Finite Group Databasebeta

← #410200

Subgroups — #410200

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#427C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C2×#1825C212yesmaximal
4#13939C212yesmaximal
5#11257C212yesmaximal, complemented
6#11256C211yesmaximal, characteristic, complemented
7#11256C211yesmaximal, characteristic, complemented
8#11232C211yesmaximal, characteristic, complemented
9#11232C211yesmaximal, characteristic, complemented
10#11231C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2212yes—
14C2×#427—24no—
15C2×#427—28no—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1850—24no—
25#1825—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1440—22no—
36#1440—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—22no—
42#1434—22no—
43C24×C4C2311yescharacteristic
44C23×C8—24no—
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—24no—
48C23×C8D414yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117D418yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#427—232no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—24no—
74C2×#107—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—24no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—22no—
85#380—24no—
86#380—24no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100#340—24no—
101#340—22no—
102#340—22no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#336—22no—
108#336—22no—
109C25C2×D411yescharacteristic
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D414yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4#4112yes—
128C23×C4#4114yes—
129C23×C4C2411yescharacteristic
130C22×C8—216no—
131C22×C8—216no—
132C22×C8—28no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—28no—
139C22×C8—28no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8#4114yes—
153C22×C8#4112yes—
154C22×C8#4112yes—
155#117—216no—
156#117—216no—
157#117—48no—
158#117—48no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×D414yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—28no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—44no—
206C2×C42—44no—
207#107—28no—
208#107—28no—
209#106—216no—
210C4⋊C8—44no—
211C4⋊C8—44no—
212C22⋊C8—44no—
213#95—22no—
214#95—22no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—82no—
221#95—82no—
222#95—82no—
223#95—82no—
224C24—21no—
225C24—22no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—21no—
231C24—22no—
232C24—21no—
233C24#12114yes—
234C24#12112yes—
235C24C22×D411yescharacteristic
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4#12014yes—
303C22×C4#12112yes—
304C22×C4#12114yes—
305C22×C4#12318yes—
306C22×C4#12414yes—
307C22×C4#13512yes—
308C22×C4#13512yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D412yes—
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4112yes—
316C22×C42−1+411yescharacteristic
317C22×C42−1+412yes—
318C22×C42−1+411yescharacteristic
319C2×C8—232no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8#12314yes—
339C2×C8#12314yes—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C22⋊C4—28no—
357C22⋊C4—44no—
358C22⋊C4—416no—
359C22⋊C4—88no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—42no—
365C42—41no—
366C42—41no—
367C42—416no—
368C23—24no—
369C23—24no—
370C23—22no—
371C23—22no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23#39414yes—
391C23C2×#12112yes—
392C23C2×#12111yescharacteristic
393C23#54711yescharacteristic
394C23#54712yes—
395C23#54711yescharacteristic
396C23C2×#13511yescharacteristic
397C23C2×#13511yescharacteristic
398C23#57512yes—
399C23#57611yescharacteristic
400C23#57712yes—
401C23#57811yescharacteristic
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#39414yes—
463C2×C4C2×#12011yescharacteristic
464C2×C4C2×#12311yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C2×C4C2×#12312yes—
467C2×C4#52512yes—
468C2×C4#52512yes—
469C2×C4C2×#12411yescharacteristic
470C2×C4#53612yes—
471C2×C4#53612yes—
472C2×C4#53611yescharacteristic
473C2×C4#53611yescharacteristic
474C2×C4#55212yes—
475C2×C4#55211yescharacteristic
476C2×C4#55211yescharacteristic
477C2×C4#55212yes—
478C2×C4#55212yes—
479C2×C4#55212yes—
480C2×C4#55212yes—
481C8—24no—
482C8—24no—
483C8—81no—
484C8—81no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22C2×#39411yescharacteristic
497C22#241312yes—
498C22#241312yes—
499C22#248611yescharacteristic
500C22#248611yescharacteristic
501C22#303512yes—
502C22#303512yes—
503C22#304512yes—
504C22#304612yes—
505C22#305011yescharacteristic
506C22#305411yescharacteristic
507C22#306011yescharacteristic
508C22#306111yescharacteristic
509C22#333611yescharacteristic
510C22#333712yes—
511C22#333911yescharacteristic
512C22#334012yes—
513C22#334111yescharacteristic
514C22#334211yescharacteristic
515C22#334712yes—
516C22#334812yes—
517C22#334912yes—
518C22#335012yes—
519C4—24no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—84no—
525C4—84no—
526C4—162no—
527C4—162no—
528C4#248412yes—
529C4#248412yes—
530C4#250311yescharacteristic
531C4#250311yescharacteristic
532C4#261411yescharacteristic
533C4#284911yescharacteristic
534C2—44no—
535C2#1507311yescharacteristic
536C2#2353911yescharacteristic
537C2#2354311yescharacteristic
538C2#2458412yes—
539C2#2458512yes—
540C2#2458612yes—
541C2#2458712yes—
542C2#2765911yescharacteristic
543C2#2766011yescharacteristic
544C2#2766411yescharacteristic
545C2#2766511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11257C2semidirect
6. #11256C2semidirect
7. #11256C2semidirect
8. #11232C2semidirect
9. #11232C2semidirect
10. #11231C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4272
C2×#18502
C2×#18252
#139392
#112572
#112562
#112322
#112312
#107212