Finite Group Databasebeta

← #6532007

Subgroups — #6532007

Order 512.

1526 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
1C2×#2582C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#27617C212yesmaximal
4#27614C211yesmaximal, characteristic
5#27611C211yesmaximal, characteristic
6#27589C212yesmaximal
7#27586C211yesmaximal, characteristic
8#27583C211yesmaximal, characteristic
9#25075C214yesmaximal, complemented
10#25074C214yesmaximal, complemented
11#25009C214yesmaximal, complemented
12#25008C214yesmaximal, complemented
13#21987C212yesmaximal
14#21822C211yesmaximal, characteristic
15#20741C212yesmaximal
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#432C2212yes—
19#2582—22no—
20#2582—22no—
21#2582C2214yes—
22#2582C2214yes—
23C2×#398—21no—
24C2×#398—21no—
25#2356—22no—
26#2356—22no—
27#2355C2212yes—
28#2355C2212yes—
29#2355C2212yes—
30#2355C2212yes—
31#2313—21no—
32#2313—21no—
33#2310—21no—
34C2×#391—21no—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#389—21no—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2283C2212yes—
44#2283C2212yes—
45#2274—21no—
46#2101C2214yes—
47#2100C2212yes—
48#2099C2212yes—
49#2097C2214yes—
50#2096C2212yes—
51#2095C2212yes—
52#2085C2214yes—
53#2084C2212yes—
54#2083C2212yes—
55#2074C2214yes—
56#2072C2212yes—
57#2072C2212yes—
58#1985C2214yes—
59#1985C2214yes—
60#1948C2214yes—
61#1947C2214yes—
62#1928C2214yes—
63#1928C2214yes—
64#1927C2214yes—
65#1927C2214yes—
66#1921C2214yes—
67#1920C2214yes—
68#1919C2214yes—
69#1918C2214yes—
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1847C2214yes—
75#1836C2214yes—
76#1824C2212yes—
77#1824C2212yes—
78#1817C2211yescharacteristic
79#1817C2211yescharacteristic
80#1817C2212yes—
81#1805C2214yes—
82#1805C2214yes—
83#1787C2212yes—
84#1787C2212yes—
85#1763C2214yes—
86#1763C2214yes—
87C23×Q8D411yescharacteristic
88C23×Q8C2311yescharacteristic
89C2×C4×Q8—22no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8D412yes—
93C2×C4×Q8D412yes—
94C2×C4×Q8C2314yes—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×#117—21no—
98C2×#117—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#432—28no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×#97C2314yes—
127#398—24no—
128#398—24no—
129#398—24no—
130#398—24no—
131#391—22no—
132#391—22no—
133#391—22no—
134#391—22no—
135#391—24no—
136#391—24no—
137#391—22no—
138#391—21no—
139#391—21no—
140#391C2312yes—
141#391C2312yes—
142#391C2314yes—
143#389—24no—
144#389—24no—
145#389—22no—
146#389—21no—
147#389—21no—
148#389C2312yes—
149#389C2312yes—
150#389C2314yes—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#384—22no—
160#384C2314yes—
161#384C2311yescharacteristic
162#384C2311yescharacteristic
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#383—22no—
166#383—22no—
167#382—21no—
168#382—21no—
169#382C2312yes—
170#382C2312yes—
171#382C2312yes—
172#382C2312yes—
173#381—22no—
174#381—22no—
175#380—21no—
176#380—21no—
177#380C2312yes—
178#380C2312yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193#376—22no—
194#376—22no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C2×D412yes—
223C22×Q8C2×D412yes—
224C22×Q8C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4×Q8—28no—
239C4×Q8—28no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—28no—
245C4×Q8—28no—
246C4×Q8—48no—
247#117—44no—
248#117—44no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—44no—
302C2×C42C2×D412yes—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D412yes—
306C2×C42#4118yes—
307C2×C42C2412yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2412yes—
311#107—44no—
312#107—44no—
313#106—44no—
314#106—44no—
315C4⋊C8—28no—
316#103—28no—
317#103—28no—
318#103—44no—
319#103—44no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322#97—28no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348C24C22×D411yescharacteristic
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q82−1+414yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4#12314yes—
402C22×C4#12314yes—
403C22×C4#13714yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4114yes—
414C22×C4C2511yescharacteristic
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—48no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C42—22no—
455C42—22no—
456C42—24no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—44no—
462C42—48no—
463C42—44no—
464C42—82no—
465C42—82no—
466C42—82no—
467C42—82no—
468C42#12318yes—
469C42C2×#4114yes—
470C42C2×#4114yes—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23C2×#12312yes—
476C23C2×#13712yes—
477C23#57512yes—
478C23C23×D411yescharacteristic
479Q8—28no—
480Q8—48no—
481Q8—48no—
482Q8—48no—
483Q8—44no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—44no—
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—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—82no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4C2×#12312yes—
542C2×C4C2×#12314yes—
543C2×C4C2×#12314yes—
544C2×C4C2×#13712yes—
545C2×C4C2×#13712yes—
546C2×C4C22×#4112yes—
547C2×C4C2×2−1+412yes—
548C2×C4#58512yes—
549C8—44no—
550C8—44no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22#309514yes—
558C22C22×#12311yescharacteristic
559C22#348611yescharacteristic
560C22#348711yescharacteristic
561C22C22×#13711yescharacteristic
562C22C2×#57511yescharacteristic
563C22#358511yescharacteristic
564C22#358611yescharacteristic
565C4—22no—
566C4—24no—
567C4—22no—
568C4—22no—
569C4—24no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#349514yes—
583C2—24no—
584C2#3567211yescharacteristic
585C2C2×#309511yescharacteristic
586C2#6052011yescharacteristic
587C2#6183011yescharacteristic
588C2#6183211yescharacteristic
589C2#6189311yescharacteristic
590C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #25075C2semidirect
10. #25074C2semidirect
11. #25009C2semidirect
12. #25008C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
#341872
#276172
#276142
#276112
#275892
#275862
#275832
#250752
#250742
#250092
#250082
#219872
#218222
#207412