Finite Group Databasebeta

← #394860

Subgroups — #394860

Order 512.

1932 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3C2×#2057C212yesmaximal, complemented
4C2×#1852C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11324C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#107—28no—
18C22×#107—24no—
19C22×#107C2212yes—
20C22×#107C2212yes—
21C22×#107C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23#2440—24no—
24#2440—24no—
25C2×#398—21no—
26C2×#395—21no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#385—21no—
31C2×#385—21no—
32C2×#384C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#2057—28no—
37#1852—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1442—21no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C23×Q8—21no—
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—24no—
56C23×C8—24no—
57C23×C8Q814yescomplemented
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8—42no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—44no—
66C2×C22⋊Q8—44no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4D414yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C2×#107—232no—
96C2×#107—24no—
97C2×#107—24no—
98C2×#107—24no—
99C2×#107—24no—
100C2×#107—24no—
101C2×#107—24no—
102C2×#107—24no—
103C2×#107—42no—
104C2×#107D418yes—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8Q814yescomplemented
108#398—28no—
109#395—28no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—24no—
115#389—42no—
116#385—28no—
117#385—44no—
118#384—22no—
119#384—22no—
120#380—22no—
121#380—22no—
122#380D414yes—
123#380Q814yescomplemented
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—42no—
140#336—24no—
141#336—24no—
142C25C2×D411yescharacteristic
143C22×Q8—24no—
144C22×Q8—44no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—82no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—82no—
167C23×C4—82no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×Q812yes—
176C23×C4C2×Q812yes—
177C23×C4C2×Q812yes—
178C23×C4#4112yes—
179C23×C4C2411yescharacteristic
180C22×C8—216no—
181C22×C8—28no—
182C22×C8—28no—
183C22×C8—28no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—216no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C2×D414yes—
196C22×C8C2×Q814yes—
197C22×C8C2×Q814yes—
198C22×C8#4114yes—
199C4⋊Q8—48no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C22⋊Q8—432no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
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—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—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×Q814yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4D814yes—
268C2×C22⋊C4Q1614yes—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277#107—216no—
278#107—216no—
279#107—48no—
280C22⋊C8—28no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—44no—
288#95—44no—
289#95—82no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24#12012yes—
295C24C22⋊Q812yes—
296C24C4⋊Q812yes—
297C24C2×D812yes—
298C24C2×Q1612yes—
299C24AGL(1,ℤ/8)12yes—
300C24#13712yes—
301C24C22×D411yescharacteristic
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—88no—
305C2×Q8—84no—
306C2×Q8—88no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
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—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—48no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—48no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—48no—
376C22×C4—84no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4#12012yes—
384C22×C4C22⋊Q814yes—
385C22×C4C22⋊Q814yes—
386C22×C4C22⋊Q812yes—
387C22×C4C4⋊Q814yes—
388C22×C4C4⋊Q814yes—
389C22×C4C4⋊Q812yes—
390C22×C4C2×D812yes—
391C22×C4C2×D812yes—
392C22×C4C2×Q1612yes—
393C22×C4C2×Q1612yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C22×Q811yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C2×C8—232no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
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×C8C22⋊Q818yescomplemented
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—88no—
445C4⋊C4—84no—
446C4⋊C4#12014yes—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—42no—
456C42—42no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—24no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23#39514yes—
474C23#44714yes—
475C23#45114yes—
476C23#47414yes—
477C23#47714yes—
478C23#50014yes—
479C23#50114yes—
480C23C2×#12011yescharacteristic
481C23C2×C22⋊Q811yescharacteristic
482C23C2×C4⋊Q811yescharacteristic
483C23D4212yes—
484C23#54712yes—
485C23C22×D811yescharacteristic
486C23C22×Q1611yescharacteristic
487C23C2×AGL(1,ℤ/8)11yescharacteristic
488C23C2×#13711yescharacteristic
489C23#57612yes—
490C23#57812yes—
491Q8—816no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—816no—
543C2×C4—816no—
544C2×C4—82no—
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#39514yes—
559C2×C4C2×#12011yescharacteristic
560C2×C4C2×C22⋊Q811yescharacteristic
561C2×C4C2×C22⋊Q812yes—
562C2×C4C2×C22⋊Q811yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52512yes—
565C2×C4C2×C4⋊Q811yescharacteristic
566C2×C4C2×C4⋊Q811yescharacteristic
567C2×C4#53411yescharacteristic
568C2×C4#53611yescharacteristic
569C2×C4#53912yes—
570C2×C4#54412yes—
571C2×C4D4212yes—
572C2×C4#54712yes—
573C2×C4D4×Q812yes—
574C2×C4D4×Q812yes—
575C2×C4#55412yes—
576C2×C4#55412yes—
577C8—28no—
578C8—44no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—28no—
586C22—28no—
587C22—28no—
588C22—28no—
589C22#205718yescomplemented
590C22C2×#39511yescharacteristic
591C22#240311yescharacteristic
592C22#240611yescharacteristic
593C22#244012yes—
594C22#244912yes—
595C22C2×#44711yescharacteristic
596C22C2×#45111yescharacteristic
597C22#300811yescharacteristic
598C22#301111yescharacteristic
599C22C2×#47411yescharacteristic
600C22C2×#47711yescharacteristic
601C22#308312yes—
602C22C2×#50011yescharacteristic
603C22C2×#50111yescharacteristic
604C22#316412yes—
605C22#326412yes—
606C22#327012yes—
607C22D4×D812yes—
608C22D4×Q1612yes—
609C22#333412yes—
610C22#333812yes—
611C22#334312yes—
612C22#334612yes—
613C4—24no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—44no—
618C4—42no—
619C4—82no—
620C4—82no—
621C4—84no—
622C4—82no—
623C4—84no—
624C4#243411yescharacteristic
625C4#244012yes—
626C4#244912yes—
627C4#258311yescharacteristic
628C4#262711yescharacteristic
629C4#284811yescharacteristic
630C2—28no—
631C2#1472011yescharacteristic
632C2C2×#205711yescharacteristic
633C2#2355511yescharacteristic
634C2#2384811yescharacteristic
635C2#2385611yescharacteristic
636C2#2408811yescharacteristic
637C2#2409811yescharacteristic
638C2#2606512yes—
639C2#2607112yes—
640C2#2633612yes—
641C2#2634212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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×#2057C2semidirect
6. #11328C2semidirect
7. #11324C2semidirect
8. #11219C2semidirect
57. C23×C8Q8semidirect
107. C2×C22⋊C8Q8semidirect
123. #380Q8semidirect
418. C2×C8C22⋊Q8semidirect
589. C22#2057semidirect

Maximal subgroups

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

subgroupindex
C23×#1072
C2×#24402
C2×#20572
C2×#18522
#141302
#113282
#113242
#112192
#107302
#107212
#107142