Finite Group Databasebeta

← #6459116

Subgroups — #6459116

Order 512.

1607 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#35309C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#27980C211yesmaximal, characteristic, complemented
4#27966C211yesmaximal, characteristic, complemented
5#27951C211yesmaximal, characteristic, complemented
6#27910C211yesmaximal, characteristic, complemented
7#27900C212yesmaximal, complemented
8#27885C212yesmaximal, complemented
9#27864C212yesmaximal, complemented
10#27858C212yesmaximal, complemented
11#24564C214yesmaximal
12#23762C212yesmaximal, complemented
13#23757C212yesmaximal, complemented
14#21585C214yesmaximal, complemented
15#21413C212yesmaximal, complemented
16#20821C212yesmaximal, complemented
17#20378C211yesmaximal, characteristic
18C2×#541C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#484C2214yes—
21C2×#408C2212yes—
22C2×#405C2212yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2851C2211yescharacteristic
26#2629C2211yescharacteristic
27#2621C2211yescharacteristic
28#2616C2212yes—
29#2468—22no—
30#2466—22no—
31#2462—22no—
32#2460—22no—
33#2355—22no—
34#2354—22no—
35#2352—22no—
36#2352—22no—
37#2323C2212yes—
38#2314C2211yescharacteristic
39#2314C2212yes—
40#2314C2212yes—
41#2311C2212yes—
42C4×C22⋊Q8C2212yes—
43#2285C2212yes—
44#2285C2212yes—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47#2274C2212yes—
48#2100C2212yes—
49#2100C2212yes—
50#2095C2212yes—
51#2095C2212yes—
52#2084C2212yes—
53#2083C2212yes—
54#2076C2214yes—
55#2075C2214yes—
56#2072C2212yes—
57#2072C2212yes—
58#2036C2214yes—
59#2035C2214yes—
60#1992C2212yes—
61#1991C2212yes—
62#1960C2212yes—
63#1956C2212yes—
64#1928C2214yes—
65#1927C2214yes—
66#1921C2214yes—
67#1918C2214yes—
68#1894C2214yes—
69#1892C2214yes—
70#1873C2214yes—
71#1854C2214yes—
72#1846C2212yes—
73#1845C2212yes—
74#1836C2212yes—
75#1836C2212yes—
76#1823C2214yes—
77#1809C2214yes—
78#1808C2214yes—
79#1776C2214yes—
80C4×#107C2214yes—
81#1763C2214yes—
82C4×#103C2214yes—
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—28no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#126—22no—
97C2×#125—22no—
98C2×#125C2311yescharacteristic
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C4×#41—28no—
103C2×C4×Q8D412yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4D412yes—
106C2×C4×D4D412yes—
107C2×C4×D4D412yes—
108C2×#117—22no—
109C2×#117—22no—
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4—22no—
115C22×C4⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#484—216no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128#408—28no—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133#405—28no—
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138#398—22no—
139#398—22no—
140#397—22no—
141#396—22no—
142#395—22no—
143#394—22no—
144#394—22no—
145#394—22no—
146#391—22no—
147#391C2312yes—
148#391C2312yes—
149#390—22no—
150#389—22no—
151#389—22no—
152#389C2312yes—
153#389C2312yes—
154#388—24no—
155#388—24no—
156#388—22no—
157#388—22no—
158#388—22no—
159#387—24no—
160#387—24no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#386—22no—
168#386—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#384—22no—
173#384C2312yes—
174#384C2311yescharacteristic
175#384C2312yes—
176#384C2312yes—
177#382—22no—
178#382—22no—
179#382C2312yes—
180#382C2311yescharacteristic
181#381—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380C2314yes—
187#380C2314yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191C4×C4⋊C4—24no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2314yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—24no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2311yescharacteristic
210#336C2311yescharacteristic
211#336C2311yescharacteristic
212#336C2311yescharacteristic
213#336C2312yes—
214#336C2312yes—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—41no—
218C2×#41—41no—
219C22×Q8C2×D411yescharacteristic
220C22×Q8C2412yes—
221C22×D4C2×D411yescharacteristic
222C22×D4C2×D411yescharacteristic
223C22×D4C2×D411yescharacteristic
224C23×C4—22no—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C2×M16—24no—
232C2×M16—24no—
233C2×M16—42no—
234C22×C8—24no—
235C22×C8—24no—
236C22×C8—42no—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C4⋊Q8—24no—
242#126—48no—
243#125—24no—
244#125—48no—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C22⋊Q8—42no—
248C22⋊Q8—42no—
249C4×Q8—28no—
250C4×Q8—28no—
251C4×D4—28no—
252C4×D4—28no—
253C4×D4—28no—
254#117—24no—
255#117—24no—
256#117—42no—
257#117—42no—
258#117—84no—
259#117—84no—
260#117#4118yes—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—84no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—28no—
314#107—28no—
315#107—44no—
316#106—44no—
317#103—28no—
318#103—28no—
319#103—44no—
320#103—44no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324C22⋊C8#4118yes—
325#97—24no—
326#97—24no—
327C4×C8—24no—
328C4×C8—24no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345C24C22×D411yescharacteristic
346C24C22×D411yescharacteristic
347C24C22×D411yescharacteristic
348#41—48no—
349#41—48no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—41no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—84no—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C2×#4114yes—
401C22×C4C2×#4114yes—
402C22×C4C2511yescharacteristic
403M16—48no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—48no—
415C2×C8C2×#4114yes—
416C2×C8C2×#4114yes—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4#12318yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—82no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4#12318yes—
455C42—24no—
456C42—24no—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—44no—
461C42—44no—
462C42—82no—
463C42—84no—
464C42—84no—
465C42C2×#4114yes—
466C42C2×#4114yes—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23C2×#12312yes—
481C23#53212yes—
482C23C23×D411yescharacteristic
483Q8—44no—
484Q8—44no—
485Q8—44no—
486D4—44no—
487D4—44no—
488D4—44no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—164no—
533C2×C4C2×#12312yes—
534C2×C4C2×#12314yes—
535C2×C4C2×#12314yes—
536C2×C4#53212yes—
537C2×C4#57814yes—
538C2×C4C22×#4112yes—
539C2×C4C22×#4112yes—
540C2×C4#58511yescharacteristic
541C2×C4#58511yescharacteristic
542C8—24no—
543C8—24no—
544C8—42no—
545C8—42no—
546C22—22no—
547C22—22no—
548C22—24no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—41no—
561C22C22×#12311yescharacteristic
562C22C2×#53211yescharacteristic
563C22#344911yescharacteristic
564C22C2×#57812yes—
565C22#358512yes—
566C4—24no—
567C4—24no—
568C4—24no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—41no—
573C4—82no—
574C4—84no—
575C4—84no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4#347512yes—
580C4#347512yes—
581C2—42no—
582C2—42no—
583C2—81no—
584C2#3481011yescharacteristic
585C2#6052612yes—
586C2#6071511yescharacteristic
587C2#6072511yescharacteristic
588C2#6099312yes—
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. #27980C2semidirect
4. #27966C2semidirect
5. #27951C2semidirect
6. #27910C2semidirect
7. #27900C2semidirect
8. #27885C2semidirect
9. #27864C2semidirect
10. #27858C2semidirect
12. #23762C2semidirect
13. #23757C2semidirect
14. #21585C2semidirect
15. #21413C2semidirect
16. #20821C2semidirect

Maximal subgroups

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

subgroupindex
#353092
#341882
#279802
#279662
#279512
#279102
#279002
#278852
#278642
#278582
#245642
#237622
#237572
#215852
#214132
#208212
#203782