Finite Group Databasebeta

← #6394853

Subgroups — #6394853

Order 512.

2186 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2#34236C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic, complemented
4#23221C212yesmaximal, complemented
5#23217C212yesmaximal, complemented
6#23190C212yesmaximal, complemented
7#23182C212yesmaximal, complemented
8#23112C214yesmaximal, complemented
9#23012C214yesmaximal, complemented
10#22047C212yesmaximal, complemented
11#22041C212yesmaximal, complemented
12#21811C211yesmaximal, characteristic, complemented
13#21454C212yesmaximal, complemented
14C4×#448C212yesmaximal, complemented
15#21395C211yesmaximal, characteristic
16#21361C211yesmaximal, characteristic, complemented
17#20535C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3206C4116yescomplemented
22C2×#492C2212yes—
23C2×#487C2212yes—
24C2×#461C2214yes—
25C2×#448C2212yes—
26C2×#448C2212yes—
27C2×#440C2214yes—
28C2×C4×SD16C2214yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2369C2211yescharacteristic
32#2369C2211yescharacteristic
33#2366C2211yescharacteristic
34#2366C2211yescharacteristic
35#2365—21no—
36#2365—22no—
37#2365—21no—
38#2365C2214yes—
39#2365C2212yes—
40#2365C2212yes—
41#2361—21no—
42#2361—21no—
43#2357C2212yes—
44#2349C2212yes—
45#2339—22no—
46#2337—22no—
47#2319—21no—
48C4×#121C2212yes—
49C4×#121C2212yes—
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#387—22no—
53#2283C2211yescharacteristic
54#2275—21no—
55C2×C4×C4⋊C4—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#1987C2212yes—
59#1982C2212yes—
60#1981C2212yes—
61#1981C2212yes—
62#1973C2214yes—
63#1968C2214yes—
64#1960C2212yes—
65#1960C2212yes—
66#1949C2212yes—
67#1949C2212yes—
68#1935C2214yes—
69#1933C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1880C2214yes—
73#1880C2214yes—
74#1877C2214yes—
75#1877C2214yes—
76#1874C2212yes—
77#1874C2212yes—
78#1865C2214yes—
79#1863C2214yes—
80#1853C2211yescharacteristic
81#1853C2211yescharacteristic
82#1852C2211yescharacteristic
83#1852C2211yescharacteristic
84#1787C2212yes—
85#1787C2212yes—
86C4×#107C2212yes—
87C4×#106C2212yes—
88#1771C2214yes—
89#1770C2212yes—
90C8×C4⋊C4C2212yes—
91#1763C2214yes—
92#1762C2214yes—
93C4×#103C2214yes—
94#1749C2212yes—
95C4×C22⋊C8C2212yes—
96C22×#41—21no—
97C23×Q8C2312yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100#532—24no—
101#532C2×C418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8C2×C418yescomplemented
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#124—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—48no—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2314yes—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—22no—
134C22×C42—21no—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#492C2×C4116yescomplemented
138#487C2×C4116yescomplemented
139#461C2×C4132yescomplemented
140#448C2×C4116yescomplemented
141#448C2×C4116yescomplemented
142#440—48no—
143C4×SD16—48no—
144#432—24no—
145#431C2×C418yescomplemented
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#106C2312yes—
149C2×#106C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2311yescharacteristic
163C2×#97C2312yes—
164C2×#97C2314yes—
165C2×C4×C8C2314yes—
166C2×C4×C8C2312yes—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2314yes—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2312yes—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—24no—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#386—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—22no—
198#383—21no—
199#382C2311yescharacteristic
200#382C2311yescharacteristic
201#381—21no—
202#381—22no—
203#380—22no—
204#380C2311yescharacteristic
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—24no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229#376—21no—
230C2×#95—22no—
231C2×#95—21no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95D412yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—28no—
244C2×#41—82no—
245C2×#41—82no—
246C22×Q8—22no—
247C22×Q8—21no—
248C22×Q8—24no—
249C22×Q8—44no—
250C22×Q8—44no—
251C22×Q8C22×C418yes—
252C22×Q8C2414yes—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4C2412yes—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C2×SD16—44no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×SD16—44no—
272C2×SD16C22×C4116yes—
273C2×SD16C22×C4116yes—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—28no—
278C4⋊Q8—28no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—22no—
282C4⋊Q8C22×C418yes—
