Finite Group Databasebeta

← #194155

Subgroups — #194155

Order 512.

1882 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic, complemented
4#21258C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic, complemented
6#8981C214yesmaximal, complemented
7#8977C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4—21no—
14#3100—24no—
15#3100C418yescomplemented
16C2×#480—21no—
17C2×#474C2212yes—
18C2×#459C2212yes—
19C2×#447—21no—
20C2×C4×D8—22no—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#417—21no—
25C2×#417—21no—
26D4×C4⋊C4—22no—
27D4×C4⋊C4—22no—
28D4×C4⋊C4—22no—
29D4×C4⋊C4C2214yes—
30C2×#389—21no—
31C2×#385—22no—
32C2×#384C2211yescharacteristic
33C2×#380—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1990C2212yescomplemented
36#1983—22no—
37#1982C2212yescomplemented
38#1981C2212yescomplemented
39#1957C2212yescomplemented
40#1874—22no—
41#1802—22no—
42#1486—24no—
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45#1279C2214yescomplemented
46C22×#41—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C24×C4—22no—
50C22×D8—22no—
51C22×M16—21no—
52#532—22no—
53#532—22no—
54#532—22no—
55#532C2×C414yescomplemented
56C2×C4⋊Q8C2311yescharacteristic
57C2×#127C2311yescharacteristic
58C2×#124—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—48no—
70C2×C4×D4C2×C414yes—
71C2×C4×D4D414yes—
72C2×C4×D4D412yes—
73C2×C4×D4C2312yes—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4—44no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D412yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#480—28no—
90#474—24no—
91#474C2×C418yescomplemented
92#459—24no—
93#459C2×C418yescomplemented
94#447—28no—
95C4×D8—216no—
96#431—22no—
97#431—22no—
98#431—22no—
99#431C2×C414yescomplemented
100C2×#107—22no—
101#423—22no—
102#423—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107#417—44no—
108#417—44no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—24no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2312yes—
120#389—24no—
121#389—24no—
122#385—28no—
123#385—28no—
124#384—21no—
125#384—21no—
126#384D412yes—
127#384C2312yes—
128#380—24no—
129#380—24no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4D412yes—
133C4×C4⋊C4C2312yes—
134C2×#95—22no—
135C2×#95—21no—
136#340—24no—
137D4⋊C8—28no—
138C25—21no—
139C2×#41—24no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C2×#41—42no—
145C22×Q8—21no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—41no—
154C22×D4C22⋊C414yes—
155C22×D4C2×D412yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—28no—
165C23×C4—48no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—48no—
171C23×C4—41no—
172C23×C4—84no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×D8—28no—
177C2×D8—44no—
178C2×D8—44no—
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22×C412yescomplemented
189#127—21no—
190#127—21no—
191#127—21no—
192#127C22×C412yescomplemented
193#124—24no—
194#124—42no—
195#121—24no—
196#121—24no—
197#121—42no—
198#121—42no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—416no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4C22⋊C418yes—
214C4×D4C22×C414yes—
215C4×D4C2×D414yes—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—416no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4C22⋊C414yes—
240C2×C4⋊C4D814yescomplemented
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D414yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—44no—
256C2×C42C22×C412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264#107—28no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—42no—
269C4⋊C8—42no—
270C4⋊C8C22×C414yescomplemented
271#103—44no—
272#103—44no—
273#102—28no—
274#102—28no—
275#102—44no—
276#102—44no—
277#102—84no—
278C22⋊C8—24no—
279C22⋊C8—42no—
280C4×C8—24no—
281C4×C8—22no—
282C4×C8—22no—
283#95—28no—
284#95—22no—
285#95—22no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—24no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—41no—
295C24#12011yescharacteristic
296#41—48no—
297#41—48no—
298#41—84no—
299#41—84no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—216no—
309C2×D4—216no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—44no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—42no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4C2×C22⋊C414yes—
332C2×D4AGL(1,ℤ/8)14yes—
