Finite Group Databasebeta

← #7196686

Subgroups — #7196686

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3356C212yesmaximal, complemented
2#59823C211yesmaximal, characteristic
3#59821C211yesmaximal, characteristic
4#30726C214yesmaximal, complemented
5#30721C214yesmaximal, complemented
6#28206C211yesmaximal, characteristic, complemented
7#28205C211yesmaximal, characteristic, complemented
8#28151C212yesmaximal, complemented
9#26151C214yesmaximal, complemented
10#26144C214yesmaximal, complemented
11#24123C212yesmaximal, complemented
12#24122C212yesmaximal, complemented
13#23801C211yesmaximal, characteristic, complemented
14#22264C211yesmaximal, characteristic
15#22246C211yesmaximal, characteristic, complemented
16C2×#556C2212yes—
17C2×#554C2212yes—
18#3356—216no—
19C2×#498C2211yescharacteristic
20C2×#485C2214yes—
21C2×#483C2214yes—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×#470C2214yes—
25C2×C4×Q16C2212yes—
26C2×C4×Q16C2212yes—
27C2×#436C2212yes—
28C2×#428C2214yes—
29#2732—22no—
30#2727—22no—
31#2552—21no—
32#2548—21no—
33#2542C2212yes—
34#2542C2212yes—
35#2538—22no—
36#2538C2212yes—
37#2538C2212yes—
38#2512—22no—
39#2454—22no—
40#2448—22no—
41#2417C2214yes—
42#2415C2214yes—
43#2366—21no—
44#2354C2212yes—
45Q8×C4⋊C4—21no—
46D4×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52#2095C2212yes—
53#2095C2212yes—
54#2084C2212yes—
55#2083C2212yes—
56#2075C2214yes—
57#2043C2212yes—
58#2043C2212yes—
59#2036C2214yes—
60#2035C2214yes—
61#2016C2212yes—
62#2016C2212yes—
63#2001C2214yes—
64#2001C2214yes—
65#2001C2214yes—
66#1990C2211yescharacteristic
67#1990C2211yescharacteristic
68#1954C2212yes—
69#1928C2214yes—
70#1927C2214yes—
71#1920C2214yes—
72#1919C2214yes—
73#1894C2214yes—
74#1894C2214yes—
75#1873C2214yes—
76#1872C2214yes—
77#1870C2214yes—
78#1870C2214yes—
79#1854C2214yes—
80#1852C2214yes—
81#1834C2211yescharacteristic
82#1831C2211yescharacteristic
83#1803C2212yes—
84#1803C2212yes—
85#1744C2211yescharacteristic
86C22×Q16C2312yes—
87C22×Q16C2312yes—
88C22×M16C2314yes—
89#556—28no—
90#556—44no—
91#554—28no—
92#554—44no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127D411yescharacteristic
98C2×#126—24no—
99C2×#125—22no—
100C2×#124—22no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117C2314yes—
117C22×C4⋊C4C2314yes—
118#498D418yescomplemented
119#485—216no—
120#483—216no—
121#477—28no—
122#475—28no—
123#470—216no—
124C4×Q16—28no—
125C4×Q16—44no—
126#436—28no—
127#428—216no—
128C2×#107C2314yes—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#399—24no—
148#399—24no—
149#398—24no—
150#397—24no—
151#396—24no—
152#396—24no—
153#396—24no—
154#394—24no—
155#391—21no—
156#391—21no—
157#391C2312yes—
158#391C2312yes—
159#390—22no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390D412yes—
165#390D412yes—
166#390D412yes—
167#389—22no—
168#389—24no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#388—24no—
180#388—24no—
181#386—24no—
182#385—24no—
183#384—22no—
184#384—22no—
185#384D411yescharacteristic
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#380—24no—
191#380C2314yes—
192#380C2314yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#336C2312yes—
209#336C2312yes—
210#336C2312yes—
211#336C2312yes—
212C22×Q8—21no—
213C22×Q8—21no—
214C22×Q8C2412yes—
215C22×Q8C2412yes—
216C22×D4C2×D412yes—
217C22×D4C2×D412yes—
218C22×D4C2×D412yes—
219C23×C4—24no—
220C23×C4C2414yes—
221C2×Q16—24no—
222C2×Q16—42no—
223C2×Q16—42no—
224C2×Q16—42no—
225C2×Q16—42no—
226C2×Q16—42no—
