Finite Group Databasebeta

← #211294

Subgroups — #211294

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22246C211yesmaximal, characteristic, complemented
2#21614C211yesmaximal, characteristic
3C4×#461C211yesmaximal, characteristic
4#20555C211yesmaximal, characteristic
5C2×#1671C211yesmaximal, characteristic
6#9729C211yesmaximal, characteristic, complemented
7#9613C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9566C211yesmaximal, characteristic, complemented
10#9217C211yesmaximal, characteristic
11#9215C211yesmaximal, characteristic
12#8646C211yesmaximal, characteristic
13#8277C211yesmaximal, characteristic, complemented
14#8236C211yesmaximal, characteristic, complemented
15C4×#333C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×C4×SD16—21no—
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1987C2211yescharacteristic
28#1981C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1927—21no—
32#1920—21no—
33#1920—21no—
34#1919—21no—
35#1919—21no—
36#1919—21no—
37#1877—21no—
38#1862—21no—
39#1834C2211yescharacteristic
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#1671C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
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#498—22no—
80#498C2×C414yescomplemented
81#476—22no—
82#476C2×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×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—21no—
117#391—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C43C2311yescharacteristic
135#340—21no—
136#340—21no—
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#336D411yescharacteristic
153#336D411yescharacteristic
154#336D411yescharacteristic
155#336D411yescharacteristic
156#334—22no—
157#334C2×C414yescomplemented
158#333—22no—
159#333C2×C414yescomplemented
160#332—22no—
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—82no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4SD1614yes—
204C4×D4C22×C412yes—
205C4×D4#4112yes—
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—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—81no—
230C2×C42—81no—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42C2411yescharacteristic
238#107—42no—
239#106—24no—
240#106—24no—
241#106—24no—
242#106—42no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253C4⋊C8C22×C412yes—
254C4⋊C8#4112yes—
255#103—24no—
256#103—42no—
257#103—42no—
258#102—24no—
259#102—42no—
260#102—42no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263#97—42no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—42no—
267C4×C8—81no—
268C4×C8—81no—
269C4×C8—81no—
270C4×C8C22×C412yes—
271C4×C8#4112yes—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95—41no—
279C24—21no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—81no—
289C2×Q8—81no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4C4×D412yes—
294C2×D4C2×SD1612yes—
295C2×D4#13512yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
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—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322SD16—48no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
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—
351C2×C8—84no—
352C2×C8—84no—
353C2×C8C4×D412yes—
354C2×C8C4×D412yes—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
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—41no—
373C4⋊C4—41no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4C4×D412yes—
379C4⋊C4C2×SD1612yes—
380C4⋊C4#13512yes—
381C22⋊C4—24no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—24no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—81no—
396C42—81no—
397C42—82no—
398C42—82no—
399C42C2×SD1612yes—
400C42#13512yes—
401C42C23×C411yescharacteristic
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C23—22no—
406C23—22no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23C2×#12111yescharacteristic
413Q8—42no—
414Q8—42no—
415Q8—82no—
416Q8—82no—
417Q8—81no—
418Q8—81no—
419D4—22no—
420D4—22no—
421D4—22no—
422D4C4×SD1614yes—
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—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—84no—
451C2×C4—84no—
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—81no—
461C2×C4—81no—
462C2×C4#46512yes—
463C2×C4#46512yes—
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C4×#4111yescharacteristic
467C2×C4#52511yescharacteristic
468C2×C4#52911yescharacteristic
469C2×C4#53211yescharacteristic
470C2×C4C22×SD1611yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4#57711yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57811yescharacteristic
478C8—22no—
479C8—22no—
480C8—41no—
481C8—41no—
482C8—42no—
483C8—42no—
484C8—42no—
485C8—42no—
486C8—81no—
487C8—81no—
488C22—24no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22C4×#12111yescharacteristic
496C22#304211yescharacteristic
497C22#304611yescharacteristic
498C22C2×#46511yescharacteristic
499C22#305011yescharacteristic
500C22#306811yescharacteristic
501C22#306911yescharacteristic
502C4—21no—
503C4—21no—
504C4—22no—
505C4—22no—
506C4—42no—
507C4—42no—
508C4—41no—
509C4—42no—
510C4—41no—
511C4—82no—
512C4—82no—
513C4—161no—
514C4—161no—
515C4C2×C4×SD1611yescharacteristic
516C4C4×#13511yescharacteristic
517C4#294711yescharacteristic
518C4#294911yescharacteristic
519C4#312011yescharacteristic
520C4#312311yescharacteristic
521C4#316711yescharacteristic
522C4#316811yescharacteristic
523C2—44no—
524C2#1939011yescharacteristic
525C2#1939411yescharacteristic
526C2#1941411yescharacteristic
527C2#1941811yescharacteristic
528C2#2151411yescharacteristic
529C2C4×#46511yescharacteristic
530C2#2155711yescharacteristic
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. #22246C2semidirect
6. #9729C2semidirect
7. #9613C2semidirect
8. #9567C2semidirect
9. #9566C2semidirect
13. #8277C2semidirect
14. #8236C2semidirect
15. C4×#333C2semidirect
55. #1671C4semidirect
80. #498C2×C4semidirect
157. #334C2×C4semidirect
159. #333C2×C4semidirect
161. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222462
#216142
C4×#4612
#205552
C2×#16712
#97292
#96132
#95672
#95662
#92172
#92152
#86462
#82772
#82362
C4×#3332