Finite Group Databasebeta

← #6469062

Subgroups — #6469062

Order 512.

1898 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34209C211yesmaximal, characteristic, complemented
2C4×#540C211yesmaximal, characteristic
3#29378C214yesmaximal, complemented
4#29257C212yesmaximal, complemented
5#28157C212yesmaximal, complemented
6#27679C212yesmaximal, complemented
7#27592C212yesmaximal, complemented
8#24209C212yesmaximal, complemented
9#23940C212yesmaximal
10#23831C212yesmaximal, complemented
11#23811C212yesmaximal, complemented
12#21504C212yesmaximal, complemented
13C4×#461C212yesmaximal, complemented
14#21462C214yesmaximal, complemented
15#20786C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#519C2212yes—
19C2×#461C2212yes—
20C2×#461C2212yes—
21C2×#448C2214yes—
22C2×#440C2214yes—
23C2×C4×SD16C2214yes—
24#2365—22no—
25#2365—22no—
26#2344—22no—
27#2344—22no—
28#2335—22no—
29#2333—22no—
30#2330C2211yescharacteristic
31#2328C2211yescharacteristic
32#2327C2212yes—
33#2327C2212yes—
34#2324C2212yes—
35#2319C2212yes—
36#2314C2211yescharacteristic
37#2310C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×#127C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×#123—22no—
43C4×#121—22no—
44C4×#121C2212yes—
45#2285C2211yescharacteristic
46C42×D4C2212yes—
47C4×#117C2211yescharacteristic
48#2101C2214yes—
49#2097C2214yes—
50#2092C2214yes—
51#2072C2214yes—
52#2043C2214yes—
53#2026C2214yes—
54#2014C2214yes—
55#2013C2214yes—
56#1981C2212yes—
57#1981C2212yes—
58#1948C2212yes—
59#1947C2212yes—
60#1928C2214yes—
61#1927C2214yes—
62#1880C2214yes—
63#1880C2214yes—
64#1877C2214yes—
65#1877C2214yes—
66#1875C2214yes—
67#1865C2214yes—
68#1863C2214yes—
69#1854C2214yes—
70#1840C2212yes—
71#1831C2212yes—
72#1818C2212yes—
73#1802C2212yes—
74#1802C2212yes—
75#1797C2212yes—
76#1787C2214yes—
77#1767C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79#1763C2214yes—
80C4×#103C2214yes—
81#1750C2212yes—
82#1746C2212yes—
83C23×Q8C2312yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86#540—28no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#127C2311yescharacteristic
93C2×#125C2311yescharacteristic
94C2×#123—22no—
95C2×#121—22no—
96C2×#121C2312yes—
97#519—44no—
98#519—44no—
99C2×C4×Q8—24no—
100C2×C4×Q8C2314yes—
101C2×C4×Q8C2314yes—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—24no—
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C42—22no—
114C22×C42C2311yescharacteristic
115#461—28no—
116#461—28no—
117#448—216no—
118#440—48no—
119C4×SD16—48no—
120C2×#107C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2312yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2312yes—
134C2×C4×C8C2312yes—
135#391—24no—
136#391C2314yes—
137#391C2314yes—
138#390—22no—
139#390—22no—
140#390C2312yes—
141#390C2312yes—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389C2312yes—
148#389C2312yes—
149#388—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#387—22no—
155#387—22no—
156#387—24no—
157#387—24no—
158#387—24no—
159#387—24no—
160#387—22no—
161#386—22no—
162#386—22no—
163#386—22no—
164#386—22no—
165#385—22no—
166#384C2311yescharacteristic
167#384C2312yes—
168#382C2311yescharacteristic
169#382C2312yes—
170#381—24no—
171#381—22no—
172#380—21no—
173#380—21no—
174#380—22no—
175#380C2312yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4D412yes—
194#376C2311yescharacteristic
195#376C2311yescharacteristic
196C43C2311yescharacteristic
197C43C2311yescharacteristic
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2314yes—
203#336C2314yes—
204#336C2312yes—
205#336C2312yes—
206#336C2314yes—
207C22×Q8—28no—
208C22×Q8—22no—
209C22×Q8—28no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8C2414yes—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4C2412yes—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2411yescharacteristic
