Finite Group Databasebeta

← #6479580

Subgroups — #6479580

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#33104C214yesmaximal, complemented
4#29911C212yesmaximal, complemented
5#29909C212yesmaximal, complemented
6#27870C214yesmaximal, complemented
7#27864C212yesmaximal, complemented
8#27858C212yesmaximal, complemented
9#27620C212yesmaximal, complemented
10#27612C212yesmaximal, complemented
11#27592C212yesmaximal, complemented
12#27586C212yesmaximal, complemented
13#21947C212yesmaximal
14#21771C212yesmaximal
15#20684C211yesmaximal, characteristic
16C2×#536C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#408C2212yes—
19#2855C2212yes—
20#2855C2212yes—
21#2625C2212yes—
22#2625C2212yes—
23#2447—22no—
24#2446—22no—
25#2419—22no—
26#2419—22no—
27#2355—22no—
28#2354—22no—
29#2352—22no—
30#2352—22no—
31#2316C2211yescharacteristic
32#2314C2212yes—
33#2314C2212yes—
34#2311C2212yes—
35#2311C2211yescharacteristic
36#2310C2211yescharacteristic
37#2296C2211yescharacteristic
38C2×#391C2211yescharacteristic
39#2285C2212yes—
40#2285C2212yes—
41#2285C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2274C2212yes—
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2097C2214yes—
48#2095C2212yes—
49#2095C2212yes—
50#2084C2212yes—
51#2083C2212yes—
52#2083C2214yes—
53#2072C2212yes—
54#2072C2212yes—
55#2072C2214yes—
56#1928C2214yes—
57#1928C2214yes—
58#1928C2214yes—
59#1927C2214yes—
60#1921C2214yes—
61#1920C2214yes—
62#1918C2214yes—
63#1918C2214yes—
64#1894C2214yes—
65#1892C2214yes—
66#1873C2214yes—
67#1872C2214yes—
68#1854C2214yes—
69#1854C2214yes—
70#1852C2214yes—
71#1852C2214yes—
72#1825C2214yes—
73#1818C2212yes—
74#1818C2212yes—
75#1817C2212yes—
76#1817C2212yes—
77#1809C2214yes—
78#1808C2214yes—
79#1787C2214yes—
80#1729C2212yes—
81#1728C2211yescharacteristic
82C2×#336C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C22×M16C2312yes—
86C23×C8C2312yes—
87#536—28no—
88C2×C4⋊Q8—21no—
89C2×#124—21no—
90C2×#123—21no—
91C2×#123—21no—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C4×#41—28no—
95C2×C4×Q8D412yes—
96C2×C4×D4D412yes—
97C2×C4×D4D412yes—
98C2×C4×D4D412yes—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121#408—28no—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2314yes—
124#400—22no—
125#400—22no—
126#398—22no—
127#398—22no—
128#397—22no—
129#397—22no—
130#397—22no—
131#397—22no—
132#391—22no—
133#391—21no—
134#391—22no—
135#391—22no—
136#391C2312yes—
137#391C2312yes—
138#391C2312yes—
139#391C2312yes—
140#390—22no—
141#390—21no—
142#389—22no—
143#389C2312yes—
144#389C2312yes—
145#389C2312yes—
146#389C2312yes—
147#388—24no—
148#388—24no—
149#387—24no—
150#387—24no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#386—22no—
156#386—22no—
157#386—21no—
158#386—21no—
159#385—22no—
160#385—22no—
161#385—22no—
162#384—22no—
163#384C2312yes—
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382C2312yes—
169#381—22no—
170#381—22no—
171#381—22no—
172#380—22no—
173#380—22no—
174#380—22no—
175#380—22no—
176#380C2314yes—
177#380C2314yes—
178#380C2312yes—
179#380C2312yes—
180C4×C4⋊C4—24no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4—24no—
188C2×#95—22no—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201#336C2312yes—
202#336C2312yes—
203C2×#41—42no—
204C2×#41—41no—
205C2×#41—41no—
206C2×#41#4118yes—
207C22×Q8—21no—
208C22×Q8—24no—
209C22×Q8—42no—
210C22×Q8—41no—
211C22×Q8—41no—
212C22×Q8C2×D411yescharacteristic
213C22×Q8C2412yes—
214C22×D4C2×D411yescharacteristic
215C22×D4C2×D411yescharacteristic
216C22×D4C2×D411yescharacteristic
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4C2×D412yes—
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C2×M16—24no—
227C2×M16—42no—
228C2×M16—42no—
229C22×C8—24no—
230C22×C8—42no—
231C22×C8—42no—
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—44no—
235#124—44no—
236#123—44no—
237#123—44no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240C4×Q8—28no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—28no—
244#117—24no—
245#117—24no—
246#117—24no—
247#117—24no—
248#117—42no—
249#117—42no—
250#117—41no—
251#117—41no—
252#117—84no—
253#117—84no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—84no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306#107—44no—
307#107—44no—
308#107—44no—
309#106—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313#103—44no—
314C22⋊C8—28no—
315C22⋊C8—28no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335C24C22×D411yescharacteristic
336C24C22×D411yescharacteristic
337C24C22×D411yescharacteristic
338#41—48no—
339#41—48no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—84no—
350C2×Q8—84no—
351C2×Q8#12314yes—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4#12314yes—
359C2×D4#12314yes—
360C2×D4#12314yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—84no—
391C22×C4—82no—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C2×#4114yes—
400C22×C4C2×#4114yes—
401C22×C4C2×#4114yes—
402C22×C4C2511yescharacteristic
403M16—84no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—84no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—44no—
455C42—42no—
456C42—42no—
457C42—84no—
458C42—84no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23C2×#12312yes—
472C23C2×#12312yes—
473C23C2×#12312yes—
474C23C23×D411yescharacteristic
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—84no—
479D4—44no—
480D4—44no—
481D4—44no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—84no—
527C2×C4—164no—
528C2×C4—162no—
529C2×C4C2×#12312yes—
530C2×C4C2×#12312yes—
531C2×C4C2×#12312yes—
532C2×C4#57814yes—
533C2×C4C22×#4112yes—
534C2×C4#58511yescharacteristic
535C2×C4#58511yescharacteristic
536C2×C4#58512yes—
537C8—44no—
538C8—44no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22C22×#12311yescharacteristic
555C22#344911yescharacteristic
556C22#344911yescharacteristic
557C22C2×#57812yes—
558C22#358512yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—82no—
567C4—84no—
568C4—82no—
569C4—84no—
570C4—82no—
571C4—82no—
572C4#349112yes—
573C4#349112yes—
574C2—42no—
575C2—42no—
576C2—42no—
577C2#3509711yescharacteristic
578C2#6052612yes—
579C2#6099312yes—
580C2#6101311yescharacteristic
581C2#6101711yescharacteristic
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. #33104C2semidirect
4. #29911C2semidirect
5. #29909C2semidirect
6. #27870C2semidirect
7. #27864C2semidirect
8. #27858C2semidirect
9. #27620C2semidirect
10. #27612C2semidirect
11. #27592C2semidirect
12. #27586C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#341882
#331042
#299112
#299092
#278702
#278642
#278582
#276202
#276122
#275922
#275862
#219472
#217712
#206842