Finite Group Databasebeta

← #242498

Subgroups — #242498

Order 512.

724 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#21974C211yesmaximal, characteristic
2#21835C211yesmaximal, characteristic
3#21392C211yesmaximal, characteristic
4#21169C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic
6C2×#1550C211yesmaximal, 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#8042C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#427—21no—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2285C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1989—21no—
29#1989—21no—
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1981C2211yescharacteristic
33#1981C2211yescharacteristic
34#1949—21no—
35#1949—21no—
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1939C2211yescharacteristic
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1873—21no—
45#1873—21no—
46#1851—21no—
47#1851—21no—
48#1842—21no—
49#1831—21no—
50#1825—21no—
51#1825—21no—
52#1763—21no—
53#1763—21no—
54#1735—21no—
55#1573—24no—
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1550—24no—
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×M16—21no—
70C23×C8—21no—
71#533—22no—
72#533—22no—
73#533—22no—
74#533Q814yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#125—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C42C2311yescharacteristic
87#432—22no—
88#432—22no—
89#431—22no—
90#431—22no—
91#427—44no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96#423—22no—
97#423—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8D414yes—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8D411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C2×#97—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C43—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#340—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#335—22no—
160#335—22no—
161#334—22no—
162#334—22no—
163#333—22no—
164#333—22no—
165#332—22no—
166#332—22no—
167C22×Q8—21no—
168C22×Q8—21no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4111yescharacteristic
172C2×M16—42no—
173C2×M16—81no—
174C2×M16—81no—
175C22×C8—24no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C2×Q812yes—
195C4⋊Q8C2×Q812yes—
196#125—24no—
197#125—42no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—42no—
205C4×Q8—42no—
206#117—24no—
207#117—22no—
208#117—22no—
209#117—41no—
210#117—41no—
211#117—42no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×Q812yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42#4111yescharacteristic
249C2×C42C2411yescharacteristic
250#107—42no—
251#107—42no—
252#106—42no—
253#106—42no—
254C4⋊C8—24no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—42no—
268C4⋊C8—81no—
269C4⋊C8—81no—
270C4⋊C8—81no—
271C4⋊C8—81no—
272C4⋊C8C2×D412yes—
273C4⋊C8C2×D412yes—
274#103—42no—
275#103—42no—
276#103—42no—
277#103—42no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—42no—
281#97—42no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—41no—
291#95—41no—
292C24#12111yescharacteristic
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—81no—
302C2×Q8—81no—
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—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4C22⋊Q812yes—
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4#12612yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333M16—84no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—48no—
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×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—81no—
390C4⋊C4—81no—
391C4⋊C4AGL(1,ℤ/8)12yes—
392C4⋊C4#13712yes—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—44no—
406C42—82no—
407C42#12112yes—
408C42#12112yes—
409C42C22⋊Q812yes—
410C42C22⋊Q812yes—
411C42C22⋊Q812yes—
412C42#12612yes—
413C42C22×D411yescharacteristic
414C42C22×Q811yescharacteristic
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23#39612yes—
424C23C2×#12111yescharacteristic
425Q8—42no—
426Q8—42no—
427Q8—82no—
428Q8—82no—
429Q8—81no—
430Q8—81no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
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—81no—
472C2×C4—81no—
473C2×C4#39612yes—
474C2×C4#39612yes—
475C2×C4#39612yes—
476C2×C4C2×#12111yescharacteristic
477C2×C4C2×#12111yescharacteristic
478C2×C4C2×C22⋊Q811yescharacteristic
479C2×C4C2×C22⋊Q811yescharacteristic
480C2×C4C2×C22⋊Q811yescharacteristic
481C2×C4C2×#12611yescharacteristic
482C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
483C2×C4C2×#13711yescharacteristic
484C2×C4#57511yescharacteristic
485C2×C4#57511yescharacteristic
486C2×C4#57711yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57811yescharacteristic
489C2×C4#57811yescharacteristic
490C8—42no—
491C8—42no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—81no—
497C8—81no—
498C8—81no—
499C8—81no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22C2×#39611yescharacteristic
508C22#304611yescharacteristic
509C22#304611yescharacteristic
510C22#305511yescharacteristic
511C22#305611yescharacteristic
512C22#306711yescharacteristic
513C22#306711yescharacteristic
514C4—21no—
515C4—21no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—81no—
523C4—82no—
524C4—82no—
525C4—161no—
526C4—161no—
527C4#303611yescharacteristic
528C4#303711yescharacteristic
529C4#305711yescharacteristic
530C4#305911yescharacteristic
531C4#307911yescharacteristic
532C4#308011yescharacteristic
533C4#308211yescharacteristic
534C4#308211yescharacteristic
535C2—42no—
536C2#1948611yescharacteristic
537C2#1948711yescharacteristic
538C2#1949811yescharacteristic
539C2#1949911yescharacteristic
540C2#2360111yescharacteristic
541C2#2361311yescharacteristic
542C2#2361711yescharacteristic
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
#219742
#218352
#213922
#211692
C2×#15732
C2×#15502
#106022
#106012
#105982
#105972
#105942
#105932
#105902
#105892
#80422