Finite Group Databasebeta

← #196697

Subgroups — #196697

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2#36625C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4C2×#1561C212yesmaximal, complemented
5#9140C212yesmaximal, complemented
6#9138C212yesmaximal, complemented
7#9132C212yesmaximal, complemented
8#9130C212yesmaximal, complemented
9#8041C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432C2212yes—
15C2×#431—21no—
16C2×#431C2212yes—
17C2×#423C2212yes—
18C2×#423C2212yes—
19C2×#407—21no—
20C2×#407—21no—
21C2×C4×M16—21no—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855—21no—
27#2855—21no—
28#2851C412yes—
29#2851C412yes—
30#2848—21no—
31#2848—21no—
32C4×C22⋊Q8—21no—
33C4×C22⋊Q8—21no—
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8—21no—
36#2296C412yes—
37#2285C2211yescharacteristic
38#2280C412yes—
39#1797—21no—
40#1797—21no—
41#1797C2212yes—
42#1797C2212yes—
43#1728—21no—
44#1561—28no—
45C2×#332C2214yescomplemented
46C2×#332C2214yescomplemented
47#1296C2214yescomplemented
48#1296C2214yescomplemented
49C24×C4C2×C411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54#533—24no—
55#533—24no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×#125C2×C412yes—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117C2×C412yes—
68C2×#117C2×C412yes—
69C22×C42—24no—
70C22×C42—24no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432—28no—
84#432—28no—
85#431—28no—
86#431—28no—
87#423—24no—
88#423—24no—
89#423—24no—
90#423D418yes—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2314yes—
94C2×C4⋊C8C2314yes—
95#407—28no—
96#407—44no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C4×M16—28no—
102C2×#97—22no—
103C2×C4×C8—22no—
104#391—21no—
105#391—21no—
106#391—21no—
107#391—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389C2×C412yes—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2×C412yes—
116#387—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#385—22no—
125#385—22no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#382C2×C412yes—
129#381—22no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380—21no—
135#380—21no—
136#380C2×C412yes—
137#380C2×C412yes—
138#380C2×C412yes—
139#380C2×C412yes—
140#379—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147#336—22no—
148#336—22no—
149#332—28no—
150#332—28no—
151C25C22⋊C411yescharacteristic
152C22×Q8—21no—
153C22×Q8—21no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
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×C4C22⋊C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C2×M16—28no—
184C2×M16—44no—
185C2×M16—44no—
186C2×M16—44no—
187C2×M16—44no—
188C2×M16—44no—
189C22×C8—24no—
190C22×C8—24no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195#125—28no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—42no—
200C22⋊Q8—44no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C4×Q8—42no—
205C4×Q8—44no—
206C4×Q8—42no—
207#117—28no—
208#117—44no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
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×C42—24no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D414yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4114yes—
266C2×C42C2411yescharacteristic
267C4⋊C8—28no—
268C4⋊C8—24no—
269C4⋊C8—24no—
270C4⋊C8—24no—
271C4⋊C8—44no—
272C4⋊C8C2×D418yes—
273C22⋊C8—28no—
274C22⋊C8—28no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277#97—28no—
278C4×C8—28no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C24C2×C22⋊C411yescharacteristic
299C2×Q8—41no—
300C2×Q8—42no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
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—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
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—416no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4#10114yes—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C4×D414yes—
380C22×C4#12012yes—
381C22×C4#12114yes—
382C22×C4#12312yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388M16—416no—
389M16—88no—
390C2×C8—28no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—44no—
394C2×C8—44no—
395C2×C8—44no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C42—22no—
421C42—22no—
422C42—21no—
423C42—21no—
424C42—22no—
425C42—22no—
426C42—21no—
427C42—21no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—44no—
435C42—44no—
436C42C4×D414yes—
437C42#12012yes—
438C42#12114yes—
439C42#12312yes—
440C42C22×D412yes—
441C42C2×#4112yes—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23#38612yes—
468C23C2×#10112yes—
469C23#41712yes—
470C23C22×C22⋊C411yescharacteristic
471Q8—82no—
472Q8—82no—
473Q8—82no—
474Q8—82no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
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—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4#38612yes—
534C2×C4#38612yes—
535C2×C4#38612yes—
536C2×C4C2×#10112yes—
537C2×C4C2×#10112yes—
538C2×C4#41312yes—
539C2×C4#41712yes—
540C2×C4C2×C4×D412yes—
541C2×C4C2×#12011yescharacteristic
542C2×C4C2×#12112yes—
543C2×C4C2×#12311yescharacteristic
544C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
545C2×C4C2×#13711yescharacteristic
546C2×C4#57512yes—
547C2×C4#57512yes—
548C2×C4#57512yes—
549C8—48no—
550C8—84no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—416no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#38611yescharacteristic
574C22C22×#10111yescharacteristic
575C22C2×#41311yescharacteristic
576C22C2×#41711yescharacteristic
577C22#289611yescharacteristic
578C22#289611yescharacteristic
579C22#289611yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4#190114yes—
594C4#300512yes—
595C4#305512yes—
596C4#305612yes—
597C4#309312yes—
598C2—42no—
599C2—42no—
600C2—42no—
601C2#1773711yescharacteristic
602C2#1773911yescharacteristic
603C2#1775111yescharacteristic
604C2#1775711yescharacteristic
605C2#2091011yescharacteristic
606C2#2097111yescharacteristic
607C2C2×#190111yescharacteristic
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×#1561C2semidirect
5. #9140C2semidirect
6. #9138C2semidirect
7. #9132C2semidirect
8. #9130C2semidirect
9. #8041C2semidirect
45. C2×#332C22semidirect
46. C2×#332C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366252
#206262
C2×#15612
#91402
#91382
#91322
#91302
#80412