Finite Group Databasebeta

← #243441

Subgroups — #243441

Order 512.

691 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21691C211yesmaximal, characteristic, complemented
2C4×#474C211yesmaximal, characteristic
3C4×#474C211yesmaximal, characteristic
4#21270C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10619C211yesmaximal, characteristic
7#10592C211yesmaximal, characteristic, complemented
8#10588C211yesmaximal, characteristic, complemented
9#10538C211yesmaximal, characteristic, complemented
10#10485C211yesmaximal, characteristic
11#10484C211yesmaximal, characteristic
12#10274C211yesmaximal, characteristic
13#8275C211yesmaximal, characteristic, complemented
14#8236C211yesmaximal, characteristic, complemented
15#8236C211yesmaximal, characteristic, complemented
16C2×#494C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#441—21no—
20C2×#441—21no—
21C2×#432—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1988C2211yescharacteristic
26#1988C2211yescharacteristic
27#1971—21no—
28#1971—21no—
29#1956C2211yescharacteristic
30#1956C2211yescharacteristic
31#1954C2211yescharacteristic
32#1954C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1924—21no—
38#1924—21no—
39#1918—21no—
40#1918—21no—
41#1862—21no—
42#1862—21no—
43#1844C2211yescharacteristic
44#1802C2211yescharacteristic
45#1802C2211yescharacteristic
46C4×#107—21no—
47C4×#107—21no—
48#1767C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×#102—21no—
52C4×#102—21no—
53#1749—21no—
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×Q16—21no—
68C22×Q16—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C22×C42—21no—
77#494—22no—
78#494—22no—
79#474—22no—
80#474—22no—
81#474—22no—
82#474—22no—
83#441—44no—
84#441—44no—
85#432—44no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×#97D411yescharacteristic
114C2×#97D411yescharacteristic
115C2×#97D411yescharacteristic
116C2×#97D411yescharacteristic
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#391—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C43C2311yescharacteristic
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#335—22no—
154#334—22no—
155#332—22no—
156#332—22no—
157#332—22no—
158#332—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C2×Q16—24no—
166C2×Q16—24no—
167C2×Q16—42no—
168C2×Q16—42no—
169C2×Q16—42no—
170C2×Q16—42no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8#4112yes—
188C4⋊Q8#4112yes—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
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—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—24no—
234#107—24no—
235#107—42no—
236#107—42no—
237#106—42no—
238#106—42no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8#4112yes—
252C4⋊C8#4112yes—
253#103—42no—
254#103—42no—
255#103—42no—
256#103—42no—
257#102—24no—
258#102—24no—
259#102—42no—
260#102—42no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265#97—22no—
266#97—22no—
267#97—41no—
268#97—42no—
269#97D814yes—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—42no—
275C4×C8—42no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—41no—
280#95—41no—
281#95—41no—
282#95—41no—
283C24—21no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4#13712yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328Q16—48no—
329Q16—48no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
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×C8C2×D812yes—
358C2×C8C2×D812yes—
359C2×C8#13512yes—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4#13712yes—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—42no—
402C42—82no—
403C42—82no—
404C42C2×D812yes—
405C42AGL(1,ℤ/8)12yes—
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23C2×#12111yescharacteristic
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—81no—
423Q8—81no—
424Q8—81no—
425Q8—81no—
426D4—22no—
427D4—22no—
428D4—41no—
429D4—41no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#45912yes—
470C2×C4#45912yes—
471C2×C4#52511yescharacteristic
472C2×C4#52511yescharacteristic
473C2×C4#52511yescharacteristic
474C2×C4#52911yescharacteristic
475C2×C4#52911yescharacteristic
476C2×C4#53211yescharacteristic
477C2×C4C22×D811yescharacteristic
478C2×C4C2×#13511yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×#13711yescharacteristic
481C2×C4C2×#13711yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57511yescharacteristic
485C8—22no—
486C8—22no—
487C8—22no—
488C8—42no—
489C8—42no—
490C8—41no—
491C8—81no—
492C8—81no—
493C8—81no—
494C8—81no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22#239211yescharacteristic
503C22C2×#45911yescharacteristic
504C22#303511yescharacteristic
505C22#305611yescharacteristic
506C22#305611yescharacteristic
507C22#306911yescharacteristic
508C22#306911yescharacteristic
509C4—21no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—42no—
514C4—41no—
515C4—41no—
516C4—81no—
517C4—82no—
518C4—82no—
519C4—81no—
520C4—161no—
521C4—161no—
522C4#310011yescharacteristic
523C4#310511yescharacteristic
524C4#311111yescharacteristic
525C4#311311yescharacteristic
526C4#311611yescharacteristic
527C4#311711yescharacteristic
528C4#316911yescharacteristic
529C4#317111yescharacteristic
530C2—82no—
531C2#1916311yescharacteristic
532C2#1916311yescharacteristic
533C2#1918111yescharacteristic
534C2#1918511yescharacteristic
535C2#2367611yescharacteristic
536C2#2370911yescharacteristic
537C2#2372011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21691C2semidirect
5. #10653C2semidirect
7. #10592C2semidirect
8. #10588C2semidirect
9. #10538C2semidirect
13. #8275C2semidirect
14. #8236C2semidirect
15. #8236C2semidirect

Maximal subgroups

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

subgroupindex
#216912
C4×#4742
#212702
#106532
#106192
#105922
#105882
#105382
#104852
#104842
#102742
#82752
#82362