Finite Group Databasebeta

← #357641

Subgroups — #357641

Order 512.

3518 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic, complemented
2C2×#2280C212yesmaximal
3C2×#2280C211yesmaximal, characteristic
4C2×#1740C212yesmaximal, complemented
5C2×#1740C211yesmaximal, characteristic, complemented
6#11070C214yesmaximal, complemented
7#11069C214yesmaximal
8C25×C4C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C22×#117C2212yes—
11C22×#117C2212yes—
12C22×#117C2211yescharacteristic
13#2317—24no—
14#2316—22no—
15#2315C2214yes—
16#2296—24no—
17#2285—22no—
18#2280—28no—
19#2280—28no—
20#2280—24no—
21#2280C418yescomplemented
22C2×#381C2214yes—
23C2×#381C2212yes—
24C2×#380—24no—
25C2×#380—22no—
26C2×#379C2214yes—
27C2×#379C2212yes—
28C2×C4×C22⋊C4—24no—
29C2×C4×C22⋊C4—22no—
30#1740—28no—
31#1740—24no—
32C2×#342C2218yes—
33C2×#342C2214yes—
34C26C2311yescharacteristic
35C24×C4—21no—
36C24×C4—22no—
37C24×C4—21no—
38C24×C4—22no—
39C24×C4—22no—
40C24×C4—24no—
41C24×C4—22no—
42C24×C4—22no—
43C24×C4—24no—
44C24×C4—24no—
45C24×C4C2×C414yes—
46C24×C4D414yes—
47C24×C4Q814yes—
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—22no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—24no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—24no—
65C2×#117—24no—
66C2×#117—24no—
67C2×#117—24no—
68C2×#117—24no—
69C2×#117C2×C414yes—
70C2×#117C2×C414yes—
71C2×#117D418yes—
72C2×#117D414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D414yes—
76C22×C4⋊C4D412yes—
77C22×C22⋊C4—28no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2312yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C42—21no—
84C22×C42—22no—
85C22×C42D414yes—
86C22×C42D412yes—
87#389—216no—
88#385—216no—
89#384—24no—
90#382—24no—
91#381—28no—
92#381—28no—
93#381—24no—
94#381C2×C418yes—
95#380—232no—
96#380—28no—
97#380—216no—
98#379—28no—
99#379—28no—
100#379—24no—
101#379C2×C418yes—
102C4×C4⋊C4—28no—
103C4×C22⋊C4—232no—
104C4×C22⋊C4—216no—
105C2×#95—28no—
106C2×#95—24no—
107#342—28no—
108#342—216no—
109C25—21no—
110C25—22no—
111C25—21no—
112C25—22no—
113C25—22no—
114C25—21no—
115C25—21no—
116C25—22no—
117C25—48no—
118C25C22×C412yes—
119C25C2×D412yes—
120C25C2×Q812yes—
121C25C2411yescharacteristic
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—28no—
133C23×C4—28no—
134C23×C4—28no—
135C23×C4—216no—
136C23×C4—28no—
137C23×C4—28no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—28no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—28no—
144C23×C4—28no—
145C23×C4—28no—
146C23×C4—216no—
147C23×C4—216no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—41no—
151C23×C4—42no—
152C23×C4—41no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—48no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—48no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—48no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C4⋊C4116yescomplemented
169C23×C4C22×C412yes—
170C23×C4C22×C412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D418yes—
173C23×C4C2×D414yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D414yes—
181C23×C4C2×Q812yes—
182C23×C4C2×Q812yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4#4114yes—
186C23×C4#4114yes—
187#117—28no—
188#117—28no—
189#117—28no—
190#117—28no—
191#117—216no—
192#117—216no—
193#117—216no—
