Finite Group Databasebeta

← #204367

Subgroups — #204367

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#21849C211yesmaximal, characteristic, complemented
3#21788C211yesmaximal, characteristic, complemented
4C2×#1569C211yesmaximal, characteristic, complemented
5C2×#1488C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic, complemented
7C2×#1453C211yesmaximal, characteristic, complemented
8#9666C211yesmaximal, characteristic, complemented
9#9665C211yesmaximal, characteristic, complemented
10#9663C211yesmaximal, characteristic, complemented
11#9661C211yesmaximal, characteristic, complemented
12#9660C211yesmaximal, characteristic, complemented
13#9659C211yesmaximal, characteristic, complemented
14#9657C211yesmaximal, characteristic, complemented
15#9655C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C4×AGL(1,ℤ/8)—24no—
20C4×AGL(1,ℤ/8)—24no—
21C2×#442C411yescharacteristic
22C2×#442C411yescharacteristic
23C2×#440C411yescharacteristic
24C2×#440C411yescharacteristic
25C2×C4×SD16C2211yescharacteristic
26C2×C4×SD16C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×C4×D8C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#428C411yescharacteristic
33C2×#423C2211yescharacteristic
34C2×#417C411yescharacteristic
35C2×#418C411yescharacteristic
36C2×C4×M16C2211yescharacteristic
37#1984C2211yescharacteristic, complemented
38#1984C2211yescharacteristic, complemented
39#1955—22no—
40#1954C2211yescharacteristic
41#1954C2211yescharacteristic
42#1954C2211yescharacteristic
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic, complemented
45#1954C2211yescharacteristic, complemented
46#1870—22no—
47#1797—22no—
48#1779—22no—
49#1569—24no—
50C2×#335C2211yescharacteristic, complemented
51C2×#335C2211yescharacteristic, complemented
52C2×#334C2211yescharacteristic, complemented
53C2×#334C2211yescharacteristic, complemented
54#1488—24no—
55#1486—24no—
56C2×Q8⋊C8C2211yescharacteristic, complemented
57C2×Q8⋊C8C2211yescharacteristic, complemented
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic, complemented
63C2×D4⋊C8C2211yescharacteristic, complemented
64#1453—24no—
65C2×#322C2211yescharacteristic, complemented
66C2×#322C2211yescharacteristic, complemented
67C2×#322C2211yescharacteristic, complemented
68C2×C82C2211yescharacteristic, complemented
69C22×#41C2×C411yescharacteristic
70C23×D4C2×C411yescharacteristic
71C24×C4—21no—
72C2×AGL(1,ℤ/8)—24no—
73C2×AGL(1,ℤ/8)—24no—
74C2×AGL(1,ℤ/8)—24no—
75C22×SD16C2×C411yescharacteristic
76C22×SD16C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78C22×D8C2×C411yescharacteristic
79C22×M16—21no—
80C22×M16C2×C411yescharacteristic
81C22×M16C2×C411yescharacteristic
82C4×#41—22no—
83C4×#41—22no—
84C4×#41—22no—
85C4×#41—22no—
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2311yescharacteristic
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4—21no—
95C2×C4×D4—22no—
96C2×C4×D4C2×C411yescharacteristic
97C2×C4×D4C2×C411yescharacteristic
98C2×C4×D4C2×C411yescharacteristic
99C2×C4×D4C2311yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×#117—21no—
103C2×#117C2×C411yescharacteristic
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C22⋊C4—21no—
106C22×C42—21no—
107C22×C42C2311yescharacteristic
108#442—24no—
109#442—24no—
110#440—24no—
111#440—24no—
112C4×SD16—22no—
113C4×SD16—22no—
114C4×SD16—22no—
115C4×SD16—22no—
116C4×D8—22no—
117C4×D8—22no—
118C4×D8—22no—
119C4×D8—22no—
120#432—22no—
121#432—22no—
122#431—22no—
123#431—22no—
124#431—22no—
125#431D414yes—
126#428—24no—
127C2×#107C2×C411yescharacteristic
128C2×#107C2×C411yescharacteristic
129C2×#106C2×C411yescharacteristic
