Finite Group Databasebeta

← #278043

Subgroups — #278043

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic
2#24838C211yesmaximal, characteristic
3#24752C211yesmaximal, characteristic
4C8×C22⋊Q8C211yesmaximal, characteristic
5C2×#1680C211yesmaximal, characteristic, complemented
6C22×#334C211yesmaximal, characteristic
7#10598C211yesmaximal, characteristic, complemented
8#10588C211yesmaximal, characteristic, complemented
9#10486C211yesmaximal, characteristic, complemented
10#10479C211yesmaximal, characteristic, complemented
11#9736C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#9131C211yesmaximal, characteristic
14#9129C211yesmaximal, characteristic
15#8647C211yesmaximal, characteristic, complemented
16C2×#487C2211yescharacteristic
17C2×#475C2211yescharacteristic
18C2×#467—21no—
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26C4×C22⋊Q8C2211yescharacteristic
27#2088—21no—
28#2068—21no—
29#2066—21no—
30#2056—21no—
31#1990C2211yescharacteristic
32#1988C2211yescharacteristic
33#1984C2211yescharacteristic
34#1981C2211yescharacteristic
35#1956C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939C2211yescharacteristic
39#1933—21no—
40#1933—21no—
41#1916—21no—
42#1892—21no—
43#1892—21no—
44#1892—21no—
45#1892—21no—
46#1850—21no—
47#1834C2211yescharacteristic
48#1807—21no—
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4—21no—
53C8×C4⋊C4C2211yescharacteristic
54C8×C22⋊C4—21no—
55#1680—24no—
56C2×#334—24no—
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×Q16—21no—
70C23×C8—21no—
71C23×C8—21no—
72C23×C8D411yescharacteristic
73C23×C8D411yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8D411yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#487—22no—
82#487—22no—
83#475—22no—
84#475—41no—
85#467—44no—
86#462—22no—
87#462—41no—
88C8×Q8—22no—
89C8×Q8—22no—
90C2×#107—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—41no—
100C2×C4⋊C8—41no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×C22⋊C8—21no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8D411yescharacteristic
123C2×C4×C8D411yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—21no—
127#391—21no—
128#391D411yescharacteristic
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389D411yescharacteristic
134#387—21no—
135#385—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380D411yescharacteristic
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—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#340—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#334—22no—
162#334—22no—
163#334—22no—
164#334D414yes—
165#332—22no—
166#332—22no—
167#332—41no—
168#332—41no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C2×Q16—42no—
176C2×Q16—42no—
177C2×Q16—81no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—41no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8—81no—
195C22×C8—81no—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C22⋊Q8—22no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213C22⋊Q8SD1614yes—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—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—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42C2411yescharacteristic
250#107—42no—
251#106—42no—
252#106—81no—
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—21no—
263C4⋊C8—42no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C4⋊C8C2×D412yes—
268C4⋊C8C2×D412yes—
269#103—42no—
270#103—42no—
271#103—42no—
272#103—42no—
273#103—42no—
274#103—42no—
275#103—42no—
276#103—42no—
277#103—81no—
278#103—81no—
279C22⋊C8—42no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—42no—
285C4×C8—42no—
286C4×C8C2×D412yes—
287C4×C8#4112yes—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—41no—
294#95—41no—
295#95—41no—
296#95—41no—
297C24#12111yescharacteristic
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8C2×SD1612yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4C2×SD1612yes—
340C22×C4#13512yes—
341C22×C4#13512yes—
342C22×C4#13712yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346Q16—84no—
347C2×C8—28no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—44no—
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—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—84no—
377C2×C8—84no—
378C2×C8#12112yes—
379C2×C8#12112yes—
380C2×C8#13512yes—
381C4⋊C4—21no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
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—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4C2×SD1612yes—
411C22⋊C4—24no—
412C22⋊C4—42no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42C22×D411yescharacteristic
426C42C2×#4111yescharacteristic
427C42H(5,𝔽2)11yescharacteristic
428C422−1+411yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23#46412yes—
434C23#46512yes—
435C23#47312yes—
436C23C2×#12111yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—41no—
440Q8—41no—
441Q8—81no—
442Q8—81no—
443Q8—81no—
444Q8—81no—
445Q8—81no—
446Q8—81no—
447Q8—81no—
448Q8—81no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4#46412yes—
493C2×C4#46512yes—
494C2×C4C2×#12111yescharacteristic
495C2×C4#53711yescharacteristic
496C2×C4D4211yescharacteristic
497C2×C4#54711yescharacteristic
498C2×C4#54811yescharacteristic
499C2×C4#55311yescharacteristic
500C2×C4C22×SD1611yescharacteristic
501C2×C4C2×#13511yescharacteristic
502C2×C4C2×#13511yescharacteristic
503C2×C4C2×#13511yescharacteristic
504C2×C4C2×#13711yescharacteristic
505C2×C4#57511yescharacteristic
506C2×C4#57711yescharacteristic
507C2×C4#57711yescharacteristic
508C8—22no—
509C8—22no—
510C8—22no—
511C8—42no—
512C8—42no—
513C8—41no—
514C8—81no—
515C8—81no—
516C8—81no—
517C8—81no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22#168014yescomplemented
525C22#241611yescharacteristic
526C22C2×#46411yescharacteristic
527C22#303511yescharacteristic
528C22C2×#46511yescharacteristic
529C22#306011yescharacteristic
530C22C2×#47311yescharacteristic
531C22#306711yescharacteristic
532C4—22no—
533C4—22no—
534C4—21no—
535C4—21no—
536C4—42no—
537C4—42no—
538C4—41no—
539C4—81no—
540C4—81no—
541C4—161no—
542C4—161no—
543C4—161no—
544C4—161no—
545C4#305011yescharacteristic
546C4#305611yescharacteristic
547C4#324811yescharacteristic
548C4#325011yescharacteristic
549C4D4×SD1611yescharacteristic
550C4#328211yescharacteristic
551C4#333511yescharacteristic
552C4#333711yescharacteristic
553C2—24no—
554C2C2×#168011yescharacteristic
555C2#1912411yescharacteristic
556C2#1921611yescharacteristic
557C2#1922811yescharacteristic
558C2#2475511yescharacteristic
559C2#2477611yescharacteristic
560C2#2480411yescharacteristic
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
5. C2×#1680C2semidirect
7. #10598C2semidirect
8. #10588C2semidirect
9. #10486C2semidirect
10. #10479C2semidirect
11. #9736C2semidirect
12. #9613C2semidirect
15. #8647C2semidirect
524. C22#1680semidirect

Maximal subgroups

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

subgroupindex
#330432
#248382
#247522
C8×C22⋊Q82
C2×#16802
C22×#3342
#105982
#105882
#104862
#104792
#97362
#96132
#91312
#91292
#86472