Finite Group Databasebeta

← #211425

Subgroups — #211425

Order 512.

712 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
1#22146C211yesmaximal, characteristic
2#22111C211yesmaximal, characteristic
3C4×#498C211yesmaximal, characteristic, complemented
4#20485C211yesmaximal, characteristic
5C2×#1691C211yesmaximal, characteristic
6#9741C211yesmaximal, characteristic, complemented
7#9618C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9565C211yesmaximal, characteristic, complemented
10#9217C211yesmaximal, characteristic
11#9215C211yesmaximal, characteristic
12#8650C211yesmaximal, characteristic
13C4×#335C211yesmaximal, characteristic, complemented
14#8237C211yesmaximal, characteristic, complemented
15#8235C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#405—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24#2293C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1987C2211yescharacteristic
28#1981C2211yescharacteristic
29#1968—21no—
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1941—21no—
33#1941—21no—
34#1941—21no—
35#1934—21no—
36#1933—21no—
37#1926—21no—
38#1919—21no—
39#1880—21no—
40#1865—21no—
41#1844C2211yescharacteristic
42#1802C2211yescharacteristic
43#1802C2211yescharacteristic
44C4×#106—21no—
45C4×#106—21no—
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48#1767C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51#1753—21no—
52C8×C22⋊C4—21no—
53#1750—21no—
54C42×C8C2211yescharacteristic
55#1691C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×M16—21no—
70C23×C8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77#498—22no—
78#498C2×C414yes—
79#476—22no—
80#476C2×C414yes—
81#461—22no—
82#461C2×C414yes—
83#436—22no—
84#436C2×C414yes—
85C8×D4—22no—
86C8×D4—22no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—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×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108#405—44no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8D411yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—21no—
121#391—21no—
122#390D411yescharacteristic
123#390D411yescharacteristic
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389D411yescharacteristic
129#389D411yescharacteristic
130#388—21no—
131#388—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C43C2311yescharacteristic
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#335—22no—
157#335C2×C414yes—
158#333—22no—
159#333C2×C414yes—
160#332—22no—
161#332C2×C414yes—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C2×SD16—24no—
167C2×SD16—81no—
168C2×SD16—81no—
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C22×C412yes—
192C4⋊Q8C22×C412yes—
193C4×Q8—81no—
194C4×Q8—81no—
195C4×Q8—82no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4C22×C412yes—
199C4×D4#4112yes—
200#117—42no—
201#117—42no—
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—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
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—41no—
226C2×C42—41no—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42C2411yescharacteristic
234#106—24no—
235#106—24no—
236#106—24no—
237#106—42no—
238#106—42no—
239#106—81no—
240#106—81no—
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⋊C8C22×C412yes—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252C4⋊C8#4112yes—
253#103—24no—
254#103—81no—
255#103—81no—
256#102—24no—
257#102—81no—
258#102—81no—
259C22⋊C8—24no—
260C22⋊C8—24no—
261#97—42no—
262#97—42no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—41no—
268C4×C8—42no—
269C4×C8—81no—
270C4×C8—81no—
271C4×C8SD1614yescomplemented
272C4×C8C22×C412yes—
273C4×C8#4112yes—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281C24—21no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—81no—
291C2×Q8—81no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4C4×D412yes—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318SD16—84no—
319M16—48no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—48no—
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×C8C4×D412yes—
347C2×C8C4×D412yes—
348C2×C8C2×SD1612yes—
349C2×C8C2×SD1612yes—
350C2×C8#13512yes—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4C4×D412yes—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—22no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—82no—
396C42—82no—
397C42C2×SD1612yes—
398C42#13712yes—
399C42C23×C411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C23—22no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23C2×#12111yescharacteristic
411Q8—42no—
412Q8—42no—
413Q8—82no—
414Q8—82no—
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—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
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#46112yes—
458C2×C4#46112yes—
459C2×C4#47112yes—
460C2×C4C2×C4×D411yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C4×#4111yescharacteristic
463C2×C4#52511yescharacteristic
464C2×C4#52911yescharacteristic
465C2×C4#53211yescharacteristic
466C2×C4C22×SD1611yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4C2×#13711yescharacteristic
469C2×C4#57511yescharacteristic
470C2×C4#57611yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C2×C4#57811yescharacteristic
474C8—22no—
475C8—22no—
476C8—42no—
477C8—41no—
478C8—42no—
479C8—42no—
480C8—42no—
481C8—81no—
482C8—81no—
483C8C4×SD1614yes—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22C4×#12111yescharacteristic
492C22C2×#46111yescharacteristic
493C22#303511yescharacteristic
494C22#305711yescharacteristic
495C22#306111yescharacteristic
496C22C2×#47111yescharacteristic
497C22#306711yescharacteristic
498C4—22no—
499C4—21no—
500C4—21no—
501C4—22no—
502C4—22no—
503C4—41no—
504C4—42no—
505C4—82no—
506C4—82no—
507C4—81no—
508C4—81no—
509C4—161no—
510C4—161no—
511C4C2×C4×SD1611yescharacteristic
512C4C4×#13711yescharacteristic
513C4#294611yescharacteristic
514C4#294711yescharacteristic
515C4#310111yescharacteristic
516C4#310711yescharacteristic
517C4#317811yescharacteristic
518C4#317911yescharacteristic
519C2—82no—
520C2#1919111yescharacteristic
521C2#1919511yescharacteristic
522C2#1922411yescharacteristic
523C2#1922911yescharacteristic
524C2C4×#46111yescharacteristic
525C2#2153511yescharacteristic
526C2C4×#47111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ 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
3. C4×#498C2semidirect
6. #9741C2semidirect
7. #9618C2semidirect
8. #9567C2semidirect
9. #9565C2semidirect
13. C4×#335C2semidirect
14. #8237C2semidirect
15. #8235C2semidirect
55. #1691C4semidirect
271. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#221462
#221112
C4×#4982
#204852
C2×#16912
#97412
#96182
#95672
#95652
#92172
#92152
#86502
C4×#3352
#82372
#82352