Finite Group Databasebeta

← #210532

Subgroups — #210532

Order 512.

855 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21682C211yesmaximal, characteristic, complemented
2C4×#459C211yesmaximal, characteristic
3#20494C211yesmaximal, characteristic, complemented
4C2×#1594C211yesmaximal, characteristic, complemented
5C2×#1586C211yesmaximal, characteristic, complemented
6#9678C211yesmaximal, characteristic, complemented
7#9605C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9542C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8951C211yesmaximal, characteristic, complemented
12#8563C211yesmaximal, characteristic, complemented
13#8263C211yesmaximal, characteristic, complemented
14#8212C211yesmaximal, characteristic, complemented
15#8200C211yesmaximal, characteristic, complemented
16C2×#459C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#435—21no—
23C2×#435—21no—
24C2×C8×D4C2211yescharacteristic
25C2×C4×M16—21no—
26C4×#127C2211yescharacteristic
27C42×D4C2211yescharacteristic
28#1982C411yescharacteristic
29#1955—21no—
30#1955—21no—
31#1954C2211yescharacteristic
32#1954C2211yescharacteristic
33#1947C411yescharacteristic
34#1925C411yescharacteristic
35#1925C411yescharacteristic
36#1879C411yescharacteristic
37#1879C411yescharacteristic
38#1842C2211yescharacteristic
39#1802C411yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1753—21no—
48#1753—21no—
49#1749—21no—
50C4×#97C2211yescharacteristic
51#1594—24no—
52#1586—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73C22×M16—21no—
74C23×C8—21no—
75C2×#127C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C22×C42—21no—
84#459—24no—
85C4×D8—22no—
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—22no—
89#436—22no—
90#436—22no—
91#436—22no—
92#436—22no—
93#436—22no—
94#436D414yescomplemented
95#435—44no—
96#435—44no—
97C8×D4—22no—
98C8×D4—22no—
99C2×#107C2×C411yescharacteristic
100C2×#107C2×C411yescharacteristic
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2×C411yescharacteristic
105C2×C4⋊C8C2×C411yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C4×M16—44no—
127C2×#97—21no—
128C2×#97D411yescharacteristic
129C2×#97D411yescharacteristic
130C2×#97D411yescharacteristic
131C2×#97D411yescharacteristic
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#390—21no—
141#390—21no—
142#390C2×C411yescharacteristic
143#390C2×C411yescharacteristic
144#384C2×C411yescharacteristic
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C43C2311yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162#334—22no—
163#334—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—22no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170#322—22no—
171#322—22no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C22×D4C22×C411yescharacteristic
176C22×D4C22×C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C2×D8—22no—
183C2×D8—22no—
184C2×D8—22no—
185C2×D8—22no—
186C2×D8—41no—
187C2×D8—41no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—42no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214#127—22no—
215#127—41no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4C2×D412yes—
235C4×D4#4112yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42C2411yescharacteristic
264#107—22no—
265#107—22no—
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—41no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8C2×D412yes—
278C4⋊C8#4112yes—
279#102—22no—
280#102—22no—
281#102—22no—
282#102—22no—
283#102—22no—
284#102—22no—
285#102—22no—
286#102M1614yescomplemented
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295#97—22no—
296#97—22no—
297#97—22no—
298#97—41no—
299#97—42no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—42no—
311C4×C8—42no—
312C4×C8C2×D412yes—
313C4×C8#4112yes—
314#95—21no—
315#95—21no—
316#95—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—81no—
341C2×D4—81no—
342C2×D4C4×D412yes—
343C2×D4C2×M1612yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4111yescharacteristic
380D8—44no—
381D8—44no—
382M16—48no—
383M16—48no—
384M16—84no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—21no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
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—84no—
424C2×C8#12112yes—
425C2×C8#12112yes—
426C2×C8C2×M1612yes—
427C2×C8AGL(1,ℤ/8)12yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4C4×D412yes—
438C4⋊C4C2×M1612yes—
439C4⋊C4#13112yes—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—42no—
463C42—42no—
464C42AGL(1,ℤ/8)12yes—
465C42AGL(1,ℤ/8)12yes—
466C42C22×D411yescharacteristic
467C42C2×#4111yescharacteristic
468C42C2×#4111yescharacteristic
469C42C2×#4111yescharacteristic
470C23—22no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×C4×D411yescharacteristic
492D4—22no—
493D4—22no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—84no—
505D4—84no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4#44212yes—
548C2×C4#44212yes—
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C4×#4111yescharacteristic
551C2×C4C2×#12111yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4#52911yescharacteristic
554C2×C4#53211yescharacteristic
555C2×C4C22×M1611yescharacteristic
556C2×C4C2×#13111yescharacteristic
557C2×C4#56811yescharacteristic
558C2×C4#56811yescharacteristic
559C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
560C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
561C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
562C2×C4#57511yescharacteristic
563C8—22no—
564C8—22no—
565C8—22no—
566C8—22no—
567C8—42no—
568C8—42no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—42no—
574C8—81no—
575C8—81no—
576C8—81no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22C4×#12111yescharacteristic
599C22#293311yescharacteristic
600C22D4×M1611yescharacteristic
601C22C2×#44211yescharacteristic
602C22C4×AGL(1,ℤ/8)11yescharacteristic
603C22#295611yescharacteristic
604C22#295711yescharacteristic
605C4—21no—
606C4—21no—
607C4—22no—
608C4—22no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—81no—
616C4D4×M1611yescharacteristic
617C4#293511yescharacteristic
618C4#296511yescharacteristic
619C4#296711yescharacteristic
620C4#303611yescharacteristic
621C4#303811yescharacteristic
622C4#310911yescharacteristic
623C4#311511yescharacteristic
624C2—82no—
625C2—82no—
626C2—161no—
627C2#1800111yescharacteristic
628C2#1801111yescharacteristic
629C2#1804411yescharacteristic
630C2#1804811yescharacteristic
631C2#2149011yescharacteristic
632C2#2150311yescharacteristic
633C2#2151911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#459C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21682C2semidirect
3. #20494C2semidirect
4. C2×#1594C2semidirect
5. C2×#1586C2semidirect
6. #9678C2semidirect
7. #9605C2semidirect
8. #9542C2semidirect
9. #9542C2semidirect
10. #8951C2semidirect
11. #8951C2semidirect
12. #8563C2semidirect
13. #8263C2semidirect
14. #8212C2semidirect
15. #8200C2semidirect
94. #436D4semidirect
286. #102M16semidirect

Maximal subgroups

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

subgroupindex
#216822
C4×#4592
#204942
C2×#15942
C2×#15862
#96782
#96052
#95422
#89512
#85632
#82632
#82122
#82002