Finite Group Databasebeta

← #6477216

Subgroups — #6477216

Order 512.

1591 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35309C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#33214C212yesmaximal, complemented
4#33056C212yesmaximal, complemented
5#30366C212yesmaximal, complemented
6#30359C212yesmaximal, complemented
7#29165C212yesmaximal, complemented
8#29159C212yesmaximal, complemented
9#27965C211yesmaximal, characteristic, complemented
10#27905C211yesmaximal, characteristic, complemented
11#27884C212yesmaximal, complemented
12#25910C211yesmaximal, characteristic, complemented
13#25814C211yesmaximal, characteristic, complemented
14#25683C212yesmaximal, complemented
15#24566C214yesmaximal
16#22089C214yesmaximal, complemented
17#20378C211yesmaximal, characteristic
18C2×#541C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#499C2212yes—
22C2×#491C2212yes—
23C2×#483C2214yes—
24C2×#408C2212yes—
25C2×#405C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2855C2212yes—
30#2851C2211yescharacteristic
31#2629C2212yes—
32#2629C2212yes—
33#2629C2211yescharacteristic
34#2621C2211yescharacteristic
35#2616C2212yes—
36#2582—22no—
37#2538—22no—
38#2468—22no—
39#2466—22no—
40#2462—22no—
41#2460—22no—
42#2417—22no—
43#2417—22no—
44#2323C2212yes—
45#2316C2212yes—
46#2314C2211yescharacteristic
47#2311C2212yes—
48C4×C22⋊Q8C2212yes—
49C2×#384C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2096C2212yes—
53#2096C2212yes—
54#2077C2214yes—
55#2077C2214yes—
56#2058C2212yes—
57#2057C2212yes—
58#2036C2214yes—
59#2035C2214yes—
60#2024C2212yes—
61#2024C2212yes—
62#1992C2212yes—
63#1992C2212yes—
64#1991C2212yes—
65#1956C2212yes—
66#1944C2214yes—
67#1941C2214yes—
68#1935C2214yes—
69#1933C2214yes—
70#1920C2214yes—
71#1919C2214yes—
72#1894C2214yes—
73#1892C2214yes—
74#1873C2214yes—
75#1865C2214yes—
76#1863C2214yes—
77#1854C2214yes—
78#1844C2212yes—
79#1836C2212yes—
80#1824C2214yes—
81#1809C2214yes—
82#1807C2214yes—
83#1750C2212yes—
84#1750C2212yes—
85#1749C2211yescharacteristic
86#1749C2211yescharacteristic
87C4×C22⋊C8C2211yescharacteristic
88C4×C22⋊C8C2211yescharacteristic
89#1728C2211yescharacteristic
90C2×#336C2211yescharacteristic
91C22×#41C2311yescharacteristic
92C22×M16C2312yes—
93C23×C8C2312yes—
94#541—44no—
95#533—44no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2312yes—
98C2×#126—22no—
99C2×#125—22no—
100C2×#125C2311yescharacteristic
101C2×#125C2312yes—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C4×#41—44no—
109C2×C4×Q8—22no—
110C2×C4×Q8D412yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4D412yes—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#499—44no—
125#491—44no—
126#483—216no—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135#408—28no—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140#405—44no—
141C2×#97C2312yes—
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#398—24no—
146#398—24no—
147#398—22no—
148#398—22no—
149#397—22no—
150#396—24no—
151#396—24no—
152#396—22no—
153#395—22no—
154#394—22no—
155#394—22no—
156#394—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391C2312yes—
161#391C2312yes—
162#390—22no—
163#389—24no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2314yes—
170#389C2314yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#386—22no—
182#386—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#384—22no—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#382C2311yescharacteristic
191#381—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—24no—
196#380C2314yes—
197#380C2314yes—
198#380C2312yes—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2311yescharacteristic
213#336C2311yescharacteristic
214#336C2311yescharacteristic
