Finite Group Databasebeta

← #6798698

Subgroups — #6798698

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36196C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic
3#33232C211yesmaximal, characteristic, complemented
4#31292C214yesmaximal, complemented
5#31288C214yesmaximal, complemented
6#29852C212yesmaximal, complemented
7#28755C211yesmaximal, characteristic, complemented
8#28518C211yesmaximal, characteristic, complemented
9#27896C212yesmaximal, complemented
10#25894C214yesmaximal
11#25861C212yesmaximal, complemented
12#25290C214yesmaximal, complemented
13#21811C212yesmaximal, complemented
14#21657C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic
16C2×D4×Q8C2212yes—
17C22×C4⋊Q8C2212yes—
18C2×#491C2211yescharacteristic
19C2×#491C2212yes—
20C2×#491C2214yes—
21C2×#479C2214yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2212yes—
26#2851C2212yes—
27#2573—22no—
28#2562C2212yes—
29#2551—22no—
30#2546C2212yes—
31#2533—22no—
32#2497C2212yes—
33#2496C2212yes—
34#2466—22no—
35#2440—22no—
36#2440—22no—
37#2434C2214yes—
38#2373—22no—
39#2370C2212yes—
40D4×C4⋊C4—22no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×#125C2211yescharacteristic
43#2283C2212yes—
44C42×D4C2211yescharacteristic
45#2096C2212yes—
46#2095C2212yes—
47#2068C2214yes—
48#2066C2214yes—
49#2043C2214yes—
50#2043C2214yes—
51#2043C2214yes—
52#2036C2214yes—
53#2036C2214yes—
54#2035C2214yes—
55#2035C2214yes—
56#1992C2211yescharacteristic
57#1981C2214yes—
58#1981C2214yes—
59#1981C2214yes—
60#1960C2212yes—
61#1960C2212yes—
62#1960C2211yescharacteristic
63#1960C2212yes—
64#1960C2212yes—
65#1956C2212yes—
66#1956C2212yes—
67#1955C2212yes—
68#1940C2212yes—
69#1933C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1923C2214yes—
73#1847C2212yes—
74#1836C2211yescharacteristic
75#1787C2214yes—
76#1787C2214yes—
77#1787C2214yes—
78#1787C2214yes—
79#1767C2211yescharacteristic
80#1763C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83C23×Q8C2314yes—
84D4×Q8—48no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—44no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2312yes—
95C2×#126—22no—
96C2×#125C2311yescharacteristic
97C2×#125C2312yes—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4D412yes—
103C2×C4×D4C2311yescharacteristic
104C22×C4⋊C4—22no—
105C22×C4⋊C4—22no—
106C22×C42C2312yes—
107#491—24no—
108#491—44no—
109#491—48no—
110#479—48no—
111#432—44no—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2311yescharacteristic
131C2×#97C2312yes—
132C2×#97C2312yes—
133C2×#97C2311yescharacteristic
134C2×C4×C8C2312yes—
135#400—22no—
136#400—22no—
137#400—22no—
138#398—22no—
139#398—22no—
140#397—22no—
141#396—22no—
142#395—24no—
143#395—24no—
144#394—22no—
145#391C2312yes—
146#391C2314yes—
147#389—24no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#385—22no—
168#385—22no—
169#385—24no—
170#385—24no—
171#384—22no—
172#384—22no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#383—22no—
177#382C2311yescharacteristic
178#382C2312yes—
179#381—22no—
180#380—22no—
181#380—22no—
182#380C2312yes—
183#380C2312yes—
184#380C2312yes—
185#380C2312yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4D412yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2311yescharacteristic
194C4×C22⋊C4—22no—
195C4×C22⋊C4D412yes—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4C2312yes—
198#376—22no—
199C43C2311yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#336C2312yes—
203#336C2314yes—
204C22×Q8—24no—
205C22×Q8—28no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—44no—
211C22×Q8C2414yes—
212C22×Q8C2412yes—
213C22×D4C2×D411yescharacteristic
214C23×C4—24no—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2412yes—
218C22×C8C2412yes—
219C22×C8C2414yes—
220C22×C8C2412yes—
221C4⋊Q8—28no—
222C4⋊Q8—24no—
223C4⋊Q8—22no—
224C4⋊Q8—24no—
225C4⋊Q8—28no—
226C4⋊Q8—48no—
227C4⋊Q8—48no—
228C4⋊Q8—44no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231C4⋊Q8—48no—
232C4⋊Q8#4114yes—
233#126—48no—
234#125—22no—
235#125—28no—
236#125—22no—
237C22⋊Q8—416no—
238C22⋊Q8—44no—
239C22⋊Q8—44no—
240C4×Q8—28no—
241C4×D4—28no—
242C4×D4—24no—
243C4×D4—48no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2412yes—
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—44no—
295#106—44no—
296C4⋊C8—24no—
297C4⋊C8—44no—
298#103—28no—
299#103—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303#103—44no—
304#103—44no—
305#103—44no—
306#103—44no—
307#103—44no—
308#103—44no—
309#103—44no—
310C22⋊C8—44no—
311C22⋊C8—44no—
312#97—22no—
313#97—22no—
314#97—24no—
315#97—24no—
316C4×C8—42no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95C2×D411yescharacteristic
334#95C2×D411yescharacteristic
335#95C2×D412yes—
336#95C2×D412yes—
337C24C22×D412yes—
338C2×Q8—22no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—84no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×Q8—88no—
353C2×Q8—88no—
354C2×Q8C2×#4114yes—
355C2×D4—24no—
356C2×D4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—84no—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C2511yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C4⋊C4—22no—
395C4⋊C4—28no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—44no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—84no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4#13718yes—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—42no—
436C42—44no—
437C42—44no—
438C42—82no—
439C42—82no—
440C42C2×#4112yes—
441C42H(5,𝔽2)14yes—
442C42H(5,𝔽2)12yes—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23C2×#13714yes—
450C23C23×D411yescharacteristic
451Q8—28no—
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—44no—
457Q8—816no—
458D4—28no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—816no—
498C2×C4#53512yes—
499C2×C4#53512yes—
500C2×C4#54614yes—
501C2×C4C2×#13714yes—
502C2×C4C2×#13714yes—
503C2×C4C22×#4111yescharacteristic
504C2×C4C2×H(5,𝔽2)12yes—
505C2×C4C2×H(5,𝔽2)11yescharacteristic
506C2×C4#58512yes—
507C8—42no—
508C8—42no—
509C8—44no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#53511yescharacteristic
518C22C2×#54611yescharacteristic
519C22#350011yescharacteristic
520C22C22×#13712yes—
521C22#358612yes—
522C4—24no—
523C4—22no—
524C4—22no—
525C4—24no—
526C4—24no—
527C4—42no—
528C4—44no—
529C4—42no—
530C4—84no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4#351412yes—
537C4#354612yes—
538C2—44no—
539C2#3895711yescharacteristic
540C2#6080011yescharacteristic
541C2#6083111yescharacteristic
542C2#6115712yes—
543C2#6257712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33232C2semidirect
4. #31292C2semidirect
5. #31288C2semidirect
6. #29852C2semidirect
7. #28755C2semidirect
8. #28518C2semidirect
9. #27896C2semidirect
11. #25861C2semidirect
12. #25290C2semidirect
13. #21811C2semidirect
14. #21657C2semidirect

Maximal subgroups

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

subgroupindex
#361962
D4×C4⋊Q82
#332322
#312922
#312882
#298522
#287552
#285182
#278962
#258942
#258612
#252902
#218112
#216572
#212712