Finite Group Databasebeta

← #205010

Subgroups — #205010

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1513C214yesmaximal, complemented
4C4×#380C211yesmaximal, characteristic, complemented
5#9716C214yesmaximal, complemented
6#9712C212yesmaximal, complemented
7#9711C212yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525C2212yes—
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407C2212yes—
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
29C2×#380C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1728—21no—
37#1513—216no—
38C2×#328C2218yes—
39C2×#327C2218yes—
40C4⋊#95—21no—
41#1434C2212yes—
42C22⋊#95—21no—
43C4×#95C2212yes—
44C22×#41C2×C412yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49#532—44no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124C2×C412yes—
53#525—28no—
54#525—28no—
55C2×#123—22no—
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2312yes—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121C2×C412yes—
65C2×#121C2312yes—
66C2×#120—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×#117C2×C412yes—
72C22×C4⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—22no—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#423—44no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91#407—28no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2314yes—
94#391C2×C412yes—
95#391C2×C412yes—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390C2×C412yes—
101#390C2×C412yes—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382C2×C412yes—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—24no—
126#380D418yescomplemented
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—44no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—44no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
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—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
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×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D414yes—
181C23×C4#4114yes—
182C23×C4C2411yescharacteristic
183C2×M16—44no—
184C2×M16—44no—
185C2×M16—44no—
186C22×C8—24no—
187C4⋊Q8—42no—
188#127—42no—
189#124—44no—
190#123—44no—
191#123—44no—
192C22⋊Q8—28no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195#121—28no—
196#121—24no—
197#121—24no—
198#121—44no—
199#121—44no—
200#121—44no—
201#121—44no—
202#121—44no—
203#121—42no—
204#121—42no—
205#120—48no—
206C4×D4—44no—
207C4×D4—48no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211#117—28no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—28no—
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—44no—
238C2×C22⋊C4—44no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
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×C42—42no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—44no—
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×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C4⋊C8—44no—
272C4⋊C8—44no—
273C22⋊C8—28no—
274C22⋊C8—44no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—24no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95C2×D418yes—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24#12114yes—
297C24C22×D411yescharacteristic
298#41—816no—
299C2×Q8—24no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×D4—24no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
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—21no—
330C22×C4—21no—
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—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—48no—
352C22×C4—48no—
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—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—416no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
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×C4C4×D414yes—
398C22×C4C4×D414yes—
399C22×C4#12114yes—
400C22×C4#12412yes—
401C22×C4#12412yes—
402C22×C4#12712yes—
403C22×C4#12712yes—
404C22×C4AGL(1,ℤ/8)12yes—
405C22×C4#13712yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C2×#4112yes—
410M16—88no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—416no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—416no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—44no—
455C42—88no—
456C42#9914yes—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—24no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23#39012yes—
491C23#39012yes—
492C23C22×C22⋊C411yescharacteristic
493C23C2×#12112yes—
494C23C2×AGL(1,ℤ/8)11yescharacteristic
495C23C2×#13711yescharacteristic
496C23#57512yes—
497Q8—84no—
498D4—84no—
499D4—84no—
500D4—88no—
501D4—88no—
502D4—84no—
503D4—84no—
504D4—84no—
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—28no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
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—21no—
533C2×C4—21no—
534C2×C4—21no—
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—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—88no—
569C2×C4#39012yes—
570C2×C4#39012yes—
571C2×C4C2×#9912yes—
572C2×C4C2×#9912yes—
573C2×C4C2×#9912yes—
574C2×C4#41012yes—
575C2×C4#41712yes—
576C2×C4#41712yes—
577C2×C4#42112yes—
578C2×C4#42112yes—
579C2×C4C2×C4×D412yes—
580C2×C4C2×#12411yescharacteristic
581C2×C4C2×#12711yescharacteristic
582C8—84no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—48no—
598C22—48no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22C2×#39011yescharacteristic
615C22C22×#9911yescharacteristic
616C22C2×#41011yescharacteristic
617C22C2×#41711yescharacteristic
618C22#289611yescharacteristic
619C22C2×#42111yescharacteristic
620C22C2×#42111yescharacteristic
621C22#303612yes—
622C22#303712yes—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—21no—
627C4—22no—
628C4—21no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—88no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4#151314yescomplemented
639C4#196414yes—
640C2—22no—
641C2—22no—
642C2—42no—
643C2—42no—
644C2—162no—
645C2C2×#151311yescharacteristic
646C2#1756311yescharacteristic
647C2#1758211yescharacteristic
648C2#1759411yescharacteristic
649C2#2120911yescharacteristic
650C2C2×#196411yescharacteristic
651C2#2125811yescharacteristic
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. #20626C2semidirect
3. C2×#1513C2semidirect
4. C4×#380C2semidirect
5. #9716C2semidirect
6. #9712C2semidirect
7. #9711C2semidirect
8. C2×#532C4semidirect
13. #2857C4semidirect
15. #2856C4semidirect
19. #2850C4semidirect
27. #2297C4semidirect
28. #2285C4semidirect
126. #380D4semidirect
638. C4#1513semidirect

Maximal subgroups

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

subgroupindex
#366242
#206262
C2×#15132
C4×#3802
#97162
#97122
#97112