Finite Group Databasebeta

← #399155

Subgroups — #399155

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#382C211yesmaximal, characteristic, complemented
2C2×#2092C212yesmaximal, complemented
3C2×#1874C211yesmaximal, characteristic
4C2×#1874C211yesmaximal, characteristic
5C2×#1853C211yesmaximal, characteristic, complemented
6C2×#1852C211yesmaximal, characteristic, complemented
7#14698C211yesmaximal, characteristic
8#11372C212yesmaximal, complemented
9#11325C212yesmaximal, complemented
10#11051C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#386—22no—
25C2×#382—22no—
26C2×#382—28no—
27C2×#382C2212yes—
28C2×#382C2212yes—
29C2×#380C2211yescharacteristic
30C22×#95—22no—
31#2092—28no—
32#1874—24no—
33#1874—24no—
34#1853—24no—
35#1852—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1437—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×D4—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—21no—
49C24×C4D412yes—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121Q814yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—24no—
81C22×C42—24no—
82C22×C42D412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—42no—
96C2×#102—42no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8D414yes—
102C2×C22⋊C8D414yes—
103#390—24no—
104#390—24no—
105#386—48no—
106#382—216no—
107#382—24no—
108#382—24no—
109#382—24no—
110#382D418yescomplemented
111#380—22no—
112#380—22no—
113C2×#95—216no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#336—24no—
124#336—24no—
125C25—21no—
126C25C2×D411yescharacteristic
127C22×D4—24no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4C2×Q812yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×Q812yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C2×D414yes—
191C22×C8C2×D414yes—
192#121—28no—
193#121—28no—
194#121—44no—
195#121—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×Q812yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C42—216no—
244C2×C42—216no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C2×D414yes—
255C2×C42#4114yes—
256#107—48no—
257#106—48no—
258#102—48no—
259#102—48no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262#95—216no—
263#95—216no—
264#95—44no—
265#95—44no—
266#95—42no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24C22⋊Q812yes—
285C24#12312yes—
286C24#12712yes—
287C24#13514yes—
288C24AGL(1,ℤ/8)14yes—
289C24C22×D411yescharacteristic
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—48no—
295C2×D4—416no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—88no—
309C2×D4—84no—
310C2×D4—84no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—216no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—416no—
352C22×C4—416no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4#12014yes—
382C22×C4C22⋊Q814yes—
383C22×C4C22⋊Q814yes—
384C22×C4C22⋊Q812yes—
385C22×C4#12314yes—
386C22×C4#12314yes—
387C22×C4#12314yes—
388C22×C4#12312yes—
389C22×C4#12712yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×Q811yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C42−1+412yes—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
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—48no—
421C4⋊C4—48no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C22⋊C4—48no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—84no—
436C42—216no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—81no—
482C23#39914yes—
483C23#47818yes—
484C23#48218yes—
485C23#49514yes—
486C23#49614yes—
487C23C2×#12011yescharacteristic
488C23C2×C22⋊Q811yescharacteristic
489C23C2×#12311yescharacteristic
490C23C2×#12311yescharacteristic
491C23C2×#12711yescharacteristic
492C23#53511yescharacteristic
493C23#53611yescharacteristic
494C23C2×#13512yes—
495C23C2×AGL(1,ℤ/8)12yes—
496C23#57512yes—
497C23#57512yes—
498D4—44no—
499D4—44no—
500D4—816no—
501D4—82no—
502D4—82no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—432no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—816no—
535C2×C4—816no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4#39914yes—
556C2×C4C2×C22⋊Q811yescharacteristic
557C2×C4C2×C22⋊Q811yescharacteristic
558C2×C4C2×#12311yescharacteristic
559C2×C4C2×#12311yescharacteristic
560C2×C4#52512yes—
561C2×C4#52512yes—
562C2×C4#53912yes—
563C2×C4#53912yes—
564C2×C4D4212yes—
565C2×C4D4212yes—
566C2×C4#54712yes—
567C2×C4#54712yes—
568C2×C4#54812yes—
569C2×C4D4×Q812yes—
570C2×C4#55312yes—
571C2×C4#55412yes—
572C8—44no—
573C8—44no—
574C22—24no—
575C22—24no—
576C22—28no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—28no—
582C22—28no—
583C22—28no—
584C22—84no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22—161no—
590C22—161no—
591C22—161no—
592C22—161no—
593C22—162no—
594C22—162no—
595C22#209218yescomplemented
596C22C2×#39911yescharacteristic
597C22#240911yescharacteristic
598C22#243311yescharacteristic
599C22#248711yescharacteristic
600C22#250111yescharacteristic
601C22#262311yescharacteristic
602C22#284911yescharacteristic
603C22#300212yes—
604C22#300612yes—
605C22C2×#47812yes—
606C22#308012yes—
607C22C2×#48212yes—
608C22#309112yes—
609C22#309312yes—
610C22#309412yes—
611C22C2×#49511yescharacteristic
612C22C2×#49611yescharacteristic
613C22#315311yescharacteristic
614C22#315311yescharacteristic
615C22#321112yes—
616C22#321312yes—
617C22#323212yes—
618C22#323412yes—
619C4—24no—
620C4—44no—
621C4—44no—
622C4—84no—
623C4—84no—
624C4—82no—
625C4—161no—
626C4—161no—
627C4#244312yes—
628C4#244312yes—
629C4#244912yes—
630C4#244912yes—
631C2—28no—
632C2—162no—
633C2#1477711yescharacteristic
634C2C2×#209211yescharacteristic
635C2#2373311yescharacteristic
636C2#2430512yes—
637C2#2572411yescharacteristic
638C2#2573111yescharacteristic
639C2#2768412yes—
640C2#2791211yescharacteristic
641C2#2791811yescharacteristic
642C2#2991212yes—
643C2#3298112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#382C2semidirect
2. C2×#2092C2semidirect
5. C2×#1853C2semidirect
6. C2×#1852C2semidirect
8. #11372C2semidirect
9. #11325C2semidirect
10. #11051C2semidirect
11. #10712C2semidirect
110. #382D4semidirect
595. C22#2092semidirect

Maximal subgroups

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

subgroupindex
C22×#3822
C2×#20922
C2×#18742
C2×#18532
C2×#18522
#146982
#113722
#113252
#110512
#107122