Finite Group Databasebeta

← #189083

Subgroups — #189083

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×M16C211yesmaximal, characteristic
2#20764C211yesmaximal, characteristic, complemented
3#20732C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5C2×#1565C211yesmaximal, characteristic, complemented
6#8656C212yesmaximal, complemented
7#8654C212yesmaximal, complemented
8#8648C212yesmaximal, complemented
9#8646C212yesmaximal, complemented
10#8045C211yesmaximal, characteristic, complemented
11#8043C211yesmaximal, characteristic, complemented
12C23×M16—21no—
13C22×C4×D4C2211yescharacteristic
14D4×M16—24no—
15D4×M16C418yescomplemented
16C2×#436C2212yes—
17C2×#435—22no—
18C2×C8×D4C2212yes—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102—21no—
23C2×#407—21no—
24C2×C4×M16C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#1836—22no—
28#1834C2212yescomplemented
29#1834C2212yescomplemented
30#1823—21no—
31#1802—21no—
32#1802—21no—
33#1802C2212yes—
34#1802C2212yes—
35#1802C2211yescharacteristic
36#1802C2211yescharacteristic
37#1802C2211yescharacteristic
38#1802C2211yescharacteristic
39#1786—21no—
40C4×#102—21no—
41C4×#102—21no—
42#1736—21no—
43#1736—21no—
44C2×#340—21no—
45#1565—24no—
46C2×#335C2212yescomplemented
47C2×#334C2212yescomplemented
48#1296C2212yescomplemented
49#1296C2212yescomplemented
50#1296C2212yescomplemented
51#1296C2212yescomplemented
52C23×D4D411yescharacteristic
53C24×C4—21no—
54C22×M16—21no—
55C22×M16—28no—
56C22×M16—22no—
57C22×M16—44no—
58C22×M16—44no—
59C22×M16D411yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8—22no—
62C23×C8—21no—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4—22no—
67C2×C4×D4C2×C414yes—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4Q814yes—
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C22⋊C4—21no—
79C22×C42C2311yescharacteristic
80#436—24no—
81#436C2×C418yescomplemented
82#435—216no—
83C8×D4—24no—
84C8×D4C2×C418yescomplemented
85#431—22no—
86#431—22no—
87#423—22no—
88#423C2×C414yes—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2312yes—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—44no—
104#407—28no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—21no—
108C4×M16C2×C414yes—
109C4×M16D414yes—
110C2×#97D412yes—
111C2×C4×C8—21no—
112C2×C4×C8C2312yes—
113#384—21no—
114#384—21no—
115#384—21no—
116#384C2312yes—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#335—24no—
135#334—24no—
136C25—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—41no—
142C22×D4—41no—
143C22×D4C4⋊C414yes—
144C22×D4C2×D411yescharacteristic
145C22×D4C2×D411yescharacteristic
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—24no—
166C2×M16—28no—
167C2×M16—44no—
168C2×M16—44no—
169C2×M16—432no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16C22⋊C414yes—
173C2×M16C22⋊C414yes—
174C22×C8—22no—
175C22×C8—28no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—44no—
180C22×C8—44no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4C4⋊C414yes—
197C4×D4C4⋊C414yes—
198C4×D4C22×C412yes—
199C4×D4C22×C412yes—
200C4×D4C2×Q812yes—
201C4×D4C2×Q812yes—
202#117—42no—
203#117—42no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C4⋊C414yes—
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—24no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—41no—
233C2×C42—41no—
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×Q812yes—
239C2×C42#4111yescharacteristic
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—24no—
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—22no—
246C4⋊C8—21no—
247C4⋊C8—22no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—42no—
251C4⋊C8C22×C414yes—
252#102—42no—
253#102—42no—
254#102—44no—
255#102—42no—
256#102—44no—
257#102—42no—
258C22⋊C8—28no—
259C22⋊C8—28no—
260C22⋊C8—42no—
261#97C22⋊C418yes—
262C4×C8—42no—
263C4×C8C22×C414yes—
264C4×C8C2×D414yes—
265#95—41no—
266#95—41no—
267C24—21no—
268C24—21no—
269C24—24no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—81no—
275C24—81no—
276C24#12111yescharacteristic
