Finite Group Databasebeta

← #6502619

Subgroups — #6502619

Order 512.

2158 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×#3189C211yesmaximal, characteristic
2C22×#467C211yesmaximal, characteristic
3#35011C211yesmaximal, characteristic
4#34212C211yesmaximal, characteristic
5#28575C214yesmaximal, complemented
6#25206C214yesmaximal, complemented
7#25082C214yesmaximal, complemented
8#24752C214yesmaximal, complemented
9#24582C212yesmaximal, complemented
10#24532C212yesmaximal, complemented
11#24482C212yesmaximal, complemented
12#24360C212yesmaximal, complemented
13#21759C211yesmaximal, characteristic
14C2×#1892C211yesmaximal, characteristic
15C2×#1850C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C24×C8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21#3189—28no—
22C2×#484C2212yes—
23C2×#484C2212yes—
24C2×#467—28no—
25C2×#467C2214yes—
26C2×#467C2214yes—
27C2×#467C2212yes—
28C2×#467C2212yes—
29C2×#462C2214yescomplemented
30C2×#451C2212yes—
31C2×#451C2212yes—
32C22×#107C2211yescharacteristic
33C22×C4⋊C8C2212yes—
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2212yes—
37C22×C22⋊C8C2211yescharacteristic
38#2857C2214yes—
39#2659—22no—
40#2583—22no—
41#2583—22no—
42#2502C2212yes—
43#2502C2212yes—
44#2434C2212yes—
45#2434C2212yes—
46#2368—22no—
47#2360C2214yes—
48#2345—22no—
49#2345C2212yes—
50Q8×C22⋊C4C2212yes—
51#2329—22no—
52#2315—21no—
53#2315—21no—
54#2315—21no—
55C4×C22⋊Q8C2214yes—
56C2×#380C2211yescharacteristic
57#2273—21no—
58C2×C4×C22⋊C4C2211yescharacteristic
59#2083C2214yescomplemented
60#2072C2214yescomplemented
61#2068C2214yescomplemented
62#2068C2214yescomplemented
63#2056C2214yescomplemented
64#2056C2214yescomplemented
65#2043C2214yescomplemented
66#2016C2214yescomplemented
67#1981C2214yescomplemented
68#1947C2214yescomplemented
69#1928C2214yescomplemented
70#1928C2214yescomplemented
71#1892C2212yes—
72#1892C2212yes—
73#1892C2214yes—
74#1892C2212yes—
75#1892C2212yes—
76#1878C2214yescomplemented
77#1878C2214yescomplemented
78#1872C2214yes—
79#1850C2212yes—
80#1850C2214yes—
81#1850C2212yes—
82#1824C2212yes—
83#1803C2214yescomplemented
84#1787C2212yes—
85C2×#340C2212yes—
86C2×#336C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×Q16—24no—
92C22×Q16—44no—
93C22×Q16C2314yes—
94C22×Q16C2312yes—
95C22×Q16C2312yes—
96C23×C8—24no—
97C23×C8—44no—
98C23×C8C2311yescharacteristic
99C23×C8C2312yes—
100C23×C8C2312yes—
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104C2×C4⋊Q8C2314yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8D418yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×C4×Q8C2314yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42C2312yes—
138#484—28no—
139#484—28no—
140#467—28no—
141#467—28no—
142#467—216no—
143#467—216no—
144#462—48no—
145#451—28no—
146#451—28no—
147C2×#107—24no—
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#107C2314yes—
151C2×C4⋊C8—44no—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—44no—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×C22⋊C8—24no—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170#398—24no—
171#398—24no—
172#395—24no—
173#395—24no—
174#391—24no—
175#391C2314yes—
176#391C2314yes—
177#391C2314yes—
178#389—24no—
179#389—24no—
180#389—24no—
181#389C2314yes—
182#387—24no—
183#387—24no—
184#387—24no—
185#385—24no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#385—24no—
195#384C2314yes—
196#380—24no—
197#380—24no—
198#380—21no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2312yes—
204C4×C4⋊C4—24no—
205C4×C4⋊C4C2314yes—
206C4×C22⋊C4—24no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—24no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213C4×C22⋊C4C2312yes—
214C2×#95—21no—
215C2×#95D412yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8—42no—
234C22×Q8—42no—
235C22×Q8—42no—
236C22×Q8—42no—
237C22×Q8C2×D414yes—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D414yes—
255C23×C4C2×D412yes—
256C23×C4#4114yes—
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×Q16—28no—
261C2×Q16—24no—
262C2×Q16—24no—
263C2×Q16—28no—
