Finite Group Databasebeta

← #196377

Subgroups — #196377

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1505C212yesmaximal, complemented
4#13753C211yesmaximal, characteristic, complemented
5#9118C214yesmaximal, complemented
6#9113C214yesmaximal, complemented
7#8022C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C2×#432—21no—
13C2×#432—28no—
14C2×#432C2212yes—
15C2×#432C2212yes—
16C22×C4⋊C8—21no—
17C22×C22⋊C8C2212yes—
18C22×#97—21no—
19#2857C411yescharacteristic, complemented
20#2857C411yescharacteristic, complemented
21#2856C411yescharacteristic, complemented
22#2856C411yescharacteristic, complemented
23#2854—21no—
24#2854C2212yes—
25#2850—21no—
26#2850—21no—
27#2847—21no—
28#2847C2212yes—
29C4×#121—21no—
30C4×#121—21no—
31C4×#121C412yescomplemented
32C4×#121C412yescomplemented
33#2297—21no—
34#2285C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36C2×#376C2212yes—
37#1505—28no—
38C2×#327C2218yes—
39#1434—21no—
40#1434C2212yes—
41C22⋊#95—21no—
42C22⋊#95C2212yes—
43C4×#95—21no—
44#1294C2218yes—
45C22×#41—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50#532—24no—
51#532—24no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55#525—48no—
56C2×#123—22no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121C2×C412yes—
65C2×#121C2×C412yes—
66C2×#121C2312yes—
67C2×#120—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×#117—21no—
73C22×C22⋊C4—22no—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42—22no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—28no—
90#432—28no—
91#432—24no—
92#432—28no—
93#432D418yescomplemented
94C2×C4⋊C8—28no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—28no—
99C2×C22⋊C8C2314yes—
100C2×#97—28no—
101C2×#97—22no—
102C2×#97—22no—
103#391—21no—
104#391—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390C2×C412yes—
110#390C2×C412yes—
111#389—21no—
112#389—21no—
113#389C2×C412yes—
114#389C2×C412yes—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#384C2×C411yescharacteristic
126#384C2×C411yescharacteristic
127#382—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380C2×C412yes—
132#380C2×C412yes—
133#380C2312yes—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2×C412yes—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139#376—24no—
140#376—24no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95C2314yes—
146#327—216no—
147C25C2×D411yescharacteristic
148C2×#41—44no—
149C2×#41—44no—
150C2×#41—44no—
151C22×Q8—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C22×C412yes—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
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×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C22×C8—28no—
189C22×C8—24no—
190C22×C8—44no—
191C22×C8—44no—
192C4⋊Q8—24no—
193#127—24no—
194#124—42no—
195#124—42no—
196#123—48no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199#121—24no—
200#121—24no—
201#121—44no—
202#121—44no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207#121—44no—
208#120—48no—
209C4×D4—28no—
210C4×D4—48no—
211C4×D4—48no—
212C4×D4—42no—
213C4×D4—42no—
214#117—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—22no—
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—44no—
238C2×C22⋊C4—44no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—28no—
256C2×C42—22no—
257C2×C42—44no—
258C2×C42—44no—
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×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D414yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4114yes—
277C4⋊C8—28no—
278C4⋊C8—28no—
279C22⋊C8—28no—
280C22⋊C8—24no—
281C22⋊C8C2×D418yes—
282#97—28no—
283#97—28no—
284#95—28no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—24no—
292#95—24no—
293#95—44no—
294#95—44no—
295#95—44no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—41no—
302C24—41no—
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305C24AGL(1,ℤ/8)14yes—
306C24C22×D411yescharacteristic
307#41—816no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×D4—24no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
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—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—416no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4#10414yes—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4#12114yes—
410C22×C4#12114yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C2×#4112yes—
415C2×C8—28no—
416C2×C8—416no—
417C2×C8—44no—
418C2×C8—44no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—416no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C42—22no—
443C42—22no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—416no—
457C42—44no—
458C42C4×D414yes—
459C42C4×D414yes—
460C42#12012yes—
461C42#12012yes—
462C42#12312yes—
463C42#12312yes—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—28no—
478C23—21no—
479C23—28no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23#41012yes—
493C23#41812yes—
494C23#41812yes—
495C23C2×#10412yes—
496C23C22×C22⋊C411yescharacteristic
497C23C2×#12112yes—
498C23C2×AGL(1,ℤ/8)12yes—
499C23#57512yes—
500Q8—84no—
501D4—84no—
502D4—84no—
503D4—88no—
504D4—88no—
505D4—88no—
506D4—84no—
507C2×C4—24no—
508C2×C4—24no—
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—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
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—21no—
536C2×C4—21no—
537C2×C4—21no—
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—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—88no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4#38612yes—
572C2×C4#38612yes—
573C2×C4#38612yes—
574C2×C4#38612yes—
575C2×C4#41012yes—
576C2×C4C2×#10412yes—
577C2×C4C2×#10412yes—
578C2×C4#42112yes—
579C2×C4C2×C4×D412yes—
580C2×C4C2×#12011yescharacteristic
581C2×C4C2×#12311yescharacteristic
582C8—48no—
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—22no—
597C22—22no—
598C22—416no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22#150514yescomplemented
611C22C2×#38611yescharacteristic
612C22C2×#41011yescharacteristic
613C22C2×#41011yescharacteristic
614C22C2×#41811yescharacteristic
615C22#289611yescharacteristic
616C22C22×#10411yescharacteristic
617C22C2×#42111yescharacteristic
618C22#305514yes—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—21no—
624C4—21no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—88no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#187114yes—
636C2—24no—
637C2—24no—
638C2—42no—
639C2—162no—
640C2C2×#150511yescharacteristic
641C2#1754311yescharacteristic
642C2#1759111yescharacteristic
643C2#1759511yescharacteristic
644C2C2×#187111yescharacteristic
645C2#2097311yescharacteristic
646C2#2098311yescharacteristic
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
1. C22×#432C2semidirect
3. C2×#1505C2semidirect
4. #13753C2semidirect
5. #9118C2semidirect
6. #9113C2semidirect
7. #8022C2semidirect
19. #2857C4semidirect
20. #2857C4semidirect
21. #2856C4semidirect
22. #2856C4semidirect
31. C4×#121C4semidirect
32. C4×#121C4semidirect
93. #432D4semidirect
610. C22#1505semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#366242
C2×#15052
#137532
#91182
#91132
#80222