Finite Group Databasebeta

← #241354

Subgroups — #241354

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic
4#20760C211yesmaximal, characteristic
5#10582C211yesmaximal, characteristic, complemented
6#10582C211yesmaximal, characteristic, complemented
7#10581C211yesmaximal, characteristic, complemented
8#10581C211yesmaximal, characteristic, complemented
9#10578C211yesmaximal, characteristic, complemented
10#10578C211yesmaximal, characteristic, complemented
11#10575C211yesmaximal, characteristic, complemented
12#10575C211yesmaximal, characteristic, complemented
13#8044C211yesmaximal, characteristic
14#7898C211yesmaximal, characteristic
15C2×#1279C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2857—21no—
20#2851—21no—
21#2666—22no—
22#2571C2211yescharacteristic
23#2570—21no—
24#2552C2211yescharacteristic
25#2552C2211yescharacteristic
26#2551C2211yescharacteristic
27#2550—21no—
28#2442—21no—
29#2440—21no—
30#2438—22no—
31#2331—21no—
32#2330—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2283C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1949—21no—
38#1949—21no—
39#1948C2211yescharacteristic
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1947C2211yescharacteristic
44#1947C2211yescharacteristic
45#1947C2211yescharacteristic
46#1947C2211yescharacteristic
47#1853—21no—
48#1852—21no—
49#1835—22no—
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1808—21no—
55#1807—21no—
56#1776—21no—
57#1776—21no—
58#1735—22no—
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71C23×C8—21no—
72C23×C8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×#125—21no—
76C2×#125—21no—
77C2×C22⋊Q8—22no—
78C2×#117—21no—
79C2×#117D412yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#432—22no—
85#432Q814yescomplemented
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8Q814yescomplemented
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—22no—
105C2×C4×C8Q814yescomplemented
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#400—22no—
109#400—22no—
110#398—22no—
111#398—22no—
112#398—22no—
113#398—22no—
114#395—22no—
115#395—22no—
116#391—22no—
117#391—22no—
118#389—22no—
119#389—22no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#387—22no—
129#387—22no—
130#387—22no—
131#387—22no—
132#385—22no—
133#385—22no—
134#384—21no—
135#384—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#383—21no—
141#382—21no—
142#382—21no—
143#382—21no—
144#382D411yescharacteristic
145#382D412yes—
146#382D411yescharacteristic
147#381—21no—
148#380—22no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380D411yescharacteristic
154#380D411yescharacteristic
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4D411yescharacteristic
162C4×C4⋊C4D411yescharacteristic
163C4×C4⋊C4C2311yescharacteristic
164C4×C4⋊C4C2311yescharacteristic
165C4×C4⋊C4C2311yescharacteristic
166C4×C4⋊C4C2311yescharacteristic
167C4×C22⋊C4—21no—
168#376—21no—
169C2×#95—21no—
170#340—21no—
171#340—21no—
172#340—21no—
173#340—21no—
174#340—22no—
175#340—22no—
176#340—22no—
177#340—22no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182C22×Q8—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C22×C8—24no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—24no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—81no—
199C22×C8—81no—
200C4⋊Q8—42no—
201C4⋊Q8—42no—
202C4⋊Q8—81no—
203C4⋊Q8—81no—
204#125—42no—
205#125—42no—
206#125—81no—
207#125—81no—
208C22⋊Q8—84no—
209#117—44no—
210#117—44no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
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—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—41no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—41no—
258C2×C42—81no—
259C2×C42—81no—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×Q812yes—
264C2×C42C2×Q812yes—
265C2×C42C2×Q812yes—
266C2×C42#4112yes—
267C2×C42C2411yescharacteristic
268#107—42no—
269#107—42no—
270#107—81no—
271#107—81no—
272#106—42no—
273#106—42no—
274#106—81no—
275#106—81no—
276C4⋊C8—24no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8C2×Q812yes—
282C4⋊C8C2×Q812yes—
283C4⋊C8C2×Q812yes—
284C4⋊C8C2×Q812yes—
285C22⋊C8—24no—
286#97—24no—
287C4×C8—24no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8C2×Q812yes—
291C4×C8C2×Q812yes—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—22no—
307#95—22no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—41no—
313#95—41no—
314C24#12011yescharacteristic
315C2×Q8—44no—
316C2×Q8—82no—
317C2×Q8—82no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4C22⋊Q812yes—
350C22×C4C22⋊Q812yes—
351C22×C4C22⋊Q812yes—
352C22×C4C4⋊Q812yes—
353C22×C4C4⋊Q812yes—
354C22×C4C4⋊Q812yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—48no—
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—41no—
377C2×C8—84no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4#13512yes—
424C4⋊C4#13512yes—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—42no—
428C22⋊C4—82no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—81no—
438C42—81no—
439C42—82no—
440C42—82no—
441C42C22⋊Q812yes—
442C42C22⋊Q812yes—
443C42C22⋊Q812yes—
444C42C4⋊Q812yes—
445C42C4⋊Q812yes—
446C42C4⋊Q812yes—
447C42C22×Q811yescharacteristic
448C42C22×Q811yescharacteristic
449C42C22×Q811yescharacteristic
450C42C2×#4111yescharacteristic
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23#39512yes—
456C23#45412yes—
457C23#45512yes—
458C23C2×#12011yescharacteristic
459Q8—164no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4#39512yes—
517C2×C4#39512yes—
518C2×C4#39512yes—
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4C2×C4⋊Q811yescharacteristic
523C2×C4C2×C4⋊Q811yescharacteristic
524C2×C4C2×C4⋊Q811yescharacteristic
525C2×C4C2×#13511yescharacteristic
526C2×C4C2×#13511yescharacteristic
527C2×C4#57511yescharacteristic
528C2×C4#57511yescharacteristic
529C2×C4#57611yescharacteristic
530C2×C4#57711yescharacteristic
531C2×C4#57711yescharacteristic
532C2×C4#57811yescharacteristic
533C8—42no—
534C8—42no—
535C8—42no—
536C8—42no—
537C8—42no—
538C8—42no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22#206114yes—
546C22C2×#39511yescharacteristic
547C22#300211yescharacteristic
548C22#300211yescharacteristic
549C22#301211yescharacteristic
550C22#301311yescharacteristic
551C22C2×#45511yescharacteristic
552C22C2×#45411yescharacteristic
553C4—22no—
554C4—21no—
555C4—21no—
556C4—22no—
557C4—22no—
558C4—42no—
559C4—42no—
560C4—81no—
561C4—82no—
562C4—81no—
563C4—81no—
564C4—82no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—161no—
569C4—161no—
570C4#307611yescharacteristic
571C4#307611yescharacteristic
572C4#308111yescharacteristic
573C4#308211yescharacteristic
574C4#316011yescharacteristic
575C4#316211yescharacteristic
576C4#316311yescharacteristic
577C4#316411yescharacteristic
578C2—24no—
579C2#1862811yescharacteristic
580C2#1862911yescharacteristic
581C2#1863811yescharacteristic
582C2#1863911yescharacteristic
583C2#2355111yescharacteristic
584C2#2355711yescharacteristic
585C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #10582C2semidirect
6. #10582C2semidirect
7. #10581C2semidirect
8. #10581C2semidirect
9. #10578C2semidirect
10. #10578C2semidirect
11. #10575C2semidirect
12. #10575C2semidirect
85. #432Q8semidirect
96. C2×C4⋊C8Q8semidirect
105. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#352392
#219622
#213652
#207602
#105822
#105812
#105782
#105752
#80442
#78982
C2×#12792