Finite Group Databasebeta

← #196628

Subgroups — #196628

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic
2#36625C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4C2×#1550C212yesmaximal, complemented
5#9137C212yesmaximal, complemented
6#9135C212yesmaximal, complemented
7#9129C212yesmaximal, complemented
8#9127C212yesmaximal, complemented
9#8045C212yesmaximal, complemented
10C23×M16—21no—
11C2×#533C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#423—21no—
14C2×#423—21no—
15C2×#423—28no—
16C2×#423C2212yes—
17C2×#423C2212yes—
18C22×C4⋊C8—21no—
19C22×C4⋊C8C2212yes—
20C22×C4⋊C8C2212yes—
21C22×C22⋊C8—21no—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855—21no—
27#2855C412yes—
28#2851—21no—
29#2851—21no—
30#2848—21no—
31#2848C412yes—
32C4×C22⋊Q8—21no—
33C4×C22⋊Q8—21no—
34C4×C22⋊Q8C412yes—
35C4×C22⋊Q8C412yes—
36#2296—21no—
37#2285C2211yescharacteristic
38#2280—21no—
39#1797—21no—
40#1797—21no—
41#1797C2212yes—
42#1797C2212yes—
43C2×#336—21no—
44#1550—28no—
45C2×#333C2214yescomplemented
46C2×#332C2214yescomplemented
47#1296C2214yescomplemented
48#1296C2214yescomplemented
49C24×C4C2×C411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C22×M16—28no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—44no—
56C22×M16—44no—
57C23×C8—22no—
58C23×C8—22no—
59#533—24no—
60#533—24no—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8C2×C412yes—
68C2×C22⋊Q8C2×C412yes—
69C2×C22⋊Q8C2×C412yes—
70C2×C4×Q8—21no—
71C2×C4×Q8C2×C412yes—
72C2×#117—21no—
73C2×#117—21no—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#423—28no—
89#423—28no—
90#423—24no—
91#423—28no—
92#423D418yes—
93C2×C4⋊C8—28no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—28no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—24no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8C2314yes—
102C2×C4⋊C8C2314yes—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—44no—
106#391—21no—
107#391—21no—
108#391C2×C412yes—
109#391C2×C412yes—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389C2×C412yes—
117#389C2×C412yes—
118#387—22no—
119#387—22no—
120#387—22no—
121#387—22no—
122#385—22no—
123#385—22no—
124#385—22no—
125#385—22no—
126#385—22no—
127#385—22no—
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#382—21no—
131#381—22no—
132#380—21no—
133#380—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380C2×C412yes—
140#380C2×C412yes—
141#380C2×C412yes—
142#379—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2×C412yes—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149#336—22no—
150#336—22no—
151#333—28no—
152#332—28no—
153C25C22⋊C411yescharacteristic
154C22×Q8—21no—
155C22×Q8C22×C412yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4SD1614yes—
177C23×C4Q1614yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C2×M16—28no—
188C2×M16—28no—
189C2×M16—432no—
190C2×M16—44no—
191C2×M16—44no—
192C2×M16—44no—
193C2×M16—44no—
194C22×C8—28no—
195C22×C8—24no—
196C22×C8—24no—
197C22×C8—44no—
198C22×C8—44no—
199C22×C8—44no—
200C22×C8—44no—
201C22×C8—44no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206#125—42no—
207#125—42no—
208C22⋊Q8—28no—
209C22⋊Q8—44no—
210C22⋊Q8—44no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213C22⋊Q8—44no—
214C22⋊Q8—44no—
215C4×Q8—28no—
216C4×Q8—44no—
217#117—42no—
218#117—42no—
219#117—42no—
220#117—42no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
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×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D414yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4114yes—
278C2×C42C2411yescharacteristic
279C4⋊C8—28no—
280C4⋊C8—28no—
281C4⋊C8—28no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C4⋊C8—44no—
286C4⋊C8C2×D418yes—
287C22⋊C8—44no—
288C22⋊C8—44no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24#10314yes—