227C2×M16—28no—
228C2×M16—44no—
229C2×M16—44no—
230C22×C8C2414yes—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—42no—
236C4⋊Q8—42no—
237C4⋊Q8C2×D414yes—
238C4⋊Q8C2×D414yes—
239#127—24no—
240#126—416no—
241#125—48no—
242#124—48no—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C22⋊Q8—416no—
246C22⋊Q8—44no—
247C22⋊Q8—44no—
248C22⋊Q8—44no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—48no—
257C4×Q8—44no—
258C4×Q8—82no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—44no—
262C4×D4—44no—
263C4×D4—48no—
264C4×D4—84no—
265#117—28no—
266#117—44no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D414yes—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—24no—
324#107—28no—
325#107—42no—
326#106—28no—
327#106C2×D4116yes—
328C4⋊C8—24no—
329#103—24no—
330#103—24no—
331#103—28no—
332#103—28no—
333#103—44no—
334#103—44no—
335#103—42no—
336#103—42no—
337C22⋊C8—28no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340C4×C8—22no—
341C4×C8C2×D414yes—
342#95—22no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—24no—
352C24C22×D414yes—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—81no—
370C2×Q8—81no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—48no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D414yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2511yescharacteristic
405Q16—48no—
406Q16—48no—
407M16—416no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8C22×D414yes—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—82no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4C22×D414yes—
452C4⋊C4C22×D414yes—
453C22⋊C4—28no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—48no—
469C42—44no—
470C42—44no—
471C42—44no—
472C42—44no—
473C42#13714yes—
474C42C22×D412yes—
475C42H(5,𝔽2)12yes—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23C23×D411yescharacteristic
484Q8—48no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—84no—
494D4—48no—
495D4—48no—
496D4—84no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—168no—
541C2×C4D4212yes—
542C2×C4D4212yes—
543C2×C4C2×#13712yes—
544C2×C4C2×#13712yes—
545C2×C4C2×#13712yes—
546C2×C4#57812yes—
547C2×C4C23×D411yescharacteristic
548C2×C4C2×H(5,𝔽2)11yescharacteristic
549C2×C4#58512yes—
550C2×C4#58512yes—
551C8—24no—
552C8—44no—
553C8—42no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22C2×D4211yescharacteristic
562C22#348511yescharacteristic
563C22#348511yescharacteristic
564C22C22×#13711yescharacteristic
565C22C2×#57811yescharacteristic
566C22#358511yescharacteristic
567C22#358611yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—44no—
572C4—44no—
573C4—41no—
574C4—41no—
575C4—44no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#327614yes—
583C4#348712yes—
584C4#353912yes—
585C2—84no—
586C2#4538711yescharacteristic
587C2C2×#327611yescharacteristic
588C2#6108011yescharacteristic
589C2#6171911yescharacteristic
590C2#6172311yescharacteristic
591C2#6180611yescharacteristic
592C2#6181011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3356C2semidirect
4. #30726C2semidirect
5. #30721C2semidirect
6. #28206C2semidirect
7. #28205C2semidirect
8. #28151C2semidirect
9. #26151C2semidirect
10. #26144C2semidirect
11. #24123C2semidirect
12. #24122C2semidirect
13. #23801C2semidirect
15. #22246C2semidirect
118. #498D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33562
#598232
#598212
#307262
#307212
#282062
#282052
#281512
#261512
#261442
#241232
#241222
#238012
#222642
#222462