented
81#1892C2212yes—
82#1892C2212yes—
83#1877C2214yes—
84#1877C2214yes—
85#1850C2212yes—
86#1850C2212yes—
87#1823C2212yes—
88#1805C2212yescomplemented
89#1805C2212yescomplemented
90#1804C2214yes—
91#1787C2212yes—
92#1768C2214yescomplemented
93#1763C2214yescomplemented
94C4×#103C2214yescomplemented
95C23×Q8C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16—24no—
99C22×SD16—28no—
100C22×SD16C2312yes—
101C22×SD16C2312yes—
102C22×SD16C2312yes—
103C22×SD16C2312yes—
104C23×C8C2311yescharacteristic
105C23×C8C2311yescharacteristic
106#533—24no—
107#533—24no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#125—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—24no—
125C2×C4×D4D418yes—
126C2×C4×D4C2314yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137C22×C22⋊C4—22no—
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140#483—28no—
141#476—24no—
142#476—24no—
143#461—216no—
144#461—24no—
145#461—24no—
146#450—28no—
147#440—28no—
148#440—28no—
149#440—28no—
150#440—28no—
151C4×SD16—28no—
152C4×SD16—28no—
153#431—24no—
154C2×#107—24no—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2314yes—
163C2×#103—24no—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#102—24no—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×C22⋊C8C2312yes—
178C2×#97—24no—
179C2×#97C2312yes—
180C2×#97C2312yes—
181C2×C4×C8C2312yes—
182#400—22no—
183#400—22no—
184#400—24no—
185#398—22no—
186#398—22no—
187#395—24no—
188#391—24no—
189#391C2314yes—
190#391C2314yes—
191#390C2314yes—
192#389—24no—
193#389C2314yes—
194#388—24no—
195#388—24no—
196#387—24no—
197#387—24no—
198#387—24no—
199#387—24no—
200#387—24no—
201#386—24no—
202#385—24no—
203#385—22no—
204#385—22no—
205#383—21no—
206#383D412yes—
207#382D412yes—
208#382C2314yes—
209#382C2312yes—
210#381—21no—
211#381—22no—
212#381—22no—
213#381—22no—
214#381—22no—
215#381—22no—
216#381D411yescharacteristic
217#381D411yescharacteristic
218#380—22no—
219#380—22no—
220#380D412yes—
221#380C2312yes—
222#379—22no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2314yes—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—24no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4C2312yes—
231#376—21no—
232#376C2312yes—
233C2×#95—22no—
234C2×#95—22no—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C25—21no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—42no—
248C22×Q8C2412yes—
249C22×Q8C2412yes—
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—42no—
254C22×D4C2×D414yes—
255C22×D4C2411yescharacteristic
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—28no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—24no—
266C23×C4—84no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×SD16—232no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—28no—
278C2×SD16—24no—
279C2×SD16—44no—
280C2×SD16#4118yes—
281C22×C8—22no—
282C22×C8—24no—
283C22×C8—22no—
284C22×C8—42no—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—28no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—44no—
295#125—44no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—44no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—44no—
316C4×D4C2×D414yes—
317C4×D4C2×D414yes—
318C4×D4C2×D414yes—
319C4×D4C2×D414yes—
320#117—44no—
321#117—84no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—28no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—84no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D414yes—
367C2×C42#4114yes—
368C2×C42C2414yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—24no—
372#107—24no—
373#106—24no—
374#106—24no—
375C4⋊C8—28no—
376C4⋊C8—22no—
377C4⋊C8—22no—
378C4⋊C8—44no—
379#103—28no—
380#103—28no—
381#103—24no—
382#103—24no—
383#103—24no—
384#103—24no—
385#103—44no—
386#103—44no—
387#102—24no—
388#102—24no—
389#102—24no—
390#102—24no—
391C22⋊C8—24no—
392#97—24no—
393#97—24no—
394C4×C8—24no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95C2×D412yes—
408#95C2×D412yes—
409#95C2×D412yes—
410#95C2×D414yes—
411#95C2×D412yes—
412#95C2×D414yes—
413C24—21no—
414C24—21no—
415C24—24no—
416C24—44no—
417C24—44no—
418C24—82no—
419C24C22×D411yescharacteristic
420C2×Q8—22no—
421C2×Q8—28no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8C2×#4114yes—
437C2×D4—28no—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—44no—
445C2×D4—48no—
446C2×D4—44no—
447C2×D4C22×D412yes—
448C2×D4C22×D412yes—
449C2×D4C2×#4114yes—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—816no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4#12114yes—
480C22×C4#12114yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4C2511yescharacteristic
491SD16—216no—
492SD16—216no—
493SD16—48no—
494SD16—48no—
495C2×C8—28no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—44no—
506C2×C8C2×#4114yes—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—84no—
528C4⋊C4C22×D412yes—
529C4⋊C4C22×D412yes—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—24no—
542C42—24no—
543C42—44no—
544C42—44no—
545C42—44no—
546C42—82no—
547C42—84no—
548C42#12114yes—
549C42#12114yes—
550C42C22×D412yes—
551C42C22×D412yes—
552C42C2×#4112yes—
553C42C2×#4112yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—22no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—48no—
565C23—42no—
566C23—84no—
567C23—816no—
568C23C2×#12112yes—
569C23#57714yes—
570C23C23×D411yescharacteristic
571Q8—24no—
572Q8—24no—
573Q8—48no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—44no—
579Q8—84no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4—24no—
584D4—48no—
585D4—42no—
586D4#52514yes—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—168no—
627C2×C4C2×#12112yes—
628C2×C4C2×#12112yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C22×#4112yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4#58512yes—
636C8—24no—
637C8—42no—
638C8—42no—
639C8—42no—
640C22—22no—
641C22—28no—
642C22—22no—
643C22—24no—
644C22—42no—
645C22—42no—
646C22—48no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—816no—
651C22—84no—
652C22#304214yes—
653C22C22×#12111yescharacteristic
654C22D4×#4112yes—
655C22C2×#57712yes—
656C22#358612yes—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4—82no—
671C4C2×#52511yescharacteristic
672C4#344911yescharacteristic
673C4#347512yes—
674C2—24no—
675C2—44no—
676C2—82no—
677C2#3473711yescharacteristic
678C2C2×#304211yescharacteristic
679C2#6042711yescharacteristic
680C2#6152014yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #35248C2semidirect
5. #24834C2semidirect
6. #24748C2semidirect
7. #24697C2semidirect
8. #24646C2semidirect
9. #24577C2semidirect
11. #24480C2semidirect
13. #23689C2semidirect
14. #21514C2semidirect
15. #21407C2semidirect
16. #21353C2semidirect
17. #20735C2semidirect
45. #2568C22semidirect
46. #2546C22semidirect
57. #2327C22semidirect
65. #2085C22semidirect
67. #2068C22semidirect
68. #2066C22semidirect
71. #2043C22semidirect
73. #1993C22semidirect
74. #1981C22semidirect
75. #1960C22semidirect
76. #1949C22semidirect
77. #1940C22semidirect
78. #1940C22semidirect
79. #1928C22semidirect
80. #1927C22semidirect
88. #1805C22semidirect
89. #1805C22semidirect
92. #1768C22semidirect
93. #1763C22semidirect
94. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C22×#4402
#352482
#341902
#248342
#247482
#246972
#246462
#245772
#245272
#244802
#243582
#236892
#215142
#214072
#213532
#207352