Finite Group Databasebeta

← #6471040

Subgroups — #6471040

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37306C211yesmaximal, characteristic
2#34276C211yesmaximal, characteristic
3#33154C212yesmaximal, complemented
4#32964C212yesmaximal, complemented
5#29976C212yesmaximal, complemented
6#29953C212yesmaximal, complemented
7#26428C214yesmaximal, complemented
8#26413C214yesmaximal, complemented
9#26353C214yesmaximal, complemented
10#26342C214yesmaximal, complemented
11#21936C211yesmaximal, characteristic
12#21771C212yesmaximal
13#20847C212yesmaximal
14C2×#536C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C2×#519C2211yescharacteristic
17C2×#477C2212yes—
18C2×#475C2212yes—
19C2×#428C2212yes—
20C22×#107C2212yes—
21C2×#408C2212yes—
22#2857C2212yes—
23#2811—21no—
24#2809—21no—
25#2804—21no—
26#2692—24no—
27#2683—21no—
28#2628C2212yes—
29#2440C2214yes—
30#2439C2214yes—
31#2362C2214yes—
32#2360C2214yes—
33#2359—24no—
34#2327—21no—
35#2326—21no—
36#2317C2211yescharacteristic
37#2311C2212yes—
38C4×C4⋊Q8—21no—
39C4×#124—21no—
40#2297C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C4×#117C2211yescharacteristic
43#2101C2214yes—
44#2095C2214yes—
45#2087C2214yes—
46#2086C2214yes—
47#2068C2214yes—
48#2068C2214yes—
49#2057C2214yes—
50#2024C2214yes—
51#1988C2212yes—
52#1988C2212yes—
53#1948C2212yes—
54#1947C2212yes—
55#1921C2214yes—
56#1920C2214yes—
57#1894C2214yes—
58#1892C2214yes—
59#1873C2214yes—
60#1872C2214yes—
61#1865C2214yes—
62#1863C2214yes—
63#1854C2214yes—
64#1854C2214yes—
65#1852C2214yes—
66#1852C2214yes—
67#1851C2214yes—
68#1850C2214yes—
69#1825C2214yes—
70#1805C2212yes—
71#1805C2212yes—
72#1798C2212yes—
73#1797C2212yes—
74#1787C2214yes—
75#1736C2212yes—
76#1735C2212yes—
77C22×#41C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C22×M16C2312yes—
80C23×C8C2312yes—
81#536—24no—
82#536Q818yescomplemented
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#124—21no—
87C2×#123—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8C2314yes—
91#520—42no—
92#520—42no—
93#519—42no—
94#519—42no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×C4×D4—21no—
101C2×C4×D4—22no—
102C2×#117—22no—
103C2×#117C2311yescharacteristic
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2312yes—
107C2×#117C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#477—44no—
113#475—44no—
114#428—44no—
115C2×#107—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2312yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128#408—28no—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131#400—22no—
132#399—22no—
133#398—24no—
134#396—24no—
135#395—22no—
136#395—24no—
137#393—22no—
138#393—24no—
139#391—22no—
140#391—21no—
141#391—21no—
142#391—22no—
143#391—24no—
144#391D411yescharacteristic
145#391D411yescharacteristic
146#391C2314yes—
147#390—22no—
148#390D411yescharacteristic
149#390D411yescharacteristic
150#389—22no—
151#389—24no—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#388—24no—
156#388—24no—
157#387—24no—
158#387—24no—
159#387—24no—
160#386—22no—
161#386—22no—
162#386D414yes—
163#385—22no—
164#385—24no—
165#385—24no—
166#384—21no—
167#384C2312yes—
168#382—21no—
169#382C2312yes—
170#380—22no—
171#380C2314yes—
172#380C2314yes—
173C4×C4⋊C4—24no—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—22no—
177C4×C22⋊C4—24no—
178C4×C22⋊C4C2314yes—
179#376—21no—
180C2×#95—22no—
181C43—21no—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192C2×#41—41no—
193C2×#41—41no—
194C2×#41—41no—
195C2×#41—41no—
196C2×#41C2×Q814yes—
197C2×#41#4114yes—
198C22×Q8—21no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—41no—
203C22×Q8—41no—
204C22×Q8—41no—
205C22×Q8—41no—
206C22×Q8C2×D411yescharacteristic
207C22×Q8C2×Q814yes—
208C22×Q8C2412yes—
209C22×D4C2×D412yes—
210C22×D4C2×D411yescharacteristic
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4C2×D412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C2×M16—24no—
219C2×M16—42no—
220C2×M16—42no—
221C22×C8—24no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8C2414yes—
225C22×C8C2414yes—
226C4⋊Q8—28no—
227C4⋊Q8—42no—
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231#124—28no—
