Finite Group Databasebeta

← #189152

Subgroups — #189152

Order 512.

1094 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2974C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#20732C211yesmaximal, characteristic
4#20646C211yesmaximal, characteristic
5C2×#1574C211yesmaximal, characteristic, complemented
6#8656C211yesmaximal, characteristic, complemented
7#8654C211yesmaximal, characteristic, complemented
8#8652C211yesmaximal, characteristic, complemented
9#8650C211yesmaximal, characteristic, complemented
10#8648C211yesmaximal, characteristic, complemented
11#8646C211yesmaximal, characteristic, complemented
12#8644C211yesmaximal, characteristic, complemented
13#8642C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic, complemented
15#8042C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17#2974—24no—
18#2974C418yescomplemented
19C2×#436C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#435—21no—
22C2×#435—21no—
23C2×C8×D4C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#418—21no—
30C2×#407—21no—
31C22×C22⋊C8—21no—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34#1949—21no—
35#1949—21no—
36#1854—21no—
37#1854—21no—
38#1845C2211yescharacteristic, complemented
39#1844C2211yescharacteristic, complemented
40#1834C2211yescharacteristic, complemented
41#1834C2211yescharacteristic, complemented
42#1823—21no—
43#1804—21no—
44#1804—21no—
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53#1786—21no—
54C4×#102—21no—
55C4×#102—21no—
56#1736—21no—
57#1574—24no—
58C2×#335C2211yescharacteristic, complemented
59C2×#335C2211yescharacteristic, complemented
60C2×#334C2211yescharacteristic, complemented
61C2×#334C2211yescharacteristic, complemented
62#1296C2211yescharacteristic, complemented
63#1296C2211yescharacteristic, complemented
64#1296C2211yescharacteristic, complemented
65#1296C2211yescharacteristic, complemented
66#1296C2211yescharacteristic, complemented
67#1296C2211yescharacteristic, complemented
68#1296C2211yescharacteristic, complemented
69#1296C2211yescharacteristic, complemented
70C23×D4D411yescharacteristic
71C24×C4—21no—
72C22×M16—21no—
73C22×M16—21no—
74C22×M16—21no—
75C23×C8—21no—
76C23×C8—21no—
77C2×C4×D4—21no—
78C2×C4×D4—22no—
79C2×C4×D4—21no—
80C2×C4×D4—22no—
81C2×C4×D4C2×C414yes—
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4Q814yes—
85C2×C4×D4C2311yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C2×#117—21no—
89C22×C4⋊C4D411yescharacteristic
90C22×C22⋊C4—21no—
91C22×C42C2311yescharacteristic
92#436—22no—
93#436—22no—
94#436C2×C414yes—
95#436C2×C414yes—
96#435—28no—
97#435—28no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4C2×C414yes—
101C8×D4C2×C414yes—
102#432—22no—
103#432C2×C414yes—
104#431—22no—
105#431—22no—
106#431C2×C414yes—
107#431D414yes—
108C2×#107—21no—
109C2×#107—21no—
110C2×#106—21no—
111C2×#106—21no—
112#423—22no—
113#423—22no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8D411yescharacteristic
116C2×C4⋊C8D411yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123#418—44no—
124C2×#102—21no—
125C2×#102—21no—
126C2×#102—21no—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×#102—21no—
131C2×#102—21no—
132#407—28no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—28no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×#97—21no—
143C2×C4×C8—21no—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382—21no—
151#382—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—21no—
171#340—21no—
172#335—22no—
173#335—22no—
174#334—22no—
175#334—22no—
176C25—21no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4C4⋊C414yes—
184C22×D4C2×D411yescharacteristic
185C22×D4C2×D411yescharacteristic
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—24no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—24no—
193C23×C4—81no—
194C23×C4—81no—
195C23×C4—81no—
196C23×C4—81no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4#4111yescharacteristic
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16—81no—
206C2×M16—81no—
207C2×M16—81no—
208C2×M16—81no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—24no—
212C22×C8—24no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—81no—
218C22×C8—81no—
219C22×C8—81no—
220C22×C8—81no—
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4C4⋊C414yes—
236C4×D4C4⋊C414yes—
237C4×D4C22×C412yes—
238C4×D4C22×C412yes—
239C4×D4C2×Q812yes—
240C4×D4C2×Q812yes—
241#117—42no—
242#117—42no—
243#117—81no—
244#117—81no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C4⋊C414yes—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C22⋊C4—24no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42C22×C412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×Q812yes—
277C2×C42#4111yescharacteristic
278C2×C42#4112yes—
279C2×C42C2411yescharacteristic
280#107—81no—
