Finite Group Databasebeta

← #192817

Subgroups — #192817

Order 512.

1002 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34243C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21862C211yesmaximal, characteristic
4#21359C211yesmaximal, characteristic, complemented
5C2×#1482C211yesmaximal, characteristic
6#8896C211yesmaximal, characteristic, complemented
7#8894C211yesmaximal, characteristic, complemented
8#8893C211yesmaximal, characteristic, complemented
9#8891C211yesmaximal, characteristic, complemented
10#8888C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic, complemented
12#8885C211yesmaximal, characteristic, complemented
13#8883C211yesmaximal, characteristic, complemented
14#8038C211yesmaximal, characteristic, complemented
15#7892C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2370—21no—
21#2367—21no—
22#2359—22no—
23#2355C2211yescharacteristic
24#2354C2211yescharacteristic
25#2352—21no—
26Q8×C4⋊C4C2211yescharacteristic
27D4×C4⋊C4—21no—
28D4×C4⋊C4C2211yescharacteristic
29#2334—22no—
30#2314—21no—
31C4×C22⋊Q8—21no—
32C4×#121—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2274—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1991—21no—
38#1958—21no—
39#1949—22no—
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1948C2211yescharacteristic
44#1875—21no—
45#1873—21no—
46#1854—22no—
47#1853—21no—
48#1853—21no—
49#1845C2211yescharacteristic
50#1844C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1803C2211yescharacteristic
54#1803C2211yescharacteristic
55#1802C2211yescharacteristic
56#1802C2211yescharacteristic
57C4×#106—21no—
58C4×#106—21no—
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
72#529—22no—
73#529—22no—
74#529C2×C414yes—
75#529Q814yes—
76C2×#126—21no—
77C2×#125D411yescharacteristic
78C2×#124D411yescharacteristic
79C2×#123—21no—
80C2×C22⋊Q8D411yescharacteristic
81C2×#121D411yescharacteristic
82C2×C4×Q8—21no—
83C2×C4×Q8C2311yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—22no—
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88C2×#117—21no—
89C2×#117—21no—
90C2×#117—21no—
91C2×#117D412yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4D411yescharacteristic
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97#431—22no—
98#431—22no—
99#431—22no—
100#431—22no—
101#431C2×C414yescomplemented
102#431C2×C414yescomplemented
103C2×#107—22no—
104C2×#106—21no—
105C2×#106—21no—
106C2×#106—21no—
107C2×#106—21no—
108C2×#106—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×#103—21no—
115C2×#103—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391—21no—
126#390—21no—
127#389—22no—
128#389—22no—
129#389—21no—
130#389—21no—
131#388—22no—
132#388—22no—
133#388—22no—
134#388—22no—
135#387—22no—
136#387—22no—
137#387—22no—
138#387—22no—
139#386—22no—
140#385—22no—
141#385—22no—
142#385—22no—
143#384—21no—
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#382—21no—
149#382—21no—
150#382D412yes—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4C2311yescharacteristic
172C4×C4⋊C4C2311yescharacteristic
173C4×C22⋊C4—22no—
174C4×C22⋊C4—22no—
175C4×C22⋊C4—22no—
176C2×#95—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—21no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207#126—24no—
208#126—24no—
209#125—22no—
210#125C4⋊C414yes—
211#124—22no—
212#124C4⋊C414yes—
213#123—24no—
214#123—24no—
215C22⋊Q8—22no—
216C22⋊Q8C4⋊C414yes—
217#121—22no—
218#121C4⋊C414yes—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—81no—
222C4×Q8—81no—
223C4×Q8C22×C412yes—
224C4×Q8C2×Q812yes—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—81no—
230C4×D4—81no—
231C4×D4—84no—
232C4×D4C22×C412yes—
233C4×D4C2×Q812yes—
234#117—24no—
235#117—24no—
236#117—42no—
237#117—42no—
238#117—81no—
239#117—81no—
240#117—82no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
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—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—41no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—41no—
294C2×C42—81no—
295C2×C42—81no—
296C2×C42C22×C412yes—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×Q812yes—
301C2×C42#4112yes—
