Finite Group Databasebeta

← #192805

Subgroups — #192805

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34384C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic, complemented
3#21981C211yesmaximal, characteristic, complemented
4#21890C211yesmaximal, characteristic
5C2×#1482C211yesmaximal, characteristic
6#8893C211yesmaximal, characteristic, complemented
7#8892C211yesmaximal, characteristic, complemented
8#8891C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic, complemented
10#8889C211yesmaximal, characteristic, complemented
11#8888C211yesmaximal, characteristic, complemented
12#8887C211yesmaximal, characteristic, complemented
13#8886C211yesmaximal, characteristic, complemented
14#8043C211yesmaximal, characteristic, complemented
15#7899C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20#2383—22no—
21#2361—21no—
22#2360—21no—
23#2355C2211yescharacteristic
24#2354C2211yescharacteristic
25#2352—21no—
26Q8×C4⋊C4C2211yescharacteristic
27D4×C4⋊C4—21no—
28D4×C4⋊C4C2211yescharacteristic
29#2330—21no—
30#2326—21no—
31#2322—22no—
32#2317—21no—
33#2314—21no—
34#2285C2211yescharacteristic
35C2×#384C2211yescharacteristic
36#2283C2211yescharacteristic
37#1989—21no—
38#1983—21no—
39#1948—22no—
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1947C2211yescharacteristic
43#1947C2211yescharacteristic
44#1895—21no—
45#1894—21no—
46#1854—21no—
47#1854—21no—
48#1853—22no—
49#1846—21no—
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1835—21no—
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1803C2211yescharacteristic
56#1803C2211yescharacteristic
57#1802C2211yescharacteristic
58#1802C2211yescharacteristic
59#1482C418yescomplemented
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72C22×M16—21no—
73#529—22no—
74#529—22no—
75#529C2×C414yes—
76#529Q814yes—
77C2×#126—21no—
78C2×#125D411yescharacteristic
79C2×#124D411yescharacteristic
80C2×#123—21no—
81C2×C22⋊Q8D411yescharacteristic
82C2×#121D411yescharacteristic
83C2×C4×Q8—21no—
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—22no—
87C2×C4×D4—21no—
88C2×C4×D4C2311yescharacteristic
89C2×#117—21no—
90C2×#117—21no—
91C2×#117—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D412yes—
96C22×C42C2311yescharacteristic
97#431—22no—
98#431—22no—
99#431C2×C414yescomplemented
100#431C2×C414yescomplemented
101C2×#107—21no—
102C2×#107—21no—
103C2×#106—22no—
104C2×#106—22no—
105C2×#106—21no—
106C2×#106—21no—
107#423—22no—
108#423—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×#103—21no—
116C2×#103—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391—21no—
125#390—21no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—21no—
131#389—21no—
132#388—22no—
133#388—22no—
134#388—22no—
135#388—22no—
136#387—22no—
137#387—22no—
138#387—22no—
139#387—22no—
140#386—22no—
141#385—22no—
142#385—22no—
143#385—22no—
144#385—22no—
145#385—22no—
146#384—21no—
147#384—21no—
148#384—21no—
149#384D412yes—
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#384C2311yescharacteristic
154#383—21no—
155#382—21no—
156#382—21no—
157#382D411yescharacteristic
158#382D411yescharacteristic
159#381—21no—
160#380—21no—
161#380—21no—
162#380—21no—
163#380—21no—
164#380D411yescharacteristic
165#380D411yescharacteristic
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2311yescharacteristic
174C4×C22⋊C4—22no—
175C4×C22⋊C4—21no—
176#376—21no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—22no—
182#340—22no—
183#340—21no—
184#340—21no—
185#340—21no—
186#340—21no—
187Q8⋊C8C2×C414yescomplemented
188Q8⋊C8C2×C414yescomplemented
189D4⋊C8C2×C414yescomplemented
190D4⋊C8C2×C414yescomplemented
191C22×Q8C2×D411yescharacteristic
192C22×D4—21no—
193C22×D4C2×D411yescharacteristic
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C2×M16—42no—
201C2×M16—81no—
202C2×M16—81no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209#126—24no—
210#126—24no—
211#125—22no—
212#125C4⋊C414yes—
213#124—22no—
214#124C4⋊C414yes—
215#123—24no—
216#123—24no—
217C22⋊Q8—22no—
218C22⋊Q8C4⋊C414yes—
219#121—22no—
220#121C4⋊C414yes—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—81no—
224C4×Q8—81no—
225C4×Q8C22×C412yes—
226C4×Q8C2×Q812yes—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—81no—
232C4×D4—81no—
233C4×D4—84no—
234C4×D4C22×C412yes—
235C4×D4C2×Q812yes—
236#117—24no—
237#117—24no—
238#117—81no—