283C4⋊Q8C22×C418yes—
284C4⋊Q8C22×C414yes—
285#127—22no—
286#127C22×C414yes—
287#124—28no—
288#121—28no—
289#121—24no—
290#121C22×C418yes—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—48no—
296C4×Q8—48no—
297C4×D4—28no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4—84no—
304C4×D4—84no—
305C4×D4C22×C418yes—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42—48no—
357C2×C42—44no—
358C2×C42C22×C414yes—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42#4114yes—
364C2×C42C2412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—24no—
369#107—42no—
370#106—24no—
371#106—42no—
372C4⋊C8—24no—
373C4⋊C8C22×C418yes—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103C22×C4116yes—
379#103C22×C4116yes—
380#102—28no—
381#102—44no—
382C22⋊C8—24no—
383C22⋊C8C22×C414yes—
384C22⋊C8C22×C414yes—
385#97—24no—
386#97—44no—
387C4×C8—44no—
388C4×C8C22×C418yes—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95C2×D412yes—
408#95C2×D412yes—
409#95C2×D412yes—
410#95C2×D412yes—
411C24—22no—
412C24C22×D411yescharacteristic
413#41—816no—
414C2×Q8—28no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—416no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—416no—
425C2×Q8—81no—
426C2×Q8—81no—
427C2×Q8C23×C418yes—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4C23×C414yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4C4×D414yes—
480C22×C4C4×D414yes—
481C22×C4C23×C412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C4C2511yescharacteristic
493SD16—416no—
494SD16—416no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8C23×C414yes—
503C2×C8C23×C414yes—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—24no—
510C4⋊C4—24no—
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—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4#13714yes—
529C4⋊C4C23×C414yes—
530C4⋊C4H(5,𝔽2)14yes—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C42—24no—
542C42—24no—
543C42—42no—
544C42—42no—
545C42—48no—
546C42—48no—
547C42—44no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—82no—
552C42—82no—
553C42C4×D414yes—
554C42C4×D414yes—
555C42C23×C412yes—
556C42C23×C412yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—24no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C2×C4×D412yes—
571C23#57712yes—
572C23#57712yes—
573C23C23×D411yescharacteristic
574Q8—28no—
575Q8—28no—
576Q8—416no—
577Q8—44no—
578Q8—44no—
579Q8—84no—
580D4—28no—
581D4—416no—
582D4—48no—
583D4—44no—
584D4—84no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—416no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—88no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4C2×C4×D412yes—
641C2×C4C2×C4×D412yes—
642C2×C4C2×C4×D412yes—
643C2×C4C2×C4×D412yes—
644C2×C4C2×C4×D412yes—
645C2×C4C2×#13714yes—
646C2×C4C2×#13712yes—
647C2×C4#57512yes—
648C2×C4C24×C411yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4#58512yes—
652C8—24no—
653C8—24no—
654C8—44no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22#294914yes—
669C22C22×C4×D411yescharacteristic
670C22#348411yescharacteristic
671C22#348711yescharacteristic
672C22C22×#13711yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57711yescharacteristic
675C22C2×#57711yescharacteristic
676C4—22no—
677C4—22no—
678C4—24no—
679C4—22no—
680C4—22no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—48no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—82no—
691C4—81no—
692C4C4×#13714yes—
693C4C4×H(5,𝔽2)12yes—
694C4#350612yes—
695C2—24no—
696C2—84no—
697C2#3426511yescharacteristic
698C2C2×C4×#13711yescharacteristic
699C2C2×#294911yescharacteristic
700C2#6159311yescharacteristic
701C2#6160411yescharacteristic
702C2#6189911yescharacteristic
703C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C2×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
3. C2×#2365C2semidirect
4. #23221C2semidirect
5. #23217C2semidirect
6. #23190C2semidirect
7. #23182C2semidirect
8. #23112C2semidirect
9. #23012C2semidirect
10. #22047C2semidirect
11. #22041C2semidirect
12. #21811C2semidirect
13. #21454C2semidirect
14. C4×#448C2semidirect
16. #21361C2semidirect
17. #20535C2semidirect
21. #3206C4semidirect
105. C2×C4⋊Q8C2×C4semidirect
137. #492C2×C4semidirect
138. #487C2×C4semidirect
139. #461C2×C4semidirect
140. #448C2×C4semidirect
141. #448C2×C4semidirect
145. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
#342362
C2×#23652
#232212
#232172
#231902
#231822
#231122
#230122
#220472
#220412
#218112
#214542
C4×#4482
#213952
#213612
#205352