Finite Group Databasebeta

← #305296

Subgroups — #305296

Order 512.

727 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26113C211yesmaximal, characteristic, complemented
2#26107C211yesmaximal, characteristic
3Q8×#102C211yesmaximal, characteristic
4D4×#103C211yesmaximal, characteristic
5#10681C211yesmaximal, characteristic, complemented
6#10602C211yesmaximal, characteristic, complemented
7#10600C211yesmaximal, characteristic, complemented
8#10353C211yesmaximal, characteristic, complemented
9#8888C211yesmaximal, characteristic, complemented
10#8886C211yesmaximal, characteristic
11#8844C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic
13#8649C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic
15#8447C211yesmaximal, characteristic
16C2×D4×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×#103—21no—
20#2571C2211yescharacteristic
21Q8×C4⋊C4C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#2101—21no—
24#2101—21no—
25#2088—21no—
26#2088—21no—
27#1990—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1988—21no—
31#1988C2211yescharacteristic
32#1981C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1892—21no—
39#1872—21no—
40#1865—21no—
41#1864—21no—
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1805—21no—
46#1804—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1787—21no—
53C4×#103C2211yescharacteristic
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×Q8—21no—
68C23×C8—21no—
69D4×Q8—22no—
70D4×Q8—22no—
71D4×Q8—41no—
72D4×Q8Q814yes—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—41no—
77C2×C4×Q8—21no—
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82C22×C4⋊C4—21no—
83C8×Q8—22no—
84C8×Q8—41no—
85C8×D4—22no—
86C8×D4—41no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—41no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—44no—
104C2×#103D411yescharacteristic
105C2×#103D411yescharacteristic
106C2×#102—21no—
107C2×#102—21no—
108C2×#102D411yescharacteristic
109C2×#102D411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#400—41no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—41no—
121#385—41no—
122#384—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#382—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—41no—
154#335—41no—
155#332—41no—
156Q8⋊C8—22no—
157D4⋊C8—22no—
158C22×Q8—24no—
159C22×Q8—21no—
160C22×Q8—41no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—81no—
165C22×Q8—81no—
166C22×Q8C2×D411yescharacteristic
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—41no—
170C23×C4—41no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—81no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—24no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—42no—
188C4⋊Q8C2×Q812yes—
189C22⋊Q8—24no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—42no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—42no—
196C4×Q8—81no—
197C4×Q8C2×Q812yes—
198C4×Q8#4112yes—
199C4×D4—24no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—81no—
205C4×D4C2×Q812yes—
206C4×D4#4112yes—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—41no—
239C2×C42C2411yescharacteristic
240#106—81no—
241#106—81no—
242#106—82no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—42no—
252C4⋊C8—81no—
253C4⋊C8—81no—
254#103—22no—
255#103—22no—
256#103—42no—
257#103—42no—
258#103—42no—
259#103—42no—
260#103—81no—
261#103—81no—
262#102—22no—
263#102—22no—
264#102—42no—
265#102—42no—
266C22⋊C8—42no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—42no—
274#95—21no—
275#95—21no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281C24—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—44no—
287C2×Q8—41no—
288C2×Q8—48no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—82no—
294C2×Q8—82no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8C22⋊Q812yes—
299C2×Q8#13512yes—
300C2×Q8AGL(1,ℤ/8)12yes—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C22⋊Q812yes—
305C2×D4#13512yes—
306C2×D4#13712yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—44no—
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—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—161no—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
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—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—84no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—44no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—82no—
391C4⋊C4—81no—
392C4⋊C4—81no—
393C4⋊C4—81no—
394C4⋊C4—82no—
395C4⋊C4—161no—
396C4⋊C4—161no—
397C4⋊C4C22⋊Q812yes—
398C4⋊C4C22⋊Q812yes—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—44no—
402C22⋊C4—82no—
403C42—22no—
404C42—22no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—82no—
410C42C22×Q811yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C42H(5,𝔽2)11yescharacteristic
414C23—22no—
415C23—22no—
416C23—22no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23D4211yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—22no—
425Q8—42no—
426Q8—42no—
427Q8—88no—
428Q8—81no—
429Q8—81no—
430Q8#47814yes—
431D4—22no—
432D4—22no—
433D4—22no—
434D4#47914yes—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—88no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—164no—
476C2×C4C2×C22⋊Q811yescharacteristic
477C2×C4C2×C22⋊Q811yescharacteristic
478C2×C4#54311yescharacteristic
479C2×C4#54611yescharacteristic
480C2×C4#54611yescharacteristic
481C2×C4#55111yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
485C2×C4C2×#13711yescharacteristic
486C2×C4#57611yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57711yescharacteristic
489C2×C4#57811yescharacteristic
490C8—42no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8—81no—
499C22—24no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22#243711yescharacteristic
507C22#326511yescharacteristic
508C22#326611yescharacteristic
509C22#328011yescharacteristic
510C22#328511yescharacteristic
511C22#328911yescharacteristic
512C22#329111yescharacteristic
513C4—24no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—82no—
522C4—81no—
523C4—161no—
524C4—161no—
525C4—161no—
526C4C2×#47811yescharacteristic
527C4C2×#47911yescharacteristic
528C4#308111yescharacteristic
529C4#308211yescharacteristic
530C4#329611yescharacteristic
531C4#330111yescharacteristic
532C4#331211yescharacteristic
533C4#331811yescharacteristic
534C2—44no—
535C2#1978011yescharacteristic
536C2#1983711yescharacteristic
537C2#1999211yescharacteristic
538C2#2007311yescharacteristic
539C2#2595211yescharacteristic
540C2#2596511yescharacteristic
541C2#2598611yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26113C2semidirect
5. #10681C2semidirect
6. #10602C2semidirect
7. #10600C2semidirect
8. #10353C2semidirect
9. #8888C2semidirect
11. #8844C2semidirect
13. #8649C2semidirect

Maximal subgroups

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

subgroupindex
#261132
#261072
Q8×#1022
D4×#1032
#106812
#106022
#106002
#103532
#88882
#88862
#88442
#88432
#86492
#86482
#84472