Finite Group Databasebeta

← #211426

Subgroups — #211426

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22147C211yesmaximal, characteristic
2#22112C211yesmaximal, characteristic, complemented
3C4×#492C211yesmaximal, characteristic
4#20516C211yesmaximal, characteristic, complemented
5C2×#1686C211yesmaximal, characteristic
6#9744C211yesmaximal, characteristic
7#9618C211yesmaximal, characteristic, complemented
8#9564C211yesmaximal, characteristic
9#9562C211yesmaximal, characteristic, complemented
10#9222C211yesmaximal, characteristic
11#9220C211yesmaximal, characteristic, complemented
12#8651C211yesmaximal, characteristic, complemented
13C4×#335C211yesmaximal, characteristic, complemented
14#8236C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic
16C2×#492C2211yescharacteristic
17C2×#475C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C4×SD16—21no—
22C2×C4×SD16—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#127C2211yescharacteristic
25#2295C2211yescharacteristic
26#1988C2211yescharacteristic
27#1982C2211yescharacteristic
28#1968—21no—
29#1968—21no—
30#1968—21no—
31#1957C2211yescharacteristic
32#1956C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1941—21no—
36#1934—21no—
37#1933—21no—
38#1927—21no—
39#1920—21no—
40#1877—21no—
41#1864—21no—
42#1844C2211yescharacteristic
43#1803C2211yescharacteristic
44#1803C2211yescharacteristic
45#1771—21no—
46#1770C2211yescharacteristic
47C8×C4⋊C4—21no—
48C8×C4⋊C4C2211yescharacteristic
49#1768—21no—
50#1767C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1726—21no—
54C42×C8C2211yescharacteristic
55#1686C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×SD16—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78#492—22no—
79#492C2×C414yes—
80#475—22no—
81#475C2×C414yescomplemented
82#460—22no—
83#460C2×C414yes—
84#446—22no—
85#446C2×C414yescomplemented
86C8×Q8—22no—
87C8×Q8—22no—
88C4×SD16—44no—
89C4×SD16—44no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8D411yescharacteristic
121C2×C4×C8D411yescharacteristic
122C2×C4×C8D411yescharacteristic
123C2×C4×C8D411yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—21no—
127#391D411yescharacteristic
128#391D411yescharacteristic
129#390—21no—
130#390—21no—
131#390—21no—
132#389—21no—
133#389—21no—
134#389D411yescharacteristic
135#389D411yescharacteristic
136#387—21no—
137#387—21no—
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4C2311yescharacteristic
144#376—21no—
145C43C2311yescharacteristic
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#335—22no—
158#335C2×C414yescomplemented
159#332—22no—
160#332C2×C414yescomplemented
161#331—22no—
162#331C2×C414yes—
163C22×Q8—21no—
164C22×Q8C2×D411yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C23×C4—41no—
169C2×SD16—24no—
170C2×SD16—24no—
171C2×SD16—24no—
172C2×SD16—42no—
173C2×SD16—42no—
174C2×SD16—42no—
175C2×SD16—42no—
176C2×SD16—81no—
177C2×SD16—81no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C22×C412yes—
192#127—21no—
193#127—21no—
194#127—21no—
195#127C22×C412yes—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8C22×C412yes—
201C4×Q8#4112yes—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—81no—
205C4×D4—81no—
206C4×D4—82no—
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—41no—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#106—24no—
236#106—42no—
237#106—42no—
238#106—81no—
239#106—81no—
240C4⋊C8—24no—
241C4⋊C8—24no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253C4⋊C8#4112yes—
254#103—24no—
255#103—42no—
256#103—42no—
257#103—81no—
258#103—81no—
259#102—24no—
260#102—42no—
261#102—42no—
262#102—81no—
263#102—81no—
264#97—24no—
265#97—42no—
266#97—42no—
267C4×C8—24no—
268C4×C8—22no—
269C4×C8—22no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—41no—
273C4×C8—42no—
274C4×C8—81no—
275C4×C8—81no—
276C4×C8SD1614yescomplemented
277C4×C8C22×C412yes—
278C4×C8#4112yes—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—41no—
287#95—41no—
288C24—41no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8C4×D412yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—81no—
316C2×D4—81no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334SD16—48no—
335SD16—48no—
336SD16—84no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
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—41no—
361C2×C8—41no—
362C2×C8C4×D412yes—
363C2×C8C4×D412yes—
364C2×C8C2×SD1612yes—
365C2×C8C2×SD1612yes—
366C2×C8#13512yes—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
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—81no—
389C4⋊C4—81no—
390C4⋊C4—82no—
391C4⋊C4C4×D412yes—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C42—22no—
396C42—22no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—82no—
408C42C2×SD1612yes—
409C42AGL(1,ℤ/8)12yes—
410C42C23×C411yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C23—42no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23C2×#12111yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—41no—
425Q8—41no—
426Q8—42no—
427Q8—42no—
428Q8—81no—
429Q8—81no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—42no—
434D4—82no—
435D4—82no—
436D4—81no—
437D4—81no—
438D4—81no—
439D4—81no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
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—42no—
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—164no—
473C2×C4#46012yes—
474C2×C4#46012yes—
475C2×C4#47112yes—
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×C4×D411yescharacteristic
478C2×C4C4×#4111yescharacteristic
479C2×C4#52511yescharacteristic
480C2×C4#52911yescharacteristic
481C2×C4#53211yescharacteristic
482C2×C4C22×SD1611yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
485C2×C4#57511yescharacteristic
486C2×C4#57611yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57711yescharacteristic
489C2×C4#57811yescharacteristic
490C8—22no—
491C8—22no—
492C8—41no—
493C8—42no—
494C8—42no—
495C8—42no—
496C8—42no—
497C8—81no—
498C8—81no—
499C8C4×SD1614yes—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22C4×#12111yescharacteristic
508C22C2×#46011yescharacteristic
509C22#303511yescharacteristic
510C22#305711yescharacteristic
511C22#306111yescharacteristic
512C22C2×#47111yescharacteristic
513C22#306711yescharacteristic
514C4—22no—
515C4—22no—
516C4—22no—
517C4—21no—
518C4—21no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—82no—
525C4—161no—
526C4—161no—
527C4C2×C4×SD1611yescharacteristic
528C4C4×AGL(1,ℤ/8)11yescharacteristic
529C4#294611yescharacteristic
530C4#294711yescharacteristic
531C4#310311yescharacteristic
532C4#310611yescharacteristic
533C4#317811yescharacteristic
534C4#317911yescharacteristic
535C2—162no—
536C2#1920211yescharacteristic
537C2#1920611yescharacteristic
538C2#1922411yescharacteristic
539C2#1922911yescharacteristic
540C2C4×#46011yescharacteristic
541C2#2153511yescharacteristic
542C2C4×#47111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
2. #22112C2semidirect
4. #20516C2semidirect
7. #9618C2semidirect
9. #9562C2semidirect
11. #9220C2semidirect
12. #8651C2semidirect
13. C4×#335C2semidirect
14. #8236C2semidirect
55. #1686C4semidirect
81. #475C2×C4semidirect
85. #446C2×C4semidirect
158. #335C2×C4semidirect
160. #332C2×C4semidirect
276. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#221472
#221122
C4×#4922
#205162
C2×#16862
#97442
#96182
#95642
#95622
#92222
#92202
#86512
C4×#3352
#82362
#82342