Finite Group Databasebeta

← #242885

Subgroups — #242885

Order 512.

673 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic, complemented
2#21867C211yesmaximal, characteristic
3#21862C211yesmaximal, characteristic
4#21850C211yesmaximal, characteristic
5#10624C211yesmaximal, characteristic, complemented
6#10623C211yesmaximal, characteristic, complemented
7#10622C211yesmaximal, characteristic, complemented
8#10621C211yesmaximal, characteristic, complemented
9#10618C211yesmaximal, characteristic, complemented
10#10617C211yesmaximal, characteristic, complemented
11#10616C211yesmaximal, characteristic, complemented
12#10615C211yesmaximal, characteristic, complemented
13#8043C211yesmaximal, characteristic, complemented
14#8042C211yesmaximal, characteristic, complemented
15#7878C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#1992—21no—
24#1991—21no—
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1959—21no—
28#1958—21no—
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1947—21no—
32#1947—21no—
33#1875—21no—
34#1874—21no—
35#1873—21no—
36#1872—21no—
37#1870—21no—
38#1870—21no—
39#1852—21no—
40#1852—21no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53#1798—21no—
54#1798—21no—
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68#529—22no—
69#529—22no—
70#529—22no—
71#529—22no—
72C2×#126—21no—
73C2×#125D411yescharacteristic
74C2×#124D411yescharacteristic
75C2×#123—21no—
76C2×C22⋊Q8D411yescharacteristic
77C2×#121D411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4C2311yescharacteristic
81C2×#117—21no—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C42C2311yescharacteristic
87#432—22no—
88#432—22no—
89#431—22no—
90#431—22no—
91#431—22no—
92#431—22no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—21no—
97#423—22no—
98#423—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
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×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8D411yescharacteristic
123C2×C22⋊C8D411yescharacteristic
124C2×#97—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#384—21no—
129#384—21no—
130#384—21no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#382—21no—
136#380—21no—
137#380—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140#340—21no—
141#340—21no—
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#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156C22×Q8C2×D411yescharacteristic
157C22×D4—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×M16—42no—
165C2×M16—81no—
166C2×M16—81no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177#126—42no—
178#126—42no—
179#125—22no—
180#125—22no—
181#124—22no—
182#124—22no—
183#123—42no—
184#123—42no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187#121—22no—
188#121—22no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—42no—
198C4×D4—42no—
199#117—42no—
200#117—42no—
201#117—81no—
202#117—81no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—41no—
235C2×C42—21no—
236C2×C42—41no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245#107—81no—
246#107—81no—
247#107—81no—
248#107—81no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—42no—
264#103—81no—
265#103—81no—
266#103—81no—
267#103—81no—
268#102—81no—
269#102—81no—
270#102—81no—
271#102—81no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—81no—
277C22⋊C8—81no—
278C22⋊C8—81no—
279C22⋊C8—81no—
280#97—42no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—42no—
286#95—41no—
287#95—41no—
288C24—21no—
289C24#12111yescharacteristic
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—81no—
304C2×D4—81no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12111yescharacteristic
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4#12312yes—
337C22×C4#12412yes—
338C22×C4#12612yes—
339C22×C4#12612yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343M16—84no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
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—41no—
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—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
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—81no—
402C4⋊C4—81no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C22⋊C4—42no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—82no—
413C42—41no—
414C42—41no—
415C42—82no—
416C42#12312yes—
417C42#12312yes—
418C42#12312yes—
419C42#12412yes—
420C42#12612yes—
421C42#12612yes—
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C42C2×#4111yescharacteristic
425C42C2×#4111yescharacteristic
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23#39712yes—
440C23C2×#12111yescharacteristic
441Q8—41no—
442Q8—41no—
443Q8—41no—
444Q8—41no—
445D4—41no—
446D4—41no—
447D4—41no—
448D4—41no—
449D4—84no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—42no—
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—41no—
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—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—164no—
494C2×C4#39712yes—
495C2×C4#39712yes—
496C2×C4#39712yes—
497C2×C4C2×#12311yescharacteristic
498C2×C4C2×#12311yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C2×C4C2×#12411yescharacteristic
501C2×C4C2×#12611yescharacteristic
502C2×C4C2×#12611yescharacteristic
503C2×C4#57611yescharacteristic
504C2×C4#57611yescharacteristic
505C2×C4#57611yescharacteristic
506C2×C4#57711yescharacteristic
507C2×C4#57711yescharacteristic
508C2×C4#57811yescharacteristic
509C2×C4#57811yescharacteristic
510C2×C4#57811yescharacteristic
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—81no—
516C8—81no—
517C8—81no—
518C8—81no—
519C8—81no—
520C8—81no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22C2×#39711yescharacteristic
536C22#304511yescharacteristic
537C22#304611yescharacteristic
538C22#305811yescharacteristic
539C22#305911yescharacteristic
540C22#306811yescharacteristic
541C22#306911yescharacteristic
542C4—22no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—161no—
551C4—161no—
552C4—161no—
553C4—161no—
554C4#309811yescharacteristic
555C4#309811yescharacteristic
556C4#309811yescharacteristic
557C4#309811yescharacteristic
558C4#309811yescharacteristic
559C4#309911yescharacteristic
560C4#313611yescharacteristic
561C4#313711yescharacteristic
562C2—42no—
563C2—161no—
564C2#1953811yescharacteristic
565C2#1953911yescharacteristic
566C2#1954611yescharacteristic
567C2#1954711yescharacteristic
568C2#2363411yescharacteristic
569C2#2363911yescharacteristic
570C2#2364711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
1. #21982C2semidirect
5. #10624C2semidirect
6. #10623C2semidirect
7. #10622C2semidirect
8. #10621C2semidirect
9. #10618C2semidirect
10. #10617C2semidirect
11. #10616C2semidirect
12. #10615C2semidirect
13. #8043C2semidirect
14. #8042C2semidirect
15. #7878C2semidirect

Maximal subgroups

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

subgroupindex
#219822
#218672
#218622
#218502
#106242
#106232
#106222
#106212
#106182
#106172
#106162
#106152
#80432
#80422
#78782