Finite Group Databasebeta

← #227002

Subgroups — #227002

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22998C211yesmaximal, characteristic, complemented
2#22962C211yesmaximal, characteristic
3Q8×#102C211yesmaximal, characteristic
4D4×#103C211yesmaximal, characteristic
5C2×#1649C211yesmaximal, characteristic
6#9823C211yesmaximal, characteristic, complemented
7#9566C211yesmaximal, characteristic, complemented
8#9561C211yesmaximal, characteristic, complemented
9#9219C211yesmaximal, characteristic
10#9218C211yesmaximal, characteristic, complemented
11#9058C211yesmaximal, characteristic, complemented
12#8884C211yesmaximal, characteristic, complemented
13#8882C211yesmaximal, characteristic
14C8×#103C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic
16C2×D4×Q8C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C4×Q16—21no—
22C2×C8×D4C2211yescharacteristic
23C22×#103—21no—
24#2365C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26D4×C4⋊C4C2211yescharacteristic
27#1990—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1988—21no—
31#1988C2211yescharacteristic
32#1981C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1942—21no—
38#1933—21no—
39#1918—21no—
40#1892—21no—
41#1878—21no—
42#1872—21no—
43#1863—21no—
44#1862—21no—
45#1845C2211yescharacteristic
46#1803C2211yescharacteristic
47#1802—21no—
48#1802C2211yescharacteristic
49#1787—21no—
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×#103C2211yescharacteristic
53C4×#102—21no—
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56#1649C418yescomplemented
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×Q8—21no—
69C22×Q16—21no—
70C23×C8—21no—
71D4×Q8—22no—
72D4×Q8—22no—
73D4×Q8—41no—
74D4×Q8C2×C414yes—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—41no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2311yescharacteristic
86C22×C4⋊C4—21no—
87#500—41no—
88#500C2×C414yescomplemented
89#474—22no—
90#474C2×C414yes—
91#462—22no—
92#462C2×C414yescomplemented
93C8×Q8—22no—
94C8×Q8—41no—
95C4×Q16—44no—
96C8×D4—22no—
97C8×D4—41no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#107—41no—
102C2×#107—41no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
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—44no—
117C2×#103D411yescharacteristic
118C2×#103D411yescharacteristic
119C2×#102—21no—
120C2×#102—21no—
121C2×#102D411yescharacteristic
122C2×#102D411yescharacteristic
123C2×C22⋊C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#391—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—41no—
135#387—41no—
136#385—41no—
137#384—21no—
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#380—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
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—21no—
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#340D411yescharacteristic
159#336—21no—
160#336—21no—
161#332—41no—
162#332C2×C414yescomplemented
163Q8⋊C8C2×C414yescomplemented
164D4⋊C8C2×C414yes—
165C82—41no—
166C22×Q8—24no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—81no—
174C22×Q8—81no—
175C22×Q8C2×D411yescharacteristic
176C22×D4C2×D411yescharacteristic
177C23×C4—21no—
178C23×C4—41no—
179C23×C4—41no—
180C2×Q16—24no—
181C2×Q16—42no—
182C2×Q16—42no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C4⋊Q8—24no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—42no—
203C4⋊Q8C22×C412yes—
204C4⋊Q8C22×C412yes—
205C22⋊Q8—24no—
206C22⋊Q8—44no—
207C22⋊Q8—44no—
208C22⋊Q8—42no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—42no—
212C4×Q8—42no—
213C4×Q8—81no—
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—81no—
223C4×D4C22×C412yes—
224C4×D4#4112yes—
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—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—81no—
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42C2411yescharacteristic
258#107—24no—
259#107—44no—
260#107—42no—
261#107—81no—
262#107—82no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—42no—
270C4⋊C8C22×C412yes—
271C4⋊C8C22×C412yes—
272#103—22no—
273#103—24no—
274#103—42no—
275#103—42no—
276#103—42no—
277#103—42no—
