Finite Group Databasebeta

← #364751

Subgroups — #364751

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#11207C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10884C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10#10730C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#123—21no—
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21#2486—24no—
22#2486C418yescomplemented
23C2×#398—21no—
24C2×#397—21no—
25D4×C4⋊C4—24no—
26D4×C4⋊C4—24no—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#385—21no—
30C2×#385—21no—
31C2×#384C2211yescharacteristic
32C2×#381—21no—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1869—24no—
39#1823—24no—
40C2×#340—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4—21no—
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—42no—
61C23×C8C2312yes—
62C23×C8C2312yes—
63C2×#123—28no—
64C2×#123—44no—
65C2×#123—44no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—28no—
71C2×C4×D4—44no—
72C2×C4×D4—44no—
73C2×C4×D4C2×C414yes—
74C2×C4×D4D414yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4C2×C414yescomplemented
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4D414yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—22no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4D411yescharacteristic
107C22×C22⋊C4D411yescharacteristic
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110C2×C22⋊C8—22no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—24no—
115C2×C22⋊C8—42no—
116#398—28no—
117#397—28no—
118#389—28no—
119#389—24no—
120#389C2×C418yescomplemented
121#385—28no—
122#385—28no—
123#384—22no—
124#384—22no—
125#381—28no—
126#380—22no—
127#380—24no—
128#380—22no—
129#380—22no—
130#380—22no—
131#380C2×C418yescomplemented
132#380C2×C414yescomplemented
133#380D414yes—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—24no—
146#340—24no—
147#340—24no—
148#340—42no—
149#340—42no—
150#336—24no—
151C25C2×D411yescharacteristic
152C25C2×D411yescharacteristic
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4C22⋊C414yes—
158C22×D4D814yes—
159C22×D4SD1614yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—82no—
194C23×C4—82no—
195C23×C4C22×C412yes—
196C23×C4C22×C412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4#4112yes—
206C23×C4#4112yes—
207C23×C4C2411yescharacteristic
208C22×C8—28no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—48no—
223C22×C8—48no—
224C22×C8—44no—
225C22×C8—44no—
226C22×C8#4114yes—
227#123—432no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—432no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4C22⋊C414yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C22×C414yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4#4114yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4C22⋊C414yes—
306C2×C22⋊C4C22⋊C414yes—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—24no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C22×C412yes—
315C2×C42C2×D412yes—
316C22⋊C8—48no—
317C22⋊C8—44no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24C2×C22⋊C412yes—
337C24C2×C22⋊C412yes—
338C24#12012yes—
339C24#12312yes—
340C24C2×D812yes—
341C24C2×D812yes—
342C24C2×SD1612yes—
343C24C2×SD1612yes—
344C24AGL(1,ℤ/8)12yes—
345C24#13712yes—
346C24C22×D411yescharacteristic
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4#102116yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4C2×C22⋊C412yes—
452C22×C4C2×C22⋊C412yes—
453C22×C4C2×C22⋊C412yes—
454C22×C4C2×C22⋊C412yes—
455C22×C4C4×D414yes—
456C22×C4C4×D414yes—
457C22×C4#12012yes—
458C22×C4#12114yes—
459C22×C4#12114yes—
460C22×C4#12312yes—
461C22×C4C2×D812yes—
462C22×C4C2×SD1612yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4111yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—416no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—48no—
512C4⋊C4—416no—
513C4⋊C4—84no—
514C4⋊C4#12314yescomplemented
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—48no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—48no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—42no—
564C23#38614yes—
565C23C2×#10214yes—
566C23C2×#10214yes—
567C23#41714yes—
568C23#44714yes—
569C23#44814yes—
570C23#48014yes—
571C23#48114yes—
572C23C22×C22⋊C411yescharacteristic
573C23C2×#12011yescharacteristic
574C23C2×#12311yescharacteristic
575C23#54612yes—
576C23#54712yes—
577C23C22×D811yescharacteristic
578C23C22×SD1611yescharacteristic
579C23C2×AGL(1,ℤ/8)11yescharacteristic
580C23C2×#13711yescharacteristic
581C23#57712yes—
582C23#57812yes—
583D4—232no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—816no—
637C2×C4—816no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—816no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4#38614yes—
654C2×C4C2×#10214yes—
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C2×C4×D411yescharacteristic
657C2×C4C4×#4112yes—
658C2×C4#52012yes—
659C2×C4C2×#12111yescharacteristic
660C2×C4C2×#12111yescharacteristic
661C2×C4C2×#12311yescharacteristic
662C2×C4#52511yescharacteristic
663C2×C4#52912yes—
664C2×C4#53411yescharacteristic
665C2×C4#53611yescharacteristic
666C2×C4#53912yes—
667C2×C4#54612yes—
668C2×C4#54612yes—
669C2×C4#54712yes—
670C2×C4#54712yes—
671C2×C4#55112yes—
672C2×C4#55212yes—
673C8—44no—
674C8—82no—
675C22—24no—
676C22—24no—
677C22—28no—
678C22—28no—
679C22—24no—
680C22—24no—
681C22—28no—
682C22—28no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—28no—
687C22—42no—
688C22—42no—
689C22—44no—
690C22—44no—
691C22—42no—
692C22—42no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22#187418yes—
697C22C2×#38611yescharacteristic
698C22#233912yes—
699C22#240511yescharacteristic
700C22#240611yescharacteristic
701C22#241912yes—
702C22C22×#10211yescharacteristic
703C22C2×#41711yescharacteristic
704C22#289812yes—
705C22C2×#44711yescharacteristic
706C22C2×#44811yescharacteristic
707C22#300911yescharacteristic
708C22#301111yescharacteristic
709C22C2×#48011yescharacteristic
710C22C2×#48111yescharacteristic
711C22#309712yes—
712C22#329412yes—
713C22#329812yes—
714C22#331012yes—
715C22#331212yes—
716C22#333412yes—
717C22#333512yes—
718C22#334312yes—
719C22#334412yes—
720C4—24no—
721C4—24no—
722C4—42no—
723C4—42no—
724C4—44no—
725C4—82no—
726C4—84no—
727C4—84no—
728C4—82no—
729C4—82no—
730C4C4×#12311yescharacteristic
731C4#232211yescharacteristic
732C4#235912yes—
733C4#242512yes—
734C4#248611yescharacteristic
735C4#253411yescharacteristic
736C2—28no—
737C2—44no—
738C2#1396811yescharacteristic
739C2C2×#187411yescharacteristic
740C2#2091311yescharacteristic
741C2#2248512yes—
742C2#2249112yes—
743C2#2403211yescharacteristic
744C2#2404111yescharacteristic
745C2#2408811yescharacteristic
746C2#2409711yescharacteristic
747C2#2493312yes—
748C2#2493712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1823C2semidirect
5. C22×#340C2semidirect
6. #11207C2semidirect
7. #11199C2semidirect
8. #10884C2semidirect
9. #10730C2semidirect
10. #10730C2semidirect
11. #10721C2semidirect
22. #2486C4semidirect
93. C22×C4⋊C4C2×C4semidirect
120. #389C2×C4semidirect
131. #380C2×C4semidirect
132. #380C2×C4semidirect
514. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×D4×C4⋊C42
C2×#18692
C2×#18232
C22×#3402
#112072
#111992
#108842
#107302
#107212