Finite Group Databasebeta

← #7187413

Subgroups — #7187413

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#59823C211yesmaximal, characteristic, complemented
3#32561C214yesmaximal, complemented
4#31654C214yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#28156C211yesmaximal, characteristic
7#27756C212yesmaximal, complemented
8#26180C214yesmaximal, complemented
9#26110C212yesmaximal, complemented
10#25367C212yesmaximal, complemented
11#24204C214yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#23001C212yesmaximal, complemented
14#22311C211yesmaximal, characteristic, complemented
15#22268C211yesmaximal, characteristic, complemented
16C2×#559C2212yes—
17C2×#554—21no—
18C2×#492C2211yescharacteristic
19C2×#463C2214yes—
20C2×#459C2212yes—
21C2×#442C2214yes—
22#2732—22no—
23#2724—22no—
24#2582C2212yes—
25#2552C2212yes—
26#2547—21no—
27#2547C2212yes—
28#2542C2212yes—
29#2538—21no—
30#2538C2214yes—
31#2486—22no—
32#2454—22no—
33#2447—22no—
34#2447C2214yes—
35#2417—22no—
36#2366C2212yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2100C2214yes—
46#2099C2212yes—
47#2096C2212yes—
48#2088C2214yes—
49#2072C2214yes—
50#2071C2212yes—
51#2071C2212yes—
52#2055C2214yes—
53#2042C2214yes—
54#2041C2212yes—
55#2024C2212yes—
56#2023C2212yes—
57#2012C2214yes—
58#2011C2212yes—
59#1985C2214yes—
60#1981C2212yes—
61#1959C2214yes—
62#1957C2211yescharacteristic
63#1956C2211yescharacteristic
64#1948C2212yes—
65#1939C2212yes—
66#1936C2214yes—
67#1935C2214yes—
68#1917C2214yes—
69#1880C2214yes—
70#1879C2214yes—
71#1879C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1865C2214yes—
75#1864C2214yes—
76#1864C2214yes—
77#1834C2211yescharacteristic
78#1831C2211yescharacteristic
79#1802C2212yes—
80#1802C2214yes—
81#1770C2212yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2314yes—
86C22×D8C2314yes—
87#559—48no—
88#554—48no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#125—22no—
95C2×#124—22no—
96C2×#124C2314yes—
97C2×#123—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2314yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116#492—24no—
117#463—48no—
118#459—28no—
119#442—48no—
120C2×#106C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×#102C2312yes—
128C2×#102C2312yes—
129C2×#102C2314yes—
130C2×#102C2312yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×#97C2314yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#399—24no—
137#398—24no—
138#398—24no—
139#397—24no—
140#397—24no—
141#397—24no—
142#396—24no—
143#396—24no—
144#391—22no—
145#391—21no—
146#391D412yes—
147#391C2312yes—
148#391C2314yes—
149#390—22no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#390C2314yes—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389D412yes—
163#389D412yes—
164#389C2312yes—
165#389C2314yes—
166#389C2314yes—
167#386—24no—
168#385—24no—
169#385—24no—
170#385—24no—
171#384D411yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2314yes—
177#381—24no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—24no—
182#380—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C43C2311yescharacteristic
190#340C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2312yes—
195#340C2312yes—
196#340C2312yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2312yes—
201#336C2312yes—
202#336C2314yes—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C2×SD16—28no—
217C2×SD16—44no—
218C2×SD16—44no—
219C2×D8—28no—
220C2×D8—28no—
221C2×D8—44no—
222C22×C8C2414yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C4⋊Q8—24no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—42no—
229#127—24no—
230#127—22no—
231#127#4114yes—
232#125—48no—
233#124—28no—
234#124—44no—
235#124—48no—
236#123—416no—
237C22⋊Q8—48no—
238C22⋊Q8—416no—
239C22⋊Q8—84no—
240C4×Q8—44no—
241C4×Q8—44no—
242C4×Q8—44no—
243C4×D4—28no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—48no—
251C4×D4—48no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4—82no—
255C4×D4—82no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—84no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2414yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#106—44no—
309C4⋊C8—24no—
310C4⋊C8—44no—
311#103—44no—
312#103—44no—
313#103—44no—
314#102—28no—
315#102—24no—
316#102—24no—
317#102—42no—
318#102—42no—
319#102—44no—
320#102—44no—
321#97—44no—
322C4×C8—22no—
323C4×C8—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95C2×D414yes—
333#95C2×D414yes—
334C24—22no—
335C24—22no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—81no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—81no—
366C2×D4C2×#4114yes—
367C2×D4H(5,𝔽2)14yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D414yes—
388C22×C4C22×D414yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2511yescharacteristic
395SD16—88no—
396D8—416no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—84no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—84no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—84no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—84no—
449C42—22no—
450C42—24no—
451C42—24no—
452C42—22no—
453C42—22no—
454C42—41no—
455C42—41no—
456C42—44no—
457C42—44no—
458C42AGL(1,ℤ/8)14yes—
459C42C2×#4112yes—
460C422−1+412yes—
461C23—24no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23C23×D411yescharacteristic
476Q8—44no—
477Q8—44no—
478Q8—84no—
479Q8—84no—
480D4—24no—
481D4—28no—
482D4—24no—
483D4—48no—
484D4—48no—
485D4—42no—
486D4—44no—
487D4—42no—
488D4—84no—
489D4—84no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—88no—
527C2×C4—88no—
528C2×C4—168no—
529C2×C4—168no—
530C2×C4#54812yes—
531C2×C4#54812yes—
532C2×C4C2×AGL(1,ℤ/8)12yes—
533C2×C4C2×AGL(1,ℤ/8)12yes—
534C2×C4C2×AGL(1,ℤ/8)12yes—
535C2×C4#57712yes—
536C2×C4C22×#4111yescharacteristic
537C2×C4C2×H(5,𝔽2)12yes—
538C2×C4C2×2−1+411yescharacteristic
539C2×C4#58512yes—
540C8—44no—
541C8—42no—
542C8—42no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22C2×#54811yescharacteristic
558C22#348411yescharacteristic
559C22#348711yescharacteristic
560C22C22×AGL(1,ℤ/8)11yescharacteristic
561C22C2×#57711yescharacteristic
562C22#358511yescharacteristic
563C22#358611yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—44no—
568C4—41no—
569C4—44no—
570C4—41no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4#337314yes—
578C4#351312yes—
579C4#355912yes—
580C2—84no—
581C2—84no—
582C2#4527111yescharacteristic
583C2C2×#337311yescharacteristic
584C2#6133311yescharacteristic
585C2#6158511yescharacteristic
586C2#6168511yescharacteristic
587C2#6189011yescharacteristic
588C2#6191611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28156C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
2. #59823C2semidirect
3. #32561C2semidirect
4. #31654C2semidirect
5. #29255C2semidirect
7. #27756C2semidirect
8. #26180C2semidirect
9. #26110C2semidirect
10. #25367C2semidirect
11. #24204C2semidirect
12. #23815C2semidirect
13. #23001C2semidirect
14. #22311C2semidirect
15. #22268C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#598232
#325612
#316542
#292552
#281562
#277562
#261802
#261102
#253672
#242042
#238152
#230012
#223112
#222682