Finite Group Databasebeta

← #6477313

Subgroups — #6477313

Order 512.

3458 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic, complemented
2C2×#3193C212yesmaximal
3C2×#2854C211yesmaximal, characteristic
4#33003C214yesmaximal, complemented
5#29638C212yesmaximal, complemented
6#29624C212yesmaximal, complemented
7#25503C218yesmaximal, complemented
8#24064C214yesmaximal, complemented
9#23933C214yesmaximal, complemented
10#21765C212yesmaximal, complemented
11#20682C211yesmaximal, characteristic, complemented
12C2×#533—22no—
13C2×#533—22no—
14C2×#533—216no—
15C2×#533C2214yes—
16C2×#533C2214yescomplemented
17C22×C4⋊Q8C2211yescharacteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic, complemented
19C22×#125—21no—
20C22×#123—21no—
21C22×C22⋊Q8—21no—
22C22×C22⋊Q8C2212yes—
23C22×#121C2211yescharacteristic
24C22×C4×Q8—21no—
25C22×C4×Q8C2212yescomplemented
26C22×#117—21no—
27C23×C42C2211yescharacteristic
28#3193—216no—
29C2×#484C2214yes—
30C2×#480C2214yes—
31C2×#468C2218yes—
32C2×#448C2214yes—
33C2×#450C2214yes—
34C2×#407C2212yes—
35#2854—21no—
36#2854—21no—
37#2854—22no—
38#2854C2214yes—
39#2854C2212yes—
40#2854C2212yes—
41C2×#390C2212yes—
42C2×#389C2212yes—
43C2×#388—22no—
44C2×#386—21no—
45C2×#380C2211yescharacteristic
46#2057C2218yescomplemented
47#2043C2218yescomplemented
48#2023C2218yescomplemented
49#2013C2218yescomplemented
50#1894C2218yescomplemented
51#1880C22116yescomplemented
52#1872C2218yes—
53#1863C22116yescomplemented
54#1823C2214yescomplemented
55#1817C2214yes—
56#1817C2212yescomplemented
57#1817C2212yescomplemented
58#1787C2214yescomplemented
59#1728C2214yescomplemented
60C23×Q8—21no—
61C23×Q8C2312yes—
62C23×D4—21no—
63C23×D4C2311yescharacteristic
64C24×C4D411yescharacteristic
65C24×C4D411yescharacteristic
66C24×C4C2311yescharacteristic
67C22×SD16C2318yes—
68C22×M16C2314yes—
69#533—232no—
70#533—232no—
71#533—232no—
72#533—232no—
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8—28no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8C2314yes—
78C2×C4⋊Q8C2314yes—
79C2×#125—216no—
80C2×#125—24no—
81C2×#125—24no—
82C2×#123—22no—
83C2×#123—24no—
84C2×#123—22no—
85C2×#123—48no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8D4116yes—
93C2×C22⋊Q8C2314yes—
94C2×C22⋊Q8C2314yes—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—28no—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C2×C4×Q8—216no—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8C2318yes—
107C2×#117—216no—
108C2×#117—24no—
109C2×#117—24no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—22no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42—28no—
125C22×C42—22no—
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D412yes—
129C22×C42D412yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#484—216no—
135#480—216no—
136#468—232no—
137#450—216no—
138#448—216no—
139C2×#107C2318yes—
140C2×#103C2318yes—
141C2×#103C23116yes—
142C2×#102C2318yes—
143#407—28no—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146#390—22no—
147#390—22no—
148#390C2314yes—
149#390C2314yes—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#388—24no—
157#388—24no—
158#386—22no—
159#386—24no—
160#386—22no—
161#380—21no—
162#380—21no—
163#380C2312yes—
164#380C2312yes—
165C2×#95—21no—
166C2×#95—22no—
167#340C23116yes—
168#336C2318yes—
169#336C2318yes—
170C25—21no—
171C25C2×D411yescharacteristic
172C22×Q8—28no—
173C22×Q8—24no—
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8—48no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8C2×D418yes—
180C22×Q8C2414yes—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—21no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4C2412yes—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D412yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D418yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C2×SD16—216no—
216C2×SD16—416no—
217C2×M16—28no—
218C2×M16—48no—
219C22×C8C2418yes—
220C4⋊Q8—216no—
221C4⋊Q8—216no—
222C4⋊Q8—216no—
223C4⋊Q8—216no—
224#125—232no—
225#125—232no—
226#123—48no—
227#123—416no—
228#123—48no—
229C22⋊Q8—232no—
230C22⋊Q8—232no—
231C22⋊Q8—216no—
232C22⋊Q8—216no—
233C22⋊Q8—28no—
234C22⋊Q8—28no—
235C22⋊Q8C2×D4116yes—
236C22⋊Q8C2×D4116yes—
237#121—28no—
238#121—28no—
239#121—44no—
240#121—44no—
241C4×Q8—232no—
242C4×Q8—232no—
243C4×Q8—232no—
244C4×Q8—216no—
