Finite Group Databasebeta

← #227574

Subgroups — #227574

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23003C211yesmaximal, characteristic, complemented
2#23001C211yesmaximal, characteristic
3#22522C211yesmaximal, characteristic, complemented
4#22487C211yesmaximal, characteristic, complemented
5C2×#1622C211yesmaximal, characteristic
6#9829C211yesmaximal, characteristic, complemented
7#9576C211yesmaximal, characteristic, complemented
8#9575C211yesmaximal, characteristic, complemented
9#9230C211yesmaximal, characteristic, complemented
10#9221C211yesmaximal, characteristic
11#9064C211yesmaximal, characteristic, complemented
12#8896C211yesmaximal, characteristic, complemented
13#8890C211yesmaximal, characteristic, complemented
14#8209C211yesmaximal, characteristic, complemented
15#8202C211yesmaximal, characteristic, complemented
16C2×#554C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#459C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#442—21no—
22C2×#440—21no—
23C2×C8×D4C2211yescharacteristic
24#2366C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26D4×C4⋊C4C2211yescharacteristic
27#1992—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1989—21no—
31#1987C2211yescharacteristic
32#1987C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1936—21no—
38#1936—21no—
39#1892—21no—
40#1880—21no—
41#1879—21no—
42#1872—21no—
43#1864—21no—
44#1862—21no—
45#1834C2211yescharacteristic
46#1824—21no—
47#1803C2211yescharacteristic
48#1802—21no—
49#1802C2211yescharacteristic
50#1770C2211yescharacteristic
51#1770C2211yescharacteristic
52#1762—21no—
53C4×#103C2211yescharacteristic
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56#1622C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—41no—
70C22×D8—21no—
71C23×C8—21no—
72#554—22no—
73#554—22no—
74#554—41no—
75#554C2×C414yescomplemented
76C2×C4⋊Q8C2311yescharacteristic
77C2×#127C2311yescharacteristic
78C2×#125—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—41no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90#492—41no—
91#492C2×C414yescomplemented
92#460—22no—
93#460C2×C414yescomplemented
94#459—22no—
95#459C2×C414yes—
96#446—22no—
97#446—41no—
98#442—44no—
99#440—44no—
100C8×D4—22no—
101C8×D4—41no—
102C2×#107—21no—
103C2×#106—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#103—21no—
110C2×#103D411yescharacteristic
111C2×#103D411yescharacteristic
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102D411yescharacteristic
120C2×#102D411yescharacteristic
121C2×C22⋊C8—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#390—21no—
129#390—21no—
130#389—21no—
131#389—21no—
132#389—41no—
133#386—41no—
134#385—41no—
135#384—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#380—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
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—41no—
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#340D411yescharacteristic
159#336—21no—
160#336—21no—
161#333C2×C414yescomplemented
162#331—41no—
163Q8⋊C8C2×C414yescomplemented
164D4⋊C8C2×C414yescomplemented
165#322—41no—
166C22×Q8—21no—
167C22×Q8C2×D411yescharacteristic
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4C2×D411yescharacteristic
172C23×C4—21no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C2×SD16—24no—
177C2×SD16—44no—
178C2×SD16—42no—
179C2×SD16—42no—
180C2×SD16—82no—
181C2×SD16—82no—
182C2×D8—24no—
183C2×D8—42no—
184C2×D8—42no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—81no—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8C22×C412yes—
201#127—21no—
202#127—21no—
203#127—21no—
204#127C22×C412yes—
205#125—24no—
206#125—42no—
207C22⋊Q8—24no—
208C22⋊Q8—44no—
209C22⋊Q8—44no—
210C22⋊Q8—42no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—42no—
214C4×Q8—81no—
215C4×Q8C22×C412yes—
216C4×Q8#4112yes—
217C4×D4—24no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—81no—
225C4×D4—81no—
226C4×D4—81no—
227C4×D4C22×C412yes—
228C4×D4#4112yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—81no—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—41no—
263C2×C42C2411yescharacteristic
264#107—42no—
265#106—42no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—42no—