130C2×#106C2×C411yescharacteristic
131#423—22no—
132#423—22no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8D412yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140#418—24no—
141#417—24no—
142C2×#103C2×C411yescharacteristic
143C2×#103C2×C411yescharacteristic
144C2×#102C2×C411yescharacteristic
145C2×#102C2×C411yescharacteristic
146C2×#102C2×C411yescharacteristic
147C2×#102C2×C411yescharacteristic
148C2×#102C2×C411yescharacteristic
149C2×#102C2×C411yescharacteristic
150C2×C22⋊C8—21no—
151C2×C22⋊C8—21no—
152C2×C22⋊C8—22no—
153C2×C22⋊C8—22no—
154C2×C22⋊C8—22no—
155C2×C22⋊C8—22no—
156C2×C22⋊C8—22no—
157C2×C22⋊C8—22no—
158C2×C22⋊C8D412yes—
159C4×M16D414yes—
160C4×M16D414yes—
161C2×#97—21no—
162C2×#97C2×C411yescharacteristic
163C2×#97C2×C411yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170#336—22no—
171#336—22no—
172#335—22no—
173#335—22no—
174#334—22no—
175#334—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178D4⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183D4⋊C8—22no—
184#322—22no—
185#322—22no—
186#322—22no—
187C82—22no—
188C25—21no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—22no—
193C2×#41—41no—
194C2×#41—41no—
195C22×Q8C22×C411yescharacteristic
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—21no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4C22×C411yescharacteristic
205C22×D4C22×C411yescharacteristic
206C22×D4C22×C411yescharacteristic
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—82no—
218C23×C4—82no—
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4#4111yescharacteristic
222AGL(1,ℤ/8)—232no—
223C2×SD16—22no—
224C2×SD16—22no—
225C2×SD16—22no—
226C2×SD16—22no—
227C2×SD16—22no—
228C2×SD16M1614yescomplemented
229C2×D8—22no—
230C2×D8—22no—
231C2×D8—22no—
232C2×D8—22no—
233C2×D8—22no—
234C2×D8M1614yescomplemented
235C2×M16—22no—
236C2×M16—22no—
237C2×M16—41no—
238C2×M16—41no—
239C2×M16—42no—
240C2×M16—82no—
241C2×M16C22⋊C414yes—
242C2×M16C22⋊C414yes—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8C22×C411yescharacteristic
250C22×C8C22×C411yescharacteristic
251C22×C8C22×C411yescharacteristic
252C22×C8C22×C411yescharacteristic
253C22×C8C2×D412yes—
254C22×C8C2×D412yes—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—42no—
280C4×D4—42no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—42no—
287#117—22no—
288#117—22no—
289#117—42no—
290#117—42no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C22×C411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42#4112yes—
320C2×C42#4111yescharacteristic
321C2×C42C2411yescharacteristic
322#107—22no—
323#107—22no—
324#106—22no—
325#106—22no—
326C4⋊C8—24no—
327C4⋊C8—24no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—42no—
337#103—22no—
338#103—22no—
339#102—22no—
340#102—22no—
341#102—22no—
342#102—22no—
343#102—22no—
344#102—22no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347C22⋊C8—42no—
348C22⋊C8—82no—
349C22⋊C8—82no—
350C22⋊C8—82no—
351C22⋊C8—82no—
352C22⋊C8—82no—
353C22⋊C8—82no—
354#97—42no—
355#97C22⋊C414yes—
356#97C22⋊C414yes—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8C2×D412yes—
364C4×C8C2×D412yes—
365C4×C8C2×D412yes—
366C4×C8C2×D412yes—
367C4×C8C2×D412yes—
368C4×C8C2×D412yes—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—42no—
373C24—42no—
374C24—44no—
375C24—42no—
376C24—82no—
377C24C4×D411yescharacteristic
378#41—28no—
379#41—44no—
380#41—44no—
381#41—44no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8C2×M1612yes—
388C2×D4—28no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4C2×M1612yes—
414C2×D4C2×M1612yes—