239#117—81no—
240#117—81no—
241#117—81no—
242C2×C4⋊C4—21no—
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—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—81no—
263C2×C4⋊C4—81no—
264C2×C4⋊C4—81no—
265C2×C4⋊C4—81no—
266C2×C4⋊C4—81no—
267C2×C4⋊C4—81no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—41no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—41no—
295C2×C42C22×C412yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×Q812yes—
300C2×C42#4112yes—
301C2×C42#4112yes—
302C2×C42C2411yescharacteristic
303#107—81no—
304#107—81no—
305#106—81no—
306#106—81no—
307#106—82no—
308#106—82no—
309C4⋊C8—24no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—42no—
317C4⋊C8C22×C412yes—
318C4⋊C8C22×C412yes—
319#103—81no—
320#103—81no—
321#102—81no—
322#102—81no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325C4×C8—24no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8C22×C412yes—
329C4×C8C22×C412yes—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—41no—
345#95—41no—
346C24—21no—
347C24#12011yescharacteristic
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8C2×C4⋊C412yes—
354C2×D4—24no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4C2×C4⋊C412yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—44no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4C2×C4⋊C412yes—
397C22×C4C4×D412yes—
398C22×C4C4×D412yes—
399C22×C4#12011yescharacteristic
400C22×C4#12011yescharacteristic
401C22×C4#12011yescharacteristic
402C22×C4C22⋊Q812yes—
403C22×C4C22⋊Q812yes—
404C22×C4#12312yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408M16—84no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4C2×C4⋊C412yes—
469C4⋊C4C2×C4⋊C412yes—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—22no—
483C42—22no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—82no—
489C42—82no—
490C42—82no—
491C42—82no—
492C42C2×C4⋊C412yes—
493C42C4×D412yes—
494C42C4×D412yes—
495C42C22⋊Q812yes—
496C42C22⋊Q812yes—
497C42#12312yes—
498C42C23×C411yescharacteristic
499C42C22×Q811yescharacteristic
500C42C2×#4111yescharacteristic
501C42C2×#4111yescharacteristic
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23#38512yes—
516C23C2×#12011yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—164no—
582C2×C4—164no—
583C2×C4#38512yes—
584C2×C4#38512yes—
585C2×C4#38512yes—
586C2×C4C22×C4⋊C411yescharacteristic
587C2×C4C2×C4×D411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C2×C22⋊Q811yescharacteristic
590C2×C4C2×C22⋊Q811yescharacteristic
591C2×C4C2×#12311yescharacteristic
592C2×C4#57611yescharacteristic
593C2×C4#57711yescharacteristic
594C2×C4#57711yescharacteristic
595C2×C4#57711yescharacteristic
596C2×C4#57711yescharacteristic
597C2×C4#57711yescharacteristic
598C2×C4#57711yescharacteristic
599C2×C4#57811yescharacteristic
600C8—42no—
601C8—42no—
602C8—42no—
603C8—42no—
604C8—81no—
605C8—81no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22C2×#38511yescharacteristic
621C22#300911yescharacteristic
622C22#301011yescharacteristic
623C22#301211yescharacteristic
624C22#301311yescharacteristic
625C22#301911yescharacteristic
626C22#302011yescharacteristic
627C4—22no—
628C4—21no—
629C4—21no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—161no—
642C4—161no—
643C4#291211yescharacteristic
644C4#291311yescharacteristic
645C4#295211yescharacteristic
646C4#295211yescharacteristic
647C4#308411yescharacteristic
648C4#308411yescharacteristic
649C4#309611yescharacteristic
650C4#309711yescharacteristic
651C2—42no—
652C2—82no—
653C2#1893111yescharacteristic
654C2#1893211yescharacteristic
655C2#1894311yescharacteristic
656C2#1894411yescharacteristic
657C2#2084911yescharacteristic
658C2#2086411yescharacteristic
659C2#2088211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
2. #21985C2semidirect
3. #21981C2semidirect
6. #8893C2semidirect
7. #8892C2semidirect
8. #8891C2semidirect
9. #8890C2semidirect
10. #8889C2semidirect
11. #8888C2semidirect
12. #8887C2semidirect
13. #8886C2semidirect
14. #8043C2semidirect
15. #7899C2semidirect
59. #1482C4semidirect
99. #431C2×C4semidirect
100. #431C2×C4semidirect
187. Q8⋊C8C2×C4semidirect
188. Q8⋊C8C2×C4semidirect
189. D4⋊C8C2×C4semidirect
190. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343842
#219852
#219812
#218902
C2×#14822
#88932
#88922
#88912
#88902
#88892
#88882
#88872
#88862
#80432
#78992