277C2×D4—28no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—24no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4C2×C4⋊C412yes—
293C2×D4C2×C4⋊C412yes—
294C2×D4AGL(1,ℤ/8)12yes—
295C2×D4#13712yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C4⋊C412yes—
337C22×C4#11712yes—
338C22×C4#12112yes—
339C22×C4#12111yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4C22⋊Q812yes—
343C22×C4C22⋊Q812yes—
344C22×C4#13512yes—
345C22×C4#13512yes—
346C22×C4AGL(1,ℤ/8)12yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351M16—28no—
352M16—28no—
353M16—416no—
354M16—48no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—416no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—84no—
373C2×C8C2×C22⋊C414yes—
374C2×C8C2×C22⋊C414yes—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—81no—
391C4⋊C4—81no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4C2×C4⋊C412yes—
397C4⋊C4C2×C4⋊C412yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—82no—
405C42—82no—
406C42C2×C22⋊C412yes—
407C42C2×C4⋊C412yes—
408C42#11712yes—
409C42#12112yes—
410C42C22⋊Q812yes—
411C42C22⋊Q812yes—
412C42C23×C411yescharacteristic
413C42C22×D411yescharacteristic
414C42C22×Q811yescharacteristic
415C42C2×#4111yescharacteristic
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—48no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—48no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—84no—
437C23—84no—
438C23—84no—
439C23—84no—
440C23—84no—
441C23#38012yes—
442C23#46312yes—
443C23#46312yes—
444C23C2×#12111yescharacteristic
445D4—22no—
446D4—22no—
447D4—22no—
448D4—22no—
449D4—22no—
450D4—48no—
451D4—41no—
452D4—41no—
453D4#42814yes—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—164no—
493C2×C4#38012yes—
494C2×C4#38012yes—
495C2×C4#38012yes—
496C2×C4#41614yes—
497C2×C4#41814yes—
498C2×C4#46312yes—
499C2×C4#46312yes—
500C2×C4#47814yes—
501C2×C4#47814yes—
502C2×C4C22×C22⋊C411yescharacteristic
503C2×C4C22×C4⋊C411yescharacteristic
504C2×C4C2×#11711yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4C2×#13511yescharacteristic
509C2×C4C2×#13511yescharacteristic
510C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
511C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
512C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
513C2×C4C2×#13711yescharacteristic
514C2×C4#57511yescharacteristic
515C2×C4#57511yescharacteristic
516C8—24no—
517C8—24no—
518C8—44no—
519C8—81no—
520C8—82no—
521C8—81no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—816no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22#180414yes—
544C22C2×#38011yescharacteristic
545C22C2×#46311yescharacteristic
546C22C2×#46311yescharacteristic
547C22#305511yescharacteristic
548C22#305611yescharacteristic
549C22#306811yescharacteristic
550C22#306911yescharacteristic
551C4—22no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—42no—
556C4—42no—
557C4—162no—
558C4—161no—
559C4—161no—
560C4#180414yes—
561C4C2×#41611yescharacteristic
562C4C2×#41811yescharacteristic
563C4C2×#42811yescharacteristic
564C4#291111yescharacteristic
565C4#305511yescharacteristic
566C4#305611yescharacteristic
567C4C2×#47811yescharacteristic
568C4C2×#47811yescharacteristic
569C2—24no—
570C2—44no—
571C2—82no—
572C2#1916611yescharacteristic
573C2#1916711yescharacteristic
574C2#1916811yescharacteristic
575C2#1916911yescharacteristic
576C2C2×#180411yescharacteristic
577C2#2066111yescharacteristic
578C2#2067911yescharacteristic
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. #20764C2semidirect
5. C2×#1565C2semidirect
6. #8656C2semidirect
7. #8654C2semidirect
8. #8648C2semidirect
9. #8646C2semidirect
10. #8045C2semidirect
11. #8043C2semidirect
15. D4×M16C4semidirect
28. #1834C22semidirect
29. #1834C22semidirect
46. C2×#335C22semidirect
47. C2×#334C22semidirect
48. #1296C22semidirect
49. #1296C22semidirect
50. #1296C22semidirect
51. #1296C22semidirect
81. #436C2×C4semidirect
84. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×M162
#207642
#207322
C2×#18022
C2×#15652
#86562
#86542
#86482
#86462
#80452
#80432