Finite Group Databasebeta

← #242632

Subgroups — #242632

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21987C211yesmaximal, characteristic
2#21973C211yesmaximal, characteristic
3#21889C211yesmaximal, characteristic
4#21860C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic
6C2×#1552C211yesmaximal, characteristic
7#10602C211yesmaximal, characteristic, complemented
8#10600C211yesmaximal, characteristic, complemented
9#10598C211yesmaximal, characteristic, complemented
10#10596C211yesmaximal, characteristic, complemented
11#10594C211yesmaximal, characteristic, complemented
12#10592C211yesmaximal, characteristic, complemented
13#10590C211yesmaximal, characteristic, complemented
14#10588C211yesmaximal, characteristic, complemented
15C2×#1296C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2283C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1988C2211yescharacteristic
26#1988C2211yescharacteristic
27#1981C2211yescharacteristic
28#1981C2211yescharacteristic
29#1956C2211yescharacteristic
30#1956C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1892—21no—
40#1892—21no—
41#1872—21no—
42#1872—21no—
43#1853—21no—
44#1852—21no—
45#1850—21no—
46#1850—21no—
47#1847—21no—
48#1843—21no—
49#1836—21no—
50#1832—21no—
51#1805—21no—
52#1805—21no—
53#1803—21no—
54#1803—21no—
55#1573—24no—
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1552—24no—
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C23×C8—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—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C42C2311yescharacteristic
87#432—22no—
88#432D414yes—
89#431—22no—
90#431—22no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—21no—
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⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—21no—
118C2×#103—21no—
119C2×#103—21no—
120C2×#103—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8D411yescharacteristic
123C2×C22⋊C8D411yescharacteristic
124C2×#97—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#384—21no—
128#384—21no—
129#384—21no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#383—21no—
133#382D411yescharacteristic
134#382D411yescharacteristic
135#381—21no—
136#380D411yescharacteristic
137#380D411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141#376—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#332—22no—
163#332—22no—
164#332—22no—
165#332—22no—
166C22×Q8—21no—
167C22×Q8—21no—
168C23×C4—41no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4111yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8—81no—
181C22×C8—81no—
182C22×C8—81no—
183C22×C8—81no—
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—42no—
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⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—41no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—41no—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×Q812yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C2×C42#4111yescharacteristic
245C2×C42C2411yescharacteristic
246#107—81no—
247#107—81no—
248#106—81no—
249#106—81no—
250C4⋊C8—22no—
251C4⋊C8—22no—
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—21no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8—81no—
265C4⋊C8—81no—
266C4⋊C8—81no—
267C4⋊C8—81no—
268C4⋊C8C2×D412yes—
269C4⋊C8C2×D412yes—
270#103—81no—
271#103—81no—
272#103—81no—
273#103—81no—
274#103—81no—
275#103—81no—
276#103—81no—
277#103—81no—
278C22⋊C8—24no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281#97—24no—
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—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294C24#12111yescharacteristic
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4C22⋊Q812yes—
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4#12612yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
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—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—84no—
371C2×C8—84no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—81no—
399C4⋊C4—81no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—44no—
411C42#12112yes—
412C42#12112yes—
413C42C22⋊Q812yes—
414C42C22⋊Q812yes—
415C42C22⋊Q812yes—
416C42#12612yes—
417C42C22×D411yescharacteristic
418C42C22×Q811yescharacteristic
419C42C2×#4111yescharacteristic
420C42C2×#4111yescharacteristic
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23#39612yes—
426C23#47112yes—
427C23#47312yes—
428C23C2×#12111yescharacteristic
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433Q8—81no—
434Q8—81no—
435Q8—81no—
436Q8—81no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
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—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—164no—
480C2×C4#39612yes—
481C2×C4#39612yes—
482C2×C4#39612yes—
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×C22⋊Q811yescharacteristic
488C2×C4C2×#12611yescharacteristic
489C2×C4#57611yescharacteristic
490C2×C4#57611yescharacteristic
491C2×C4#57711yescharacteristic
492C2×C4#57711yescharacteristic
493C2×C4#57711yescharacteristic
494C2×C4#57711yescharacteristic
495C2×C4#57811yescharacteristic
496C2×C4#57811yescharacteristic
497C8—42no—
498C8—42no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—81no—
504C8—81no—
505C8—81no—
506C8—81no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22#206414yes—
514C22C2×#39611yescharacteristic
515C22#304511yescharacteristic
516C22#304611yescharacteristic
517C22#306211yescharacteristic
518C22#306311yescharacteristic
519C22C2×#47111yescharacteristic
520C22C2×#47311yescharacteristic
521C4—21no—
522C4—21no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—82no—
527C4—82no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—161no—
533C4—161no—
534C4#304511yescharacteristic
535C4#304611yescharacteristic
536C4#306211yescharacteristic
537C4#306311yescharacteristic
538C4#308311yescharacteristic
539C4#308311yescharacteristic
540C4#308411yescharacteristic
541C4#308411yescharacteristic
542C2—24no—
543C2#1958111yescharacteristic
544C2#1958311yescharacteristic
545C2#1958811yescharacteristic
546C2#1959011yescharacteristic
547C2C2×#206411yescharacteristic
548C2#2361611yescharacteristic
549C2#2361611yescharacteristic
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. #10600C2semidirect
9. #10598C2semidirect
10. #10596C2semidirect
11. #10594C2semidirect
12. #10592C2semidirect
13. #10590C2semidirect
14. #10588C2semidirect

Maximal subgroups

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

subgroupindex
#219872
#219732
#218892
#218602
C2×#15732
C2×#15522
#106022
#106002
#105982
#105962
#105942
#105922
#105902
#105882
C2×#12962