281#107—81no—
282#106—81no—
283#106—81no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—42no—
295C4⋊C8C22⋊C414yes—
296C4⋊C8C22⋊C414yes—
297C4⋊C8C22×C412yes—
298C4⋊C8C22×C412yes—
299#102—42no—
300#102—42no—
301#102—42no—
302#102—42no—
303#102—42no—
304#102—42no—
305#102—42no—
306#102—42no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—24no—
312C22⋊C8—42no—
313C22⋊C8C22⋊C414yes—
314C22⋊C8C22⋊C414yes—
315#97—24no—
316C4×C8—42no—
317C4×C8C22×C412yes—
318C4×C8C22×C412yes—
319C4×C8C2×D412yes—
320C4×C8C2×D412yes—
321#95—41no—
322#95—41no—
323#95—41no—
324#95—41no—
325C24—21no—
326C24—21no—
327C24—24no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—81no—
333C24—81no—
334C24#12111yescharacteristic
335C2×D4—28no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—24no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4C2×C4⋊C412yes—
351C2×D4C2×C4⋊C412yes—
352C2×D4AGL(1,ℤ/8)12yes—
353C2×D4#13712yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C4⋊C412yes—
397C22×C4#11712yes—
398C22×C4#12112yes—
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4C22⋊Q812yes—
403C22×C4C22⋊Q812yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4111yescharacteristic
407M16—84no—
408M16—84no—
409M16—84no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—84no—
437C2×C8—84no—
438C2×C8C2×C22⋊C412yes—
439C2×C8C2×C22⋊C412yes—
440C2×C8C2×C22⋊C412yes—
441C2×C8C2×C22⋊C412yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4C2×C4⋊C412yes—
466C4⋊C4C2×C4⋊C412yes—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C42—82no—
476C42—82no—
477C42C2×C22⋊C412yes—
478C42C2×C4⋊C412yes—
479C42#11712yes—
480C42#12112yes—
481C42C22⋊Q812yes—
482C42C22⋊Q812yes—
483C42C23×C411yescharacteristic
484C42C22×D411yescharacteristic
485C42C22×Q811yescharacteristic
486C42C2×#4111yescharacteristic
487C23—21no—
488C23—21no—
489C23—22no—
490C23—21no—
491C23—22no—
492C23—21no—
493C23—22no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—48no—
498C23—48no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23#38012yes—
515C23C2×#12111yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4—22no—
520D4—22no—
521D4—48no—
522D4—41no—
523D4—41no—
524D4#42814yes—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4#38012yes—
573C2×C4#38012yes—
574C2×C4#38012yes—
575C2×C4C22×C22⋊C411yescharacteristic
576C2×C4C22×C4⋊C411yescharacteristic
577C2×C4C2×#11711yescharacteristic
578C2×C4C2×#12111yescharacteristic
579C2×C4C2×C22⋊Q811yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
582C2×C4C2×#13711yescharacteristic
583C2×C4#57511yescharacteristic
584C2×C4#57511yescharacteristic
585C2×C4#57611yescharacteristic
586C2×C4#57611yescharacteristic
587C2×C4#57711yescharacteristic
588C2×C4#57711yescharacteristic
589C8—22no—
590C8—22no—
591C8—22no—
592C8—22no—
593C8—42no—
594C8—42no—
595C8—81no—
596C8—81no—
597C8—81no—
598C8—81no—
599C22—22no—
600C22—22no—
601C22—24no—
602C22—22no—
603C22—41no—
604C22—42no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—42no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—816no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#38011yescharacteristic
622C22#304111yescharacteristic
623C22#304211yescharacteristic
624C22#305511yescharacteristic
625C22#305611yescharacteristic
626C22#306711yescharacteristic
627C22#306711yescharacteristic
628C4—22no—
629C4—22no—
630C4—21no—
631C4—21no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—161no—
636C4—161no—
637C4—161no—
638C4—161no—
639C4#289711yescharacteristic
640C4#289711yescharacteristic
641C4C2×#42811yescharacteristic
642C4#291111yescharacteristic
643C4#305511yescharacteristic
644C4#305611yescharacteristic
645C4#308111yescharacteristic
646C4#308111yescharacteristic
647C2—42no—
648C2—44no—
649C2—82no—
650C2#1946011yescharacteristic
651C2#1946111yescharacteristic
652C2#1946611yescharacteristic
653C2#1946711yescharacteristic
654C2#2065011yescharacteristic
655C2#2066111yescharacteristic
656C2#2067811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21984C2semidirect
5. C2×#1574C2semidirect
6. #8656C2semidirect
7. #8654C2semidirect
8. #8652C2semidirect
9. #8650C2semidirect
10. #8648C2semidirect
11. #8646C2semidirect
12. #8644C2semidirect
13. #8642C2semidirect
14. #8046C2semidirect
15. #8042C2semidirect
18. #2974C4semidirect
38. #1845C22semidirect
39. #1844C22semidirect
40. #1834C22semidirect
41. #1834C22semidirect
58. C2×#335C22semidirect
59. C2×#335C22semidirect
60. C2×#334C22semidirect
61. C2×#334C22semidirect
62. #1296C22semidirect
63. #1296C22semidirect
64. #1296C22semidirect
65. #1296C22semidirect
66. #1296C22semidirect
67. #1296C22semidirect
68. #1296C22semidirect
69. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29742
#219842
#207322
#206462
C2×#15742
#86562
#86542
#86522
#86502
#86482
#86462
#86442
#86422
#80462
#80422