232#123—216no—
233C22⋊Q8—28no—
234C22⋊Q8—88no—
235C22⋊Q8—84no—
236C22⋊Q8C2×Q8116yes—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—42no—
240C4×Q8—42no—
241C4×Q8—44no—
242C4×Q8—44no—
243C4×Q8—44no—
244C4×D4—44no—
245C4×D4—48no—
246#117—22no—
247#117—22no—
248#117—22no—
249#117—24no—
250#117—41no—
251#117—41no—
252#117—48no—
253#117—42no—
254#117—42no—
255#117—42no—
256#117C2×Q814yes—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4C2×D411yescharacteristic
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2411yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4C2×D414yes—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D414yes—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2412yes—
307C2×C42C2412yes—
308#107—28no—
309#107—44no—
310#107—44no—
311#106—28no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316#103—44no—
317#103—44no—
318#103—44no—
319#103—44no—
320C22⋊C8—28no—
321C22⋊C8—28no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24C22×D412yes—
336C24C22×D411yescharacteristic
337#41—44no—
338#41—44no—
339#41—44no—
340#41—44no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—82no—
357C2×Q8—84no—
358C2×Q8—82no—
359C2×Q8—82no—
360C2×Q8C22⋊Q814yes—
361C2×Q8C22×Q814yes—
362C2×D4—41no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4C22⋊Q818yes—
367C2×D4C22⋊Q814yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—82no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—82no—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×Q812yes—
402C22×C4C22×Q814yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4114yes—
405C22×C4C2511yescharacteristic
406M16—48no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—48no—
414C4⋊C4—216no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—84no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—84no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4C22×Q814yes—
439C4⋊C4H(5,𝔽2)12yes—
440C4⋊C4H(5,𝔽2)12yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—216no—
444C22⋊C4—28no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—82no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C42—24no—
452C42—24no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—41no—
458C42—41no—
459C42—44no—
460C42—44no—
461C42—44no—
462C42—44no—
463C42—82no—
464C42—82no—
465C23—24no—
466C23—22no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—24no—
472C23—22no—
473C23C2×C22⋊Q814yes—
474C23C2×C22⋊Q812yes—
475C23C23×D411yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—44no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482Q8—164no—
483Q8—164no—
484D4—48no—
485D4—44no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—84no—
527C2×C4—164no—
528C2×C4—162no—
529C2×C4—164no—
530C2×C4C2×C22⋊Q812yes—
531C2×C4C2×C22⋊Q814yes—
532C2×C4#57814yes—
533C2×C4C23×Q811yescharacteristic
534C2×C4C22×#4111yescharacteristic
535C2×C4C2×H(5,𝔽2)11yescharacteristic
536C2×C4C2×H(5,𝔽2)11yescharacteristic
537C2×C4#58511yescharacteristic
538C2×C4#58511yescharacteristic
539C8—44no—
540C8—44no—
541C22—28no—
542C22—24no—
543C22—22no—
544C22—24no—
545C22—42no—
546C22—44no—
547C22—41no—
548C22—42no—
549C22—42no—
550C22—41no—
551C22C22×C22⋊Q811yescharacteristic
552C22#348411yescharacteristic
553C22#348511yescharacteristic
554C22C2×#57812yes—
555C22#358512yes—
556C4—24no—
557C4—24no—
558C4—22no—
559C4—42no—
560C4—44no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—84no—
567C4—82no—
568C4—84no—
569C4#345911yescharacteristic
570C4#345911yescharacteristic
571C4#350111yescharacteristic
572C4#350111yescharacteristic
573C2—42no—
574C2—44no—
575C2#3498511yescharacteristic
576C2#6049812yes—
577C2#6166511yescharacteristic
578C2#6167311yescharacteristic
579C2#6175412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33154C2semidirect
4. #32964C2semidirect
5. #29976C2semidirect
6. #29953C2semidirect
7. #26428C2semidirect
8. #26413C2semidirect
9. #26353C2semidirect
10. #26342C2semidirect
82. #536Q8semidirect

Maximal subgroups

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

subgroupindex
#373062
#342762
#331542
#329642
#299762
#299532
#264282
#264132
#263532
#263422
#219362
#217712
#208472