302C2×C42#4112yes—
303C2×C42C2411yescharacteristic
304#107—82no—
305#106—42no—
306#106—42no—
307#106—42no—
308#106—42no—
309#106—82no—
310C4⋊C8—24no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8C22×C412yes—
318C4⋊C8C22×C412yes—
319#103—81no—
320#103—81no—
321#102—81no—
322#102—81no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325C22⋊C8—42no—
326C4×C8—24no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—42no—
330C4×C8C22×C412yes—
331C4×C8C22×C412yes—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—21no—
339#95—21no—
340#95—22no—
341#95—22no—
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—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4C2×C4⋊C412yes—
400C22×C4C4×D412yes—
401C22×C4C4×D412yes—
402C22×C4#12011yescharacteristic
403C22×C4#12011yescharacteristic
404C22×C4#12011yescharacteristic
405C22×C4C22⋊Q812yes—
406C22×C4C22⋊Q812yes—
407C22×C4#12312yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
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—
427C2×C8—41no—
428C2×C8—41no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
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—44no—
447C4⋊C4—42no—
448C4⋊C4—42no—
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—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4C2×C4⋊C412yes—
468C4⋊C4C2×C4⋊C412yes—
469C4⋊C4#13512yes—
470C4⋊C4#13512yes—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—22no—
483C42—22no—
484C42—44no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—81no—
490C42—81no—
491C42—82no—
492C42—82no—
493C42—82no—
494C42—82no—
495C42C2×C4⋊C412yes—
496C42C4×D412yes—
497C42C4×D412yes—
498C42C22⋊Q812yes—
499C42C22⋊Q812yes—
500C42#12312yes—
501C42C23×C411yescharacteristic
502C42C22×Q811yescharacteristic
503C42C2×#4111yescharacteristic
504C42C2×#4111yescharacteristic
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23#38512yes—
519C23C2×#12011yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524D4—22no—
525D4—22no—
526D4—48no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
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—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—164no—
587C2×C4#38512yes—
588C2×C4#38512yes—
589C2×C4#38512yes—
590C2×C4C22×C4⋊C411yescharacteristic
591C2×C4C2×C4×D411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4C2×C22⋊Q811yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C2×C4C2×#13511yescharacteristic
597C2×C4C2×#13511yescharacteristic
598C2×C4#57511yescharacteristic
599C2×C4#57511yescharacteristic
600C2×C4#57711yescharacteristic
601C2×C4#57711yescharacteristic
602C2×C4#57711yescharacteristic
603C2×C4#57711yescharacteristic
604C8—42no—
605C8—42no—
606C8—42no—
607C8—42no—
608C8—81no—
609C8—81no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22C2×#38511yescharacteristic
625C22#300211yescharacteristic
626C22#300211yescharacteristic
627C22#300911yescharacteristic
628C22#301011yescharacteristic
629C22#301911yescharacteristic
630C22#302011yescharacteristic
631C4—22no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—82no—
643C4—82no—
644C4—81no—
645C4—161no—
646C4—161no—
647C4#291211yescharacteristic
648C4#291211yescharacteristic
649C4C4×#13511yescharacteristic
650C4#294311yescharacteristic
651C4#307611yescharacteristic
652C4#307611yescharacteristic
653C4#309911yescharacteristic
654C4#309911yescharacteristic
655C2—42no—
656C2—82no—
657C2#1886711yescharacteristic
658C2#1886811yescharacteristic
659C2#1887911yescharacteristic
660C2#1888011yescharacteristic
661C2#2084811yescharacteristic
662C2#2086411yescharacteristic
663C2#2087211yescharacteristic
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. #21984C2semidirect
4. #21359C2semidirect
6. #8896C2semidirect
7. #8894C2semidirect
8. #8893C2semidirect
9. #8891C2semidirect
10. #8888C2semidirect
11. #8886C2semidirect
12. #8885C2semidirect
13. #8883C2semidirect
14. #8038C2semidirect
15. #7892C2semidirect
59. #1482C4semidirect
101. #431C2×C4semidirect
102. #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
#342432
#219842
#218622
#213592
C2×#14822
#88962
#88942
#88932
#88912
#88882
#88862
#88852
#88832
#80382
#78922