306C24C2×C22⋊C411yescharacteristic
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309C2×Q8—24no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
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—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
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—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—48no—
374C22×C4—48no—
375C22×C4—416no—
376C22×C4#10314yes—
377C22×C4#10314yes—
378C22×C4#10314yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C4×D414yes—
389C22×C4#12012yes—
390C22×C4#12114yes—
391C22×C4#12312yes—
392C22×C4C2×SD1612yes—
393C22×C4C2×SD1612yes—
394C22×C4C2×SD1612yes—
395C22×C4C2×Q1612yes—
396C22×C4C2×Q1612yes—
397C22×C4C2×Q1612yes—
398C22×C4AGL(1,ℤ/8)12yes—
399C22×C4AGL(1,ℤ/8)12yes—
400C22×C4#13712yes—
401C22×C4#13712yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405M16—416no—
406M16—416no—
407C2×C8—28no—
408C2×C8—416no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—88no—
415C4⋊C4—24no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C42—22no—
443C42—22no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—44no—
457C42—44no—
458C42C4×D414yes—
459C42#12012yes—
460C42#12114yes—
461C42#12312yes—
462C42C22×D412yes—
463C42C2×#4112yes—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—28no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—28no—
478C23—21no—
479C23—21no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23#38612yes—
486C23#41312yes—
487C23C2×#10312yes—
488C23C2×#10312yes—
489C23C2×#10312yes—
490C23#41712yes—
491C23#41712yes—
492C23C22×C22⋊C411yescharacteristic
493Q8—84no—
494Q8—82no—
495Q8—82no—
496C2×C4—28no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—28no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
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—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4#38612yes—
555C2×C4#38612yes—
556C2×C4#38612yes—
557C2×C4#41312yes—
558C2×C4C2×#10312yes—
559C2×C4C2×#10312yes—
560C2×C4C2×#10312yes—
561C2×C4#41712yes—
562C2×C4#41712yes—
563C2×C4#45014yes—
564C2×C4#45114yes—
565C2×C4#48314yes—
566C2×C4#48414yes—
567C2×C4C2×C4×D412yes—
568C2×C4C2×#12011yescharacteristic
569C2×C4C2×#12112yes—
570C2×C4C2×#12311yescharacteristic
571C2×C4C22×SD1611yescharacteristic
572C2×C4C22×Q1611yescharacteristic
573C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
574C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
575C2×C4C2×#13711yescharacteristic
576C2×C4C2×#13711yescharacteristic
577C2×C4#57512yes—
578C8—48no—
579C8—84no—
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—22no—
595C22—416no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22#155014yescomplemented
601C22#187214yes—
602C22C2×#38611yescharacteristic
603C22C2×#41311yescharacteristic
604C22C2×#41311yescharacteristic
605C22C22×#10311yescharacteristic
606C22C2×#41711yescharacteristic
607C22C2×#41711yescharacteristic
608C22#289611yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#187214yes—
623C4C2×#45011yescharacteristic
624C4C2×#45111yescharacteristic
625C4#305512yes—
626C4#305612yes—
627C4C2×#48311yescharacteristic
628C4C2×#48411yescharacteristic
629C2—24no—
630C2—24no—
631C2—42no—
632C2C2×#155011yescharacteristic
633C2#1770811yescharacteristic
634C2#1775211yescharacteristic
635C2#1775311yescharacteristic
636C2C2×#187211yescharacteristic
637C2#2097111yescharacteristic
638C2#2098911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1550C2semidirect
5. #9137C2semidirect
6. #9135C2semidirect
7. #9129C2semidirect
8. #9127C2semidirect
9. #8045C2semidirect
45. C2×#333C22semidirect
46. C2×#332C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect
600. C22#1550semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
#366252
C2×#17972
C2×#15502
#91372
#91352
#91292
#91272
#80452