215#336C2311yescharacteristic
216#336C2312yes—
217#336C2312yes—
218C2×#41—24no—
219C2×#41—22no—
220C2×#41—22no—
221C2×#41—42no—
222C22×Q8—22no—
223C22×Q8C2×D411yescharacteristic
224C22×Q8C2412yes—
225C22×D4C2×D411yescharacteristic
226C22×D4C2×D411yescharacteristic
227C22×D4C2×D411yescharacteristic
228C23×C4—22no—
229C23×C4C2×D412yes—
230C23×C4C2×D412yes—
231C23×C4C2×D412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×M16—24no—
236C2×M16—24no—
237C2×M16—42no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8—42no—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C4⋊Q8—24no—
246C4⋊Q8—44no—
247#126—48no—
248#125—24no—
249#125—24no—
250#125—24no—
251#125—48no—
252C22⋊Q8—28no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—44no—
256C22⋊Q8—48no—
257C22⋊Q8—84no—
258C22⋊Q8—84no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—48no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—44no—
265#117—24no—
266#117—24no—
267#117—42no—
268#117—44no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—84no—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#107—44no—
322#106—28no—
323#106—28no—
324#106—44no—
325#103—28no—
326#103—28no—
327#103—44no—
328#103—44no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8#4118yes—
333#97—24no—
334#97—42no—
335C4×C8—24no—
336C4×C8—42no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352C24C22×D411yescharacteristic
353C24C22×D411yescharacteristic
354C24C22×D411yescharacteristic
355#41—48no—
356#41—48no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—41no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—84no—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C2×#4114yes—
410C22×C42−1+412yes—
411C22×C42−1+412yes—
412C22×C4C2511yescharacteristic
413M16—48no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—48no—
425C2×C8C2×#4114yes—
426C2×C8C2×#4114yes—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—42no—
472C42—42no—
473C42—44no—
474C42—44no—
475C42—82no—
476C42H(5,𝔽2)14yes—
477C23—22no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23#53212yes—
491C23#53612yes—
492C23C23×D411yescharacteristic
493Q8—44no—
494Q8—44no—
495Q8—44no—
496Q8—88no—
497D4—44no—
498D4—44no—
499D4—44no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—164no—
545C2×C4#53212yes—
546C2×C4#53414yes—
547C2×C4#53612yes—
548C2×C4#57714yes—
549C2×C4C22×#4112yes—
550C2×C4C2×H(5,𝔽2)12yes—
551C2×C4C2×2−1+411yescharacteristic
552C2×C4C2×2−1+411yescharacteristic
553C8—24no—
554C8—24no—
555C8—42no—
556C8—42no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—24no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—42no—
571C22—42no—
572C22C2×#53211yescharacteristic
573C22C2×#53411yescharacteristic
574C22C2×#53611yescharacteristic
575C22C2×#57712yes—
576C22#358612yes—
577C4—22no—
578C4—22no—
579C4—24no—
580C4—24no—
581C4—41no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—84no—
586C4—82no—
587C4—82no—
588C4—84no—
589C4—82no—
590C4#352012yes—
591C4#352012yes—
592C2—42no—
593C2—42no—
594C2—81no—
595C2#3508511yescharacteristic
596C2#6071211yescharacteristic
597C2#6072611yescharacteristic
598C2#6076312yes—
599C2#6088912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33214C2semidirect
4. #33056C2semidirect
5. #30366C2semidirect
6. #30359C2semidirect
7. #29165C2semidirect
8. #29159C2semidirect
9. #27965C2semidirect
10. #27905C2semidirect
11. #27884C2semidirect
12. #25910C2semidirect
13. #25814C2semidirect
14. #25683C2semidirect
16. #22089C2semidirect

Maximal subgroups

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

subgroupindex
#353092
#352802
#332142
#330562
#303662
#303592
#291652
#291592
#279652
#279052
#278842
#259102
#258142
#256832
#245662
#220892
#203782