Finite Group Databasebeta

← #242520

Subgroups — #242520

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21972C211yesmaximal, characteristic
2#21835C211yesmaximal, characteristic
3#21392C211yesmaximal, characteristic
4#21362C211yesmaximal, characteristic
5C2×#1572C211yesmaximal, characteristic
6C2×#1542C211yesmaximal, characteristic
7#10602C211yesmaximal, characteristic, complemented
8#10601C211yesmaximal, characteristic, complemented
9#10598C211yesmaximal, characteristic, complemented
10#10597C211yesmaximal, characteristic, complemented
11#10594C211yesmaximal, characteristic, complemented
12#10593C211yesmaximal, characteristic, complemented
13#10590C211yesmaximal, characteristic, complemented
14#10589C211yesmaximal, characteristic, complemented
15#8044C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2285C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1989—21no—
28#1989—21no—
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1981C2211yescharacteristic
32#1981C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1873—21no—
42#1873—21no—
43#1854—21no—
44#1854—21no—
45#1850—21no—
46#1850—21no—
47#1842—21no—
48#1831—21no—
49#1825—21no—
50#1825—21no—
51C4×#107—21no—
52C4×#106—21no—
53#1763—21no—
54#1763—21no—
55#1572—24no—
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1542—24no—
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C23×C8—21no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533Q814yes—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117D411yescharacteristic
84C2×#117D411yescharacteristic
85C22×C42C2311yescharacteristic
86#432—22no—
87#432—22no—
88#431—22no—
89#431—22no—
90#431—22no—
91#431D414yes—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8D411yescharacteristic
118C2×C22⋊C8D411yescharacteristic
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#382—21no—
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136#376—21no—
137C43—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#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#335—22no—
159#335—22no—
160#334—22no—
161#334—22no—
162#333—22no—
163#333—22no—
164#332—22no—
165#332—22no—
166C22×Q8—21no—
167C22×Q8—21no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—42no—
178C22×C8—81no—
179C22×C8—81no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8C2×Q812yes—
191C4⋊Q8C2×Q812yes—
192#125—24no—
193#125—42no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—42no—
201C4×Q8—42no—
202#117—22no—
203#117—22no—
204#117—24no—
205#117—41no—
206#117—41no—
207#117—42no—
208#117—82no—
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—21no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×Q812yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245#107—42no—
246#107—42no—
247#106—42no—
248#106—42no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—42no—
262C4⋊C8—81no—
263C4⋊C8—81no—
264C4⋊C8—81no—
265C4⋊C8—81no—
266C4⋊C8C2×D412yes—
267C4⋊C8C2×D412yes—
268#103—42no—
269#103—42no—
270#103—42no—
271#103—42no—
272C22⋊C8—24no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—42no—
276#97—42no—
277C4×C8—24no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—41no—
287#95—41no—
288C24#12111yescharacteristic
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—81no—
298C2×Q8—81no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4#12612yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
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—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—84no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4#13512yes—
387C4⋊C4#13512yes—
388C22⋊C4—24no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—82no—
392C42—24no—
393C42—42no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—44no—
399C42—81no—
400C42—81no—
401C42—82no—
402C42#12112yes—
403C42#12112yes—
404C42C22⋊Q812yes—
405C42C22⋊Q812yes—
406C42C22⋊Q812yes—
407C42#12612yes—
408C42C22×D411yescharacteristic
409C42C22×Q811yescharacteristic
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23#39612yes—
419C23C2×#12111yescharacteristic
420Q8—42no—
421Q8—42no—
422Q8—82no—
423Q8—82no—
424Q8—81no—
425Q8—81no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#39612yes—
469C2×C4#39612yes—
470C2×C4#39612yes—
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×C22⋊Q811yescharacteristic
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4C2×#12611yescharacteristic
477C2×C4C2×#13511yescharacteristic
478C2×C4C2×#13511yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57711yescharacteristic
483C2×C4#57811yescharacteristic
484C2×C4#57811yescharacteristic
485C8—42no—
486C8—42no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—81no—
492C8—81no—
493C8—81no—
494C8—81no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22C2×#39611yescharacteristic
503C22#304611yescharacteristic
504C22#304611yescharacteristic
505C22#305011yescharacteristic
506C22#305411yescharacteristic
507C22#306811yescharacteristic
508C22#306911yescharacteristic
509C4—21no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—42no—
515C4—42no—
516C4—82no—
517C4—82no—
518C4—81no—
519C4—81no—
520C4—161no—
521C4—161no—
522C4#303511yescharacteristic
523C4#303511yescharacteristic
524C4#305711yescharacteristic
525C4#305911yescharacteristic
526C4#307611yescharacteristic
527C4#307611yescharacteristic
528C4#308211yescharacteristic
529C4#308211yescharacteristic
530C2—42no—
531C2#1941811yescharacteristic
532C2#1941911yescharacteristic
533C2#1945411yescharacteristic
534C2#1945511yescharacteristic
535C2#2360411yescharacteristic
536C2#2360911yescharacteristic
537C2#2361311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. #10602C2semidirect
8. #10601C2semidirect
9. #10598C2semidirect
10. #10597C2semidirect
11. #10594C2semidirect
12. #10593C2semidirect
13. #10590C2semidirect
14. #10589C2semidirect

Maximal subgroups

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

subgroupindex
#219722
#218352
#213922
#213622
C2×#15722
C2×#15422
#106022
#106012
#105982
#105972
#105942
#105932
#105902
#105892
#80442