Finite Group Databasebeta

← #354285

Subgroups — #354285

Order 512.

2254 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
1C2×#2322C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3C2×#1890C212yesmaximal
4C2×C4×#99C211yesmaximal, characteristic
5#13776C212yesmaximal, complemented
6#10936C212yesmaximal, complemented
7#10918C212yesmaximal, complemented
8#10745C212yesmaximal
9C4×#342C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C2×#518C2212yes—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#99C2212yes—
15#2322—24no—
16#2322C418yescomplemented
17C2×#389—21no—
18C2×#388—21no—
19C2×#387—21no—
20C2×#386C2211yescharacteristic
21C2×#385—21no—
22C2×#385C2211yescharacteristic
23C2×#381C2211yescharacteristic
24C2×#381C2214yes—
25#2275—24no—
26#2274—21no—
27#2273—21no—
28#2273C2212yes—
29C42×D4—24no—
30C4×#117—21no—
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35#1890C4116yescomplemented
36C4×#99—24no—
37C2×#342C2212yes—
38C2×#342C2214yes—
39C2×#342C2214yes—
40#1436—24no—
41#1436—24no—
42C22⋊#95C2214yes—
43C23×D4C2311yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#123—22no—
49C2×#123—24no—
50C2×#123C2×C414yes—
51C2×#123C2×C418yes—
52#518—24no—
53#518—28no—
54#518C2×C418yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—28no—
58C2×C4×D4—24no—
59C2×C4×D4C2×C414yes—
60C2×C4×D4D414yes—
61C2×#117—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C414yescomplemented
71C22×C4⋊C4D411yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2311yescharacteristic
78C22×C22⋊C4C2311yescharacteristic
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2311yescharacteristic
81C22×C42—22no—
82C22×C42—21no—
83C22×C42—22no—
84C22×C42D411yescharacteristic
85C22×C42D414yes—
86C2×#99—24no—
87C2×#99—24no—
88C2×#99C2×C418yes—
89#389—28no—
90#388—28no—
91#387—28no—
92#386—22no—
93#386C2×C414yes—
94#385—28no—
95#385—22no—
96#385C2×C414yescomplemented
97#381—22no—
98#381—28no—
99#381C2×C414yescomplemented
100#381C2×C4116yescomplemented
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—28no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—28no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—28no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—24no—
117C4×C22⋊C4C2×C414yescomplemented
118#376—22no—
119C2×#95—22no—
120C2×#95—28no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—28no—
128C2×#95D414yes—
129C43—22no—
130#342—24no—
131#342—28no—
132#342C2×C4116yescomplemented
133C25C2×D411yescharacteristic
134C25C2411yescharacteristic
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—24no—
138C22×D4C22×C412yes—
139C22×D4C22×C414yes—
140C22×D4C2×D412yes—
141C22×D4#4114yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C22⋊C414yes—
163C23×C4C22×C412yes—
164C23×C4C22×C414yes—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D418yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4#4114yes—
177#123—28no—
178#123—216no—
179#123C42116yescomplemented
180C4×D4—232no—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—232no—
184C4×D4—28no—
185C4×D4—28no—
186#117—28no—
187#117—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C22⋊C414yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4C22⋊C414yes—
232C2×C22⋊C4C22×C418yes—
233C2×C22⋊C4C22×C412yes—
234C2×C22⋊C4C22×C412yes—
235C2×C22⋊C4C22×C418yes—
236C2×C22⋊C4C22×C414yes—
237C2×C22⋊C4C22×C412yes—
238C2×C22⋊C4C2×D412yes—
239C2×C22⋊C4#4118yes—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—44no—
250C2×C42—84no—
251C2×C42—84no—
252C2×C42—82no—
253C2×C42—84no—
254C2×C42C22⋊C414yes—
255#99—216no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—232no—
264#95—24no—
265#95—24no—
266#95—44no—
267#95—88no—
268C24—21no—
269C24—21no—
270C24—22no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24C2×C22⋊C412yes—
276C24C4×D414yes—
277C24C23×C411yescharacteristic
278C24C23×C412yes—
279C24C22×D411yescharacteristic
280C24C22×D412yes—
281C24C22×D411yescharacteristic
282C24C2×#4112yes—
283C24C2×#4112yes—
284C24C2×#4114yes—
285C2×D4—24no—
286C2×D4—28no—
287C2×D4—24no—
288C2×D4—28no—
289C2×D4—24no—
290C2×D4—28no—
291C2×D4—28no—
