Finite Group Databasebeta

← #366757

Subgroups — #366757

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic, complemented
4C2×#1752C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#14137C211yesmaximal, characteristic
7#13904C211yesmaximal, characteristic, complemented
8#11377C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#10927C212yesmaximal, complemented
11#10863C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#480—24no—
17C2×#480—24no—
18C2×#480C418yescomplemented
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2212yes—
26C2×#390C2212yes—
27C2×#386—22no—
28C2×#381—22no—
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1893—24no—
32#1853—24no—
33#1752—24no—
34C2×#340—24no—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#336C2212yes—
41#1445—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×D4—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—42no—
55C23×C8D414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121C2×C414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4C2×C414yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4D412yes—
86C22×C42C2312yes—
87#480—232no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107C2×C418yescomplemented
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—42no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102C2×C418yescomplemented
101C2×C22⋊C8C2×C414yes—
102C2×C22⋊C8D414yes—
103C2×C22⋊C8D414yes—
104C2×#97—24no—
105C2×#97—24no—
106C2×#97—42no—
107#390—24no—
108#390—42no—
109#386—48no—
110#381—48no—
111#380—22no—
112#380—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—24no—
118C2×#95—21no—
119C2×#95D412yes—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—24no—
125#336—24no—
126C25—21no—
127C25C2×D411yescharacteristic
128C22×D4—24no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4C22×C412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—82no—
161C23×C4C22×C412yes—
162C23×C4C2×D414yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—28no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—28no—
183C22×C8—44no—
184C22×C8—48no—
185C22×C8—48no—
186C22×C8C22×C414yes—
187C22×C8C2×D412yes—
188C22×C8C2×D414yes—
189C22×C8C2×D414yes—
190C22×C8C2×D412yes—
191C22×C8#4114yes—
192#121—28no—
193#121—28no—
194#121—44no—
195#121—44no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4#4114yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—84no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251#107—216no—
252#106—48no—
253#102—216no—
254#102—44no—
255#102—44no—
256C22⋊C8C22⋊C4116yescomplemented
257#97—48no—
258#95—24no—
259#95—48no—
260#95—48no—
261#95—48no—
262#95—42no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24C4×D412yes—
277C24#12112yes—
278C24#12712yes—
279C24AGL(1,ℤ/8)14yes—
280C24AGL(1,ℤ/8)14yes—
281C24C22×D411yescharacteristic
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—48no—
287C2×D4—416no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—88no—
303C2×D4—84no—
304C2×D4—84no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—48no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—48no—
345C22×C4—416no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—816no—
355C22×C4—84no—
356C22×C4—88no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4C2×C22⋊C414yes—
361C22×C4C4×D414yes—
362C22×C4C4×D412yes—
363C22×C4#12114yes—
364C22×C4#12112yes—
365C22×C4#12414yes—
366C22×C4#12712yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+412yes—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—416no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8C2×C22⋊C418yes—
394C2×C8#12314yes—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—816no—
418C22⋊C4—84no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C23—28no—
424C23—28no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—28no—
433C23—28no—
434C23—24no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—84no—
440C23—84no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—82no—
446C23—81no—
447C23—81no—
448C23—82no—
449C23—84no—
450C23—84no—
451C23—82no—
452C23#39014yes—
453C23#44218yes—
454C23#46918yes—
455C23#49614yes—
456C23#49614yes—
457C23C2×C4×D411yescharacteristic
458C23C2×#12111yescharacteristic
459C23C2×#12711yescharacteristic
460C23#54612yes—
461C23#54712yes—
462C23C2×AGL(1,ℤ/8)12yes—
463C23C2×AGL(1,ℤ/8)12yes—
464C23#57712yes—
465C23#57812yes—
466D4—42no—
467D4—44no—
468D4—42no—
469D4—816no—
470D4—81no—
471D4—81no—
472D4—82no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—816no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—816no—
504C2×C4—84no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—84no—
515C2×C4#39014yes—
516C2×C4C22×C22⋊C411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C4×#4112yes—
519C2×C4#52012yes—
520C2×C4C2×#12111yescharacteristic
521C2×C4C2×#12311yescharacteristic
522C2×C4#52511yescharacteristic
523C2×C4C2×#12411yescharacteristic
524C2×C4#52912yes—
525C2×C4#53411yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#53912yes—
528C2×C4#54612yes—
529C2×C4#54612yes—
530C2×C4#54712yes—
531C2×C4#54712yes—
532C2×C4#55112yes—
533C2×C4#55212yes—
534C8—28no—
535C8—24no—
536C8—42no—
537C22—24no—
538C22—24no—
539C22—216no—
540C22—216no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—88no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—161no—
550C22—161no—
551C22—164no—
552C22—161no—
553C22—161no—
554C22#197218yes—
555C22C2×#39011yescharacteristic
556C22#235912yes—
557C22#242512yes—
558C22#250111yescharacteristic
559C22#253911yescharacteristic
560C22C2×#44212yes—
561C22#294712yes—
562C22C2×#46912yes—
563C22#306312yes—
564C22C2×#49611yescharacteristic
565C22C2×#49611yescharacteristic
566C22#315411yescharacteristic
567C22#315611yescharacteristic
568C22#330814yes—
569C22#331014yes—
570C22#334314yes—
571C22#334514yes—
572C4—24no—
573C4—84no—
574C4—84no—
575C4—84no—
576C4—82no—
577C4—161no—
578C4—161no—
579C4C4×#12311yescharacteristic
580C4#232211yescharacteristic
581C4#233912yes—
582C4#241211yescharacteristic
583C4#241311yescharacteristic
584C4#248812yes—
585C2—28no—
586C2—162no—
587C2#1399611yescharacteristic
588C2C2×#197211yescharacteristic
589C2#2123411yescharacteristic
590C2#2293814yes—
591C2#2525014yes—
592C2#2793411yescharacteristic
593C2#2793911yescharacteristic
594C2#2834411yescharacteristic
595C2#2834611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
3. C2×#1853C2semidirect
4. C2×#1752C2semidirect
5. C22×#340C2semidirect
7. #13904C2semidirect
8. #11377C2semidirect
9. #11121C2semidirect
10. #10927C2semidirect
11. #10863C2semidirect
18. C2×#480C4semidirect
90. C2×#107C2×C4semidirect
100. C2×#102C2×C4semidirect
256. C22⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C2×#18932
C2×#18532
C2×#17522
C22×#3402
#141372
#139042
#113772
#111212
#109272
#108632