Finite Group Databasebeta

← #6456320

Subgroups — #6456320

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic
3#34946C211yesmaximal, characteristic, complemented
4#34141C211yesmaximal, characteristic, complemented
5#25452C212yesmaximal, complemented
6#25367C212yesmaximal, complemented
7#25321C212yesmaximal, complemented
8#25275C212yesmaximal, complemented
9#25212C214yesmaximal, complemented
10#25088C214yesmaximal, complemented
11#23680C214yesmaximal, complemented
12C4×#463C214yesmaximal, complemented
13#21385C211yesmaximal, characteristic, complemented
14#21349C211yesmaximal, characteristic, complemented
15C2×C4×#102C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×C4×#41—21no—
19C22×C4×D4C2211yescharacteristic
20#3167—28no—
21C2×#498C2211yescharacteristic, complemented
22C2×#492C2211yescharacteristic, complemented
23C2×#487C2211yescharacteristic, complemented
24C2×#486C2211yescharacteristic, complemented
25#3100—28no—
26C2×#480C2212yes—
27C2×#474C2212yes—
28C2×#465C2214yes—
29C2×#463C2214yes—
30C2×#459C2212yes—
31C2×#447C2212yes—
32C2×C4×SD16C2214yescomplemented
33C2×C4×D8C2214yes—
34C2×#431C2211yescharacteristic
35C22×#102C2212yes—
36C22×C4×C8C2211yescharacteristic
37#2563C2212yes—
38#2562—22no—
39#2561—22no—
40#2547C2212yes—
41#2503C2212yes—
42#2435C2212yes—
43#2398—22no—
44#2367—22no—
45#2366C2212yescomplemented
46#2365C2212yescomplemented
47#2364—22no—
48#2361C2214yes—
49C4×#124C2214yes—
50#2297—21no—
51#2293C2212yescomplemented
52#2290—22no—
53#2280—21no—
54#2280—21no—
55C42×D4C2212yescomplemented
56C4×#117C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2085C2214yescomplemented
59#2073C2214yescomplemented
60#2068C2214yescomplemented
61#2065C2214yes—
62#2055C2214yescomplemented
63#2054C2214yes—
64#2042C2214yes—
65#2012C2214yes—
66#1987C2212yescomplemented
67#1983C2214yes—
68#1981C2212yescomplemented
69#1949C2214yescomplemented
70#1939C2212yescomplemented
71#1939C2212yescomplemented
72#1928C2214yescomplemented
73#1926C2214yes—
74#1879C2214yes—
75#1877C2214yescomplemented
76#1872C2214yes—
77#1850C2214yes—
78#1804C2214yescomplemented
79C4×C4⋊C8C2214yescomplemented
80C4×#103C2212yescomplemented
81C4×#103C2212yescomplemented
82C4×#102C2212yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85C4×#102C2212yes—
86C2×#340C2212yes—
87C22×#41—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C22×D8C2314yes—
92C23×C8C2312yes—
93#533—24no—
94#533—24no—
95#532—24no—
96#532D418yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#125—21no—
102C2×#124—21no—
103C2×#124C2314yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×#121—21no—
107C2×#121C2312yes—
108C4×#41—48no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4—24no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2314yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117—22no—
125C2×#117D411yescharacteristic
126C2×#117D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—21no—
130C22×C42—21no—
131C22×C42—22no—
132C22×C42C2311yescharacteristic
133#498—24no—
134#492—24no—
135#487—24no—
136#486—24no—
137#480—28no—
138#474—28no—
139#465—216no—
140#463—216no—
141#459—28no—
142#447—28no—
143C4×SD16—216no—
144C4×D8—216no—
145#431—24no—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102—28no—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C4×C8—24no—
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168#400—24no—
169#397—24no—
170#394—24no—
171#393—24no—
172#391—22no—
173#391C2314yes—
174#390—22no—
175#390C2314yes—
176#390C2314yes—
177#389C2314yes—
178#388—24no—
179#388—24no—
180#388—24no—
181#388—24no—
182#387—24no—
183#386—24no—
184#386—24no—
185#385—24no—
186#382C2314yes—
187#381—22no—
188#381—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#379—22no—
194#379—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—24no—
204C4×C22⋊C4—24no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4C2312yes—
210#376D412yes—
211C2×#95—21no—
212C43C2312yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C2×#41—28no—
223C2×#41—44no—
224C2×#41—44no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8C2412yes—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—42no—
236C22×D4C2412yes—
237C22×D4C2412yes—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—24no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—28no—
251C2×SD16—28no—
252C2×SD16—44no—
253C2×D8—28no—
254C2×D8—28no—
255C2×D8—44no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—42no—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8C2×D414yes—
268#127—22no—
269#127C2×D414yes—
270#125—44no—
271#124—28no—
272#124—28no—
273#124—44no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—44no—
277#121—28no—
278#121—24no—
279#121C2×D418yes—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—48no—
283C4×Q8—44no—
284C4×D4—28no—
