Finite Group Databasebeta

← #194182

Subgroups — #194182

Order 512.

1404 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#21885C211yesmaximal, characteristic, complemented
4#21860C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#8983C212yesmaximal, complemented
7#8981C212yesmaximal, complemented
8#8975C212yesmaximal, complemented
9#8973C212yesmaximal, complemented
10C2×#1296C211yesmaximal, characteristic, complemented
11#7893C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#520—21no—
14C2×C4×#41C2211yescharacteristic
15#3076—24no—
16#3076C418yescomplemented
17C2×#479—21no—
18C2×#477C2211yescharacteristic
19C2×#475C2211yescharacteristic
20C2×#478—21no—
21C2×#476C2211yescharacteristic
22C2×#474C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#427—21no—
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C2×#416—21no—
28Q8×C4⋊C4—21no—
29Q8×C4⋊C4C2212yes—
30D4×C4⋊C4—22no—
31D4×C4⋊C4—21no—
32D4×C4⋊C4—22no—
33D4×C4⋊C4C2212yes—
34#2317—22no—
35#2317—21no—
36#2285—21no—
37#2285C2211yescharacteristic
38C2×#384—21no—
39C2×#384C2211yescharacteristic
40#1990C2212yes—
41#1987C2212yes—
42#1981C2212yes—
43#1956C2212yes—
44#1892—22no—
45#1872—22no—
46#1803—22no—
47#1803—21no—
48#1803—21no—
49#1477—24no—
50C2×Q8⋊C8C2212yes—
51C2×D4⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×#41D411yescharacteristic
57C23×C8—21no—
58C23×C8—21no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533C2×C414yescomplemented
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68#520—44no—
69#520—44no—
70C4×#41—22no—
71C4×#41—22no—
72C4×#41C2×C414yes—
73C4×#41D414yes—
74C2×C4×Q8—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—24no—
82C2×C4×D4—22no—
83C2×C4×D4D411yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C2×#117—22no—
86C2×#117—21no—
87C2×#117—21no—
88C2×#117—21no—
89C2×#117D411yescharacteristic
90C22×C4⋊C4—22no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4D412yes—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98#479—28no—
99#478—28no—
100#477—22no—
101#477C2×C414yescomplemented
102#476—22no—
103#476C2×C414yescomplemented
104#475—22no—
105#475C2×C414yescomplemented
106#474—22no—
107#474C2×C414yescomplemented
108#431—22no—
109#431—22no—
110#427—28no—
111C2×#107—21no—
112C2×#107—21no—
113C2×#106—21no—
114C2×#106—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8C2×C414yescomplemented
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125#416—28no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—24no—
129C2×#103—22no—
130C2×#103—22no—
131C2×#102—21no—
132C2×#102—21no—
133C2×C22⋊C8—21no—
134C2×C4×C8C2312yes—
135#389—24no—
136#389—22no—
137#389—24no—
138#389—22no—
139#385—24no—
140#385—24no—
141#385—24no—
142#384—22no—
143#384—22no—
144#384—22no—
145#384—21no—
146#384—21no—
147#384D412yes—
148#384D411yescharacteristic
149#384D411yescharacteristic
150#384C2312yes—
151#382—22no—
152#382—21no—
153#380—22no—
154#380—21no—
155#380—22no—
156#380D412yes—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2312yes—
161#340—24no—
162#340—22no—
163#340—22no—
164Q8⋊C8—24no—
165D4⋊C8—24no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41C22⋊C414yes—
170C2×#41SD1614yes—
171C2×#41Q1614yes—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8C2×D411yescharacteristic
175C22×D4—21no—
176C22×D4C2×D411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C22×C8—24no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—81no—
194C22×C8—81no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8C22×C412yes—
202C4⋊Q8C22×C412yes—
203#125—24no—
204#125—42no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—42no—
208C22⋊Q8—42no—
209C4×Q8—22no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×Q8C22⋊C414yes—
219C4×Q8C22×C412yes—
220C4×Q8C2×D412yes—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—22no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—416no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4C22⋊C414yes—
232C4×D4C22×C412yes—
233C4×D4C2×D412yes—
234#117—22no—
235#117—24no—
236#117—24no—
237#117—24no—
238#117—48no—
239#117—42no—
240#117—42no—
241#117—42no—
242#117C22⋊C414yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—81no—
270C2×C4⋊C4—81no—
271C2×C4⋊C4—84no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2×D414yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—41no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—41no—
291C2×C42C22×C412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42#4112yes—
