Finite Group Databasebeta

← #6484716

Subgroups — #6484716

Order 512.

2034 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C212yesmaximal
2#36645C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic
4#33108C214yesmaximal
5#29756C214yesmaximal
6#28748C212yesmaximal, complemented
7#28687C212yesmaximal, complemented
8#28653C214yesmaximal, complemented
9#28283C214yesmaximal, complemented
10#28205C214yesmaximal, complemented
11#21860C212yesmaximal
12#20808C211yesmaximal, characteristic
13C2×#533C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8—21no—
16#3229—216no—
17C2×#498C2214yes—
18C2×#487C2214yes—
19C2×#483C2218yes—
20C2×#475C2218yes—
21C2×#431C2212yes—
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2862C2212yes—
25#2859—21no—
26#2859—21no—
27#2857—21no—
28#2857—21no—
29#2857C2212yes—
30#2857C2212yes—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43C2×#391—21no—
44C2×#389—21no—
45C2×#389C2212yes—
46#2285—21no—
47#2285—21no—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#2084C2218yes—
53#2066C22116yes—
54#2043C2218yes—
55#1990C2214yes—
56#1981C2218yes—
57#1956C2214yes—
58#1948C2218yes—
59#1927C22116yes—
60#1872C2218yes—
61#1842C2212yes—
62#1831C2212yes—
63#1803C2218yes—
64#1797C2212yes—
65#1797C2212yes—
66C4×C4⋊C8C2212yes—
67C4×C4⋊C8C2212yes—
68C23×Q8—21no—
69#533—28no—
70#533—28no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8D418yes—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×#125—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8C2314yes—
86C2×C4×Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—48no—
92C2×C4×Q8C2314yes—
93C2×#117—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C42—22no—
101C22×C42D411yescharacteristic
102C22×C42D411yescharacteristic
103C22×C42D411yescharacteristic
104C22×C42D411yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#498—216no—
109#487—216no—
110#483—232no—
111#475—232no—
112#431—28no—
113C2×#106C23116yes—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2314yes—
116C2×C4⋊C8C2314yes—
117C2×#103C23116yes—
118C2×#103C23116yes—
119C2×C22⋊C8C2314yes—
120C2×C4×C8C2314yes—
121#391—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#391C2314yes—
128#391C2314yes—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389C2314yes—
144#389C2314yes—
145#384—21no—
146#384—21no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384—21no—
152#384—21no—
153#384C2312yes—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#382—22no—
158#382—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C2×#95—22no—
176C2×#95—22no—
177C43D411yescharacteristic
178C43D411yescharacteristic
179C43D411yescharacteristic
180C43D411yescharacteristic
181C43C2311yescharacteristic
182C43C2311yescharacteristic
183C43C2311yescharacteristic
184C43C2311yescharacteristic
185#340C23116yes—
186#336C2314yes—
187#336C2314yes—
188C22×Q8—28no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8C2414yes—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2411yescharacteristic
204C22×C8C2418yes—
205C4⋊Q8—28no—
206C4⋊Q8—28no—
207C4⋊Q8—24no—
208C4⋊Q8—24no—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—28no—
214C4⋊Q8—44no—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
217C4⋊Q8C2×D418yes—
218C4⋊Q8C2×D418yes—
219#125—48no—
220C22⋊Q8—28no—
221C22⋊Q8—28no—
222C22⋊Q8—48no—
223C4×Q8—28no—
224C4×Q8—28no—
225C4×Q8—48no—
226C4×Q8—48no—
227C4×Q8—48no—
228C4×Q8—48no—
229C4×Q8—48no—
230#117—48no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2414yes—
264C2×C22⋊C4—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—48no—
274C2×C42SD1618yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D414yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4114yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305#106—232no—
306C4⋊C8—28no—
307C4⋊C8—28no—
308C4⋊C8—44no—
309#103—232no—
310#103—416no—
311C22⋊C8—28no—
312C4×C8—28no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321C24C22×D411yescharacteristic
322C2×Q8—28no—
323C2×Q8—28no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—28no—
327C2×Q8—28no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—432no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4#12114yes—
361C22×C4#12114yes—
362C22×C4C2×SD1614yes—
363C22×C4C2×SD1614yes—
364C22×C4C2×SD1614yes—
365C22×C4#13714yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4112yes—
381C22×C4C2511yescharacteristic
382C2×C8—28no—
383C2×C8—28no—
384C2×C8—28no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4C22×D418yes—
413C22⋊C4—48no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—48no—
433C42—48no—
434C42—48no—
435C42#12114yes—
436C42#12114yes—
437C42C2×SD1614yes—
438C42C2×SD1614yes—
439C42C2×SD1614yes—
440C42C2×SD1614yes—
441C42#13714yes—
442C42C22×D412yes—
443C42C22×D412yes—
444C42C2×#4112yes—
445C42C2×#4112yes—
446C422−1+414yes—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23C2×#12112yes—
452C23C22×SD1612yes—
453C23C2×#13712yes—
454C23C23×D411yescharacteristic
455Q8—416no—
456Q8—416no—
457Q8—48no—
458Q8—48no—
459Q8—88no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
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—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—416no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—88no—
501C2×C4#461116yes—
502C2×C4C2×#12112yes—
503C2×C4C2×#12112yes—
504C2×C4C2×#12112yes—
505C2×C4C2×#12112yes—
506C2×C4C2×#12112yes—
507C2×C4#53612yes—
508C2×C4#53612yes—
509C2×C4C22×SD1612yes—
510C2×C4C22×SD1612yes—
511C2×C4C22×SD1612yes—
512C2×C4C22×SD1612yes—
513C2×C4C22×SD1612yes—
514C2×C4C22×SD1612yes—
515C2×C4C2×#13712yes—
516C2×C4C2×#13712yes—
517C2×C4C2×#13712yes—
518C2×C4C2×#13712yes—
519C2×C4C2×#13712yes—
520C2×C4#57512yes—
521C2×C4C23×D411yescharacteristic
522C2×C4C22×#4111yescharacteristic
523C2×C4C2×2−1+412yes—
524C2×C4#58512yes—
525C8—48no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22C2×#46114yes—
533C22C22×#12111yescharacteristic
534C22C2×#53611yescharacteristic
535C22#344911yescharacteristic
536C22C23×SD1611yescharacteristic
537C22C22×#13711yescharacteristic
538C22C22×#13711yescharacteristic
539C22C2×#57511yescharacteristic
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—48no—
550C4—84no—
551C4—84no—
552C4—84no—
553C4C2×#46114yes—
554C4C2×#46114yes—
555C4#303714yes—
556C4#322914yescomplemented
557C4#349712yes—
558C4#350712yes—
559C2—24no—
560C2#3517811yescharacteristic
561C2C22×#46111yescharacteristic
562C2C2×#303711yescharacteristic
563C2C2×#322911yescharacteristic
564C2#6086511yescharacteristic
565C2#6094111yescharacteristic
566C2#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
6. #28748C2semidirect
7. #28687C2semidirect
8. #28653C2semidirect
9. #28283C2semidirect
10. #28205C2semidirect
556. C4#3229semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#366452
C2×C4×C4⋊Q82
#331082
#297562
#287482
#286872
#286532
#282832
#282052
#218602
#208082