245C4×Q8—216no—
246#117—232no—
247#117—232no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2414yes—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—48no—
287C2×C42—216no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2412yes—
304C2×C42C2412yes—
305#107—216no—
306#103—216no—
307#103—416no—
308#102—216no—
309C22⋊C8—28no—
310C22⋊C8—28no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—24no—
315C24—22no—
316C24—24no—
317C24—22no—
318C24—22no—
319C24—42no—
320C24—42no—
321C24—44no—
322C24—42no—
323C24—42no—
324C24—44no—
325C24#12014yes—
326C24C22×D411yescharacteristic
327C24C22×D411yescharacteristic
328C24C22×D411yescharacteristic
329C2×Q8—216no—
330C2×Q8—28no—
331C2×Q8—24no—
332C2×Q8—216no—
333C2×Q8—416no—
334C2×Q8—48no—
335C2×Q8—432no—
336C2×Q8—48no—
337C2×Q8—416no—
338C2×Q8—48no—
339C2×Q8C22×D418yes—
340C2×D4—24no—
341C2×D4—28no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—42no—
350C2×D4—88no—
351C2×D4—816no—
352C22×C4—232no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—416no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4#12014yes—
397C22×C4#12014yes—
398C22×C4#12014yes—
399C22×C4AGL(1,ℤ/8)14yes—
400C22×C4#13718yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D414yes—
406C22×C4C22×D414yes—
407C22×C4C22×D414yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C2511yescharacteristic
412SD16—816no—
413M16—88no—
414C2×C8—28no—
415C2×C8—216no—
416C4⋊C4—216no—
417C4⋊C4—216no—
418C4⋊C4—232no—
419C4⋊C4—216no—
420C4⋊C4—216no—
421C4⋊C4—216no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—42no—
429C4⋊C4—48no—
430C4⋊C4C22×D418yes—
431C22⋊C4—216no—
432C22⋊C4—216no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C42—232no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—24no—
450C23—22no—
451C23—22no—
452C23—21no—
453C23—22no—
454C23—21no—
455C23—22no—
456C23—28no—
457C23—28no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—44no—
462C23—48no—
463C23—48no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23—44no—
468C23—42no—
469C23—84no—
470C23—84no—
471C23—88no—
472C23—88no—
473C23—84no—
474C23C2×#12012yes—
475C23C2×#12012yes—
476C23C2×#12012yes—
477C23C2×AGL(1,ℤ/8)12yes—
478C23C2×#13714yes—
479C23#57512yes—
480C23C23×D411yescharacteristic
481Q8—216no—
482Q8—416no—
483Q8—432no—
484Q8—48no—
485D4—48no—
486D4—84no—
487D4—88no—
488D4—88no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—216no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—216no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—416no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—416no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—88no—
529C2×C4—84no—
530C2×C4C2×#12012yes—
531C2×C4C2×#12012yes—
532C2×C4C2×#12012yes—
533C2×C4C2×AGL(1,ℤ/8)14yes—
534C2×C4C2×#13714yes—
535C2×C4C2×#13714yes—
536C2×C4C23×D412yes—
537C2×C4#58514yes—
538C8—48no—
539C22—22no—
540C22—22no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—416no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—84no—
555C22—84no—
556C22—82no—
557C22—82no—
558C22—168no—
559C22#300414yes—
560C22#300614yes—
561C22C22×#12011yescharacteristic
562C22#344912yes—
563C22C22×AGL(1,ℤ/8)11yescharacteristic
564C22C22×#13712yes—
565C22C2×#57511yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—24no—
570C4—24no—
571C4—24no—
572C4—48no—
573C4—42no—
574C4—48no—
575C4—84no—
576C4—82no—
577C4#348514yes—
578C2—24no—
579C2—24no—
580C2—42no—
581C2—162no—
582C2#3508811yescharacteristic
583C2C2×#300411yescharacteristic
584C2C2×#300611yescharacteristic
585C2#6095012yes—
586C2#6096912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#533C2semidirect
4. #33003C2semidirect
5. #29638C2semidirect
6. #29624C2semidirect
7. #25503C2semidirect
8. #24064C2semidirect
9. #23933C2semidirect
10. #21765C2semidirect
11. #20682C2semidirect
16. C2×#533C22semidirect
17. C22×C4⋊Q8C22semidirect
18. C22×C4⋊Q8C22semidirect
25. C22×C4×Q8C22semidirect
46. #2057C22semidirect
47. #2043C22semidirect
48. #2023C22semidirect
49. #2013C22semidirect
50. #1894C22semidirect
51. #1880C22semidirect
53. #1863C22semidirect
54. #1823C22semidirect
56. #1817C22semidirect
57. #1817C22semidirect
58. #1787C22semidirect
59. #1728C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#31932
C2×#28542
#330032
#296382
#296242
#255032
#240642
#239332
#217652
#206822