194#117—44no—
195#117—44no—
196#117—48no—
197#117—44no—
198#117—44no—
199#117—44no—
200#117—44no—
201#117—48no—
202#117—48no—
203#117—44no—
204#117—44no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22⋊C418yescomplemented
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—232no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—216no—
237C2×C22⋊C4—416no—
238C2×C22⋊C4—432no—
239C2×C22⋊C4C22×C414yes—
240C2×C22⋊C4C22×C414yes—
241C2×C22⋊C4C2×D418yes—
242C2×C22⋊C4C2×D414yes—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—28no—
248C2×C42—24no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—44no—
253C2×C42—42no—
254C2×C42—44no—
255C2×C42—42no—
256C2×C42C22⋊C418yescomplemented
257#95—232no—
258#95—216no—
259#95—216no—
260C24—216no—
261C24—28no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—22no—
266C24—22no—
267C24—24no—
268C24—22no—
269C24—24no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—48no—
275C24—48no—
276C24—48no—
277C24—44no—
278C24—48no—
279C24—44no—
280C24—416no—
281C24—48no—
282C24—48no—
283C24—48no—
284C24—416no—
285C24—416no—
286C24—41no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—41no—
291C24—42no—
292C24—42no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24C2×C4⋊C414yes—
297C24C23×C411yescharacteristic
298C24C22×D411yescharacteristic
299C24C22×D412yes—
300C24C22×D411yescharacteristic
301C24C22×D411yescharacteristic
302C24C22×D412yes—
303C24C22×Q811yescharacteristic
304C24C2×#4111yescharacteristic
305C24C2×#4111yescharacteristic
306C24C2×#4112yes—
307C24C2×#4112yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—216no—
323C22×C4—216no—
324C22×C4—232no—
325C22×C4—232no—
326C22×C4—24no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—48no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4—48no—
350C22×C4—44no—
351C22×C4—48no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—416no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—416no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—416no—
364C22×C4—416no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4C2×C22⋊C418yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C4⋊C414yes—
374C22×C4C2×C4⋊C414yes—
375C22×C4#11714yes—
376C22×C4C4×D414yes—
377C22×C4C4×D414yes—
378C22×C4C4×D418yes—
379C22×C4#12014yes—
380C22×C4#12014yes—
381C22×C4#12011yescharacteristic
382C22×C4#12012yes—
383C22×C4#12011yescharacteristic
384C22×C4#12112yes—
385C22×C4#12112yes—
386C22×C4#12114yes—
387C22×C4#12114yes—
388C22×C4#12114yes—
389C22×C4#12114yes—
390C22×C4#12114yes—
391C22×C4#12118yes—
392C22×C4C22⋊Q812yes—
393C22×C4C22⋊Q812yes—
394C22×C4C22⋊Q814yes—
395C22×C4C22⋊Q814yes—
396C22×C4C22⋊Q814yes—
397C22×C4C22⋊Q818yes—
398C22×C4#12318yes—
399C22×C4#12312yes—
400C22×C4#12312yes—
401C22×C4#12318yes—
402C22×C4#12414yes—
403C22×C4#12414yes—
404C22×C4#12412yes—
405C22×C4#12412yes—
406C22×C4#12618yes—
407C22×C4#12712yes—
408C22×C4#12712yes—
409C4⋊C4—216no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—416no—
414C4⋊C4—816no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—432no—
421C22⋊C4—464no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C42—28no—
430C42—28no—
431C42—48no—
432C42—48no—
433C42—48no—
434C42—416no—
435C23—28no—
436C23—24no—
437C23—216no—
438C23—28no—
439C23—28no—
440C23—28no—
