Finite Group Databasebeta

← #243570

Subgroups — #243570

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22270C211yesmaximal, characteristic, complemented
2#22008C211yesmaximal, characteristic
3#21613C211yesmaximal, characteristic
4C4×#474C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10622C211yesmaximal, characteristic
7#10600C211yesmaximal, characteristic, complemented
8#10592C211yesmaximal, characteristic, complemented
9#10538C211yesmaximal, characteristic, complemented
10#10485C211yesmaximal, characteristic
11#10485C211yesmaximal, characteristic
12#10274C211yesmaximal, characteristic
13#8278C211yesmaximal, characteristic, complemented
14#8237C211yesmaximal, characteristic, complemented
15C4×#332C211yesmaximal, characteristic, complemented
16C2×#494C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#1988C2211yescharacteristic
23#1988C2211yescharacteristic
24#1956C2211yescharacteristic
25#1956C2211yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1923—21no—
33#1923—21no—
34#1921—21no—
35#1918—21no—
36#1878—21no—
37#1878—21no—
38#1878—21no—
39#1878—21no—
40#1864—21no—
41#1862—21no—
42#1834C2211yescharacteristic
43#1817—21no—
44#1802C2211yescharacteristic
45#1802C2211yescharacteristic
46#1797—21no—
47#1776—21no—
48C4×#107—21no—
49#1767C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52#1762—21no—
53C4×#102—21no—
54#1744C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
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—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#391—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C43C2311yescharacteristic
131#340—21no—
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#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—22no—
151#335—22no—
152#332—22no—
153#332—22no—
154#332—22no—
155#332—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C2×Q16—24no—
163C2×Q16—24no—
164C2×Q16—81no—
165C2×Q16—81no—
166C2×Q16—81no—
167C2×Q16—81no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8#4112yes—
185C4⋊Q8#4112yes—
186C4×Q8—81no—
187C4×Q8—81no—
188C4×Q8—81no—
189C4×Q8—81no—
190C4×D4—22no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4Q1614yescomplemented
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—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—81no—
222C2×C42—81no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#107—24no—
231#107—24no—
232#107—42no—
233#106—42no—
234C4⋊C8—21no—
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—81no—
245C4⋊C8—81no—
246C4⋊C8#4112yes—
247C4⋊C8#4112yes—
248#103—81no—
249#103—81no—
250#103—81no—
251#103—81no—
252#102—24no—
253#102—24no—
254#102—42no—
255#102—42no—
256C22⋊C8—81no—
257C22⋊C8—81no—
258C22⋊C8—81no—
259C22⋊C8—81no—
260#97—42no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266C4×C8—81no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24—21no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4C2×Q1612yes—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319Q16—84no—
320Q16—84no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
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×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
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×Q1612yes—
377C22⋊C4—24no—
378C22⋊C4—42no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—81no—
392C42—81no—
393C42—82no—
394C42—82no—
395C42C2×Q1612yes—
396C42#13512yes—
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—42no—
412Q8—42no—
413Q8—81no—
414Q8—81no—
415Q8—81no—
416Q8—81no—
417D4—22no—
418D4—22no—
419D4—41no—
420D4—41no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—84no—
445C2×C4—84no—
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—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4#46712yes—
461C2×C4#46712yes—
462C2×C4#52511yescharacteristic
463C2×C4#52511yescharacteristic
464C2×C4#52511yescharacteristic
465C2×C4#52911yescharacteristic
466C2×C4#52911yescharacteristic
467C2×C4#53211yescharacteristic
468C2×C4C22×Q1611yescharacteristic
469C2×C4C2×#13511yescharacteristic
470C2×C4#57511yescharacteristic
471C2×C4#57511yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57611yescharacteristic
475C2×C4#57711yescharacteristic
476C8—42no—
477C8—42no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—81no—
483C8—81no—
484C8—81no—
485C8—81no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22#239211yescharacteristic
494C22#304111yescharacteristic
495C22#304511yescharacteristic
496C22C2×#46711yescharacteristic
497C22#305411yescharacteristic
498C22#306911yescharacteristic
499C22#306911yescharacteristic
500C4—21no—
501C4—21no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—41no—
506C4—41no—
507C4—81no—
508C4—82no—
509C4—82no—
510C4—81no—
511C4—161no—
512C4—161no—
513C4#310211yescharacteristic
514C4#310211yescharacteristic
515C4#310411yescharacteristic
516C4#310411yescharacteristic
517C4#311811yescharacteristic
518C4#312211yescharacteristic
519C4#317611yescharacteristic
520C4#317911yescharacteristic
521C2—82no—
522C2#1938711yescharacteristic
523C2#1941111yescharacteristic
524C2#1942411yescharacteristic
525C2#1944911yescharacteristic
526C2#2368811yescharacteristic
527C2#2369811yescharacteristic
528C2#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. #22270C2semidirect
5. #10656C2semidirect
7. #10600C2semidirect
8. #10592C2semidirect
9. #10538C2semidirect
13. #8278C2semidirect
14. #8237C2semidirect
15. C4×#332C2semidirect
197. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#222702
#220082
#216132
C4×#4742
#106562
#106222
#106002
#105922
#105382
#104852
#102742
#82782
#82372
C4×#3322