Finite Group Databasebeta

← #192633

Subgroups — #192633

Order 512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#117C211yesmaximal, characteristic
2#21364C211yesmaximal, characteristic
3#20824C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic
6#8896C211yesmaximal, characteristic, complemented
7#8895C211yesmaximal, characteristic, complemented
8#8892C211yesmaximal, characteristic, complemented
9#8891C211yesmaximal, characteristic, complemented
10#8888C211yesmaximal, characteristic, complemented
11#8887C211yesmaximal, characteristic, complemented
12#8884C211yesmaximal, characteristic, complemented
13#8883C211yesmaximal, characteristic, complemented
14#8042C211yesmaximal, characteristic
15#7903C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#419—21no—
20C2×C4×M16C2211yescharacteristic
21#2355—21no—
22#2355—21no—
23#2355C2211yescharacteristic
24#2355C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26Q8×C4⋊C4C2211yescharacteristic
27#2345—22no—
28#2345—21no—
29Q8×C22⋊C4—21no—
30#2295—22no—
31#2295—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35C42×Q8—21no—
36C4×#117—21no—
37C4×#117C2211yescharacteristic
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1854—21no—
43#1854—21no—
44#1845C2211yescharacteristic
45#1844C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1809—22no—
49#1805—21no—
50#1805—21no—
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1803C2211yescharacteristic
54#1803C2211yescharacteristic
55#1776—22no—
56#1776—21no—
57#1776—21no—
58#1736—21no—
59#1491C418yescomplemented
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72C23×Q8D411yescharacteristic
73C22×M16—21no—
74C22×M16—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8C2×C414yes—
82C2×C4×Q8D411yescharacteristic
83C2×C4×Q8D411yescharacteristic
84C2×C4×Q8Q814yes—
85C2×C4×Q8C2311yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×#117—21no—
88C2×#117—21no—
89C2×#117D411yescharacteristic
90C2×#117D411yescharacteristic
91C22×C4⋊C4—21no—
92C22×C4⋊C4D411yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95#432—22no—
96#432C2×C414yes—
97C2×#107—22no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#106—22no—
101C2×#106—21no—
102C2×#106—21no—
103#423—22no—
104#423—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110#419—44no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×C22⋊C8—21no—
116C4×M16—22no—
117C4×M16C2×C414yes—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—22no—
123#391—22no—
124#391—21no—
125#391—21no—
126#389—21no—
127#389—21no—
128#389—22no—
129#389—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#387—22no—
135#387—22no—
136#384—21no—
137#384—21no—
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#384C2311yescharacteristic
142#382—21no—
143#382—21no—
144#382—21no—
145#382—21no—
146#380—22no—
147#380—22no—
148#380—21no—
149#380—21no—
150#380—21no—
151#380D412yes—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
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⋊C4D411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4D412yes—
168C4×C4⋊C4C2311yescharacteristic
169C4×C4⋊C4C2311yescharacteristic
170C4×C22⋊C4—22no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174#376—22no—
175#376—21no—
176C43—22no—
177C43—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182#340—22no—
183#340—22no—
184#340—21no—
185#340—21no—
186#340—21no—
187#340—21no—
188Q8⋊C8C2×C414yes—
189Q8⋊C8C2×C414yes—
190Q8⋊C8C2×C414yes—
191Q8⋊C8C2×C414yes—
192C22×Q8—21no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—41no—
197C22×Q8—41no—
198C22×Q8C4⋊C414yes—
199C22×Q8C2×D411yescharacteristic
200C22×Q8C2×D411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C2×M16—24no—
207C2×M16—42no—
208C2×M16—42no—
209C2×M16—42no—
210C2×M16—42no—
211C2×M16—42no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—42no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8C4⋊C414yes—
233C4×Q8C4⋊C414yes—
234C4×Q8C22×C412yes—
235C4×Q8C22×C412yes—
236C4×Q8C2×Q812yes—
237C4×Q8C2×Q812yes—
238#117—22no—
239#117—22no—
240#117—41no—
241#117—41no—
242#117—42no—
243#117—42no—
244#117—42no—
245#117—42no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C4⋊C414yes—
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—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—41no—
281C2×C42—24no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—24no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—41no—