333C2×D4#13714yes—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—416no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—416no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—88no—
372C22×C4—832no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4C2×C22⋊C412yes—
376C22×C4C4×D412yes—
377C22×C4C4×D412yes—
378C22×C4#12011yescharacteristic
379C22×C4#12011yescharacteristic
380C22×C4#12011yescharacteristic
381C22×C4#12112yes—
382C22×C4#12112yes—
383C22×C4#12312yes—
384C22×C4C2×D812yes—
385C22×C4#13512yes—
386C22×C4AGL(1,ℤ/8)12yes—
387C22×C4AGL(1,ℤ/8)12yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391D8—416no—
392M16—84no—
393C2×C8—24no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—44no—
411C4⋊C4—416no—
412C4⋊C4—44no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—82no—
421C4⋊C4C2×C22⋊C414yes—
422C4⋊C4C2×D814yes—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—28no—
426C22⋊C4—416no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C42—44no—
430C42—44no—
431C42C2×C22⋊C412yes—
432C42C4×D412yes—
433C42C4×D412yes—
434C42#12112yes—
435C42#12112yes—
436C42#12312yes—
437C42C23×C411yescharacteristic
438C42C22×D411yescharacteristic
439C42C2×#4111yescharacteristic
440C42C2×#4111yescharacteristic
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—216no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—48no—
450C23—48no—
451C23—48no—
452C23—48no—
453C23—48no—
454C23—48no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23#38612yes—
467C23#44712yes—
468C23#44912yes—
469C23C2×#12011yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—81no—
473Q8—81no—
474D4—216no—
475D4—24no—
476D4—24no—
477D4—24no—
478D4—44no—
479D4—44no—
480D4—42no—
481D4—42no—
482D4—82no—
483D4—88no—
484D4—82no—
485D4—81no—
486D4—81no—
487D4#41718yescomplemented
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—416no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—832no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—88no—
524C2×C4#38612yes—
525C2×C4#38612yes—
526C2×C4#38612yes—
527C2×C4C4×D814yes—
528C2×C4#44214yes—
529C2×C4#44712yes—
530C2×C4#44912yes—
531C2×C4#45914yes—
532C2×C4#46314yes—
533C2×C4C22×C22⋊C411yescharacteristic
534C2×C4C2×C4×D411yescharacteristic
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×#12311yescharacteristic
539C2×C4C22×D811yescharacteristic
540C2×C4C2×#13511yescharacteristic
541C2×C4C2×AGL(1,ℤ/8)12yes—
542C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
543C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
544C2×C4C2×#13712yes—
545C8—44no—
546C8—82no—
547C8—82no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—28no—
555C22—432no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—161no—
566C22—161no—
567C22—161no—
568C22—161no—
569C22#187914yes—
570C22C2×#38611yescharacteristic
571C22C2×#44711yescharacteristic
572C22C2×#44911yescharacteristic
573C22#300311yescharacteristic
574C22#300411yescharacteristic
575C22#301911yescharacteristic
576C22#302011yescharacteristic
577C4—22no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—44no—
582C4—44no—
583C4—84no—
584C4—82no—
585C4—82no—
586C4#187914yes—
587C4C2×#41712yes—
588C4C2×C4×D811yescharacteristic
589C4C2×#44211yescharacteristic
590C4C2×#45911yescharacteristic
591C4C2×#46311yescharacteristic
592C4#309212yes—
593C2—24no—
594C2—48no—
595C2—162no—
596C2#1874611yescharacteristic
597C2#1875211yescharacteristic
598C2#1877611yescharacteristic
599C2#1878211yescharacteristic
600C2#2090911yescharacteristic
601C2C2×#187911yescharacteristic
602C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3100C2
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. C2×D4×C4⋊C4C2semidirect
3. #21859C2semidirect
4. #21258C2semidirect
5. C2×#1486C2semidirect
6. #8981C2semidirect
7. #8977C2semidirect
8. #8043C2semidirect
9. #7892C2semidirect
15. #3100C4semidirect
35. #1990C22semidirect
37. #1982C22semidirect
38. #1981C22semidirect
39. #1957C22semidirect
44. #1296C22semidirect
45. #1279C22semidirect
55. #532C2×C4semidirect
91. #474C2×C4semidirect
93. #459C2×C4semidirect
99. #431C2×C4semidirect
188. C4⋊Q8C22×C4semidirect
192. #127C22×C4semidirect
240. C2×C4⋊C4D8semidirect
270. C4⋊C8C22×C4semidirect
487. D4#417semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
C2×D4×C4⋊C42
#218592
#212582
C2×#14862
#89812
#89772
#80432
#78922