292C2×D4—24no—
293C2×D4C2×C4214yes—
294C2×D4#11714yes—
295C2×D4C4×D418yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—21no—
310C22×C4—28no—
311C22×C4—21no—
312C22×C4—216no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—28no—
321C22×C4—24no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—48no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—82no—
363C22×C4—84no—
364C22×C4C2×C4214yescomplemented
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C4×D414yes—
371C22×C4C4×D4116yes—
372C22×C4C4×D414yes—
373C22×C4C4×D414yes—
374C22×C4#12011yescharacteristic
375C22×C4#12012yes—
376C22×C4#12011yescharacteristic
377C22×C4#12114yes—
378C22×C4#12114yes—
379C22×C4#12312yes—
380C22×C4#12312yes—
381C4⋊C4—216no—
382C4⋊C4—216no—
383C4⋊C4—28no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—84no—
388C22⋊C4—216no—
389C22⋊C4—22no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—28no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—28no—
396C22⋊C4—28no—
397C22⋊C4—216no—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—24no—
401C22⋊C4—28no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—84no—
406C22⋊C4C2×C4214yescomplemented
407C42—216no—
408C42—24no—
409C42—24no—
410C42—48no—
411C42—84no—
412C42—84no—
413C42—84no—
414C42—84no—
415C42—88no—
416C23—24no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—22no—
423C23—24no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—44no—
429C23—41no—
430C23—42no—
431C23—48no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—48no—
436C23—42no—
437C23—42no—
438C23—81no—
439C23—81no—
440C23—84no—
441C23C4×C22⋊C414yes—
442C23C22×C4211yescharacteristic
443C23C22×C22⋊C411yescharacteristic
444C23C2×#11711yescharacteristic
445C23C2×C4×D412yes—
446C23C2×C4×D412yes—
447C23C2×C4×D414yes—
448C23C2×C4×D412yes—
449C23C4×#4112yes—
450C23C4×#4114yes—
451C23C2×#12011yescharacteristic
452C23C2×#12112yes—
453C23C2×#12311yescharacteristic
454C23#52514yes—
455C23#52512yes—
456C23#52914yes—
457C23#53212yes—
458D4—28no—
459D4—28no—
460D4—216no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—216no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—48no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—416no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—416no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—48no—
508C2×C4—84no—
509C2×C4—81no—
510C2×C4—88no—
511C2×C4—81no—
512C2×C4—816no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—88no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4C4×C22⋊C414yes—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4#38614yes—
524C22—21no—
525C22—22no—
526C22—22no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—21no—
531C22—22no—
532C22—28no—
533C22—24no—
534C22—28no—
535C22—24no—
536C22—24no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—44no—
541C22—44no—
542C22—42no—
543C22—41no—
544C22—44no—
545C22—42no—
546C22—42no—
547C22—48no—
548C22—82no—
549C22—84no—
550C22C2×C4×C22⋊C411yescharacteristic
551C22C42×D412yes—
552C22C2×#38611yescharacteristic
553C22#229012yes—
554C22#229711yescharacteristic
555C22C4×#12012yes—
556C22C4×#12114yes—
557C22C4×#12312yes—
558C22#302612yes—
559C22#302812yes—
560C4—24no—
561C4—22no—
562C4—22no—
563C4—24no—
564C4—44no—
565C4—44no—
566C4—42no—
567C4—42no—
568C4—416no—
569C4—82no—
570C4—88no—
571C4—82no—
572C2—21no—
573C2—21no—
574C2—24no—
575C2—22no—
576C2—22no—
577C2—22no—
578C2—44no—
579C2C4×#38611yescharacteristic
580C2#2096011yescharacteristic
581C2#2096211yescharacteristic
582C2#2148114yes—
subgroupG/H
center, socleC23C2×#120
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13776C2semidirect
6. #10936C2semidirect
7. #10918C2semidirect
9. C4×#342C2semidirect
16. #2322C4semidirect
35. #1890C4semidirect
70. C22×C4⋊C4C2×C4semidirect
96. #385C2×C4semidirect
99. #381C2×C4semidirect
100. #381C2×C4semidirect
117. C4×C22⋊C4C2×C4semidirect
132. #342C2×C4semidirect
179. #123C42semidirect
364. C22×C4C2×C42semidirect
406. C22⋊C4C2×C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
C4×#5182
C2×#18902
C2×C4×#992
#137762
#109362
#109182
#107452
C4×#3422