299C2×C42C22×C412yes—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×Q812yes—
304C2×C42#4112yes—
305C2×C42#4112yes—
306C2×C42C2411yescharacteristic
307#107—42no—
308#107—44no—
309#107—42no—
310#106—42no—
311#106—44no—
312#106—42no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—42no—
320C4⋊C8C22×C412yes—
321C4⋊C8C22×C412yes—
322#103—42no—
323#103—42no—
324#103—42no—
325#103—42no—
326C22⋊C8—24no—
327#97—24no—
328#97—24no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8C22×C412yes—
332C4×C8C22×C412yes—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—21no—
343#95—22no—
344#95—22no—
345#95—21no—
346#95—21no—
347#95—41no—
348#95—41no—
349C24#12011yescharacteristic
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—24no—
355C2×Q8—28no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8C2×C4⋊C412yes—
366C2×Q8C2×C4⋊C412yes—
367C2×Q8AGL(1,ℤ/8)12yes—
368C2×Q8#13712yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
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×C4C2×C4⋊C412yes—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4#12011yescharacteristic
401C22×C4#12011yescharacteristic
402C22×C4#12011yescharacteristic
403C22×C4C22⋊Q812yes—
404C22×C4C22⋊Q812yes—
405C22×C4#12312yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409M16—48no—
410M16—84no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
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—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
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—41no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4C2×C4⋊C412yes—
469C4⋊C4C2×C4⋊C412yes—
470C4⋊C4#13712yes—
471C4⋊C4#13712yes—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—82no—
476C42—24no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—24no—
482C42—24no—
483C42—42no—
484C42—44no—
485C42—44no—
486C42—44no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—81no—
492C42—81no—
493C42—82no—
494C42—82no—
495C42—82no—
496C42C2×C4⋊C412yes—
497C42C4×D412yes—
498C42C4×D412yes—
499C42C22⋊Q812yes—
500C42C22⋊Q812yes—
501C42#12312yes—
502C42C23×C411yescharacteristic
503C42C22×Q811yescharacteristic
504C42C2×#4111yescharacteristic
505C42C2×#4111yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#38512yes—
513C23C2×#12011yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—48no—
520Q8—41no—
521Q8—41no—
522Q8#42814yes—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
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—22no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
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—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4#38512yes—
580C2×C4#38512yes—
581C2×C4#38512yes—
582C2×C4C22×C4⋊C411yescharacteristic
583C2×C4C2×C4×D411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4C2×#12311yescharacteristic
588C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
589C2×C4C2×#13711yescharacteristic
590C2×C4C2×#13711yescharacteristic
591C2×C4C2×#13711yescharacteristic
592C2×C4#57511yescharacteristic
593C2×C4#57511yescharacteristic
594C2×C4#57511yescharacteristic
595C2×C4#57511yescharacteristic
596C8—24no—
597C8—42no—
598C8—42no—
599C8—42no—
600C8—81no—
601C8—81no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22C2×#38511yescharacteristic
610C22#300411yescharacteristic
611C22#300511yescharacteristic
612C22#300611yescharacteristic
613C22#300711yescharacteristic
614C22#301811yescharacteristic
615C22#301811yescharacteristic
616C4—22no—
617C4—24no—
618C4—21no—
619C4—21no—
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4—82no—
631C4—161no—
632C4—161no—
633C4C2×#42811yescharacteristic
634C4#291111yescharacteristic
635C4C4×#13711yescharacteristic
636C4C4×#13711yescharacteristic
637C4#307811yescharacteristic
638C4#307811yescharacteristic
639C4#309411yescharacteristic
640C4#309511yescharacteristic
641C2—42no—
642C2#1855611yescharacteristic
643C2#1855711yescharacteristic
644C2#1855811yescharacteristic
645C2#1855911yescharacteristic
646C2#2084611yescharacteristic
647C2#2086311yescharacteristic
648C2#2087411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #8896C2semidirect
7. #8895C2semidirect
8. #8892C2semidirect
9. #8891C2semidirect
10. #8888C2semidirect
11. #8887C2semidirect
12. #8884C2semidirect
13. #8883C2semidirect
59. #1491C4semidirect

Maximal subgroups

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

subgroupindex
Q8×#1172
#213642
#208242
#206472
C2×#14912
#88962
#88952
#88922
#88912
#88882
#88872
#88842
#88832
#80422
#79032