285C4×D4—28no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—416no—
293C4×D4—48no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4C2×D418yes—
297#117—24no—
298#117—24no—
299#117—48no—
300#117—44no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—48no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D414yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D414yes—
349C2×C42#4114yes—
350C2×C42C2414yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#106—28no—
355C4⋊C8—28no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358#103—28no—
359#103—28no—
360#103—44no—
361#102—28no—
362#102—28no—
363#102—24no—
364#102—24no—
365#102—28no—
366#102—24no—
367#102#4118yescomplemented
368C22⋊C8—24no—
369C4×C8—24no—
370C4×C8—22no—
371C4×C8—22no—
372C4×C8—24no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95C2×D414yes—
383#95C2×D414yes—
384C24—22no—
385C24—22no—
386C24—44no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—82no—
391C24C22×D411yescharacteristic
392#41—432no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—82no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—82no—
423C2×D4C22×D414yes—
424C2×D4C2×#4114yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—84no—
451C22×C4—82no—
452C22×C4—816no—
453C22×C4#12114yes—
454C22×C4#12114yes—
455C22×C4#13514yes—
456C22×C4AGL(1,ℤ/8)14yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4112yes—
466C22×C4C2511yescharacteristic
467SD16—416no—
468D8—416no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—48no—
478C2×C8C2×#4114yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4C22×D414yes—
499C4⋊C4C2×#4114yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—24no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—44no—
519C42—48no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—44no—
524C42#12114yes—
525C42#12114yes—
526C42C22×D412yes—
527C42C22×D412yes—
528C42C2×#4112yes—
529C42C2×#4112yes—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—48no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—88no—
546C23—88no—
547C23—84no—
548C23—84no—
549C23C2×#12112yes—
550C23C2×#13512yes—
551C23C2×AGL(1,ℤ/8)12yes—
552C23C23×D411yescharacteristic
553Q8—48no—
554Q8—42no—
555Q8—42no—
556Q8—44no—
557Q8—84no—
558D4—24no—
559D4—24no—
560D4—28no—
561D4—48no—
562D4—416no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—48no—
568D4—84no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—416no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—816no—
600C2×C4—88no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4#463116yescomplemented
605C2×C4C2×#12112yes—
606C2×C4C2×#12112yes—
607C2×C4C2×#12112yes—
608C2×C4C2×#12112yes—
609C2×C4C2×#12112yes—
610C2×C4C2×#13512yes—
611C2×C4C2×#13512yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C23×D411yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4112yes—
617C2×C4#58512yes—
618C8—24no—
619C8—42no—
620C8—44no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—24no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—816no—
632C22—82no—
633C22—84no—
634C22—82no—
635C22—82no—
636C22—84no—
637C22—82no—
638C22C2×#46314yes—
639C22C22×#12111yescharacteristic
640C22D4×#4112yes—
641C22C22×#13511yescharacteristic
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22#358412yes—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4C2×#46314yes—
656C4C2×#46314yes—
657C4#344411yescharacteristic
658C4#345511yescharacteristic
659C4D4×#4112yes—
660C2—24no—
661C2—84no—
662C2—84no—
663C2#3474311yescharacteristic
664C2C22×#46311yescharacteristic
665C2#6042211yescharacteristic
666C2#6142812yes—
667C2#6143712yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#3100C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3167C2semidirect
3. #34946C2semidirect
4. #34141C2semidirect
5. #25452C2semidirect
6. #25367C2semidirect
7. #25321C2semidirect
8. #25275C2semidirect
9. #25212C2semidirect
10. #25088C2semidirect
11. #23680C2semidirect
12. C4×#463C2semidirect
13. #21385C2semidirect
14. #21349C2semidirect
15. C2×C4×#102C2semidirect
21. C2×#498C22semidirect
22. C2×#492C22semidirect
23. C2×#487C22semidirect
24. C2×#486C22semidirect
32. C2×C4×SD16C22semidirect
45. #2366C22semidirect
46. #2365C22semidirect
51. #2293C22semidirect
55. C42×D4C22semidirect
58. #2085C22semidirect
59. #2073C22semidirect
60. #2068C22semidirect
62. #2055C22semidirect
66. #1987C22semidirect
68. #1981C22semidirect
69. #1949C22semidirect
70. #1939C22semidirect
71. #1939C22semidirect
72. #1928C22semidirect
75. #1877C22semidirect
78. #1804C22semidirect
79. C4×C4⋊C8C22semidirect
80. C4×#103C22semidirect
81. C4×#103C22semidirect
367. #102#41semidirect
604. C2×C4#463semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
C2×#31002
#349462
#341412
#254522
#253672
#253212
#252752
#252122
#250882
#236802
C4×#4632
#213852
#213492
C2×C4×#1022