441C23—216no—
442C23—216no—
443C23—24no—
444C23—28no—
445C23—48no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—48no—
456C23—416no—
457C23—48no—
458C23—44no—
459C23—416no—
460C23—48no—
461C23—44no—
462C23—48no—
463C23—44no—
464C23—416no—
465C23—48no—
466C23—48no—
467C23—48no—
468C23—416no—
469C23—48no—
470C23—48no—
471C23—416no—
472C23—416no—
473C23—416no—
474C23—416no—
475C23—416no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—48no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23C22×C22⋊C411yescharacteristic
487C23C22×C4⋊C411yescharacteristic
488C23C2×#11711yescharacteristic
489C23C2×C4×D412yes—
490C23C2×C4×D412yes—
491C23C2×#12012yes—
492C23C2×#12011yescharacteristic
493C23C2×#12112yes—
494C23C2×#12111yescharacteristic
495C23C2×#12112yes—
496C23C2×#12111yescharacteristic
497C23C2×#12112yes—
498C23C2×C22⋊Q812yes—
499C23C2×C22⋊Q811yescharacteristic
500C23C2×C22⋊Q811yescharacteristic
501C23C2×C22⋊Q812yes—
502C23C2×#12311yescharacteristic
503C23C2×#12312yes—
504C23C2×#12312yes—
505C23C2×#12412yes—
506C23C2×#12411yescharacteristic
507C23C2×#12612yes—
508C23C2×#12711yescharacteristic
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—432no—
525C2×C4—48no—
526C2×C4—416no—
527C2×C4—48no—
528C2×C4—416no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—416no—
532C2×C4—44no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—88no—
536C2×C4—84no—
537C2×C4—88no—
538C2×C4—88no—
539C2×C4#38014yes—
540C2×C4#38014yes—
541C2×C4#38518yes—
542C2×C4#38518yes—
543C2×C4#38612yes—
544C2×C4#38612yes—
545C2×C4#38614yes—
546C2×C4#38818yes—
547C2×C4#38818yes—
548C2×C4#39014yes—
549C2×C4#39014yes—
550C2×C4#39212yes—
551C2×C4#39214yes—
552C2×C4#39212yes—
553C2×C4#39314yes—
554C2×C4#39312yes—
555C2×C4#39312yes—
556C2×C4#39418yes—
557C2×C4#39418yes—
558C2×C4#39618yes—
559C2×C4#39618yes—
560C2×C4#39714yes—
561C2×C4#39714yes—
562C2×C4#39914yes—
563C2×C4#39914yes—
564C22—24no—
565C22—24no—
566C22—22no—
567C22—22no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—24no—
576C22—28no—
577C22—216no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—416no—
589C22—48no—
590C22—44no—
591C22—48no—
592C22—416no—
593C22—48no—
594C22—48no—
595C22—416no—
596C22—44no—
597C22—48no—
598C22—48no—
599C22—48no—
600C22C2×#38011yescharacteristic
601C22C2×#38512yes—
602C22C2×#38611yescharacteristic
603C22C2×#38812yes—
604C22C2×#39011yescharacteristic
605C22C2×#39211yescharacteristic
606C22C2×#39311yescharacteristic
607C22C2×#39412yes—
608C22C2×#39612yes—
609C22C2×#39711yescharacteristic
610C22C2×#39911yescharacteristic
611C22#302514yes—
612C4—44no—
613C4—48no—
614C4—88no—
615C4—84no—
616C4—88no—
617C4#143914yes—
618C4#143914yes—
619C4#143918yes—
620C2—22no—
621C2—21no—
622C2—21no—
623C2—22no—
624C2—22no—
625C2—21no—
626C2—21no—
627C2—22no—
628C2—48no—
629C2C2×#143911yescharacteristic
630C2#2095912yes—
631C2#2344312yes—
632C2#2346712yes—
subgroupG/H
centerC22×C4#120
commutatorC24C23×C4
Frattini, cosocleC25C24
socleC23C2×#120
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34052C2semidirect
4. C2×#1740C2semidirect
5. C2×#1740C2semidirect
6. #11070C2semidirect
21. #2280C4semidirect
168. C23×C4C4⋊C4semidirect
218. C2×C4⋊C4C22⋊C4semidirect
256. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#22802
C2×#17402
#110702
#110692