Finite Group Databasebeta

← #207252

Subgroups — #207252

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C2×C4×#95C211yesmaximal, characteristic
4#9910C214yesmaximal, complemented
5#9909C212yesmaximal, complemented
6#9908C212yesmaximal, complemented
7#8020C214yesmaximal
8C2×#533C411yescharacteristic, complemented
9C23×C42—21no—
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407C2212yes—
13#2857C411yescharacteristic, complemented
14#2857C411yescharacteristic, complemented
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2855—21no—
18#2855—21no—
19#2851C412yescomplemented
20#2851C2212yes—
21#2848—21no—
22#2848—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27#2296C412yescomplemented
28#2285C411yescharacteristic, complemented
29#2280C2212yes—
30C22×#95—21no—
31C22×#95C2212yes—
32#1797—21no—
33#1797—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1728—21no—
37C4×#95—21no—
38C4×#95—21no—
39C4×#95C2212yes—
40C4×#95C2212yes—
41#1294C2218yes—
42#1294C2218yes—
43C24×C4—22no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48#533—44no—
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×C4⋊Q8C2×C411yescharacteristic, complemented
51C2×#125C2×C412yescomplemented
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×#117C2×C412yes—
61C2×#117C2×C412yes—
62C22×C42—28no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—44no—
66C22×C42—44no—
67C22×C42—44no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#423—44no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83#407—28no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8C2314yes—
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yescomplemented
95#389C2×C412yescomplemented
96#389C2×C412yescomplemented
97#389C2×C412yescomplemented
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384C2×C411yescharacteristic, complemented
109#384C2×C411yescharacteristic, complemented
110#382C2×C412yescomplemented
111#381—22no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380C2×C412yes—
119#380C2×C412yes—
120#380C2312yes—
121#380C2312yes—
122#379—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95—28no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—24no—
134C2×#95Q818yescomplemented
135C2×#95C2314yes—
136#336—22no—
137#336—22no—
138C25C2×D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—28no—
146C23×C4—28no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×Q814yes—
176C23×C4#4114yes—
177C23×C4C2411yescharacteristic
178C2×M16—44no—
179C2×M16—44no—
180C2×M16—44no—
181C22×C8—24no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184#125—44no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C4×Q8—44no—
193C4×Q8—44no—
194#117—44no—
195#117—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—28no—
221C2×C42—28no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—432no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C4⋊C8—44no—
256C4⋊C8—44no—
257C22⋊C8—28no—
258C22⋊C8—44no—
259#95—28no—
260#95—28no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—28no—
270#95—22no—
271#95—22no—
272#95—24no—
273#95C2×Q818yes—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24C2×C22⋊C411yescharacteristic
282C24C2×C22⋊C411yescharacteristic
283C24C22⋊Q814yes—
284C24C22×D411yescharacteristic
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C22×C4—216no—
292C22×C4—216no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
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—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—232no—
316C22×C4—28no—
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—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—41no—
348C22×C4—416no—
349C22×C4—44no—
350C22×C4—416no—
351C22×C4—416no—
352C22×C4—416no—
353C22×C4—416no—
354C22×C4—44no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4#9914yes—
360C22×C4#10414yes—
361C22×C4#10414yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C4×Q814yes—
373C22×C4C4×Q814yes—
374C22×C4C22⋊Q814yes—
375C22×C4#12412yes—
376C22×C4#12412yes—
377C22×C4C4⋊Q812yes—
378C22×C4C4⋊Q812yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4#13712yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q812yes—
384C22×C4C2×#4112yes—
385M16—88no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—44no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—416no—
429C42—416no—
430C42—44no—
431C42—44no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—28no—
436C23—28no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—28no—
453C23—28no—
454C23#39112yes—
455C23#39112yes—
456C23C2×#9912yes—
457C23#41012yes—
458C23C2×#10412yes—
459C23C2×#10412yes—
460C23C22×C22⋊C411yescharacteristic
461C23C2×C22⋊Q812yes—
462C23C2×AGL(1,ℤ/8)11yescharacteristic
463C23C2×#13711yescharacteristic
464C23#57512yes—
465Q8—84no—
466Q8—84no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—24no—
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—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—216no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—464no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—48no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4#39112yes—
535C2×C4#39112yes—
536C2×C4C2×#9912yes—
537C2×C4C2×#9912yes—
538C2×C4#41712yes—
539C2×C4#41712yes—
540C2×C4C2×#10412yes—
541C2×C4C2×#10412yes—
542C2×C4C2×#10412yes—
543C2×C4C2×#10412yes—
544C2×C4C2×C4×Q812yes—
545C2×C4C2×#12411yescharacteristic
546C2×C4C2×C4⋊Q811yescharacteristic
547C8—84no—
548C22—216no—
549C22—216no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—21no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22#199414yes—
571C22#199614yes—
572C22C2×#39111yescharacteristic
573C22C22×#9911yescharacteristic
574C22C2×#41011yescharacteristic
575C22C2×#41711yescharacteristic
576C22#289611yescharacteristic
577C22C22×#10411yescharacteristic
578C22C22×#10411yescharacteristic
579C22#307712yes—
580C22#307812yes—
581C4—21no—
582C4—22no—
583C4—21no—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—416no—
592C4—41no—
593C4—41no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#150414yes—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—24no—
603C2C2×#150411yescharacteristic
604C2#1754111yescharacteristic
605C2#1754211yescharacteristic
606C2#1754611yescharacteristic
607C2#2131211yescharacteristic
608C2C2×#199411yescharacteristic
609C2C2×#199611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9910C2semidirect
5. #9909C2semidirect
6. #9908C2semidirect
8. C2×#533C4semidirect
13. #2857C4semidirect
14. #2857C4semidirect
19. #2851C4semidirect
27. #2296C4semidirect
28. #2285C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×C4⋊Q8C2×C4semidirect
51. C2×#125C2×C4semidirect
94. #389C2×C4semidirect
95. #389C2×C4semidirect
96. #389C2×C4semidirect
97. #389C2×C4semidirect
108. #384C2×C4semidirect
109. #384C2×C4semidirect
110. #382C2×C4semidirect
134. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
#206262
C2×C4×#952
#99102
#99092
#99082
#80202