Finite Group Databasebeta

← #243298

Subgroups — #243298

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#498C211yesmaximal, characteristic, complemented
2C4×#476C211yesmaximal, characteristic
3C4×#461C211yesmaximal, characteristic
4#21264C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10619C211yesmaximal, characteristic
7#10598C211yesmaximal, characteristic, complemented
8#10597C211yesmaximal, characteristic, complemented
9#10544C211yesmaximal, characteristic, complemented
10#10480C211yesmaximal, characteristic
11#10477C211yesmaximal, characteristic
12#10271C211yesmaximal, characteristic
13C4×#334C211yesmaximal, characteristic, complemented
14C4×#332C211yesmaximal, characteristic, complemented
15C4×#333C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×C4×SD16—21no—
20C2×#431—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1981C2211yescharacteristic
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1941—21no—
33#1941—21no—
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1927—21no—
37#1923—21no—
38#1923—21no—
39#1920—21no—
40#1877—21no—
41#1864—21no—
42#1844C2211yescharacteristic
43#1802C2211yescharacteristic
44#1802C2211yescharacteristic
45C4×#106—21no—
46C4×#106—21no—
47C4×#106—21no—
48C4×C4⋊C8C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×#103—21no—
52C4×#102—21no—
53C4×C22⋊C8—21no—
54C42×C8C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C22×C42—21no—
76#498—22no—
77#498—22no—
78#476—22no—
79#476—22no—
80#461—22no—
81#461—22no—
82C4×SD16—44no—
83#431—44no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8D411yescharacteristic
113C2×C4×C8D411yescharacteristic
114C2×C4×C8D411yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#391—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C43C2311yescharacteristic
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—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#334—22no—
153#334—22no—
154#333—22no—
155#333—22no—
156#332—22no—
157#332—22no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C2×SD16—24no—
165C2×SD16—42no—
166C2×SD16—42no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8#4112yes—
184C4⋊Q8#4112yes—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—82no—
188C4×D4—22no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4SD1614yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42C2411yescharacteristic
228#106—24no—
229#106—24no—
230#106—24no—
231#106—42no—
232#106—42no—
233#106—42no—
234#106—42no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8#4112yes—
248C4⋊C8#4112yes—
249#103—24no—
250#103—42no—
251#103—42no—
252#102—24no—
253#102—42no—
254#102—42no—
255C22⋊C8—42no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—41no—
266C4×C8—42no—
267C4×C8—42no—
268C4×C8—42no—
269C4×C8SD1614yescomplemented
270#95—21no—
271#95—21no—
272#95—21no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277C24—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4C2×SD1612yes—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320SD16—48no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8C2×SD1612yes—
349C2×C8C2×SD1612yes—
350C2×C8#13512yes—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4C2×SD1612yes—
377C22⋊C4—24no—
378C22⋊C4—42no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—42no—
390C42—42no—
391C42—82no—
392C42—82no—
393C42C2×SD1612yes—
394C42C2×SD1612yes—
395C42#13512yes—
396C42#13712yes—
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C42C2×#4111yescharacteristic
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23C2×#12111yescharacteristic
409Q8—42no—
410Q8—42no—
411Q8—82no—
412Q8—82no—
413Q8—81no—
414Q8—81no—
415D4—22no—
416D4—22no—
417D4—41no—
418D4—41no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—48no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—84no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4#46112yes—
456C2×C4#46112yes—
457C2×C4#46512yes—
458C2×C4#46512yes—
459C2×C4#47112yes—
460C2×C4#52511yescharacteristic
461C2×C4#52511yescharacteristic
462C2×C4#52511yescharacteristic
463C2×C4#52911yescharacteristic
464C2×C4#52911yescharacteristic
465C2×C4#53211yescharacteristic
466C2×C4C22×SD1611yescharacteristic
467C2×C4C22×SD1611yescharacteristic
468C2×C4C2×#13511yescharacteristic
469C2×C4C2×#13511yescharacteristic
470C2×C4C2×#13711yescharacteristic
471C2×C4#57511yescharacteristic
472C2×C4#57511yescharacteristic
473C2×C4#57511yescharacteristic
474C8—22no—
475C8—22no—
476C8—22no—
477C8—42no—
478C8—42no—
479C8—41no—
480C8—81no—
481C8—81no—
482C8—81no—
483C8—81no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22#239211yescharacteristic
492C22C2×#46111yescharacteristic
493C22#303511yescharacteristic
494C22C2×#46511yescharacteristic
495C22#305411yescharacteristic
496C22C2×#47111yescharacteristic
497C22#306711yescharacteristic
498C4—21no—
499C4—21no—
500C4—22no—
501C4—22no—
502C4—42no—
503C4—41no—
504C4—81no—
505C4—82no—
506C4—82no—
507C4—81no—
508C4—161no—
509C4—161no—
510C4#167114yes—
511C4#310111yescharacteristic
512C4#310211yescharacteristic
513C4#310211yescharacteristic
514C4#310311yescharacteristic
515C4#310311yescharacteristic
516C4#310811yescharacteristic
517C4#316711yescharacteristic
518C4#317211yescharacteristic
519C2—82no—
520C2C2×#167111yescharacteristic
521C2#1912411yescharacteristic
522C2#1913711yescharacteristic
523C2#1915911yescharacteristic
524C2#2367711yescharacteristic
525C2#2370011yescharacteristic
526C2#2371811yescharacteristic
subgroupG/H
centerC22×C4#121
commutator, Frattini, cosocleC2×C42C24
socleC23C2×#121
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#498C2semidirect
5. #10654C2semidirect
7. #10598C2semidirect
8. #10597C2semidirect
9. #10544C2semidirect
13. C4×#334C2semidirect
14. C4×#332C2semidirect
15. C4×#333C2semidirect
269. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
C4×#4982
C4×#4762
C4×#4612
#212642
#106542
#106192
#105982
#105972
#105442
#104802
#104772
#102712
C4×#3342
C4×#3322
C4×#3332