Finite Group Databasebeta

← #364260

Subgroups — #364260

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#13880C211yesmaximal, characteristic
6#11228C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#10849C212yesmaximal, complemented
9#10807C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2345—24no—
18#2345C418yescomplemented
19C2×#389C2212yes—
20C2×#387—22no—
21C2×#385—21no—
22C2×#385—21no—
23C2×#384C2212yes—
24C2×#380—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1850—24no—
31#1824—24no—
32C2×#340—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1444—22no—
45C4⋊#95C2212yes—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—24no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4C2×C414yes—
70C22×C4⋊C4D414yes—
71C22×C4⋊C4Q814yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42—22no—
83C22×C42C2312yes—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—42no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90#389—24no—
91#389C2×C418yescomplemented
92#387—216no—
93#385—44no—
94#385—44no—
95#384—24no—
96#384—42no—
97#380—22no—
98#380—28no—
99#380—22no—
100#380—22no—
101#380C2×C414yes—
102C4×C4⋊C4—24no—
103C4×C4⋊C4C2×C418yescomplemented
104C4×C22⋊C4—28no—
105C4×C22⋊C4C2×C414yes—
106C4×C22⋊C4D414yes—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95D412yes—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—42no—
123#340—42no—
124#336—24no—
125C25C2×D411yescharacteristic
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×Q812yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C22×C8—28no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—48no—
174C22×C8—48no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8#4114yes—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C4⋊C4116yescomplemented
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×Q812yes—
229C2×C4⋊C4C2×Q812yes—
230C2×C4⋊C4#4114yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4C22⋊C414yes—
240C2×C22⋊C4C22⋊C414yes—
241C2×C22⋊C4SD1614yes—
242C2×C22⋊C4Q1614yes—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—28no—
248C2×C42—82no—
249C2×C42—82no—
250C2×C42C22×C414yes—
251C2×C42C2×D414yes—
252C4⋊C8—48no—
253C22⋊C8—44no—
254#95—28no—
255#95—28no—
256#95—24no—
257#95—44no—
258#95—42no—
259#95—44no—
260#95C22⋊C418yes—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24C2×C22⋊C412yes—
266C24#12112yes—
267C24C22⋊Q812yes—
268C24C2×SD1612yes—
269C24C2×Q1612yes—
270C24AGL(1,ℤ/8)12yes—
271C24#13712yes—
272C24C22×D411yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—216no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—48no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—88no—
336C22×C4—88no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—88no—
340C22×C4—88no—
341C22×C4—84no—
342C22×C4—84no—
343C22×C4C2×C22⋊C414yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C414yes—
346C22×C4C2×C4⋊C414yes—
347C22×C4#11714yes—
348C22×C4#12114yes—
349C22×C4#12112yes—
350C22×C4C22⋊Q814yes—
351C22×C4C22⋊Q812yes—
352C22×C4C2×SD1614yes—
353C22×C4C2×Q1614yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4C2×#4112yes—
362C22×C4H(5,𝔽2)12yes—
363C22×C42−1+412yes—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—416no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—216no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—48no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—88no—
407C4⋊C4C2×C4⋊C414yes—
408C4⋊C4C2×C4⋊C414yes—
409C22⋊C4—28no—
410C22⋊C4—48no—
411C22⋊C4—84no—
412C22⋊C4#103116yes—
413C42—24no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—88no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23#38014yes—
435C23C2×#10314yes—
436C23#41714yes—
437C23#46014yes—
438C23#46214yes—
439C23#47514yes—
440C23#47714yes—
441C23C22×C22⋊C411yescharacteristic
442C23C2×#12111yescharacteristic
443C23C2×C22⋊Q811yescharacteristic
444C23#54612yes—
445C23#54712yes—
446C23C22×SD1611yescharacteristic
447C23C22×Q1611yescharacteristic
448C23C2×AGL(1,ℤ/8)11yescharacteristic
449C23C2×#13711yescharacteristic
450C23#57612yes—
451C23#57712yes—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—816no—
492C2×C4—816no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4#38014yes—
512C2×C4C2×#10318yes—
513C2×C4C22×C4⋊C411yescharacteristic
514C2×C4C2×#11711yescharacteristic
515C2×C4C4×#4112yes—
516C2×C4#52012yes—
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×C22⋊Q811yescharacteristic
519C2×C4#52511yescharacteristic
520C2×C4#52511yescharacteristic
521C2×C4#53312yes—
522C2×C4#53511yescharacteristic
523C2×C4#53611yescharacteristic
524C2×C4#54412yes—
525C2×C4#54612yes—
526C2×C4#54712yes—
527C2×C4#55112yes—
528C2×C4#55112yes—
529C2×C4#55212yes—
530C2×C4#55212yes—
531C8—44no—
532C8—81no—
533C8—81no—
534C22—24no—
535C22—24no—
536C22—28no—
537C22—24no—
538C22—28no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—28no—
543C22—28no—
544C22#180318yes—
545C22C2×#38011yescharacteristic
546C22#233912yes—
547C22#241111yescharacteristic
548C22#241311yescharacteristic
549C22#243812yes—
550C22C22×#10311yescharacteristic
551C22C2×#41711yescharacteristic
552C22#289712yes—
553C22C2×#46011yescharacteristic
554C22C2×#46211yescharacteristic
555C22#304111yescharacteristic
556C22#304211yescharacteristic
557C22C2×#47511yescharacteristic
558C22C2×#47711yescharacteristic
559C22#308112yes—
560C22#329512yes—
561C22#329912yes—
562C22#331812yes—
563C22#332012yes—
564C22#333512yes—
565C22#333812yes—
566C22#334512yes—
567C22#334612yes—
568C4—24no—
569C4—28no—
570C4—44no—
571C4—44no—
572C4—44no—
573C4—84no—
574C4—84no—
575C4—161no—
576C4—161no—
577C4#229611yescharacteristic
578C4#231711yescharacteristic
579C4#234312yes—
580C4#241111yescharacteristic
581C4#241311yescharacteristic
582C4#243812yes—
583C2—28no—
584C2#1396211yescharacteristic
585C2C2×#180311yescharacteristic
586C2#2065011yescharacteristic
587C2#2248712yes—
588C2#2249312yes—
589C2#2447811yescharacteristic
590C2#2448211yescharacteristic
591C2#2457811yescharacteristic
592C2#2458211yescharacteristic
593C2#2599212yes—
594C2#2599412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11228C2semidirect
7. #11207C2semidirect
8. #10849C2semidirect
9. #10807C2semidirect
18. #2345C4semidirect
91. #389C2×C4semidirect
103. C4×C4⋊C4C2×C4semidirect
222. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C2×#18502
C2×#18242
C22×#3402
#138802
#112282
#112072
#108492
#108072
#107302
#107212