Finite Group Databasebeta

← #217830

Subgroups — #217830

Order 512.

785 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21488C211yesmaximal, characteristic
2C4×#448C211yesmaximal, characteristic
3#21303C211yesmaximal, characteristic, complemented
4#9666C211yesmaximal, characteristic, complemented
5#9660C211yesmaximal, characteristic, complemented
6#9559C211yesmaximal, characteristic, complemented
7#9555C211yesmaximal, characteristic, complemented
8#9544C211yesmaximal, characteristic, complemented
9#9544C211yesmaximal, characteristic, complemented
10#8949C211yesmaximal, characteristic, complemented
11#8949C211yesmaximal, characteristic, complemented
12#8944C211yesmaximal, characteristic
13#8832C211yesmaximal, characteristic, complemented
14#8311C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#448C411yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×#436—21no—
21C2×#436C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C4×#121C2211yescharacteristic
25#1985—21no—
26#1985—21no—
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1984C2211yescharacteristic
31#1954C2211yescharacteristic
32#1939C2211yescharacteristic
33#1924—21no—
34#1880C411yescharacteristic
35#1880C411yescharacteristic
36#1874C411yescharacteristic
37#1870—21no—
38#1865C411yescharacteristic
39#1865C411yescharacteristic
40#1853C411yescharacteristic
41#1797C2211yescharacteristic
42#1787C411yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46#1753—21no—
47C4×C22⋊C8C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49#1728—21no—
50#1728—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×Q8C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16—21no—
72C22×M16D411yescharacteristic
73C22×M16D411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—41no—
78C2×C4×Q8—41no—
79C2×C4×Q8—41no—
80C2×C4×Q8—41no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4C2311yescharacteristic
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#448—24no—
92C4×SD16—41no—
93C4×SD16—41no—
94C4×SD16—41no—
95C4×SD16—41no—
96#436—41no—
97#436—41no—
98#436—44no—
99C2×#106C2×C411yescharacteristic
100C2×#106C2×C411yescharacteristic
101#423—41no—
102#423—41no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8C2×C411yescharacteristic
123C2×C22⋊C8C2×C411yescharacteristic
124C4×M16—41no—
125C4×M16—41no—
126C2×#97D411yescharacteristic
127C2×#97D411yescharacteristic
128C2×C4×C8—21no—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#390C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#388—21no—
138#386—21no—
139#380C2×C411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336C2×C411yescharacteristic
159#336C2×C411yescharacteristic
160#335—41no—
161#335—41no—
162Q8⋊C8—41no—
163Q8⋊C8—41no—
164Q8⋊C8—41no—
165Q8⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168#322—41no—
169#322—41no—
170C22×Q8—21no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8C22×C411yescharacteristic
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4111yescharacteristic
187C2×SD16—22no—
188C2×SD16—22no—
189C2×SD16—22no—
190C2×SD16—41no—
191C2×SD16—41no—
192C2×SD16—41no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—41no—
197C2×M16—41no—
198C2×M16—41no—
199C2×M16—42no—
200C2×M16—42no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—42no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219#121—22no—
220#121—41no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×Q8—42no—
234C4×Q8—42no—
235C4×Q8—42no—
236C4×Q8—42no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—42no—
242C4×D4—42no—
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—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42C2411yescharacteristic
278#106—41no—
279#106—41no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—42no—
289C4⋊C8—42no—
290#103—22no—
291#103—22no—
292#103—41no—
293#103—41no—
294#103—41no—
295#103M1614yescomplemented
296#102—41no—
297#102—41no—
298C22⋊C8—22no—
299C22⋊C8—41no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305#97—41no—
306#97—41no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—42no—
320#95—21no—
321#95—21no—
322#95—21no—
323C24—21no—
324C24C4×D411yescharacteristic
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8C2×M1612yes—
343C2×Q8#13112yes—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—81no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
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—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4#13712yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394SD16—44no—
395SD16—44no—
396M16—44no—
397M16—44no—
398M16—84no—
399M16—84no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8C2×M1612yes—
441C2×C8#13712yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4C2×M1612yes—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—42no—
473C42—42no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42H(5,𝔽2)11yescharacteristic
481C42H(5,𝔽2)11yescharacteristic
482C42H(5,𝔽2)11yescharacteristic
483C42H(5,𝔽2)11yescharacteristic
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#44012yes—
498C23C2×C4×D411yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8—48no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—84no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—48no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—84no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4#44012yes—
561C2×C4#51811yescharacteristic
562C2×C4#51811yescharacteristic
563C2×C4#53411yescharacteristic
564C2×C4#53511yescharacteristic
565C2×C4#53811yescharacteristic
566C2×C4#53811yescharacteristic
567C2×C4C22×M1611yescharacteristic
568C2×C4C2×#13111yescharacteristic
569C2×C4#56811yescharacteristic
570C2×C4#56811yescharacteristic
571C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
572C2×C4C2×#13711yescharacteristic
573C2×C4C2×#13711yescharacteristic
574C2×C4#57511yescharacteristic
575C8—22no—
576C8—22no—
577C8—42no—
578C8—41no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—41no—
583C8—41no—
584C8—81no—
585C8—81no—
586C8—81no—
587C8—81no—
588C8—81no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#232111yescharacteristic
604C22#293311yescharacteristic
605C22D4×M1611yescharacteristic
606C22C2×#44011yescharacteristic
607C22C4×#13711yescharacteristic
608C22#295611yescharacteristic
609C22#295711yescharacteristic
610C4—22no—
611C4—21no—
612C4—21no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4#297411yescharacteristic
623C4#297511yescharacteristic
624C4#297611yescharacteristic
625C4#297811yescharacteristic
626C4#319311yescharacteristic
627C4#319611yescharacteristic
628C4#321311yescharacteristic
629C4#321811yescharacteristic
630C2—42no—
631C2—161no—
632C2#1800411yescharacteristic
633C2#1801211yescharacteristic
634C2#1804011yescharacteristic
635C2#1804911yescharacteristic
636C2#2203811yescharacteristic
637C2#2204711yescharacteristic
638C2#2206211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21303C2semidirect
4. #9666C2semidirect
5. #9660C2semidirect
6. #9559C2semidirect
7. #9555C2semidirect
8. #9544C2semidirect
9. #9544C2semidirect
10. #8949C2semidirect
11. #8949C2semidirect
13. #8832C2semidirect
14. #8311C2semidirect
15. #8147C2semidirect
295. #103M16semidirect

Maximal subgroups

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

subgroupindex
#214882
C4×#4482
#213032
#96662
#96602
#95592
#95552
#95442
#89492
#89442
#88322
#83112
#81472