273C4⋊C8C22×C412yes—
274C4⋊C8C22×C412yes—
275#103—22no—
276#103—41no—
277#103—81no—
278#102—22no—
279#102—22no—
280#102—24no—
281#102—24no—
282#102—42no—
283#102—42no—
284#102—42no—
285#102—81no—
286#102—81no—
287C22⋊C8—42no—
288#97—42no—
289#97—42no—
290#97—42no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8C22×C412yes—
297#95—21no—
298#95—21no—
299#95—41no—
300#95—41no—
301C24—21no—
302C24—41no—
303C2×Q8—21no—
304C2×Q8—22no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—41no—
310C2×Q8—81no—
311C2×Q8—81no—
312C2×Q8C4×D412yes—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—81no—
331C2×D4—81no—
332C2×D4C4×D412yes—
333C2×D4AGL(1,ℤ/8)12yes—
334C2×D4#13712yes—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—44no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—161no—
357C22×C4—161no—
358C22×C4—161no—
359C22×C4—161no—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363SD16—48no—
364SD16—88no—
365D8—48no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—84no—
393C2×C8AGL(1,ℤ/8)12yes—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—161no—
425C4⋊C4C4×D412yes—
426C4⋊C4C4×D412yes—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—161no—
434C42—22no—
435C42—22no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—82no—
441C42C23×C411yescharacteristic
442C42C2×#4111yescharacteristic
443C42C2×#4111yescharacteristic
444C42H(5,𝔽2)11yescharacteristic
445C23—22no—
446C23—22no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23D4211yescharacteristic
460Q8—22no—
461Q8—22no—
462Q8—41no—
463Q8—41no—
464Q8—82no—
465Q8—161no—
466D4—22no—
467D4—22no—
468D4—22no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—82no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—81no—
478D4—161no—
479D4#44014yescomplemented
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—164no—
517C2×C4—164no—
518C2×C4—164no—
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4#51811yescharacteristic
522C2×C4#54611yescharacteristic
523C2×C4#54611yescharacteristic
524C2×C4#55111yescharacteristic
525C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
526C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
527C2×C4C2×#13711yescharacteristic
528C2×C4#57511yescharacteristic
529C2×C4#57711yescharacteristic
530C2×C4#57711yescharacteristic
531C2×C4#57711yescharacteristic
532C2×C4#57811yescharacteristic
533C8—22no—
534C8—42no—
535C8—42no—
536C8—42no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—81no—
541C8—81no—
542C22—24no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—161no—
553C22—161no—
554C22—161no—
555C22—161no—
556C22#235811yescharacteristic
557C22#326511yescharacteristic
558C22#327011yescharacteristic
559C22#327211yescharacteristic
560C22#327511yescharacteristic
561C22#328811yescharacteristic
562C22#328811yescharacteristic
563C4—21no—
564C4—21no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—82no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—161no—
574C4—161no—
575C4C2×#44011yescharacteristic
576C4C4×AGL(1,ℤ/8)11yescharacteristic
577C4#294711yescharacteristic
578C4#295211yescharacteristic
579C4#330611yescharacteristic
580C4#330711yescharacteristic
581C4#330811yescharacteristic
582C4#332011yescharacteristic
583C2—44no—
584C2—162no—
585C2#2019311yescharacteristic
586C2#2019511yescharacteristic
587C2#2020911yescharacteristic
588C2#2021111yescharacteristic
589C2#2285511yescharacteristic
590C2#2286611yescharacteristic
591C2#2289111yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23003C2semidirect
3. #22522C2semidirect
4. #22487C2semidirect
6. #9829C2semidirect
7. #9576C2semidirect
8. #9575C2semidirect
9. #9230C2semidirect
11. #9064C2semidirect
12. #8896C2semidirect
13. #8890C2semidirect
14. #8209C2semidirect
15. #8202C2semidirect
56. #1622C4semidirect
75. #554C2×C4semidirect
91. #492C2×C4semidirect
93. #460C2×C4semidirect
161. #333C2×C4semidirect
163. Q8⋊C8C2×C4semidirect
164. D4⋊C8C2×C4semidirect
479. D4#440semidirect

Maximal subgroups

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

subgroupindex
#230032
#230012
#225222
#224872
C2×#16222
#98292
#95762
#95752
#92302
#92212
#90642
#88962
#88902
#82092
#82022