Finite Group Databasebeta

← #204581

Subgroups — #204581

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C2×#1504C214yesmaximal, complemented
4C2×#1434C211yesmaximal, characteristic, complemented
5#9684C212yesmaximal, complemented
6#9683C212yesmaximal, complemented
7#9681C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525C2212yes—
10C23×C42C2211yescharacteristic
11C2×#407C2212yes—
12C2×C4×M16—21no—
13#2857C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2856C411yescharacteristic, complemented
16#2856C2211yescharacteristic
17#2854—21no—
18#2854—21no—
19#2850C412yescomplemented
20#2850C2212yes—
21#2847—21no—
22#2847—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27#2297C412yescomplemented
28#2285C411yescharacteristic, complemented
29C22×#95—21no—
30C22×#95—21no—
31C22×#95C2212yes—
32#1749—21no—
33#1749C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8C2212yes—
36#1728—21no—
37#1504—216no—
38C2×#328C2218yes—
39C2×#327C2218yes—
40#1434—21no—
41#1434—21no—
42#1434C2212yes—
43#1434C2212yes—
44C22×#41C2×C412yes—
45C24×C4—22no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50#532—44no—
51C2×C4⋊Q8C2×C411yescharacteristic, complemented
52C2×#127C2×C411yescharacteristic, complemented
53C2×#124C2×C412yescomplemented
54#525—28no—
55#525—28no—
56C2×#123—22no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2312yes—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121C2×C412yes—
66C2×#121C2312yes—
67C2×#120—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×#117C2×C412yes—
73C22×C42—44no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#407—24no—
87#407D418yescomplemented
88C2×C22⋊C8—22no—
89C2×C22⋊C8C2314yes—
90C4×M16—44no—
91C2×#97—22no—
92C2×C4×C8—22no—
93#391C2×C412yescomplemented
94#391C2×C412yescomplemented
95#390—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390C2×C412yescomplemented
100#390C2×C412yescomplemented
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#384C2×C411yescharacteristic, complemented
116#384C2×C411yescharacteristic, complemented
117#382C2×C412yescomplemented
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C2×#95—28no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—28no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—44no—
136C2×#95C2314yes—
137#336—22no—
138#336—22no—
139#328—216no—
140#327—216no—
141C25C2×D411yescharacteristic
142C2×#41—28no—
143C2×#41—44no—
144C2×#41—44no—
145C22×Q8C22×C412yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22×C412yes—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—28no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D414yes—
187C23×C4#4114yes—
188C23×C4C2411yescharacteristic
189C2×M16—28no—
190C2×M16—44no—
191C2×M16—44no—
192C22×C8—24no—
193C4⋊Q8—42no—
194#127—42no—
195#124—44no—
196#123—48no—
197C22⋊Q8—28no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200#121—28no—
201#121—24no—
202#121—24no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207#121—44no—
208#121—44no—
209#120—44no—
210#120—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—48no—
214C4×D4—44no—
215C4×D4—44no—
216#117—28no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C22⋊C8—28no—
268C22⋊C8—24no—
269C22⋊C8C2×D418yes—
270#97—44no—
271C4×C8—44no—
272#95—28no—
273#95—28no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—28no—
281#95—44no—
282#95—44no—
283#95—44no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24C2×C22⋊C411yescharacteristic
292C24C2×C22⋊C411yescharacteristic
293C24#12412yes—
294C24#12712yes—
295C24C22×D411yescharacteristic
296#41—816no—
297C2×Q8—24no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×D4—24no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—42no—
318C2×D4—42no—
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—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
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—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—41no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—416no—
374C22×C4—416no—
375C22×C4—44no—
376C22×C4—416no—
377C22×C4—416no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—416no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4#9914yes—
390C22×C4#10414yes—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C4×D414yes—
402C22×C4C4×D414yes—
403C22×C4#12114yes—
404C22×C4#12114yes—
405C22×C4#12412yes—
406C22×C4#12712yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C2×#4112yes—
411M16—416no—
412C2×C8—28no—
413C2×C8—44no—
414C2×C8—44no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—44no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—24no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23#39012yes—
484C23#39012yes—
485C23C2×#9912yes—
486C23#41012yes—
487C23C2×#10412yes—
488C23#42112yes—
489C23C22×C22⋊C411yescharacteristic
490C23C2×#12411yescharacteristic
491C23C2×#12711yescharacteristic
492C23#57512yes—
493C23#57512yes—
494Q8—84no—
495D4—84no—
496D4—84no—
497D4—84no—
498D4—84no—
499D4—84no—
500D4—88no—
501D4—88no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—88no—
563C2×C4—832no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4#39012yes—
569C2×C4#39012yes—
570C2×C4C2×#9912yes—
571C2×C4C2×#9912yes—
572C2×C4C2×#10412yes—
573C2×C4C2×#10412yes—
574C2×C4C2×C4×D412yes—
575C2×C4C2×#12112yes—
576C8—48no—
577C22—216no—
578C22—216no—
579C22—21no—
580C22—21no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#196414yes—
607C22#198014yes—
608C22C2×#39011yescharacteristic
609C22C22×#9911yescharacteristic
610C22C2×#41011yescharacteristic
611C22#289611yescharacteristic
612C22#289611yescharacteristic
613C22C22×#10411yescharacteristic
614C22C2×#42111yescharacteristic
615C22#314212yes—
616C22#315312yes—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—88no—
630C4—82no—
631C4—82no—
632C4—82no—
633C2—22no—
634C2—22no—
635C2—24no—
636C2—24no—
637C2—162no—
638C2#1754111yescharacteristic
639C2#1754211yescharacteristic
640C2#1758711yescharacteristic
641C2#1759311yescharacteristic
642C2#2120111yescharacteristic
643C2C2×#196411yescharacteristic
644C2C2×#198011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
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
2. C4×#407C2semidirect
3. C2×#1504C2semidirect
4. C2×#1434C2semidirect
5. #9684C2semidirect
6. #9683C2semidirect
7. #9681C2semidirect
8. C2×#532C4semidirect
13. #2857C4semidirect
15. #2856C4semidirect
19. #2850C4semidirect
27. #2297C4semidirect
28. #2285C4semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×#127C2×C4semidirect
53. C2×#124C2×C4semidirect
87. #407D4semidirect
93. #391C2×C4semidirect
94. #391C2×C4semidirect
99. #390C2×C4semidirect
100. #390C2×C4semidirect
115. #384C2×C4semidirect
116. #384C2×C4semidirect
117. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#366242
C4×#4072
C2×#15042
C2×#14342
#96842
#96832
#96812