Finite Group Databasebeta

← #211278

Subgroups — #211278

Order 512.

747 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22218C211yesmaximal, characteristic, complemented
2#21613C211yesmaximal, characteristic
3C4×#461C211yesmaximal, characteristic
4#20555C211yesmaximal, characteristic
5C2×#1679C211yesmaximal, characteristic
6#9736C211yesmaximal, characteristic, complemented
7#9612C211yesmaximal, characteristic, complemented
8#9565C211yesmaximal, characteristic, complemented
9#9562C211yesmaximal, characteristic, complemented
10#9219C211yesmaximal, characteristic
11#9217C211yesmaximal, characteristic
12#8642C211yesmaximal, characteristic
13#8277C211yesmaximal, characteristic, complemented
14#8236C211yesmaximal, characteristic, complemented
15C4×#332C211yesmaximal, characteristic, complemented
16C2×#487C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×C4×SD16—21no—
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1988C2211yescharacteristic
27#1981C2211yescharacteristic
28#1956C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1927—21no—
32#1927—21no—
33#1927—21no—
34#1921—21no—
35#1877—21no—
36#1865—21no—
37#1865—21no—
38#1864—21no—
39#1844C2211yescharacteristic
40#1802C2211yescharacteristic
41#1802C2211yescharacteristic
42#1776—21no—
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48#1762—21no—
49C4×#103—21no—
50C8×C22⋊C4—21no—
51C8×C22⋊C4—21no—
52C4×C22⋊C8—21no—
53#1744C2211yescharacteristic
54C2×#336—21no—
55#1679C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C23×C8—21no—
70C23×C8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C42—21no—
79#487—22no—
80#487C2×C414yescomplemented
81#474—22no—
82#474C2×C414yes—
83#461—22no—
84#461C2×C414yes—
85C4×SD16—44no—
86C8×D4—22no—
87C8×D4—22no—
88C8×D4—22no—
89C8×D4C2×C414yes—
90C2×#107—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
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×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—21no—
121#391—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C43C2311yescharacteristic
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#336D411yescharacteristic
153#336D411yescharacteristic
154#336D411yescharacteristic
155#336D411yescharacteristic
156#334—22no—
157#334C2×C414yescomplemented
158#332—22no—
159#332—22no—
160#332C2×C414yescomplemented
161#332C2×C414yescomplemented
162C22×Q8—21no—
163C22×Q8—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C2×SD16—24no—
169C2×SD16—42no—
170C2×SD16—42no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—81no—
180C22×C8—81no—
181C22×C8—81no—
182C22×C8—81no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22×C412yes—
194C4⋊Q8C22×C412yes—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—81no—
198C4×Q8—81no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4SD1614yes—
205C4×D4C22×C412yes—
206C4×D4#4112yes—
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—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—81no—
231C2×C42—81no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42C2411yescharacteristic
239#107—24no—
240#106—42no—
241#106—42no—
242#106—81no—
243#106—81no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8C22×C412yes—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255C4⋊C8#4112yes—
256#103—24no—
257#103—24no—
258#103—24no—
259#103—42no—
260#103—42no—
261#103—81no—
262#103—81no—
263#102—24no—
264#102—42no—
265#102—42no—
266C22⋊C8—24no—
267C22⋊C8—24no—
268#97—42no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—42no—
272C4×C8—81no—
273C4×C8—81no—
274C4×C8—81no—
275C4×C8C22×C412yes—
276C4×C8#4112yes—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282#95—41no—
283#95—41no—
284C24—21no—
285C2×Q8—22no—
286C2×Q8—22no—
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×Q8—41no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4C4×D412yes—
301C2×D4C2×SD1612yes—
302C2×D4#13512yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329SD16—48no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
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×C8—41no—
358C2×C8—84no—
359C2×C8—84no—
360C2×C8C4×D412yes—
361C2×C8C4×D412yes—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
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⋊C4C4×D412yes—
386C4⋊C4C2×SD1612yes—
387C4⋊C4#13512yes—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—24no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—81no—
403C42—81no—
404C42—82no—
405C42—82no—
406C42C2×SD1612yes—
407C42#13512yes—
408C42C23×C411yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C23—22no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23C2×#12111yescharacteristic
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—42no—
424Q8—81no—
425Q8—81no—
426Q8—81no—
427Q8—81no—
428D4—22no—
429D4—22no—
430D4—22no—
431D4C4×SD1614yes—
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—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—84no—
460C2×C4—84no—
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—81no—
470C2×C4—81no—
471C2×C4#46512yes—
472C2×C4#46512yes—
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4C4×#4111yescharacteristic
476C2×C4#52511yescharacteristic
477C2×C4#52911yescharacteristic
478C2×C4#53211yescharacteristic
479C2×C4C22×SD1611yescharacteristic
480C2×C4C2×#13511yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4#57611yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57711yescharacteristic
487C8—22no—
488C8—22no—
489C8—41no—
490C8—42no—
491C8—41no—
492C8—42no—
493C8—42no—
494C8—42no—
495C8—81no—
496C8—81no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22C4×#12111yescharacteristic
505C22#304211yescharacteristic
506C22#304511yescharacteristic
507C22C2×#46511yescharacteristic
508C22#305011yescharacteristic
509C22#306911yescharacteristic
510C22#306911yescharacteristic
511C4—21no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—41no—
518C4—42no—
519C4—41no—
520C4—82no—
521C4—82no—
522C4—161no—
523C4—161no—
524C4C2×C4×SD1611yescharacteristic
525C4C4×#13511yescharacteristic
526C4#294611yescharacteristic
527C4#295211yescharacteristic
528C4#311811yescharacteristic
529C4#312511yescharacteristic
530C4#316711yescharacteristic
531C4#316811yescharacteristic
532C2—44no—
533C2#1939011yescharacteristic
534C2#1939511yescharacteristic
535C2#1940811yescharacteristic
536C2#1941311yescharacteristic
537C2#2151011yescharacteristic
538C2C4×#46511yescharacteristic
539C2#2155611yescharacteristic
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
1. #22218C2semidirect
6. #9736C2semidirect
7. #9612C2semidirect
8. #9565C2semidirect
9. #9562C2semidirect
13. #8277C2semidirect
14. #8236C2semidirect
15. C4×#332C2semidirect
55. #1679C4semidirect
80. #487C2×C4semidirect
157. #334C2×C4semidirect
160. #332C2×C4semidirect
161. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222182
#216132
C4×#4612
#205552
C2×#16792
#97362
#96122
#95652
#95622
#92192
#92172
#86422
#82772
#82362
C4×#3322