Finite Group Databasebeta

← #6630458

Subgroups — #6630458

Order 512.

2457 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3#33151C214yesmaximal, complemented
4#33111C214yesmaximal, complemented
5#33099C212yesmaximal, complemented
6#33095C212yesmaximal, complemented
7#32982C212yesmaximal, complemented
8#32978C212yesmaximal
9#21579C214yesmaximal, complemented
10#21578C214yesmaximal, complemented
11#21391C212yesmaximal, complemented
12#21363C212yesmaximal, complemented
13#20715C211yesmaximal, characteristic, complemented
14C2×#529C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C23×C42—21no—
17C23×C42C2211yescharacteristic
18C2×#483C2214yes—
19C2×#480C2214yes—
20C2×#432C2212yes—
21#2855C2211yescharacteristic
22#2854C2211yescharacteristic
23#2853—21no—
24#2852—22no—
25#2851C2212yes—
26#2850C2212yes—
27#2849—21no—
28#2848C2211yescharacteristic
29#2847C2211yescharacteristic
30C4×C22⋊Q8C2212yes—
31C4×#123—21no—
32C4×#121C2212yes—
33C4×#120—21no—
34#2290—22no—
35#2283C2211yescharacteristic
36C2×C4×C4⋊C4—22no—
37C2×C4×C4⋊C4—21no—
38C2×C4×C4⋊C4—22no—
39C2×C4×C4⋊C4—21no—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2214yes—
42C2×C4×C4⋊C4C2214yes—
43C2×C4×C4⋊C4C2212yes—
44C22×#95—21no—
45#1989C2214yescomplemented
46#1983C2214yescomplemented
47#1949C2218yescomplemented
48#1928C2218yescomplemented
49#1926C2218yescomplemented
50#1921C2218yescomplemented
51#1919C2218yescomplemented
52#1874C2214yes—
53#1874C2214yes—
54#1872C2214yes—
55#1872C2214yes—
56#1853C2214yes—
57#1853C2214yes—
58#1852C2214yes—
59#1852C2214yes—
60#1823C2214yes—
61#1823C2214yes—
62#1817C2212yes—
63#1797C2214yes—
64#1779C2212yes—
65#1776C2218yescomplemented
66#1763C2214yescomplemented
67#1762C2214yescomplemented
68#1749C2214yes—
69C24×C4—22no—
70C24×C4D411yescharacteristic
71C24×C4D411yescharacteristic
72C24×C4C2311yescharacteristic
73#529—28no—
74C2×#126—21no—
75C2×#125C2311yescharacteristic
76C2×#124C2311yescharacteristic
77C2×#123—21no—
78C2×#123—21no—
79C2×C22⋊Q8C2312yes—
80C2×C22⋊Q8C2311yescharacteristic
81C2×#121C2312yes—
82C2×#121C2311yescharacteristic
83C2×#120—21no—
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—22no—
87C2×C4×D4C2311yescharacteristic
88C2×#117—21no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—216no—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C42—28no—
97C22×C42—24no—
98C22×C42—24no—
99C22×C42—22no—
100C22×C42—22no—
101C22×C42—48no—
102C22×C42D412yes—
103C22×C42D411yescharacteristic
104C22×C42D412yes—
105C22×C42D411yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110#483—216no—
111#480—216no—
112#432—28no—
113C2×#107C2318yes—
114C2×#106C2318yes—
115C2×C4⋊C8C2314yes—
116C2×#103C2318yes—
117C2×#102C2318yes—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2314yes—
122#391C2312yes—
123#390C2312yes—
124#389C2312yes—
125#389C2312yes—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#386—22no—
135#386—21no—
136#386—22no—
137#386—22no—
138#386—22no—
139#386—21no—
140#385—22no—
141#385—21no—
142#385—22no—
143#385—22no—
144#385—22no—
145#385—21no—
146#383—21no—
147#382C2312yes—
148#381—21no—
149#381—21no—
150#380C2312yes—
151#380C2312yes—
152#380C2311yescharacteristic
153#380C2311yescharacteristic
154#379—21no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4C2318yes—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2312yes—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—22no—
169C4×C22⋊C4—22no—
170#376—21no—
171C2×#95—24no—
172C2×#95—24no—
173C2×#95—22no—
174C2×#95—22no—
175#340C2318yes—
176#340C2318yes—
177#340C2318yes—
178#340C2318yes—
179#336C2318yes—
180C25C2×D411yescharacteristic
181C22×Q8C2411yescharacteristic
182C22×D4—21no—
183C22×D4—22no—
184C22×D4C2411yescharacteristic
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—216no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—21no—
196C23×C4—44no—
197C23×C4—48no—
198C23×C4—48no—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C22×C8C2418yes—
212#126—44no—
213#125—24no—
214#124—24no—
215#123—44no—
216#123—44no—
217C22⋊Q8—24no—
218C22⋊Q8—28no—
219#121—24no—
220#121—28no—
221#120—44no—
222C4×Q8—24no—
223C4×D4—24no—
224C4×D4—48no—
225C4×D4—44no—
