Finite Group Databasebeta

← #416951

Subgroups — #416951

Order 512.

1452 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic, complemented
3C2×#1836C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5C2×#1762C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#11189C212yesmaximal
10#11189C212yesmaximal
11#10850C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18#2580—24no—
19#2580—24no—
20C2×#399—21no—
21C2×#393—21no—
22C2×#391—21no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#385—21no—
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1985—24no—
33#1836—24no—
34#1802—24no—
35#1762—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—21no—
45#1441—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C23×Q8D411yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#97—22no—
97C2×#97—22no—
98C2×#97D414yes—
99#399—28no—
100#393—28no—
101#391—44no—
102#390—22no—
103#390—22no—
104#390—22no—
105#390—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#385—28no—
111#385—28no—
112#384—22no—
113#384—22no—
114#382—22no—
115#382D414yes—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#336—24no—
134#336—24no—
135C25—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—41no—
142C22×Q8—41no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4C2×D412yes—
152C22×D4#4114yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×D414yes—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8—44no—
187C4×Q8—44no—
188C4×D4—28no—
189C4×D4—28no—
190C4×D4—44no—
191C4×D4—44no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4#4114yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270C4⋊C8—44no—
271#102—216no—
272#102—48no—
273#97—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—44no—
279#95—42no—
280#95—44no—
281#95—82no—
282#95—82no—
283C24—22no—
284C24—24no—
285C24—41no—
286C24—41no—
287C24—42no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—42no—
293C24—42no—
294C24C22×D411yescharacteristic
295C2×Q8—28no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4#12414yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4#12012yes—
383C22×C4#12012yes—
384C22×C4#12112yes—
385C22×C4#12112yes—
386C22×C4#12312yes—
387C22×C4#12312yes—
388C22×C4AGL(1,ℤ/8)12yes—
389C22×C4#13712yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C42−1+412yes—
399C22×C42−1+412yes—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—84no—
453C42#12114yes—
454C42#12114yes—
455C42#12314yes—
456C42#12414yes—
457C23—24no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—81no—
471C23—81no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—81no—
478C23—81no—
479C23—82no—
480C23C2×#12011yescharacteristic
481C23C2×#12111yescharacteristic
482C23C2×#12311yescharacteristic
483C23#54812yes—
484C23#54812yes—
485C23C2×AGL(1,ℤ/8)11yescharacteristic
486C23C2×#13711yescharacteristic
487C23#57511yescharacteristic
488C23#57511yescharacteristic
489C23#57612yes—
490C23#57712yes—
491Q8—416no—
492D4—24no—
493D4—24no—
494D4—42no—
495D4—44no—
496D4—44no—
497D4—42no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—816no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4#39414yes—
563C2×C4#39414yes—
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12311yescharacteristic
567C2×C4#52512yes—
568C2×C4C2×#12411yescharacteristic
569C2×C4C2×#12411yescharacteristic
570C2×C4#52912yes—
571C2×C4#53211yescharacteristic
572C2×C4#53612yes—
573C2×C4#53611yescharacteristic
574C2×C4#53611yescharacteristic
575C2×C4#54112yes—
576C2×C4#54712yes—
577C2×C4#54812yes—
578C2×C4#55212yes—
579C2×C4#55212yes—
580C2×C4#55212yes—
581C2×C4#55612yes—
582C8—44no—
583C8—82no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—84no—
588C22—84no—
589C22—82no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22—82no—
595C22C2×#39411yescharacteristic
596C22#242112yes—
597C22#243211yescharacteristic
598C22#243611yescharacteristic
599C22#253612yes—
600C22#300611yescharacteristic
601C22#300711yescharacteristic
602C22#300811yescharacteristic
603C22#300911yescharacteristic
604C22#305511yescharacteristic
605C22#305611yescharacteristic
606C22#306212yes—
607C22#309211yescharacteristic
608C22#309311yescharacteristic
609C22#309612yes—
610C22#336612yes—
611C22#337012yes—
612C22#337012yes—
613C22#337112yes—
614C22#337212yes—
615C22#337312yes—
616C22#337512yes—
617C22#337612yes—
618C4—24no—
619C4—24no—
620C4—42no—
621C4—42no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—81no—
626C4—81no—
627C4—82no—
628C4—162no—
629C4#241512yes—
630C4#248912yes—
631C4#249911yescharacteristic
632C4#250311yescharacteristic
633C4#262111yescharacteristic
634C4#285011yescharacteristic
635C2—84no—
636C2#1523611yescharacteristic
637C2#2348211yescharacteristic
638C2#2348311yescharacteristic
639C2#2500212yes—
640C2#2500312yes—
641C2#2565911yescharacteristic
642C2#2566011yescharacteristic
643C2#2594511yescharacteristic
644C2#2594611yescharacteristic
645C2#2819712yes—
646C2#2820012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1985C2semidirect
3. C2×#1836C2semidirect
6. #13926C2semidirect
7. #11293C2semidirect
8. #11292C2semidirect
11. #10850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#19852
C2×#18362
C2×#18022
C2×#17622
#139262
#112932
#112922
#111892
#108502