Finite Group Databasebeta

← #196524

Subgroups — #196524

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic
4C2×#1528C212yesmaximal, complemented
5#9122C214yesmaximal, complemented
6#9121C214yesmaximal, complemented
7#8033C212yesmaximal, complemented
8C23×M16—21no—
9C22×#125C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#432C2212yes—
12C2×C4×M16—21no—
13C2×C4×M16—21no—
14C2×C4×M16—28no—
15C2×C4×M16C2212yes—
16C2×C4×M16C2212yes—
17C22×#97C2212yes—
18C22×C4×C8—21no—
19#2851—21no—
20#2851—21no—
21#2851—21no—
22#2851—21no—
23#2851C412yes—
24#2851C412yes—
25#2851C2212yes—
26#2851C2212yes—
27C2×#389C411yescharacteristic
28C2×#389C411yescharacteristic
29C2×#387—21no—
30C2×#387—21no—
31C2×#385—21no—
32C2×#385—21no—
33C2×#382C2211yescharacteristic
34C2×#380C411yescharacteristic
35C2×#380C411yescharacteristic
36#1817—21no—
37#1797—21no—
38#1797—21no—
39#1779—21no—
40#1749C2212yes—
41#1749C2212yes—
42#1528—28no—
43C2×#330C2218yescomplemented
44#1295C2218yescomplemented
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C22×M16—28no—
49C22×M16—28no—
50C22×M16—28no—
51C22×M16—22no—
52C22×M16—22no—
53C23×C8—22no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—24no—
57C2×#125—24no—
58C2×#125C2×C412yes—
59C2×#125C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#432—24no—
85#432—24no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C4×M16—28no—
91C4×M16—28no—
92C4×M16—24no—
93C4×M16—28no—
94C4×M16D418yes—
95C2×#97—22no—
96C2×#97—28no—
97C2×#97C2314yes—
98C2×#97C2314yes—
99C2×C4×C8—28no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#385—22no—
125#385—22no—
126#382—21no—
127#382—21no—
128#382C2×C412yes—
129#382C2312yes—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380C2×C412yes—
135#380C2×C412yes—
136#380C2×C412yes—
137#380C2×C412yes—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144#336—22no—
145#336—22no—
146#330—216no—
147C25C22⋊C411yescharacteristic
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C2×M16—264no—
181C2×M16—28no—
182C2×M16—28no—
183C2×M16—28no—
184C2×M16—28no—
185C2×M16—28no—
186C2×M16—28no—
187C22×C8—28no—
188C22×C8—28no—
189C22×C8—28no—
190C22×C8—24no—
191C22×C8—24no—
192#125—28no—
193#125—24no—
194#125—24no—
195#125—42no—
196#125—44no—
197#125—42no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D414yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4114yes—
262C2×C42C2411yescharacteristic
263C4⋊C8—44no—
264C22⋊C8—44no—
265C22⋊C8—44no—
266C22⋊C8—44no—
267#97—28no—
268#97—24no—
269#97—24no—
270#97—24no—
271#97C2×D418yes—
272C4×C8—28no—
273C4×C8—28no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—41no—
291C24—41no—
292C24#10414yes—
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C22×C4—216no—
297C22×C4—216no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—416no—
354C22×C4—416no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4#10114yes—
368C22×C4#10414yes—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C4×D414yes—
379C22×C4#12012yes—
380C22×C4#12114yes—
381C22×C4#12312yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385M16—232no—
386M16—232no—
387C2×C8—232no—
388C2×C8—28no—
389C2×C8—28no—
390C2×C8—28no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—44no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C42—22no—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42C4×D414yes—
430C42#12012yes—
431C42#12114yes—
432C42#12312yes—
433C42C22×D412yes—
434C42C2×#4112yes—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—28no—
449C23—21no—
450C23—28no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23#38612yes—
457C23C2×#10112yes—
458C23C2×#10412yes—
459C23C2×#10412yes—
460C23C2×#10412yes—
461C23C2×#10412yes—
462C23#42112yes—
463C23C22×C22⋊C411yescharacteristic
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4#38612yes—
526C2×C4#38612yes—
527C2×C4#38612yes—
528C2×C4C2×#10112yes—
529C2×C4C2×#10112yes—
530C2×C4#41312yes—
531C2×C4C2×#10412yes—
532C2×C4C2×#10412yes—
533C2×C4#42112yes—
534C2×C4C2×C4×D412yes—
535C2×C4C2×#12011yescharacteristic
536C2×C4C2×#12112yes—
537C2×C4C2×#12311yescharacteristic
538C8—216no—
539C8—84no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—416no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22#152814yescomplemented
561C22#189614yes—
562C22C2×#38611yescharacteristic
563C22C22×#10111yescharacteristic
564C22C2×#41311yescharacteristic
565C22C22×#10411yescharacteristic
566C22C22×#10411yescharacteristic
567C22C2×#42111yescharacteristic
568C22C2×#42111yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4#190114yes—
586C2—24no—
587C2—24no—
588C2—42no—
589C2C2×#152811yescharacteristic
590C2#1764211yescharacteristic
591C2#1764811yescharacteristic
592C2#1765011yescharacteristic
593C2#2090211yescharacteristic
594C2C2×#189611yescharacteristic
595C2C2×#190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1528C2semidirect
5. #9122C2semidirect
6. #9121C2semidirect
7. #8033C2semidirect
43. C2×#330C22semidirect
44. #1295C22semidirect
560. C22#1528semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
C2×#28512
#207152
C2×#15282
#91222
#91212
#80332