226#117—44no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4#41116yes—
244C2×C4⋊C4C2411yescharacteristic
245C2×C4⋊C4C2411yescharacteristic
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2418yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—28no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—28no—
274C2×C42—416no—
275C2×C42—416no—
276C2×C42—48no—
277C2×C42—48no—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4118yes—
286C2×C42C2412yes—
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289#107—216no—
290#106—216no—
291C4⋊C8—28no—
292#103—216no—
293#102—216no—
294C22⋊C8—28no—
295C22⋊C8—28no—
296C22⋊C8—28no—
297#97—28no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—21no—
302#95—21no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312C24—22no—
313C24—24no—
314C24—22no—
315C24—24no—
316C24—21no—
317C24AGL(1,ℤ/8)14yes—
318C24C22×D411yescharacteristic
319C24C22×D412yes—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—42no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—44no—
330C22×C4—232no—
331C22×C4—232no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—22no—
353C22×C4—216no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—416no—
370C22×C4—464no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4#12314yes—
379C22×C4#12314yes—
380C22×C4#13714yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C2×#4114yes—
390C22×C4C2×#4118yes—
391C22×C4C2511yescharacteristic
392C2×C8—28no—
393C2×C8—28no—
394C2×C8—28no—
395C4⋊C4—216no—
396C4⋊C4—216no—
397C4⋊C4—216no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4C2×#41116yes—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C42—24no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—48no—
431C42—48no—
432C42—416no—
433C42—44no—
434C42—42no—
435C42#12314yes—
436C42#12314yes—
437C42C2×#4114yes—
438C42C2×#4114yes—
439C23—28no—
440C23—28no—
441C23—216no—
442C23—28no—
443C23—216no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—41no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23C2×#12312yes—
456C23#52514yes—
457C23C2×AGL(1,ℤ/8)12yes—
458C23C2×AGL(1,ℤ/8)14yes—
459C23C2×#13712yes—
460C23#57514yes—
461C23C23×D411yescharacteristic
462Q8—44no—
463D4—44no—
464D4—84no—
465D4—88no—
466C2×C4—216no—
467C2×C4—28no—
468C2×C4—216no—
469C2×C4—216no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—28no—
484C2×C4—216no—
485C2×C4—22no—
486C2×C4—216no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—432no—
494C2×C4—432no—
495C2×C4—432no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—84no—
512C2×C4C2×#12312yes—
513C2×C4C2×#12312yes—
514C2×C4C2×#12312yes—
515C2×C4C2×#12312yes—
516C2×C4C2×#12312yes—
517C2×C4#52514yes—
518C2×C4C2×#13714yes—
519C2×C4C22×#4112yes—
520C2×C4C22×#4114yes—
521C8—48no—
522C22—232no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—22no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—82no—
537C22—82no—
538C22#309214yes—
539C22#311418yes—
540C22C22×#12311yescharacteristic
541C22C2×#52512yes—
542C22C22×AGL(1,ℤ/8)11yescharacteristic
543C22C22×#13711yescharacteristic
544C22C2×#57512yes—
545C4—22no—
546C4—24no—
547C4—22no—
548C4—22no—
549C4—24no—
550C4—22no—
551C4—22no—
552C4—42no—
553C4—416no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—82no—
558C4—82no—
559C4#344414yes—
560C2—24no—
561C2—28no—
562C2—161no—
563C2#3662611yescharacteristic
564C2C2×#309211yescharacteristic
565C2#6051111yescharacteristic
566C2C2×#311412yes—
567C2#6056012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2847C22
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
2. C22×C4×C4⋊C4C2semidirect
3. #33151C2semidirect
4. #33111C2semidirect
5. #33099C2semidirect
6. #33095C2semidirect
7. #32982C2semidirect
9. #21579C2semidirect
10. #21578C2semidirect
11. #21391C2semidirect
12. #21363C2semidirect
13. #20715C2semidirect
45. #1989C22semidirect
46. #1983C22semidirect
47. #1949C22semidirect
48. #1928C22semidirect
49. #1926C22semidirect
50. #1921C22semidirect
51. #1919C22semidirect
65. #1776C22semidirect
66. #1763C22semidirect
67. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#366212
C22×C4×C4⋊C42
#331512
#331112
#330992
#330952
#329822
#329782
#215792
#215782
#213912
#213632
#207152