Finite Group Databasebeta

← #163435

Subgroups — #163435

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#7715C214yesmaximal, complemented
5#7681C214yesmaximal, complemented
6#7643C212yesmaximal, complemented
7#7641C212yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#423—21no—
11C22×C4⋊C8—21no—
12C2×#407—21no—
13C22×C22⋊C8—21no—
14#2855—21no—
15#2855—21no—
16#2855—21no—
17#2855—21no—
18#2855C412yes—
19#2855C412yes—
20#2855C2212yes—
21#2855C2212yes—
22C2×#391C411yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389C411yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#387—21no—
27C2×#387—21no—
28C2×#385—21no—
29C2×#381—21no—
30C2×#380C411yescharacteristic
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1728C2212yes—
40C2×#336C2212yes—
41#1272C2218yescomplemented
42#1271C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—22no—
48C23×C8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—44no—
52C2×C22⋊Q8C2×C412yes—
53C2×C22⋊Q8C2×C412yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C42—44no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#423—44no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—44no—
81#407—44no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—44no—
87#391—21no—
88#391—21no—
89#391—21no—
90#391—21no—
91#391C2×C412yes—
92#391C2×C412yes—
93#391C2×C412yes—
94#391C2312yes—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2312yes—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#381—22no—
116#381—22no—
117#381—22no—
118#381—22no—
119#380—21no—
120#380—21no—
121#380C2×C412yes—
122#380C2×C412yes—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130#336—22no—
131#336—22no—
132#336C2314yes—
133#336C2314yes—
134C25C22⋊C411yescharacteristic
135C22×Q8—21no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8C22×C412yes—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4SD1614yes—
163C23×C4Q1614yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C2×M16—44no—
174C2×M16—44no—
175C2×M16—44no—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—44no—
179C22×C8—44no—
180C22×C8—44no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2411yescharacteristic
238C4⋊C8—44no—
239C4⋊C8—44no—
240C4⋊C8—44no—
241C4⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—44no—
244C22⋊C8—44no—
245C22⋊C8—44no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24#10314yes—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270C2×Q8—24no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—88no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—416no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4#9914yes—
347C22×C4#10314yes—
348C22×C4#10314yes—
349C22×C4#10314yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×SD1612yes—
360C22×C4C2×SD1612yes—
361C22×C4C2×SD1612yes—
362C22×C4C2×Q1612yes—
363C22×C4C2×Q1612yes—
364C22×C4C2×Q1612yes—
365C22×C4AGL(1,ℤ/8)12yes—
366C22×C4#13712yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370M16—88no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—88no—
378C4⋊C4—24no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—28no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—28no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23C2×#9912yes—
433C23#41312yes—
434C23C2×#10312yes—
435C23C2×#10312yes—
436C23C2×#10312yes—
437C23#41712yes—
438C23#42112yes—
439C23C22×C22⋊C411yescharacteristic
440Q8—84no—
441Q8—84no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4C2×#9912yes—
509C2×C4C2×#9912yes—
510C2×C4C2×#10312yes—
511C2×C4C2×#10312yes—
512C2×C4C2×#10312yes—
513C2×C4#41712yes—
514C2×C4#42112yes—
515C2×C4#56812yes—
516C2×C4C22×SD1611yescharacteristic
517C2×C4C22×Q1611yescharacteristic
518C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
519C2×C4C2×#13711yescharacteristic
520C8—84no—
521C8—84no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—416no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22#152014yes—
543C22#155014yes—
544C22#287911yescharacteristic
545C22C22×#9911yescharacteristic
546C22C2×#41311yescharacteristic
547C22C22×#10311yescharacteristic
548C22C2×#41711yescharacteristic
549C22C2×#42111yescharacteristic
550C22C2×#42111yescharacteristic
551C4—21no—
552C4—21no—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—21no—
557C4—21no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C2—24no—
570C2—24no—
571C2—42no—
572C2#1739211yescharacteristic
573C2#1748611yescharacteristic
574C2#1749211yescharacteristic
575C2C2×#152011yescharacteristic
576C2#1759211yescharacteristic
577C2#1765911yescharacteristic
578C2C2×#155011yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7715C2semidirect
5. #7681C2semidirect
6. #7643C2semidirect
7. #7641C2semidirect
41. #1272C22semidirect
42. #1271C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
#206262
C2×#17972
#77152
#76812
#76432
#76412