415C2×D4C2×M1612yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—88no—
453C22×C4—82no—
454C22×C4C2×C22⋊C412yes—
455C22×C4C4×D412yes—
456C22×C4C4×D411yescharacteristic
457C22×C4C4×D411yescharacteristic
458C22×C4C4×D411yescharacteristic
459C22×C4#12112yes—
460C22×C4#12112yes—
461C22×C4#12412yes—
462C22×C4#12712yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4111yescharacteristic
466SD16—28no—
467SD16—28no—
468D8—28no—
469D8—28no—
470M16—28no—
471M16—44no—
472M16—84no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8C2×C22⋊C412yes—
498C2×C8C2×C22⋊C412yes—
499C2×C8C2×C22⋊C412yes—
500C2×C8C2×C22⋊C412yes—
501C2×C8C2×M1612yes—
502C2×C8C2×M1612yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C42—41no—
526C42—41no—
527C42—42no—
528C42—42no—
529C42C2×C22⋊C412yes—
530C42C4×D412yes—
531C42#12112yes—
532C42#12112yes—
533C42#12412yes—
534C42#12712yes—
535C42C22×D411yescharacteristic
536C42C22×D411yescharacteristic
537C42C22×D411yescharacteristic
538C42C2×#4111yescharacteristic
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—48no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—84no—
560C23—84no—
561C23—88no—
562C23—88no—
563C23#39012yes—
564C23C2×C4×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4—22no—
574D4—22no—
575D4—48no—
576D4—48no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4#39012yes—
623C2×C4#39012yes—
624C2×C4#39012yes—
625C2×C4C22×C22⋊C411yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×#12411yescharacteristic
630C2×C4C2×#12711yescharacteristic
631C2×C4C22×M1611yescharacteristic
632C2×C4C22×M1611yescharacteristic
633C2×C4#56811yescharacteristic
634C2×C4#56811yescharacteristic
635C2×C4#57612yes—
636C2×C4#57712yes—
637C8—22no—
638C8—22no—
639C8—22no—
640C8—42no—
641C8—42no—
642C8—42no—
643C8—42no—
644C8—81no—
645C8—81no—
646C8—81no—
647C8—81no—
648C8#40714yes—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—816no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22C2×#39011yescharacteristic
671C22D4×M1611yescharacteristic
672C22D4×M1611yescharacteristic
673C22#294612yes—
674C22#295512yes—
675C4—22no—
676C4—22no—
677C4—21no—
678C4—22no—
679C4—21no—
680C4—42no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4C2×#40711yescharacteristic
686C4#287911yescharacteristic
687C4D4×M1611yescharacteristic
688C4D4×M1611yescharacteristic
689C4#306212yes—
690C4#315411yescharacteristic
691C4#315511yescharacteristic
692C2—42no—
693C2—82no—
694C2—82no—
695C2—82no—
696C2#1805012yes—
697C2#1805212yes—
698C2#2119011yescharacteristic
699C2#2123511yescharacteristic
700C2#2124211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×AGL(1,ℤ/8)C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21849C2semidirect
3. #21788C2semidirect
4. C2×#1569C2semidirect
5. C2×#1488C2semidirect
6. C2×#1486C2semidirect
7. C2×#1453C2semidirect
8. #9666C2semidirect
9. #9665C2semidirect
10. #9663C2semidirect
11. #9661C2semidirect
12. #9660C2semidirect
13. #9659C2semidirect
14. #9657C2semidirect
15. #9655C2semidirect
37. #1984C22semidirect
38. #1984C22semidirect
44. #1954C22semidirect
45. #1954C22semidirect
50. C2×#335C22semidirect
51. C2×#335C22semidirect
52. C2×#334C22semidirect
53. C2×#334C22semidirect
56. C2×Q8⋊C8C22semidirect
57. C2×Q8⋊C8C22semidirect
62. C2×D4⋊C8C22semidirect
63. C2×D4⋊C8C22semidirect
65. C2×#322C22semidirect
66. C2×#322C22semidirect
67. C2×#322C22semidirect
68. C2×C82C22semidirect
228. C2×SD16M16semidirect
234. C2×D8M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×AGL(1,ℤ/8)2
#218492
#217882
C2×#15692
C2×#14882
C2×#14862
C2×#14532
#96662
#96652
#96632
#96612
#96602
#96592
#96572
#96552