Finite Group Databasebeta

← #392901

Subgroups — #392901

Order 512.

1940 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#518C211yesmaximal, characteristic
2C2×#1867C212yesmaximal
3C2×#1799C211yesmaximal, characteristic
4#14149C212yesmaximal, complemented
5#13960C211yesmaximal, characteristic
6#10918C214yesmaximal, complemented
7#10846C212yesmaximal, complemented
8#10745C212yesmaximal
9C2×#518C2211yescharacteristic
10C2×#518C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#99C2212yes—
13C2×#386C2212yes—
14C2×#385C2214yes—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17#2275—24no—
18#2274—21no—
19#2273—21no—
20#2273C2212yes—
21C42×D4—24no—
22C4×#117—21no—
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4C2212yes—
25C22×#95C2211yescharacteristic
26#1867—28no—
27#1799C418yescomplemented
28C2×#342C2214yes—
29C2×#342C2214yes—
30C2×#342C2212yes—
31#1441—24no—
32#1441—21no—
33#1439—21no—
34#1436—24no—
35#1434—21no—
36C22⋊#95—21no—
37C22⋊#95C2214yes—
38C22⋊#95C2212yes—
39C23×D4C2311yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2312yes—
42C24×C4C2311yescharacteristic
43#518—22no—
44#518—22no—
45#518—24no—
46#518—24no—
47#518D414yes—
48#518Q814yes—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4Q814yes—
56C2×#117—21no—
57C2×#117—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2312yes—
71C22×C42—22no—
72C22×C42D411yescharacteristic
73C22×C42D414yes—
74C2×#99—24no—
75C2×#99—24no—
76C2×#99C2×C418yes—
77#386—24no—
78#386—24no—
79#385—28no—
80#385—28no—
81#380—24no—
82#380—24no—
83#380C2×C418yescomplemented
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—24no—
100#376—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—24no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—24no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—24no—
114C2×#95D414yes—
115C43—22no—
116#342—28no—
117#342—28no—
118#342C2×C418yescomplemented
119C25C2×D411yescharacteristic
120C25C2411yescharacteristic
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4C22×C412yes—
126C22×D4C2×D412yes—
127C22×D4C2×Q812yes—
128C22×D4C2×Q812yes—
129C22×D4#4112yes—
130C22×D4#4112yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—44no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C4⋊C414yescomplemented
148C23×C4C22×C412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D414yes—
153C23×C4C2×D414yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×Q812yes—
159C23×C4#4114yes—
160C4×D4—216no—
161C4×D4—216no—
162C4×D4—28no—
163C4×D4—28no—
164C4×D4—216no—
165C4×D4—28no—
166C4×D4—28no—
167C4×D4—216no—
168#117—24no—
169#117—24no—
170#117—24no—
171#117—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—84no—
181C2×C4⋊C4—84no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C4⋊C414yescomplemented
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4C4⋊C414yescomplemented
215C2×C22⋊C4C22×C418yes—
216C2×C22⋊C4C22×C412yes—
217C2×C22⋊C4C2×D414yes—
218C2×C22⋊C4C2×Q812yes—
219C2×C22⋊C4C2×Q814yes—
220C2×C22⋊C4#4114yes—
221C2×C22⋊C4#4114yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—84no—
232C2×C42—84no—
233C2×C42C4⋊C414yescomplemented
234#99—216no—
235#95—24no—
236#95—24no—
237#95—28no—
238#95—24no—
239#95—24no—
240#95—44no—
241#95—48no—
242#95—41no—
243#95—42no—
244#95—41no—
245#95—48no—
246#95—44no—
247#95—42no—
248#95—84no—
249#95—82no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24C2×C4⋊C412yes—
259C24#12112yes—
260C24C22⋊Q812yes—
261C24C23×C411yescharacteristic
262C24C22×D411yescharacteristic
263C24C22×D412yes—
264C24C22×D411yescharacteristic
265C24C22×Q811yescharacteristic
266C24C22×Q811yescharacteristic
267C24C2×#4112yes—