223C2×SD16—28no—
224C2×SD16—28no—
225C2×SD16—28no—
226C2×SD16—28no—
227C2×SD16—44no—
228C2×SD16—44no—
229C22×C8C2414yes—
230C22×C8C2414yes—
231C4⋊Q8—28no—
232C4⋊Q8—28no—
233C4⋊Q8—28no—
234C4⋊Q8—24no—
235C4⋊Q8#4118yes—
236#127—24no—
237#125—24no—
238#123—216no—
239#121—216no—
240#121—24no—
241#121#4118yes—
242C4×Q8—28no—
243C4×Q8—28no—
244C4×Q8—44no—
245C4×Q8—44no—
246C4×Q8—416no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—84no—
254C4×D4—84no—
255#117—24no—
256#117—22no—
257#117—22no—
258#117—416no—
259#117—42no—
260#117—42no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
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—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D412yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—84no—
301C2×C42C2×D412yes—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—44no—
314#106—44no—
315C4⋊C8—24no—
316C4⋊C8—24no—
317#103—28no—
318#103—28no—
319#103—28no—
320#103—28no—
321#103—44no—
322#103—44no—
323#102—44no—
324#102—44no—
325C22⋊C8—28no—
326#97—24no—
327C4×C8—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—22no—
341#95C2×D412yes—
342#95C2×D412yes—
343#95C2×D412yes—
344#95C2×D412yes—
345C24—22no—
346C24C22×D411yescharacteristic
347C2×Q8—28no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—416no—
354C2×Q8—416no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8C2×#4118yes—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4C2×#4114yes—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4114yes—
404C22×C4C2511yescharacteristic
405SD16—416no—
406SD16—416no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4C2×#4114yes—
435C22⋊C4—28no—
436C22⋊C4—24no—
437C22⋊C4—28no—
438C22⋊C4—24no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—21no—
455C42—21no—
456C42—42no—
457C42—42no—
458C42—44no—
459C42—48no—
460C42—48no—
461C42—44no—
462C42—82no—
463C42—82no—
464C42C2×#4114yes—
465C42H(5,𝔽2)12yes—
466C42H(5,𝔽2)12yes—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23#52512yes—
479C23C23×D411yescharacteristic
480Q8—28no—
481Q8—28no—
482Q8—416no—
483Q8—44no—
484Q8—44no—
485D4—24no—
486D4—24no—
487D4—42no—
488D4—42no—
489D4—88no—
490D4—88no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—44no—
514C2×C4—416no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—88no—
530C2×C4—88no—
531C2×C4—88no—
532C2×C4#52512yes—
533C2×C4#52512yes—
534C2×C4#52512yes—
535C2×C4#53514yes—
536C2×C4#57714yes—
537C2×C4C22×#4112yes—
538C2×C4C22×#4112yes—
539C2×C4C2×H(5,𝔽2)11yescharacteristic
540C2×C4C2×H(5,𝔽2)11yescharacteristic
541C8—44no—
542C8—44no—
543C22—22no—
544C22—24no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22C2×#52511yescharacteristic
556C22C2×#52511yescharacteristic
557C22C2×#53511yescharacteristic
558C22C2×#57712yes—
559C22#358612yes—
560C4—22no—
561C4—22no—
562C4—24no—
563C4—41no—
564C4—48no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—44no—
570C4—82no—
571C4—84no—
572C4—82no—
573C4#312614yes—
574C4#348112yes—
575C4#348112yes—
576C2—42no—
577C2—84no—
578C2#3496311yescharacteristic
579C2C2×#312611yescharacteristic
580C2#6056611yescharacteristic
581C2#6059112yes—
582C2#6083612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34209C2semidirect
3. #29378C2semidirect
4. #29257C2semidirect
5. #28157C2semidirect
6. #27679C2semidirect
7. #27592C2semidirect
8. #24209C2semidirect
10. #23831C2semidirect
11. #23811C2semidirect
12. #21504C2semidirect
13. C4×#461C2semidirect
14. #21462C2semidirect
15. #20786C2semidirect

Maximal subgroups

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

subgroupindex
#342092
C4×#5402
#293782
#292572
#281572
#276792
#275922
#242092
#239402
#238312
#238112
#215042
C4×#4612
#214622
#207862