Finite Group Databasebeta

← #212690

Subgroups — #212690

Order 512.

717 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21713C211yesmaximal, characteristic, complemented
2#21682C211yesmaximal, characteristic
3C4×#486C211yesmaximal, characteristic
4#9874C211yesmaximal, characteristic, complemented
5#9644C211yesmaximal, characteristic
6#9558C211yesmaximal, characteristic, complemented
7#9558C211yesmaximal, characteristic, complemented
8#9554C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic, complemented
10#8784C211yesmaximal, characteristic, complemented
11#8210C211yesmaximal, characteristic, complemented
12#8206C211yesmaximal, characteristic, complemented
13#8202C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×#486C411yescharacteristic
17C2×#446—21no—
18C2×#436C2211yescharacteristic
19C2×#435—21no—
20C2×#435—21no—
21C2×C8×D4C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24#1990C411yescharacteristic
25#1985—21no—
26#1957C411yescharacteristic
27#1955—21no—
28#1955—21no—
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1940—21no—
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1934C411yescharacteristic
35#1934C411yescharacteristic
36#1934C411yescharacteristic
37#1934C411yescharacteristic
38#1842C2211yescharacteristic
39#1842C2211yescharacteristic
40#1834C411yescharacteristic
41#1771—21no—
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57C2×C82C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×#127C2×C411yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76#486—24no—
77#446—44no—
78#436—22no—
79#436—22no—
80#435—44no—
81#435—44no—
82C8×D4—22no—
83C8×D4—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×#97D411yescharacteristic
112C2×#97D411yescharacteristic
113C2×C4×C8—21no—
114C2×C4×C8C2×C411yescharacteristic
115C2×C4×C8C2×C411yescharacteristic
116C2×C4×C8C2×C411yescharacteristic
117C2×C4×C8C2×C411yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391—21no—
125#390—21no—
126#390—21no—
127#390C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#389—21no—
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#384C2×C411yescharacteristic
133#384C2×C411yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C43C2311yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
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—
152D4⋊C8—22no—
153D4⋊C8—22no—
154D4⋊C8—22no—
155D4⋊C8—22no—
156#322—22no—
157#322—22no—
158C82—22no—
159C82—22no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22×C411yescharacteristic
164C22×D4C22×C411yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—42no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—22no—
188C4⋊Q8—41no—
189#127—22no—
190#127—41no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247#102—22no—
248#102—22no—
249#102—22no—
250#102—22no—
251#102—22no—
252#102—41no—
253#102—41no—
254#102M1614yescomplemented
255C22⋊C8—42no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261#97—22no—
262#97—22no—
263#97—42no—
264#97D814yes—
265#97SD1614yes—
266C4×C8—22no—
267C4×C8—22no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—41no—
277C4×C8—41no—
278C4×C8—42no—
279C4×C8#4112yes—
280C4×C8#4112yes—
281C4×C8#4112yes—
282C4×C8#4112yes—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287C24—21no—
288C24—21no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—81no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—81no—
308C2×D4—81no—
309C2×D4C2×M1612yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335M16—84no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—84no—
371C2×C8#12412yes—
372C2×C8#12412yes—
373C2×C8C2×M1612yes—
374C2×C8#13112yes—
375C2×C8C2×D812yes—
376C2×C8C2×D812yes—
377C2×C8C2×SD1612yes—
378C2×C8C2×SD1612yes—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—81no—
397C4⋊C4C2×M1612yes—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42C2×D812yes—
417C42C2×SD1612yes—
418C42#13512yes—
419C42#13512yes—
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23C2×C4×D411yescharacteristic
439Q8—84no—
440D4—22no—
441D4—22no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—41no—
448D4—84no—
449D4—84no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4C4×D812yes—
487C2×C4C4×D812yes—
488C2×C4C4×SD1612yes—
489C2×C4C4×SD1612yes—
490C2×C4C4×#4111yescharacteristic
491C2×C4C4×#4111yescharacteristic
492C2×C4C2×#12411yescharacteristic
493C2×C4#52911yescharacteristic
494C2×C4#52911yescharacteristic
495C2×C4#53211yescharacteristic
496C2×C4C22×M1611yescharacteristic
497C2×C4C2×#13111yescharacteristic
498C2×C4#56811yescharacteristic
499C2×C4#56811yescharacteristic
500C2×C4C22×D811yescharacteristic
501C2×C4C22×SD1611yescharacteristic
502C2×C4C2×#13511yescharacteristic
503C2×C4C2×#13511yescharacteristic
504C8—22no—
505C8—22no—
506C8—22no—
507C8—22no—
508C8—22no—
509C8—42no—
510C8—42no—
511C8—41no—
512C8—41no—
513C8—81no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8#48614yescomplemented
518C22—42no—
519C22—42no—
520C22—42no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22C4×#12411yescharacteristic
533C22D4×M1611yescharacteristic
534C22#293511yescharacteristic
535C22C2×C4×D811yescharacteristic
536C22C2×C4×SD1611yescharacteristic
537C22#295711yescharacteristic
538C22#295711yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4#296511yescharacteristic
552C4#296711yescharacteristic
553C4#296811yescharacteristic
554C4#297011yescharacteristic
555C4C2×#48611yescharacteristic
556C4#313111yescharacteristic
557C4#316711yescharacteristic
558C4#316911yescharacteristic
559C2—82no—
560C2—161no—
561C2M16×D811yescharacteristic
562C2M16×SD1611yescharacteristic
563C2#1806611yescharacteristic
564C2#1806711yescharacteristic
565C2#2163211yescharacteristic
566C2C4×#48611yescharacteristic
567C2#2165911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
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. #21713C2semidirect
4. #9874C2semidirect
6. #9558C2semidirect
7. #9558C2semidirect
8. #9554C2semidirect
9. #9554C2semidirect
10. #8784C2semidirect
11. #8210C2semidirect
12. #8206C2semidirect
13. #8202C2semidirect
14. C8×#102C2semidirect
15. C8×#97C2semidirect
254. #102M16semidirect
517. C8#486semidirect

Maximal subgroups

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

subgroupindex
#217132
#216822
C4×#4862
#98742
#96442
#95582
#95542
#87842
#82102
#82062
#82022
C8×#1022
C8×#972