Finite Group Databasebeta

← #161950

Subgroups — #161950

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C2×#1817C211yesmaximal, characteristic, complemented
4#7708C212yesmaximal, complemented
5#7704C212yesmaximal, complemented
6#7654C214yesmaximal, complemented
7#7599C214yesmaximal, complemented
8C2×#532C411yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C22×C22⋊C8—21no—
13#2857C411yescharacteristic
14#2857C2211yescharacteristic
15#2856C411yescharacteristic
16#2856C2211yescharacteristic
17#2854—21no—
18#2854C2212yes—
19#2850—21no—
20#2850—21no—
21#2847—21no—
22#2847C412yes—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121C412yes—
26C4×#121C2212yes—
27#2297—21no—
28#2285C411yescharacteristic
29#1817—21no—
30#1817—21no—
31#1817C2212yes—
32#1817C2212yes—
33#1817C2212yes—
34#1797—21no—
35#1797C2212yes—
36#1779—21no—
37#1749—21no—
38#1749C2212yes—
39C2×#336—21no—
40C2×#336C2212yes—
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41—21no—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C23×C8—22no—
48#532—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#124—21no—
52#525—48no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×#120—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117—21no—
70C22×C42—44no—
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—44no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—44no—
91C2×#97—22no—
92#391—21no—
93#391—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390C2×C412yes—
99#390C2312yes—
100#389—21no—
101#389—21no—
102#389C2×C412yes—
103#389C2312yes—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#382—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380C2×C412yes—
121#380C2×C412yes—
122#380C2×C412yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C412yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336C2314yes—
132#336C2314yes—
133C25C22⋊C411yescharacteristic
134C2×#41—44no—
135C2×#41—44no—
136C2×#41—44no—
137C22×Q8—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C22×C412yes—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
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×C4C22⋊C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—44no—
175C22×C8—44no—
176C22×C8—44no—
177C4⋊Q8—42no—
178#127—42no—
179#124—44no—
180#123—48no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183#121—28no—
184#121—44no—
185#121—44no—
186#121—44no—
187#121—44no—
188#121—44no—
189#121—44no—
190#121—44no—
191#120—48no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4—48no—
195C4×D4—48no—
196#117—44no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2411yescharacteristic
247C4⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253#97—44no—
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—22no—
265C24—22no—
266C24—41no—
267C24—41no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271#41—816no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—42no—
292C2×D4—42no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
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—22no—
308C22×C4—22no—
309C22×C4—22no—
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—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—416no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—416no—
351C22×C4—416no—
352C22×C4—44no—
353C22×C4#9914yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4#13114yes—
364C22×C4AGL(1,ℤ/8)14yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—88no—
375C4⋊C4—24no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C22⋊C4—44no—
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—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—44no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—28no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23#40812yes—
444C23C2×#9912yes—
445C23#41312yes—
446C23#41812yes—
447C23#42112yes—
448C23C22×C22⋊C411yescharacteristic
449Q8—84no—
450D4—84no—
451D4—84no—
452D4—84no—
453D4—88no—
454D4—88no—
455D4—88no—
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—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—21no—
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—21no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—88no—
512C2×C4#40812yes—
513C2×C4C2×#9912yes—
514C2×C4C2×#9912yes—
515C2×C4#41812yes—
516C2×C4#42112yes—
517C2×C4C2×#13112yes—
518C2×C4C2×AGL(1,ℤ/8)12yes—
519C2×C4#57512yes—
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—416no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22#151014yes—
551C22C2×#40811yescharacteristic
552C22C22×#9911yescharacteristic
553C22C2×#41311yescharacteristic
554C22C2×#41811yescharacteristic
555C22#289611yescharacteristic
556C22C2×#42111yescharacteristic
557C22C2×#42111yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—82no—
571C4—82no—
572C4—82no—
573C2—24no—
574C2—42no—
575C2—42no—
576C2—162no—
577C2#1738111yescharacteristic
578C2#1746311yescharacteristic
579C2#1748211yescharacteristic
580C2C2×#151011yescharacteristic
581C2#1758211yescharacteristic
582C2#1767711yescharacteristic
583C2#1773111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
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
2. #20715C2semidirect
3. C2×#1817C2semidirect
4. #7708C2semidirect
5. #7704C2semidirect
6. #7654C2semidirect
7. #7599C2semidirect
41. #1271C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#366242
#207152
C2×#18172
#77082
#77042
#76542
#75992