Finite Group Databasebeta

← #6434459

Subgroups — #6434459

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#30132C214yesmaximal, complemented
4#30078C212yesmaximal, complemented
5#30042C212yesmaximal, complemented
6#29951C212yesmaximal
7#29908C212yesmaximal
8#29715C212yesmaximal
9#29627C212yesmaximal
10#23823C212yesmaximal, complemented
11#23805C212yesmaximal, complemented
12#23685C214yesmaximal, complemented
13#21455C214yesmaximal
14#20778C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8C2212yes—
17C2×#451C2214yes—
18C2×#441C2218yes—
19#2864—21no—
20#2863—21no—
21#2863C2212yes—
22#2862C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855—21no—
27#2855C2212yes—
28#2851—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×#125—21no—
40C4×C22⋊Q8—21no—
41C4×C22⋊Q8C2212yes—
42#2295—21no—
43#2295C2212yes—
44C2×#391C2212yes—
45C2×#389—21no—
46C2×#389—21no—
47C2×#384C2211yescharacteristic
48#2283—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4—21no—
51C2×C43C2211yescharacteristic
52#2099C2214yes—
53#2096C2214yes—
54#2096C2214yes—
55#2084C2214yes—
56#2072C2214yes—
57#2072C2214yes—
58#2066C2218yes—
59#2056C2218yes—
60#2043C2214yes—
61#2043C2214yes—
62#2024C2214yes—
63#2016C2214yes—
64#2016C2214yes—
65#2016C2214yes—
66#1878C2218yes—
67#1872C2214yes—
68#1865C2218yes—
69#1853C2214yes—
70#1839C2212yes—
71#1831C2212yes—
72#1817C2212yes—
73#1797C2212yes—
74#1787C2214yes—
75#1767C2212yes—
76#1763C2218yes—
77#1749C2212yes—
78#1745C2212yes—
79C23×Q8C2312yes—
80C22×Q16C2318yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×#125—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8C2312yes—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—21no—
94C2×C4×Q8—48no—
95C2×C4×Q8C2314yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2314yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2312yes—
104C22×C42—22no—
105C22×C42D411yescharacteristic
106C22×C42D411yescharacteristic
107C22×C42D411yescharacteristic
108C22×C42D411yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#451—216no—
113#441—416no—
114C2×#106C2318yes—
115C2×C4⋊C8C2314yes—
116C2×#103C2318yes—
117C2×#103C2318yes—
118C2×#103C2318yes—
119C2×#103C2318yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2314yes—
122#391—22no—
123#391—22no—
124#391—21no—
125#391—21no—
126#391—21no—
127#391—21no—
128#391C2314yes—
129#391C2314yes—
130#391C2312yes—
131#391C2312yes—
132#391C2312yes—
133#391C2312yes—
134#389—21no—
135#389—21no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—21no—
145#389—21no—
146#389—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#385—22no—
163#385—22no—
164#384—21no—
165#384—21no—
166#384C2312yes—
167#384C2312yes—
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#383—22no—
171#382—21no—
172#382—21no—
173#382—21no—
174#381—22no—
175#380—21no—
176#380—21no—
177#380—21no—
178#380C2312yes—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—22no—
194#376—22no—
195C2×#95—22no—
196C2×#95—22no—
197C43D411yescharacteristic
198C43D411yescharacteristic
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2318yes—
206#340C2318yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211C22×Q8—22no—
212C22×Q8—21no—
213C22×Q8—24no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8#4118yes—
217C22×Q8C2412yes—
218C22×Q8C2414yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2411yescharacteristic
228C2×Q16—216no—
229C2×Q16—48no—
230C2×Q16—48no—
231C22×C8C2418yes—
232C4⋊Q8—28no—
233C4⋊Q8—28no—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—42no—
237C4⋊Q8—42no—
238#125—44no—
239C22⋊Q8—28no—
240C22⋊Q8—44no—
241C4×Q8—216no—
242C4×Q8—28no—
243C4×Q8—48no—
244C4×Q8—44no—
245C4×Q8—48no—
246C4×Q8—48no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—48no—
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×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#106—48no—
324C4⋊C8—44no—
325#103—216no—
326#103—48no—
327#103—48no—
328#103—48no—
329C22⋊C8—28no—
330#97—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349C24C22×D411yescharacteristic
350C2×Q8—28no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—28no—
354C2×Q8—24no—
355C2×Q8—28no—
356C2×Q8—24no—
357C2×Q8—416no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—416no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8C2×#4118yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4#13714yes—
396C22×C4#13714yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4114yes—
411C22×C4C2511yescharacteristic
412Q16—432no—
413C2×C8—28no—
414C2×C8—28no—
415C2×C8—28no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—48no—
473C42—44no—
474C42—48no—
475C42—44no—
476C42#13714yes—
477C42H(5,𝔽2)14yes—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23#52512yes—
483C23C2×#13712yes—
484C23C2×#13712yes—
485C23C23×D411yescharacteristic
486Q8—216no—
487Q8—416no—
488Q8—48no—
489Q8—48no—
490Q8—84no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
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—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—28no—
517C2×C4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—416no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—88no—
546C2×C4#52512yes—
547C2×C4#53512yes—
548C2×C4#53512yes—
549C2×C4C2×#13712yes—
550C2×C4C2×#13712yes—
551C2×C4C2×#13712yes—
552C2×C4C2×#13712yes—
553C2×C4C2×#13712yes—
554C2×C4C2×#13712yes—
555C2×C4C2×#13712yes—
556C2×C4#57512yes—
557C2×C4C22×#4112yes—
558C2×C4C2×H(5,𝔽2)12yes—
559C2×C4#58512yes—
560C8—48no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22#311314yes—
568C22C2×#52511yescharacteristic
569C22C2×#53511yescharacteristic
570C22#344911yescharacteristic
571C22C22×#13711yescharacteristic
572C22C22×#13711yescharacteristic
573C22C22×#13711yescharacteristic
574C22C2×#57511yescharacteristic
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—48no—
587C4—41no—
588C4—41no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4#345914yes—
595C2—24no—
596C2#3459311yescharacteristic
597C2C2×#311311yescharacteristic
598C2#6056011yescharacteristic
599C2#6080011yescharacteristic
600C2#6080411yescharacteristic
601C2#6096011yescharacteristic
602C2#6097111yescharacteristic
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
3. #30132C2semidirect
4. #30078C2semidirect
5. #30042C2semidirect
10. #23823C2semidirect
11. #23805C2semidirect
12. #23685C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×C4⋊Q82
#301322
#300782
#300422
#299512
#299082
#297152
#296272
#238232
#238052
#236852
#214552
#207782