Finite Group Databasebeta

← #394887

Subgroups — #394887

Order 512.

1868 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
1C22×#476C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic, complemented
4C2×#1862C212yesmaximal
5#15128C211yesmaximal, characteristic, complemented
6#11327C212yesmaximal, complemented
7#11325C212yesmaximal, complemented
8#11228C211yesmaximal, characteristic, complemented
9#11202C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#476—28no—
17C2×#476—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C2×#395—21no—
24#2349—24no—
25#2349—24no—
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#385—21no—
30C2×#385—21no—
31C2×#382C2211yescharacteristic
32C2×#380—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C22×#95—21no—
35#2057—24no—
36#1862—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4—21no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—42no—
71C2×C4×D4—42no—
72C2×C4×D4D414yes—
73C2×C4×D4Q814yescomplemented
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#476—232no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—42no—
95C2×#106—24no—
96C2×#106—24no—
97C2×#106—24no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—22no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102D418yes—
105C2×#102Q818yescomplemented
106#395—44no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—24no—
111#389—42no—
112#389D414yes—
113#385—28no—
114#385—28no—
115#382—22no—
116#382—22no—
117#380—28no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C2×#95—28no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132#340—24no—
133#340—24no—
134#340—24no—
135#340—42no—
136#336—24no—
137#336—22no—
138#336—24no—
139C25—21no—
140C22×Q8—44no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—82no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C2×D412yes—
151C22×D4C2×D414yes—
152C22×D4C2×Q812yes—
153C22×D4C2×Q814yes—
154C22×D4#4112yes—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—82no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
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×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8C2×D414yes—
195C22×C8C2×Q814yes—
196C4⋊Q8—28no—
197C4⋊Q8—48no—
198C4⋊Q8—44no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—28no—
203C4×D4—28no—
204C4×D4—44no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—44no—
208C4×D4—44no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
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—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×Q814yes—
248C2×C4⋊C4C2×Q812yes—
249C2×C4⋊C4#4112yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×Q812yes—
270C2×C42#4112yes—
271#107—48no—
272#106—216no—
273C4⋊C8—28no—
274#102—216no—
275#102—216no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—24no—
281#95—22no—
282#95—44no—
283#95—44no—
284#95—42no—
285#95—42no—
286#95—82no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—42no—
297C24—42no—
298C24C22×D411yescharacteristic
299C2×Q8—48no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—88no—
303C2×Q8—88no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4#12014yes—
318C2×D4C22⋊Q818yes—
319C2×D4C22⋊Q814yes—
320C2×D4C4⋊Q814yes—
321C2×D4C4⋊Q818yes—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—216no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—48no—
381C22×C4—84no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—88no—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4C22⋊Q812yes—
394C22×C4C22⋊Q812yes—
395C22×C4#13512yes—
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4#13712yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×Q811yescharacteristic
402C22×C4C22×Q812yes—
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C4⋊C4C22⋊Q814yes—
446C22⋊C4—216no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—24no—
456C42—41no—
457C42—41no—
458C42—42no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—28no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23C2×#12011yescharacteristic
483C23C2×C22⋊Q811yescharacteristic
484C23#52511yescharacteristic
485C23D4212yes—
486C23D4×Q812yes—
487C23C2×#13511yescharacteristic
488C23C2×AGL(1,ℤ/8)11yescharacteristic
489C23C2×#13711yescharacteristic
490C23#57511yescharacteristic
491C23#57612yes—
492C23#57712yes—
493Q8—816no—
494D4—24no—
495D4—24no—
496D4—24no—
497D4—28no—
498D4—28no—
499D4#39518yescomplemented
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—816no—
549C2×C4—816no—
550C2×C4—816no—
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—82no—
563C2×C4#45214yes—
564C2×C4C2×#12011yescharacteristic
565C2×C4C2×C22⋊Q811yescharacteristic
566C2×C4C2×C22⋊Q812yes—
567C2×C4C2×C22⋊Q811yescharacteristic
568C2×C4#52511yescharacteristic
569C2×C4#52511yescharacteristic
570C2×C4C2×C4⋊Q811yescharacteristic
571C2×C4C2×C4⋊Q812yes—
572C2×C4#53411yescharacteristic
573C2×C4#53611yescharacteristic
574C2×C4#53912yes—
575C2×C4#54412yes—
576C2×C4D4212yes—
577C2×C4#54712yes—
578C2×C4#54712yes—
579C2×C4D4×Q812yes—
580C2×C4#55412yes—
581C2×C4#55412yes—
582C8—44no—
583C8—44no—
584C22—28no—
585C22—48no—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—48no—
592C22—48no—
593C22—44no—
594C22—44no—
595C22#240311yescharacteristic
596C22#243411yescharacteristic
597C22#244012yes—
598C22#244912yes—
599C22#284811yescharacteristic
600C22C2×#45211yescharacteristic
601C22#300611yescharacteristic
602C22#300811yescharacteristic
603C22#300911yescharacteristic
604C22#307611yescharacteristic
605C22#307911yescharacteristic
606C22#308112yes—
607C22#310611yescharacteristic
608C22#311611yescharacteristic
609C22#311812yes—
610C22#326412yes—
611C22#326612yes—
612C22#327312yes—
613C22#328212yes—
614C22#338012yes—
615C22#338412yes—
616C22#338512yes—
617C22#338812yes—
618C4—24no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—44no—
624C4—84no—
625C4—81no—
626C4—82no—
627C4—82no—
628C4—84no—
629C4C2×#39511yescharacteristic
630C4#240611yescharacteristic
631C4#244012yes—
632C4#244912yes—
633C4#258311yescharacteristic
634C4#262711yescharacteristic
635C2—48no—
636C2#1472011yescharacteristic
637C2#2385211yescharacteristic
638C2#2386111yescharacteristic
639C2#2583911yescharacteristic
640C2#2585111yescharacteristic
641C2#2605112yes—
642C2#2605812yes—
643C2#2638012yes—
644C2#2639612yes—
645C2#3295811yescharacteristic
646C2#3296211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2057C2semidirect
5. #15128C2semidirect
6. #11327C2semidirect
7. #11325C2semidirect
8. #11228C2semidirect
9. #11202C2semidirect
73. C2×C4×D4Q8semidirect
105. C2×#102Q8semidirect
499. D4#395semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#23492
C2×#20572
C2×#18622
#151282
#113272
#113252
#112282
#112022
#107882