268C24C2×#4111yescharacteristic
269C24C2×#4111yescharacteristic
270C24C2×#4112yes—
271C24C2×#4112yes—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4C4×D414yes—
285C2×D4C4×Q814yes—
286C2×D4#12514yes—
287C2×D4C4⋊Q814yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—88no—
350C22×C4—84no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C2×C4⋊C412yes—
356C22×C4C2×C4⋊C412yes—
357C22×C4C2×C4⋊C412yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4C2×C4⋊C412yes—
360C22×C4C4×D414yes—
361C22×C4C4×D414yes—
362C22×C4#12011yescharacteristic
363C22×C4#12012yes—
364C22×C4#12011yescharacteristic
365C22×C4#12112yes—
366C22×C4#12118yes—
367C22×C4C22⋊Q812yes—
368C22×C4C22⋊Q818yes—
369C22×C4C22⋊Q814yes—
370C22×C4C22⋊Q814yes—
371C22×C4#12312yes—
372C22×C4#12312yes—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—88no—
377C4⋊C4—816no—
378C4⋊C4—84no—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—88no—
399C42—28no—
400C42—28no—
401C42—24no—
402C42—24no—
403C42—48no—
404C42—84no—
405C42—84no—
406C42—84no—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—42no—
429C23—44no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—81no—
436C23—81no—
437C23—82no—
438C23—82no—
439C23#38914yes—
440C23C22×C4⋊C411yescharacteristic
441C23C2×C4×D412yes—
442C23C2×C4×D411yescharacteristic
443C23C2×C4×Q811yescharacteristic
444C23C4×#4112yes—
445C23C2×#12011yescharacteristic
446C23C2×#12111yescharacteristic
447C23C2×#12112yes—
448C23C2×C22⋊Q811yescharacteristic
449C23C2×C22⋊Q812yes—
450C23C2×C22⋊Q812yes—
451C23C2×#12311yescharacteristic
452C23#52512yes—
453C23#52511yescharacteristic
454C23#52512yes—
455C23#52511yescharacteristic
456C23C2×#12511yescharacteristic
457C23#52912yes—
458C23#52912yes—
459C23C2×C4⋊Q811yescharacteristic
460C23#53212yes—
461C23#53312yes—
462C23#53312yes—
463D4—28no—
464D4—28no—
465D4—28no—
466D4—28no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—84no—
508C2×C4—81no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—81no—
512C2×C4—88no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4#38512yes—
532C2×C4#38512yes—
533C2×C4#38514yes—
534C2×C4#38914yes—
535C22—22no—
536C22—21no—
537C22—21no—
538C22—22no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—24no—
544C22—22no—
545C22—24no—
546C22—22no—
547C22—24no—
548C22—24no—
549C22—22no—
550C22—22no—
551C22—24no—
552C22—41no—
553C22—41no—
554C22—42no—
555C22—41no—
556C22—42no—
557C22—42no—
558C22—41no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22C2×#38511yescharacteristic
571C22C2×#38911yescharacteristic
572C22#229611yescharacteristic
573C22C4×#12112yes—
574C22C4×C22⋊Q812yes—
575C22#284711yescharacteristic
576C22#284811yescharacteristic
577C22#285112yes—
578C22#285411yescharacteristic
579C22#285511yescharacteristic
580C22#285712yes—
581C22#302512yes—
582C22#302612yes—
583C4—24no—
584C4—22no—
585C4—22no—
586C4—24no—
587C4—42no—
588C4—42no—
589C4—48no—
590C4—48no—
591C4—88no—
592C4—84no—
593C4—162no—
594C2—22no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—22no—
600C2—22no—
601C2—22no—
602C2—44no—
603C2#1469711yescharacteristic
604C2#2088811yescharacteristic
605C2#2088911yescharacteristic
606C2#3292812yes—
607C2#3297712yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ 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
4. #14149C2semidirect
6. #10918C2semidirect
7. #10846C2semidirect
27. #1799C4semidirect
83. #380C2×C4semidirect
118. #342C2×C4semidirect
147. C23×C4C4⋊C4semidirect
183. C2×C4⋊C4C4⋊C4semidirect
214. C2×C22⋊C4C4⋊C4semidirect
233. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5182
C2×#18672
C2×#17992
#141492
#139602
#109182
#108462
#107452