Finite Group Databasebeta

← #162700

Subgroups — #162700

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C2×#1728C211yesmaximal, characteristic, complemented
4#7710C214yesmaximal, complemented
5#7690C214yesmaximal, complemented
6#7609C212yesmaximal, complemented
7#7606C212yesmaximal, complemented
8C23×M16—21no—
9C22×#124C411yescharacteristic
10C22×C22⋊Q8C411yescharacteristic
11C22×#121C411yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#432—21no—
14#2850—21no—
15#2850—21no—
16#2850—21no—
17#2850—21no—
18#2850C412yes—
19#2850C412yes—
20#2850C2212yes—
21#2850C2212yes—
22C2×#391C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#388—21no—
25C2×#388—21no—
26C2×#386—21no—
27C2×#386—21no—
28C2×#382C411yescharacteristic
29#1817C2212yes—
30#1797—21no—
31#1797C2212yescomplemented
32#1779—21no—
33#1749—21no—
34#1749C2212yescomplemented
35#1728—21no—
36#1728—21no—
37#1728C2212yes—
38#1728C2212yescomplemented
39C2×#336—21no—
40C2×#336C2212yescomplemented
41#1272C2218yescomplemented
42#1270C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—28no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—44no—
53C22×M16—44no—
54C23×C8—22no—
55C2×#124—21no—
56C2×#124—21no—
57C2×#124—44no—
58C2×#124C2×C412yes—
59C2×#124C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—44no—
63C2×C22⋊Q8C2×C412yes—
64C2×C22⋊Q8C2×C412yes—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—44no—
68C2×#121C2×C412yes—
69C2×#121C2×C412yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—44no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—44no—
93C2×C4⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×#97—22no—
98#391—21no—
99#391—21no—
100#391C2×C412yes—
101#391C2312yes—
102#390—21no—
103#390—21no—
104#390C2×C412yes—
105#390C2312yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#382—21no—
123#382—21no—
124#382C2×C412yes—
125#382C2×C412yes—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336C2314yes—
135#336C2314yes—
136C25—21no—
137C25C22⋊C411yescharacteristic
138C22×Q8—21no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8C22×C412yes—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4C22×C412yes—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
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×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—28no—
185C2×M16—28no—
186C2×M16—432no—
187C2×M16—44no—
188C2×M16—44no—
189C2×M16—44no—
190C2×M16—44no—
191C22×C8—28no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—44no—
195C22×C8—44no—
196#124—44no—
197#124—44no—
198#124—44no—
199#124—44no—
200C22⋊Q8—44no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207#121—44no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2411yescharacteristic
264C4⋊C8—44no—
265C22⋊C8—44no—
266C22⋊C8—44no—
267C22⋊C8—44no—
268#97—44no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24#10414yes—
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C2×Q8—24no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—88no—
302C2×D4—24no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—88no—
315C2×D4—816no—
316C22×C4—216no—
317C22×C4—216no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
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—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
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—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—416no—
375C22×C4—416no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4#9914yes—
386C22×C4#10014yes—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4#13712yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401M16—416no—
402M16—416no—
403C2×C8—28no—
404C2×C8—416no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
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—28no—
460C23—28no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—84no—
478C23—84no—
479C23—84no—
480C23C2×#9912yes—
481C23C2×#10012yes—
482C23#41712yes—
483C23C2×#10412yes—
484C23C2×#10412yes—
485C23C2×#10412yes—
486C23#42112yes—
487C23C22×C22⋊C411yescharacteristic
488Q8—84no—
489Q8—84no—
490D4—84no—
491D4—84no—
492D4—88no—
493D4—88no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
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—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
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—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
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—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4C2×#9912yes—
556C2×C4C2×#9912yes—
557C2×C4C2×#10012yes—
558C2×C4C2×#10012yes—
559C2×C4#41712yes—
560C2×C4#56812yes—
561C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
562C2×C4C2×#13711yescharacteristic
563C2×C4#57512yes—
564C8—48no—
565C8—84no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—416no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—168no—
594C22#150414yes—
595C22#152714yes—
596C22#287911yescharacteristic
597C22C22×#9911yescharacteristic
598C22C22×#10011yescharacteristic
599C22C2×#41711yescharacteristic
600C22#289611yescharacteristic
601C22C22×#10411yescharacteristic
602C22C2×#42111yescharacteristic
603C4—21no—
604C4—21no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—82no—
617C4—82no—
618C4—82no—
619C2—24no—
620C2—24no—
621C2—42no—
622C2—162no—
623C2#1739611yescharacteristic
624C2#1749211yescharacteristic
625C2#1750211yescharacteristic
626C2C2×#150411yescharacteristic
627C2#1755911yescharacteristic
628C2C2×#152711yescharacteristic
629C2#1775011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
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×#1728C2semidirect
4. #7710C2semidirect
5. #7690C2semidirect
6. #7609C2semidirect
7. #7606C2semidirect
31. #1797C22semidirect
34. #1749C22semidirect
38. #1728C22semidirect
40. C2×#336C22semidirect
41. #1272C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#207152
C2×#17282
#77102
#76902
#76092
#76062