Finite Group Databasebeta

← #6490687

Subgroups — #6490687

Order 512.

1662 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#33151C214yesmaximal
4#30095C212yesmaximal, complemented
5#30094C212yesmaximal, complemented
6#29278C214yesmaximal, complemented
7#29223C212yesmaximal, complemented
8#28692C212yesmaximal, complemented
9#27966C212yesmaximal
10#27613C212yesmaximal
11#27610C212yesmaximal
12#25816C212yesmaximal
13#22250C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8—21no—
17C2×#501C2214yes—
18#2864—21no—
19#2864—21no—
20#2859C2211yescharacteristic
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2851—21no—
25#2851—21no—
26#2851C2212yes—
27#2851C2212yes—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8C2212yes—
33C4×C4⋊Q8C2212yes—
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×#125—21no—
37C4×#125—21no—
38C4×#125C2212yes—
39C4×#125C2212yes—
40#2296—21no—
41#2296—21no—
42C2×#391—21no—
43C2×#389—21no—
44C2×#389C2212yes—
45#2285C2211yescharacteristic
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2212yes—
50C2×C43C2211yescharacteristic
51#2100C2214yes—
52#2100C2214yes—
53#2100C2214yes—
54#2099C2214yes—
55#2084C2214yes—
56#2083C2214yes—
57#2058C2214yes—
58#2057C2214yes—
59#1993C2214yes—
60#1992C2214yes—
61#1989C2218yes—
62#1949C2218yes—
63#1921C2218yes—
64#1919C2218yes—
65#1919C2218yes—
66#1918C2218yes—
67#1853C2214yes—
68#1852C2214yes—
69#1839C2212yes—
70#1836C2214yes—
71#1831C2212yes—
72#1823C2214yes—
73#1823C2214yes—
74#1817C2212yes—
75#1797C2212yes—
76#1767C2212yes—
77#1749C2212yes—
78#1745C2212yes—
79C23×Q8—21no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#125—21no—
86C2×#125—21no—
87C2×#125C2312yes—
88C2×#125C2312yes—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—48no—
94C2×#117—21no—
95C2×#117—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C42—22no—
103C22×C42D411yescharacteristic
104C22×C42D411yescharacteristic
105C22×C42D411yescharacteristic
106C22×C42D411yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#501—216no—
111C2×#107C2318yes—
112C2×#106C2318yes—
113C2×C4⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2314yes—
116#391—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389C2312yes—
139#389C2312yes—
140#389C2312yes—
141#389C2312yes—
142#389C2314yes—
143#389C2314yes—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#385—22no—
153#385—22no—
154#385—22no—
155#385—22no—
156#384—21no—
157#384—21no—
158#384C2312yes—
159#384C2312yes—
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#384C2311yescharacteristic
163#384C2311yescharacteristic
164#382—21no—
165#382—21no—
166#382—21no—
167#382—21no—
168#382C2312yes—
169#382C2312yes—
170#380—21no—
171#380—21no—
172#380C2312yes—
173#380C2312yes—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2314yes—
190C2×#95—22no—
191C2×#95—22no—
192C43D411yescharacteristic
193C43D411yescharacteristic
194C43D411yescharacteristic
195C43D411yescharacteristic
196C43C2311yescharacteristic
197C43C2311yescharacteristic
198C43C2311yescharacteristic
199C43C2311yescharacteristic
200#340C2318yes—
201#340C2318yes—
202#340C2318yes—
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210C22×Q8—28no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—44no—
214C22×Q8—44no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2411yescharacteristic
224C22×C8C2418yes—
225C4⋊Q8—28no—
226C4⋊Q8—28no—
227C4⋊Q8—28no—
228C4⋊Q8—24no—
229C4⋊Q8#4118yes—
230#125—28no—
231#125—28no—
232#125—44no—
233#125—44no—
234C4×Q8—48no—
235C4×Q8—48no—
236C4×Q8—48no—
237C4×Q8—48no—
238#117—44no—
239#117—44no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2414yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—48no—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—216no—
321#106—216no—
322C4⋊C8—28no—
323C22⋊C8—28no—
324#97—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341C24C22×D411yescharacteristic
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—432no—
347C2×Q8—44no—
348C2×Q8—44no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4AGL(1,ℤ/8)14yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C42−1+414yes—
395C22×C4C2511yescharacteristic
396C2×C8—28no—
397C2×C8—28no—
398C2×C8—28no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4C2×#4118yes—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—48no—
460C42—48no—
461C42—44no—
462C42—44no—
463C42#13714yes—
464C42C2×#4114yes—
465C422−1+414yes—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23#53612yes—
471C23C2×AGL(1,ℤ/8)12yes—
472C23#57512yes—
473C23C23×D411yescharacteristic
474Q8—416no—
475Q8—416no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—28no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—416no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—88no—
531C2×C4#52512yes—
532C2×C4#52512yes—
533C2×C4#53612yes—
534C2×C4#53612yes—
535C2×C4#53612yes—
536C2×C4C2×AGL(1,ℤ/8)12yes—
537C2×C4C2×AGL(1,ℤ/8)12yes—
538C2×C4C2×#13712yes—
539C2×C4C2×#13712yes—
540C2×C4C2×#13712yes—
541C2×C4#57512yes—
542C2×C4C22×#4112yes—
543C2×C4C2×2−1+412yes—
544C2×C4C2×2−1+412yes—
545C8—48no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22#323414yes—
553C22C2×#52511yescharacteristic
554C22C2×#53611yescharacteristic
555C22C2×#53611yescharacteristic
556C22C22×AGL(1,ℤ/8)11yescharacteristic
557C22C22×#13711yescharacteristic
558C22C2×#57511yescharacteristic
559C22C2×#57511yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—48no—
572C4—41no—
573C4—41no—
574C4—82no—
575C4—82no—
576C4—84no—
577C4—82no—
578C4—82no—
579C4#352314yes—
580C2—24no—
581C2#3524811yescharacteristic
582C2#6055811yescharacteristic
583C2#6056011yescharacteristic
584C2C2×#323411yescharacteristic
585C2#6085411yescharacteristic
586C2#6086011yescharacteristic
587C2#6086511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #30095C2semidirect
5. #30094C2semidirect
6. #29278C2semidirect
7. #29223C2semidirect
8. #28692C2semidirect
13. #22250C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C2×C4×C4⋊Q82
#331512
#300952
#300942
#292782
#292232
#286922
#279662
#276132
#276102
#258162
#222502
#207782