Finite Group Databasebeta

← #323938

Subgroups — #323938

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29878C211yesmaximal, characteristic, complemented
2#24708C211yesmaximal, characteristic
3#24701C211yesmaximal, characteristic
4#22133C211yesmaximal, characteristic
5#10655C211yesmaximal, characteristic, complemented
6#10621C211yesmaximal, characteristic, complemented
7#10543C211yesmaximal, characteristic, complemented
8#10278C211yesmaximal, characteristic, complemented
9#9568C211yesmaximal, characteristic, complemented
10#9567C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic, complemented
12#9215C211yesmaximal, characteristic, complemented
13#9136C211yesmaximal, characteristic, complemented
14#9128C211yesmaximal, characteristic, complemented
15#7878C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#476C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2087—21no—
27#2087—21no—
28#2077—21no—
29#2077—21no—
30#2076—21no—
31#2076—21no—
32#2034—21no—
33#2033—21no—
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1954—21no—
39#1954C2211yescharacteristic
40#1948C2211yescharacteristic
41#1947C2211yescharacteristic
42#1943—21no—
43#1941—21no—
44#1939C2211yescharacteristic
45#1936—21no—
46#1934—21no—
47#1923—21no—
48#1869—21no—
49#1850—21no—
50#1834C2211yescharacteristic
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53#1797C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×#121D411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#500—41no—
77#500—41no—
78#476—22no—
79#476—22no—
80#476—41no—
81#476—41no—
82C8×D4—41no—
83C8×D4—41no—
84#432—22no—
85#432—41no—
86#431—22no—
87#431—41no—
88#431—41no—
89#431—41no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—21no—
123#391—21no—
124#390D411yescharacteristic
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389D411yescharacteristic
130#388—21no—
131#386—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138#380D411yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—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#340—21no—
153#340—21no—
154#340—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#333—22no—
159#333—22no—
160#333—41no—
161#333—41no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×D4—21no—
165C22×D4C2×D411yescharacteristic
166C23×C4—21no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C22⋊Q8—82no—
192C22⋊Q8—82no—
193#121—22no—
194#121—22no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—81no—
200C4×D4—81no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
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—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—42no—
234#107—42no—
235#107—42no—
236#106—42no—
237#106—42no—
238#106—42no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—81no—
252C4⋊C8#4112yes—
253C4⋊C8#4112yes—
254#102—42no—
255#102—42no—
256#102—42no—
257#102—42no—
258C22⋊C8—41no—
259C22⋊C8—41no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263C22⋊C8—81no—
264C22⋊C8—81no—
265C22⋊C8—81no—
266#97—42no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—42no—
272C4×C8—81no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280C24—21no—
281C24#12111yescharacteristic
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×Q8—81no—
289C2×Q8—81no—
290C2×D4—24no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—81no—
297C2×D4—81no—
298C2×D4#13512yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
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#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
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—84no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—81no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4#13512yes—
392C22⋊C4—24no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—81no—
396C22⋊C4—81no—
397C22⋊C4—81no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—41no—
407C42—41no—
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42H(5,𝔽2)11yescharacteristic
411C42H(5,𝔽2)11yescharacteristic
412C23—21no—
413C23—21no—
414C23—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23C2×#12111yescharacteristic
427Q8—161no—
428Q8—161no—
429Q8—161no—
430Q8—161no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435D4—84no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4#53211yescharacteristic
483C2×C4#53411yescharacteristic
484C2×C4#54611yescharacteristic
485C2×C4#54611yescharacteristic
486C2×C4#55311yescharacteristic
487C2×C4#55311yescharacteristic
488C2×C4C2×#13511yescharacteristic
489C2×C4C2×#13511yescharacteristic
490C2×C4#57711yescharacteristic
491C2×C4#57711yescharacteristic
492C2×C4#57711yescharacteristic
493C2×C4#57811yescharacteristic
494C2×C4#57811yescharacteristic
495C2×C4#57811yescharacteristic
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—42no—
501C8—42no—
502C8—81no—
503C8—81no—
504C8—81no—
505C8—81no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—41no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22#254311yescharacteristic
521C22#304211yescharacteristic
522C22#304611yescharacteristic
523C22#305011yescharacteristic
524C22#305911yescharacteristic
525C22#306811yescharacteristic
526C22#306811yescharacteristic
527C4—21no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4—161no—
536C4—161no—
537C4—161no—
538C4—161no—
539C4#317411yescharacteristic
540C4#317811yescharacteristic
541C4#320011yescharacteristic
542C4#320111yescharacteristic
543C4#330011yescharacteristic
544C4#330011yescharacteristic
545C4#330211yescharacteristic
546C4#330211yescharacteristic
547C2—81no—
548C2—82no—
549C2#1939411yescharacteristic
550C2#1941911yescharacteristic
551C2#1951511yescharacteristic
552C2#1954711yescharacteristic
553C2#2855411yescharacteristic
554C2#2858311yescharacteristic
555C2#2859511yescharacteristic
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. #29878C2semidirect
5. #10655C2semidirect
6. #10621C2semidirect
7. #10543C2semidirect
8. #10278C2semidirect
9. #9568C2semidirect
10. #9567C2semidirect
11. #9215C2semidirect
12. #9215C2semidirect
13. #9136C2semidirect
14. #9128C2semidirect
15. #7878C2semidirect

Maximal subgroups

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

subgroupindex
#298782
#247082
#247012
#221332
#106552
#106212
#105432
#102782
#95682
#95672
#92152
#91362
#91282
#78782