297C2×C42#4112yes—
298C2×C42C2411yescharacteristic
299#107—24no—
300#107—24no—
301#106—24no—
302#106—24no—
303C4⋊C8—24no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—42no—
309C4⋊C8—42no—
310C4⋊C8C22×C412yes—
311C4⋊C8C22×C412yes—
312#103—24no—
313#103—24no—
314#103—82no—
315#103—82no—
316#103—84no—
317#102—24no—
318#102—24no—
319C22⋊C8—42no—
320C4×C8—22no—
321C4×C8—22no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326C24—21no—
327C24#12011yescharacteristic
328#41—28no—
329#41—28no—
330#41—28no—
331#41#103116yescomplemented
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8C2×C22⋊C412yes—
346C2×Q8C2×SD1612yes—
347C2×Q8C2×Q1612yes—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4C2×C22⋊C412yes—
355C2×D4C2×SD1612yes—
356C2×D4C2×Q1612yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C2×C22⋊C412yes—
390C22×C4C4×D412yes—
391C22×C4C4×D412yes—
392C22×C4#12011yescharacteristic
393C22×C4#12011yescharacteristic
394C22×C4#12011yescharacteristic
395C22×C4#12112yes—
396C22×C4#12112yes—
397C22×C4#12312yes—
398C22×C4C2×SD1612yes—
399C22×C4C2×Q1612yes—
400C22×C4AGL(1,ℤ/8)12yes—
401C22×C4#13712yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—48no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—84no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—82no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4C2×C22⋊C412yes—
454C4⋊C4C2×C22⋊C412yes—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—24no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C42—22no—
463C42—22no—
464C42—24no—
465C42—24no—
466C42—48no—
467C42—42no—
468C42—42no—
469C42—44no—
470C42C2×C22⋊C412yes—
471C42C4×D412yes—
472C42C4×D412yes—
473C42#12112yes—
474C42#12112yes—
475C42#12312yes—
476C42C23×C411yescharacteristic
477C42C22×D411yescharacteristic
478C42C2×#4111yescharacteristic
479C42C2×#4111yescharacteristic
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23#38612yes—
492C23#45012yes—
493C23#45112yes—
494C23C2×#12011yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—22no—
498Q8—44no—
499Q8—42no—
500Q8—42no—
501Q8—82no—
502Q8—81no—
503Q8—81no—
504Q8C2×#10314yes—
505D4—216no—
506D4—22no—
507D4—22no—
508D4—22no—
509D4C2×#10314yes—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—48no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—164no—
553C2×C4—168no—
554C2×C4#38612yes—
555C2×C4#38612yes—
556C2×C4#38612yes—
557C2×C4C2×#10314yes—
558C2×C4#41714yes—
559C2×C4#45012yes—
560C2×C4#45112yes—
561C2×C4#48314yes—
562C2×C4#48414yes—
563C2×C4C22×C22⋊C411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12311yescharacteristic
569C2×C4C22×SD1611yescharacteristic
570C2×C4C22×Q1611yescharacteristic
571C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
572C2×C4C2×#13711yescharacteristic
573C2×C4#57611yescharacteristic
574C2×C4#57711yescharacteristic
575C2×C4#57711yescharacteristic
576C2×C4#57811yescharacteristic
577C8—42no—
578C8—42no—
579C8—82no—
580C8—82no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—24no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22#187214yes—
595C22C2×#38611yescharacteristic
596C22C2×#45011yescharacteristic
597C22C2×#45111yescharacteristic
598C22#301211yescharacteristic
599C22#301311yescharacteristic
600C22#301911yescharacteristic
601C22#302011yescharacteristic
602C4—24no—
603C4—22no—
604C4—21no—
605C4—22no—
606C4—21no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—84no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4#187214yes—
615C4C22×#10311yescharacteristic
616C4C2×#41711yescharacteristic
617C4#295111yescharacteristic
618C4#295211yescharacteristic
619C4#304511yescharacteristic
620C4#304611yescharacteristic
621C4C2×#48311yescharacteristic
622C4C2×#48411yescharacteristic
623C2—24no—
624C2—44no—
625C2#1887111yescharacteristic
626C2#1887311yescharacteristic
627C2#1888311yescharacteristic
628C2#1888511yescharacteristic
629C2C2×#187211yescharacteristic
630C2#2093711yescharacteristic
631C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21885C2semidirect
4. #21860C2semidirect
6. #8983C2semidirect
7. #8981C2semidirect
8. #8975C2semidirect
9. #8973C2semidirect
10. C2×#1296C2semidirect
11. #7893C2semidirect
16. #3076C4semidirect
62. #533C2×C4semidirect
101. #477C2×C4semidirect
103. #476C2×C4semidirect
105. #475C2×C4semidirect
107. #474C2×C4semidirect
121. C2×C4⋊C8C2×C4semidirect
331. #41#103semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
#341832
#218852
#218602
C2×#14772
#89832
#89812
#89752
#89732
C2×#12962
#78932