Finite Group Databasebeta

← #211028

Subgroups — #211028

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23697C211yesmaximal, characteristic
2#21488C211yesmaximal, characteristic, complemented
3#21486C211yesmaximal, characteristic, complemented
4C2×#1586C211yesmaximal, characteristic, complemented
5C2×#1453C211yesmaximal, characteristic, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9593C211yesmaximal, characteristic, complemented
8#9558C211yesmaximal, characteristic, complemented
9#9558C211yesmaximal, characteristic, complemented
10#9074C211yesmaximal, characteristic, complemented
11#9074C211yesmaximal, characteristic, complemented
12#8600C211yesmaximal, characteristic, complemented
13#8289C211yesmaximal, characteristic, complemented
14#8208C211yesmaximal, characteristic, complemented
15#8200C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436—21no—
18C2×#436C2211yescharacteristic
19C2×#435—21no—
20C2×C8×D4C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2087C411yescharacteristic
27#2076C411yescharacteristic
28#2065C411yescharacteristic
29#2065C411yescharacteristic
30#2054C411yescharacteristic
31#2034C411yescharacteristic
32#2034C411yescharacteristic
33#1999C411yescharacteristic
34#1954C2211yescharacteristic
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1924—21no—
40#1870—21no—
41#1870—21no—
42#1797C2211yescharacteristic
43#1797C2211yescharacteristic
44C4×#106C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1753—21no—
50#1752—21no—
51#1586—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58#1453—24no—
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73C23×C8—21no—
74C2×#121C2×C411yescharacteristic
75C2×#121C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C4×D8—22no—
82C4×D8—22no—
83#436—22no—
84#436—22no—
85#436—44no—
86#435—44no—
87C8×D4—22no—
88C8×D4—22no—
89#431—22no—
90#431—22no—
91#431—22no—
92#431—22no—
93C2×#107C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95C2×#106C2×C411yescharacteristic
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2×C411yescharacteristic
100C2×C4⋊C8C2×C411yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8C2×C411yescharacteristic
121C2×C22⋊C8C2×C411yescharacteristic
122C4×M16—22no—
123C4×M16—22no—
124C2×#97—21no—
125C2×#97D411yescharacteristic
126C2×#97D411yescharacteristic
127C2×C4×C8—21no—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134#390C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#388—21no—
139#388—21no—
140#386—21no—
141#386—21no—
142#380C2×C411yescharacteristic
143#380C2×C411yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#335—22no—
161#334—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166#322—22no—
167#322—22no—
168#322D414yescomplemented
169C82—22no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C2×D8—22no—
180C2×D8—22no—
181C2×D8—41no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—41no—
187C2×M16—41no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—42no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208#121—22no—
209#121—22no—
210#121—41no—
211#121—41no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42C2411yescharacteristic
252#107—22no—
253#106—22no—
254#106—41no—
255C4⋊C8—24no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268#102—22no—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—41no—
273#102—41no—
274C22⋊C8—24no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8—42no—
285#97—22no—
286#97—22no—
287#97—42no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—42no—
295C4×C8C2×D412yes—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298C4×C8#4112yes—
299C4×C8#4112yes—
300C4×C8#4112yes—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307C24—21no—
308C24—21no—
309C24C4×D411yescharacteristic
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—81no—
326C2×D4—81no—
327C2×D4#13112yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
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×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368D8—44no—
369M16—44no—
370M16—44no—
371M16—84no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—84no—
409C2×C8C4×D412yes—
410C2×C8C4×D412yes—
411C2×C8#12112yes—
412C2×C8#12112yes—
413C2×C8AGL(1,ℤ/8)12yes—
414C2×C8AGL(1,ℤ/8)12yes—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4#13112yes—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42C22×D411yescharacteristic
452C42C2×#4111yescharacteristic
453C42C2×#4111yescharacteristic
454C42C2×#4111yescharacteristic
455C23—22no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23C2×C4×D411yescharacteristic
477D4—22no—
478D4—22no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—84no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4C2×C4×D411yescharacteristic
535C2×C4C4×#4111yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4#52511yescharacteristic
538C2×C4#52911yescharacteristic
539C2×C4#53211yescharacteristic
540C2×C4C2×#13111yescharacteristic
541C2×C4C2×#13111yescharacteristic
542C2×C4#56811yescharacteristic
543C2×C4#56811yescharacteristic
544C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
545C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
546C2×C4#57511yescharacteristic
547C2×C4#57511yescharacteristic
548C8—22no—
549C8—22no—
550C8—22no—
551C8—22no—
552C8—22no—
553C8—22no—
554C8—42no—
555C8—42no—
556C8—41no—
557C8—41no—
558C8—81no—
559C8—81no—
560C8—81no—
561C8—81no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—41no—
567C22—42no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C4×#12111yescharacteristic
584C22#293311yescharacteristic
585C22#293311yescharacteristic
586C22C4×AGL(1,ℤ/8)11yescharacteristic
587C22C4×AGL(1,ℤ/8)11yescharacteristic
588C22#295711yescharacteristic
589C22#295711yescharacteristic
590C4—21no—
591C4—21no—
592C4—41no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4#293311yescharacteristic
602C4#293511yescharacteristic
603C4#296611yescharacteristic
604C4#296711yescharacteristic
605C4#305511yescharacteristic
606C4#305511yescharacteristic
607C4#318311yescharacteristic
608C4#318411yescharacteristic
609C2—81no—
610C2—82no—
611C2—161no—
612C2#1800611yescharacteristic
613C2#1800711yescharacteristic
614C2#1800811yescharacteristic
615C2#1801111yescharacteristic
616C2#2149611yescharacteristic
617C2#2153811yescharacteristic
618C2#2154511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
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
2. #21488C2semidirect
3. #21486C2semidirect
4. C2×#1586C2semidirect
5. C2×#1453C2semidirect
6. #9657C2semidirect
7. #9593C2semidirect
8. #9558C2semidirect
9. #9558C2semidirect
10. #9074C2semidirect
11. #9074C2semidirect
12. #8600C2semidirect
13. #8289C2semidirect
14. #8208C2semidirect
15. #8200C2semidirect
168. #322D4semidirect

Maximal subgroups

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

subgroupindex
#236972
#214882
#214862
C2×#15862
C2×#14532
#96572
#95932
#95582
#90742
#86002
#82892
#82082
#82002