278#103—81no—
279#103—81no—
280#103—81no—
281#103D814yes—
282#102—24no—
283#102—22no—
284#102—42no—
285#102Q1614yescomplemented
286C22⋊C8—42no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—42no—
293C4×C8—42no—
294C4×C8—81no—
295C4×C8C22×C412yes—
296#95—21no—
297#95—21no—
298#95—41no—
299#95—41no—
300#95—41no—
301#95—41no—
302C24—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—44no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—48no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×Q8—82no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8C4×D412yes—
326C2×Q8C2×D812yes—
327C2×Q8#13512yes—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4C4×D412yes—
332C2×D4C2×Q1612yes—
333C2×D4#13512yes—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—44no—
344C22×C4—41no—
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×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360Q16—48no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—84no—
387C2×C8C2×D812yes—
388C2×C8C2×Q1612yes—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—44no—
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—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C4⋊C4—161no—
422C4⋊C4C4×D412yes—
423C4⋊C4C4×D412yes—
424C4⋊C4C2×D812yes—
425C4⋊C4C2×Q1612yes—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—44no—
429C22⋊C4—82no—
430C42—22no—
431C42—22no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—82no—
437C42—82no—
438C42C23×C411yescharacteristic
439C42C2×#4111yescharacteristic
440C42C2×#4111yescharacteristic
441C42H(5,𝔽2)11yescharacteristic
442C23—22no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23D4211yescharacteristic
450Q8—22no—
451Q8—22no—
452Q8—22no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—42no—
457Q8—88no—
458Q8—81no—
459Q8—81no—
460Q8—81no—
461Q8—81no—
462Q8C4×D814yes—
463D4—22no—
464D4—22no—
465D4—22no—
466D4C4×Q1614yescomplemented
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—88no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—164no—
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4#51811yescharacteristic
511C2×C4#54611yescharacteristic
512C2×C4#54611yescharacteristic
513C2×C4#55111yescharacteristic
514C2×C4C22×D811yescharacteristic
515C2×C4C22×Q1611yescharacteristic
516C2×C4C2×#13511yescharacteristic
517C2×C4C2×#13511yescharacteristic
518C2×C4#57611yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57711yescharacteristic
521C2×C4#57811yescharacteristic
522C8—24no—
523C8—22no—
524C8—22no—
525C8—42no—
526C8—42no—
527C8—41no—
528C8—41no—
529C8—81no—
530C8—81no—
531C22—24no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22#235811yescharacteristic
539C22D4×D811yescharacteristic
540C22#328211yescharacteristic
541C22#328211yescharacteristic
542C22D4×Q1611yescharacteristic
543C22#329111yescharacteristic
544C22#329311yescharacteristic
545C4—24no—
546C4—21no—
547C4—21no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—81no—
553C4—81no—
554C4—82no—
555C4—82no—
556C4—161no—
557C4—161no—
558C4C2×C4×D811yescharacteristic
559C4C2×C4×Q1611yescharacteristic
560C4#294611yescharacteristic
561C4#294711yescharacteristic
562C4#329411yescharacteristic
563C4#329711yescharacteristic
564C4#329911yescharacteristic
565C4#330011yescharacteristic
566C2—44no—
567C2#1997011yescharacteristic
568C2D8×Q1611yescharacteristic
569C2#1999811yescharacteristic
570C2#2000111yescharacteristic
571C2#2287011yescharacteristic
572C2#2287111yescharacteristic
573C2#2289411yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #22998C2semidirect
6. #9823C2semidirect
7. #9566C2semidirect
8. #9561C2semidirect
10. #9218C2semidirect
11. #9058C2semidirect
12. #8884C2semidirect
14. C8×#103C2semidirect
56. #1649C4semidirect
88. #500C2×C4semidirect
92. #462C2×C4semidirect
162. #332C2×C4semidirect
163. Q8⋊C8C2×C4semidirect
285. #102Q16semidirect
466. D4C4×Q16semidirect

Maximal subgroups

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

subgroupindex
#229982
#229622
Q8×#1022
D4×#1032
C2×#16492
#98232
#95662
#95612
#92192
#92182
#90582
#88842
#88822
C8×#1032
C8×#1022