264C2×Q16—44no—
265C2×Q16—416no—
266C2×Q16—44no—
267C2×Q16—44no—
268C22×C8—22no—
269C22×C8—24no—
270C22×C8—22no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—42no—
278C22×C8—42no—
279C22×C8—42no—
280C22×C8—42no—
281C22×C8—42no—
282C22×C8—416no—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C22⋊Q8C2×D418yes—
306C22⋊Q8C2×D418yes—
307C4×Q8—44no—
308C4×Q8—44no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4Q1618yescomplemented
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42C2×D414yes—
363C2×C42C2414yes—
364#107—28no—
365#107—24no—
366#107—24no—
367C4⋊C8—28no—
368C4⋊C8—44no—
369#103—28no—
370#103—24no—
371#103—24no—
372#103—28no—
373#103—28no—
374#103—24no—
375#103—24no—
376#103—44no—
377#103—44no—
378#103—44no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D414yes—
393C24—42no—
394C24—42no—
395C24#12114yes—
396C24C2×Q1614yes—
397C24#13714yes—
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—48no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8—82no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8—84no—
426C2×Q8—84no—
427C2×Q8C22×D414yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4#12114yes—
468C22×C4#12114yes—
469C22×C4#12114yes—
470C22×C4C2×Q1618yescomplemented
471C22×C4C22×D414yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2511yescharacteristic
483Q16—416no—
484Q16—84no—
485Q16—84no—
486C2×C8—28no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—48no—
497C2×C8—48no—
498C2×C8—44no—
499C2×C8—82no—
500C2×C8—82no—
501C2×C8H(5,𝔽2)14yes—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—82no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4C22×D414yes—
526C22⋊C4—28no—
527C22⋊C4—28no—
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⋊C4C2×Q1618yes—
537C22⋊C4C2×Q1618yescomplemented
538C42—24no—
539C42—24no—
540C42—44no—
541C42—84no—
542C42—84no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—44no—
550C23—44no—
551C23—48no—
552C23—48no—
553C23#462116yescomplemented
554C23C2×#12112yes—
555C23C2×#12112yes—
556C23C2×#12112yes—
557C23C22×Q1612yes—
558C23C22×Q1612yes—
559C23C22×Q1612yes—
560C23C2×#13712yes—
561C23C2×#13712yes—
562C23C2×#13712yes—
563C23C23×D411yescharacteristic
564Q8—24no—
565Q8—24no—
566Q8—44no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—44no—
572Q8—44no—
573Q8—84no—
574Q8—84no—
575Q8—84no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—164no—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C22×Q1618yes—
624C2×C4C23×D411yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4#58512yes—
628C8—24no—
629C8—42no—
630C8—42no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—416no—
645C22C2×#46214yes—
646C22C2×#46214yes—
647C22C2×#46214yes—
648C22C22×#12111yescharacteristic
649C22#349612yes—
650C22C23×Q1611yescharacteristic
651C22C22×#13711yescharacteristic
652C22#358412yes—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—28no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4#348411yescharacteristic
666C4#348511yescharacteristic
667C4#350512yes—
668C2—24no—
669C2—24no—
670C2—24no—
671C2#3535811yescharacteristic
672C2C22×#46211yescharacteristic
673C2#6042211yescharacteristic
674C2#6224612yes—
675C2#6225412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28575C2semidirect
6. #25206C2semidirect
7. #25082C2semidirect
8. #24752C2semidirect
9. #24582C2semidirect
10. #24532C2semidirect
11. #24482C2semidirect
12. #24360C2semidirect
29. C2×#462C22semidirect
59. #2083C22semidirect
60. #2072C22semidirect
61. #2068C22semidirect
62. #2068C22semidirect
63. #2056C22semidirect
64. #2056C22semidirect
65. #2043C22semidirect
66. #2016C22semidirect
67. #1981C22semidirect
68. #1947C22semidirect
69. #1928C22semidirect
70. #1928C22semidirect
76. #1878C22semidirect
77. #1878C22semidirect
83. #1803C22semidirect
353. C2×C22⋊C4Q16semidirect
470. C22×C4C2×Q16semidirect
537. C22⋊C4C2×Q16semidirect
553. C23#462semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C22×#4672
#350112
#342122
#285752
#252062
#250822
#247522
#245822
#245322
#244822
#243602
#217592
C2×#18922
C2×#18502