Finite Group Databasebeta

← #6475428

Subgroups — #6475428

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35333C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#33218C212yesmaximal, complemented
4#33217C212yesmaximal, complemented
5#30303C212yesmaximal, complemented
6#30301C212yesmaximal, complemented
7#29309C214yesmaximal, complemented
8#29307C212yesmaximal, complemented
9#29306C212yesmaximal, complemented
10#27955C212yesmaximal, complemented
11#27953C212yesmaximal, complemented
12#25819C212yesmaximal, complemented
13#25817C212yesmaximal, complemented
14#21945C214yesmaximal
15#20375C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#501C2212yes—
20C2×#501C2212yes—
21C2×#408C2212yes—
22C2×C4×M16C2211yescharacteristic
23C22×#97C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2851C2212yes—
27#2851C2212yes—
28#2629C2211yescharacteristic
29#2629C2211yescharacteristic
30#2629C2211yescharacteristic
31#2629C2211yescharacteristic
32#2629C2212yes—
33#2629C2212yes—
34#2582—22no—
35#2579—22no—
36#2579—22no—
37#2566—22no—
38#2565—22no—
39#2538—22no—
40#2417—22no—
41#2417—22no—
42#2317C2212yes—
43#2316C2212yes—
44#2311C2212yes—
45#2311C2212yes—
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2084C2214yes—
49#2084C2214yes—
50#2083C2214yes—
51#2083C2214yes—
52#2059C2214yes—
53#2059C2214yes—
54#2059C2214yes—
55#2059C2214yes—
56#1993C2212yes—
57#1993C2212yes—
58#1992C2212yes—
59#1992C2212yes—
60#1944C2214yes—
61#1943C2214yes—
62#1942C2214yes—
63#1941C2214yes—
64#1854C2214yes—
65#1854C2214yes—
66#1854C2214yes—
67#1854C2214yes—
68#1853C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1852C2214yes—
72#1847C2212yes—
73#1847C2212yes—
74#1809C2214yes—
75#1809C2214yes—
76#1808C2214yes—
77#1807C2214yes—
78#1750C2212yes—
79#1750C2212yes—
80#1749C2212yes—
81#1749C2212yes—
82#1729C2212yes—
83C22×#41C2311yescharacteristic
84C22×M16C2312yes—
85C23×C8C2312yes—
86#533—42no—
87#533—42no—
88C2×C4⋊Q8C2312yes—
89C2×#125—22no—
90C2×#125—42no—
91C2×#125—42no—
92C2×#125C2312yes—
93C2×#125C2311yescharacteristic
94C2×#125C2311yescharacteristic
95C2×#125C2312yes—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C4×#41—42no—
100C4×#41—42no—
101C2×C4×Q8—22no—
102C2×C4×Q8D412yes—
103C2×C4×D4D412yes—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#501—44no—
119#501—44no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128#408Q8116yescomplemented
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131C4×M16—42no—
132C2×#97—42no—
133C2×#97C2312yes—
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#398—24no—
142#398—24no—
143#396—22no—
144#396—22no—
145#396—22no—
146#396—22no—
147#396—24no—
148#396—24no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#390—22no—
154#390—22no—
155#389—24no—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2312yes—
160#389C2312yes—
161#389C2314yes—
162#389C2312yes—
163#389C2312yes—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—22no—
179#384—22no—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2311yescharacteristic
183#384C2312yes—
184#382—22no—
185#382C2312yes—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—24no—
190#380C2312yes—
191#380C2314yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#380C2314yes—
196C4×C4⋊C4—22no—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207C2×#41—41no—
208C2×#41—41no—
209C2×#41—42no—
210C2×#41C2×Q818yes—
211C22×Q8—22no—
212C22×Q8C2×D411yescharacteristic
213C22×D4C2×D411yescharacteristic
214C22×D4C2×D411yescharacteristic
215C22×D4C2×D411yescharacteristic
216C23×C4—22no—
217C23×C4—22no—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C2×M16—42no—
225C2×M16—42no—
226C2×M16C2×Q818yes—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8C2×Q818yes—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234#125—24no—
235#125—24no—
236#125—41no—
237#125—41no—
238#125—41no—
239#125—41no—
240#125—42no—
241#125—42no—
242#125—48no—
243C22⋊Q8—48no—
244C22⋊Q8—84no—
245C22⋊Q8—84no—
246C4×Q8—44no—
247C4×Q8—48no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—44no—
251#117—41no—
252#117—41no—
253#117—41no—
254#117—41no—
255#117—41no—
256#117—41no—
257#117—44no—
258#117—42no—
259#117—42no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
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—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—84no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#107—44no—
320#107—44no—
321#107—44no—
322#107—44no—
323#106—44no—
324#106—44no—
325#106—44no—
326#106—44no—
327C22⋊C8C2×Q8116yes—
328C22⋊C8C2×Q8116yes—
329#97—24no—
330#97—24no—
331#97—42no—
332C4×C8—42no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347C24C22×D411yescharacteristic
348C24C22×D411yescharacteristic
349C24C22×D411yescharacteristic
350#41—48no—
351#41—48no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8C4⋊Q814yes—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4C4⋊Q814yes—
365C2×D4C4⋊Q814yes—
366C2×D4C4⋊Q814yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—84no—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×Q814yes—
404C22×C4C22×Q814yes—
405C22×C4C22×Q814yes—
406C22×C4C2511yescharacteristic
407M16—48no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—48no—
413C2×C8C22×Q818yes—
414C2×C8C22×Q818yes—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—44no—
468C42—44no—
469C42H(5,𝔽2)12yes—
470C42H(5,𝔽2)12yes—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23C2×C4⋊Q812yes—
484C23C2×C4⋊Q812yes—
485C23C2×C4⋊Q812yes—
486C23C23×D411yescharacteristic
487Q8—44no—
488Q8—88no—
489D4—44no—
490D4—44no—
491D4—44no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—48no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—88no—
537C2×C4—164no—
538C2×C4C2×C4⋊Q812yes—
539C2×C4C2×C4⋊Q812yes—
540C2×C4C2×C4⋊Q812yes—
541C2×C4#53414yes—
542C2×C4C23×Q812yes—
543C2×C4C2×H(5,𝔽2)11yescharacteristic
544C2×C4C2×H(5,𝔽2)11yescharacteristic
545C2×C4#58512yes—
546C8—28no—
547C8—28no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×C4⋊Q811yescharacteristic
564C22C2×#53411yescharacteristic
565C22#344911yescharacteristic
566C22#358512yes—
567C22#358612yes—
568C4—24no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—82no—
577C4—84no—
578C4—82no—
579C4—84no—
580C4—82no—
581C4#350112yes—
582C4#350112yes—
583C2—42no—
584C2—42no—
585C2—42no—
586C2#3501111yescharacteristic
587C2#6066812yes—
588C2#6075512yes—
589C2#6101911yescharacteristic
590C2#6102011yescharacteristic
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. #33218C2semidirect
4. #33217C2semidirect
5. #30303C2semidirect
6. #30301C2semidirect
7. #29309C2semidirect
8. #29307C2semidirect
9. #29306C2semidirect
10. #27955C2semidirect
11. #27953C2semidirect
12. #25819C2semidirect
13. #25817C2semidirect
128. #408Q8semidirect

Maximal subgroups

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

subgroupindex
#353332
#352802
#332182
#332172
#303032
#303012
#293092
#293072
#293062
#279552
#279532
#258192
#258172
#219452
#203752