Finite Group Databasebeta

← #227537

Subgroups — #227537

Order 512.

726 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23102C211yesmaximal, characteristic, complemented
2#22968C211yesmaximal, characteristic
3#22541C211yesmaximal, characteristic
4#22504C211yesmaximal, characteristic
5C2×#1652C211yesmaximal, characteristic
6#9828C211yesmaximal, characteristic, complemented
7#9573C211yesmaximal, characteristic, complemented
8#9570C211yesmaximal, characteristic, complemented
9#9227C211yesmaximal, characteristic
10#9218C211yesmaximal, characteristic, complemented
11#9059C211yesmaximal, characteristic, complemented
12#8894C211yesmaximal, characteristic
13#8891C211yesmaximal, characteristic, complemented
14#8208C211yesmaximal, characteristic
15#8203C211yesmaximal, characteristic, complemented
16C2×#556C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22#2365C2211yescharacteristic
23#2355C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1988C2211yescharacteristic
28#1983—21no—
29#1981C2211yescharacteristic
30#1971—21no—
31#1958—21no—
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1943—21no—
37#1927—21no—
38#1921—21no—
39#1894—21no—
40#1878—21no—
41#1872—21no—
42#1864—21no—
43#1834C2211yescharacteristic
44#1825—21no—
45#1805—21no—
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50#1763—21no—
51#1762—21no—
52C4×#103—21no—
53C4×#103C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1736—21no—
56#1652C418yescomplemented
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—21no—
69C22×M16—21no—
70#556—22no—
71#556—22no—
72#556—41no—
73#556C2×C414yes—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#126—41no—
77C2×#124—21no—
78C2×C22⋊Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C2×#117—21no—
85C22×C4⋊C4—21no—
86#498—41no—
87#498C2×C414yescomplemented
88#474—22no—
89#474C2×C414yes—
90#462—22no—
91#462C2×C414yescomplemented
92C8×Q8—22no—
93C8×Q8—41no—
94#436—22no—
95#436—41no—
96C2×#107—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—41no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103D411yescharacteristic
114C2×#103D411yescharacteristic
115C2×#102—21no—
116C2×#102—21no—
117C2×#102D411yescharacteristic
118C2×#102D411yescharacteristic
119C2×C22⋊C8—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—21no—
127#391—21no—
128#389—21no—
129#389—21no—
130#387—41no—
131#387—41no—
132#385—41no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#382—21no—
136#380—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—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—41no—
153#340D411yescharacteristic
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#336—21no—
158#336—21no—
159#332—41no—
160#332C2×C414yescomplemented
161Q8⋊C8C2×C414yescomplemented
162D4⋊C8C2×C414yes—
163#322—41no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8C2×D411yescharacteristic
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—41no—
170C2×Q16—24no—
171C2×Q16—81no—
172C2×Q16—81no—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—81no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C22×C412yes—
192C4⋊Q8C22×C412yes—
193#126—44no—
194#126—44no—
195#124—24no—
196#124—42no—
197C22⋊Q8—24no—
198C22⋊Q8—42no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—24no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—81no—
205C4×Q8C22×C412yes—
206C4×Q8#4112yes—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—42no—
210C4×D4—81no—
211C4×D4C22×C412yes—
212C4×D4#4112yes—
213#117—24no—
214#117—42no—
215C2×C4⋊C4—21no—
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—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—81no—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42C2411yescharacteristic
248#107—24no—
249#106—44no—
250#106—81no—
251#106—81no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—42no—
259C4⋊C8C22×C412yes—
260C4⋊C8C22×C412yes—
261#103—24no—
262#103—41no—
263#103—41no—
264#103—42no—
265#103—42no—
266#103—42no—
267#103—81no—
268#103—81no—
269#103—81no—
270#103—81no—
271#102—24no—
272#102—22no—
273#102—41no—
274#102—81no—
275C22⋊C8—42no—
276#97—81no—
277#97—81no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—42no—
284C4×C8C22×C412yes—
285#95—21no—
286#95—21no—
287#95—41no—
288#95—41no—
289#95—41no—
290#95—41no—
291#95—41no—
292#95—41no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—41no—
302C2×Q8—41no—
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×Q8C4×D412yes—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4C4×D412yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—161no—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341Q16—84no—
342M16—84no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
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#13712yes—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
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—44no—
387C4⋊C4—41no—
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—81no—
400C4⋊C4—81no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4C4×D412yes—
404C4⋊C4C4×D412yes—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—82no—
410C42—22no—
411C42—22no—
412C42—44no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—82no—
418C42—161no—
419C42C23×C411yescharacteristic
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C42H(5,𝔽2)11yescharacteristic
423C23—22no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23D4211yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—41no—
434Q8—41no—
435Q8—42no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—81no—
440Q8—81no—
441Q8—81no—
442Q8—81no—
443D4—22no—
444D4—22no—
445D4—41no—
446D4—41no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—164no—
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4#51811yescharacteristic
490C2×C4#54611yescharacteristic
491C2×C4#54611yescharacteristic
492C2×C4#55111yescharacteristic
493C2×C4C2×#13711yescharacteristic
494C2×C4#57511yescharacteristic
495C2×C4#57511yescharacteristic
496C2×C4#57511yescharacteristic
497C2×C4#57611yescharacteristic
498C2×C4#57711yescharacteristic
499C2×C4#57711yescharacteristic
500C2×C4#57711yescharacteristic
501C8—22no—
502C8—42no—
503C8—42no—
504C8—42no—
505C8—41no—
506C8—41no—
507C8—41no—
508C8—81no—
509C8—81no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22#235811yescharacteristic
518C22#326711yescharacteristic
519C22#326811yescharacteristic
520C22#327511yescharacteristic
521C22#327811yescharacteristic
522C22#328911yescharacteristic
523C22#328911yescharacteristic
524C4—21no—
525C4—21no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—82no—
534C4—82no—
535C4—161no—
536C4—161no—
537C4#294311yescharacteristic
538C4C4×#13711yescharacteristic
539C4#294611yescharacteristic
540C4#295211yescharacteristic
541C4#330711yescharacteristic
542C4#331611yescharacteristic
543C4#331711yescharacteristic
544C4#331911yescharacteristic
545C2—82no—
546C2#2025111yescharacteristic
547C2#2025211yescharacteristic
548C2#2026711yescharacteristic
549C2#2026811yescharacteristic
550C2#2286011yescharacteristic
551C2#2286711yescharacteristic
552C2#2289211yescharacteristic
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. #23102C2semidirect
6. #9828C2semidirect
7. #9573C2semidirect
8. #9570C2semidirect
10. #9218C2semidirect
11. #9059C2semidirect
13. #8891C2semidirect
15. #8203C2semidirect
56. #1652C4semidirect
87. #498C2×C4semidirect
91. #462C2×C4semidirect
160. #332C2×C4semidirect
161. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#231022
#229682
#225412
#225042
C2×#16522
#98282
#95732
#95702
#92272
#92182
#90592
#88942
#88912
#82082
#82032