Finite Group Databasebeta

← #196416

Subgroups — #196416

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1514C212yesmaximal, complemented
4#14683C211yesmaximal, characteristic, complemented
5#9118C214yesmaximal, complemented
6#9113C214yesmaximal, complemented
7#8018C212yesmaximal, 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×#383C2212yes—
36#1514—28no—
37C2×#327C2218yes—
38#1434—21no—
39#1434—21no—
40#1434C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C22⋊#95C2212yes—
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×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#432—28no—
89#432—28no—
90#432—24no—
91#432—28no—
92#432D418yescomplemented
93C2×C4⋊C8—28no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—28no—
98C2×C22⋊C8C2314yes—
99C2×#97—28no—
100C2×#97—22no—
101C2×#97—22no—
102#391—21no—
103#391—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#390C2×C412yes—
109#390C2×C412yes—
110#389—21no—
111#389—21no—
112#389C2×C412yes—
113#389C2×C412yes—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#383—24no—
127#383—24no—
128#382—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380C2×C412yes—
133#380C2×C412yes—
134#380C2312yes—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2×C412yes—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95C2314yes—
145#327—216no—
146C25C2×D411yescharacteristic
147C2×#41—44no—
148C2×#41—44no—
149C2×#41—44no—
150C22×Q8—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C22×C412yes—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
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×C4C2411yescharacteristic
187C22×C8—28no—
188C22×C8—24no—
189C22×C8—44no—
190C22×C8—44no—
191C4⋊Q8—24no—
192#127—24no—
193#124—42no—
194#124—42no—
195#123—48no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198#121—24no—
199#121—24no—
200#121—44no—
201#121—44no—
202#121—44no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207#120—48no—
208C4×D4—28no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—42no—
212C4×D4—42no—
213#117—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
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—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
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×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×D414yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4114yes—
273C4⋊C8—28no—
274C4⋊C8—28no—
275C22⋊C8—28no—
276C22⋊C8—24no—
277C22⋊C8C2×D418yes—
278#97—28no—
279#97—28no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—41no—
299C24—41no—
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C24AGL(1,ℤ/8)14yes—
303C24C22×D411yescharacteristic
304#41—816no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×D4—24no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—44no—
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—42no—
325C2×D4—42no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
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—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
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—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—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—416no—
383C22×C4—48no—
384C22×C4—416no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
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×C4#12114yes—
407C22×C4#12114yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C2×#4112yes—
412C2×C8—28no—
413C2×C8—416no—
414C2×C8—44no—
415C2×C8—44no—
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—
425C4⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—88no—
439C42—22no—
440C42—22no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—44no—
454C42C4×D414yes—
455C42C4×D414yes—
456C42#12012yes—
457C42#12012yes—
458C42#12312yes—
459C42#12312yes—
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—28no—
474C23—21no—
475C23—28no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23#41012yes—
489C23#41812yes—
490C23#41812yes—
491C23#42112yes—
492C23C22×C22⋊C411yescharacteristic
493C23C2×#12112yes—
494C23C2×AGL(1,ℤ/8)12yes—
495C23#57512yes—
496Q8—84no—
497D4—84no—
498D4—84no—
499D4—84no—
500D4—88no—
501D4—88no—
502D4—88no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
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—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
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—44no—
539C2×C4—44no—
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—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
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—88no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4#38612yes—
570C2×C4#38612yes—
571C2×C4#38612yes—
572C2×C4#38612yes—
573C2×C4#41012yes—
574C2×C4#42112yes—
575C2×C4C2×C4×D412yes—
576C2×C4C2×#12011yescharacteristic
577C2×C4C2×#12311yescharacteristic
578C8—48no—
579C22—22no—
580C22—22no—
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—416no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#151414yescomplemented
607C22C2×#38611yescharacteristic
608C22C2×#41011yescharacteristic
609C22C2×#41011yescharacteristic
610C22C2×#41811yescharacteristic
611C22#289611yescharacteristic
612C22C2×#42111yescharacteristic
613C22C2×#42111yescharacteristic
614C22#305514yes—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—21no—
621C4—21no—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—164no—
632C2—24no—
633C2—24no—
634C2—42no—
635C2—162no—
636C2C2×#151411yescharacteristic
637C2#1756011yescharacteristic
638C2#1758611yescharacteristic
639C2#1759511yescharacteristic
640C2#2090211yescharacteristic
641C2#2097311yescharacteristic
642C2#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×#1514C2semidirect
4. #14683C2semidirect
5. #9118C2semidirect
6. #9113C2semidirect
7. #8018C2semidirect
19. #2857C4semidirect
20. #2857C4semidirect
21. #2856C4semidirect
22. #2856C4semidirect
31. C4×#121C4semidirect
32. C4×#121C4semidirect
92. #432D4semidirect
606. C22#1514semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#366242
C2×#15142
#146832
#91182
#91132
#80182