Finite Group Databasebeta

← #6374383

Subgroups — #6374383

Order 512.

3209 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#34057C211yesmaximal, characteristic
4#21828C214yesmaximal, complemented
5#21827C214yesmaximal, complemented
6#21826C214yesmaximal, complemented
7#21825C214yesmaximal, complemented
8#21333C214yesmaximal, complemented
9C2×#1996C214yesmaximal, complemented
10#20689C214yesmaximal
11#3501—24no—
12#3501—24no—
13#3501—24no—
14#3501—24no—
15C2×#554C418yescomplemented
16C2×#554C414yescomplemented
17C2×D4×Q8C414yescomplemented
18C2×#544C2212yes—
19C2×#543C2212yes—
20C2×#543C2212yes—
21C2×#533C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C2×#518C2212yes—
25C22×C4×D4C2211yescharacteristic
26#2353—24no—
27#2352—24no—
28#2318—24no—
29#2315C2212yes—
30#2313C2212yes—
31#2313C2212yes—
32#2312C2212yes—
33#2312C2212yes—
34#2308C2212yes—
35#2308C2212yes—
36#2290—24no—
37#2283C2212yes—
38#2283C2212yes—
39#2280C2212yes—
40#2280C2212yes—
41C2×#380C2212yes—
42#2273C2212yes—
43C2×C4×C22⋊C4C2212yes—
44#1996C22116yes—
45#1996C22116yes—
46#1996C22116yes—
47#1996C22116yes—
48#1820C22116yes—
49#1819C22116yes—
50C2×#342C22116yes—
51C23×Q8C2×C412yes—
52C23×Q8C2×C412yescomplemented
53C23×D4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56#554—232no—
57#554—232no—
58#554—216no—
59#554—216no—
60D4×Q8—216no—
61D4×Q8—216no—
62#544—28no—
63#544—28no—
64#543—28no—
65#543—28no—
66#543—28no—
67#543—28no—
68#533—24no—
69#533—24no—
70#533—24no—
71#533—24no—
72C2×C4⋊Q8C2×C412yescomplemented
73C2×C4⋊Q8C2×C412yescomplemented
74C2×C4⋊Q8C2×C418yescomplemented
75C2×#125C2×C414yescomplemented
76C2×#125C2×C418yescomplemented
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8C2×C414yes—
86C2×C22⋊Q8C2×C414yes—
87C2×C22⋊Q8C2×C418yes—
88C2×C22⋊Q8C2×C418yes—
89C2×C22⋊Q8C2×C414yescomplemented
90C2×C22⋊Q8C2×C414yescomplemented
91C2×C22⋊Q8C2×C418yescomplemented
92C2×C22⋊Q8C2×C418yescomplemented
93C2×C22⋊Q8C2314yes—
94C2×C22⋊Q8C2314yes—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97#518—28no—
98#518—28no—
99C2×C4×Q8C2×C414yescomplemented
100C2×C4×Q8C2×C414yescomplemented
101C2×C4×D4C2×C418yes—
102C2×C4×D4C2×C418yes—
103C2×C4×D4C2×C414yes—
104C2×C4×D4C2×C414yes—
105C2×C4×D4D414yes—
106C2×C4×D4D414yes—
107C2×C4×D4D414yes—
108C2×C4×D4D414yes—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117C2314yes—
112C22×C4⋊C4—22no—
113C22×C4⋊C4C2×C414yes—
114C22×C4⋊C4C2×C412yes—
115C22×C4⋊C4C2×C414yescomplemented
116C22×C4⋊C4C2×C412yescomplemented
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42—22no—
125C22×C42C2311yescharacteristic
126#388—28no—
127#388—28no—
128#387—28no—
129#387—28no—
130#386—28no—
131#386—28no—
132#385—28no—
133#383—22no—
134#383—22no—
135#382—22no—
136#382—22no—
137#381—22no—
138#381—22no—
139#381C2314yes—
140#381C2314yes—
141#380—22no—
142#380—22no—
143#380—22no—
144#380—22no—
145#380C2314yes—
146#380C2314yes—
147#380C2314yes—
148#380C2314yes—
149#379C2314yes—
150#379C2314yes—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—22no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4C2314yes—
159C4×C22⋊C4C2314yes—
160C4×C22⋊C4C2314yes—
161C4×C22⋊C4C2314yes—
162#376—22no—
163#376—22no—
164C2×#95—22no—
165C2×#95—22no—
166#342C23164yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—84no—
174C22×Q8—84no—
175C22×Q8C22×C414yes—
176C22×Q8C22×C414yes—
177C22×Q8C22×C414yes—
178C22×Q8C22×C414yes—
179C22×D4—28no—
180C22×D4C22×C414yes—
181C22×D4C2×D414yes—
182C22×D4C2×D414yes—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—82no—
186C23×C4—82no—
187C23×C4C22×C414yes—
188C23×C4C22×C414yes—
189C23×C4C22×C414yes—
190C23×C4C22×C414yes—
191C23×C4C22×C412yes—
192C23×C4C22×C412yes—
193C23×C4C22×C412yes—
194C23×C4C22×C412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C4⋊Q8—232no—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213#125—232no—
214#125—28no—
215#125—28no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—28no—
220C22⋊Q8—28no—
221C22⋊Q8—28no—
222C22⋊Q8—28no—
223C22⋊Q8—28no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230C22⋊Q8—28no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233C22⋊Q8—28no—
234C22⋊Q8—232no—
235C22⋊Q8—232no—
236C22⋊Q8—232no—
237C22⋊Q8—232no—
238C4×Q8—28no—
239C4×Q8—28no—
240C4×Q8—28no—
241C4×Q8—28no—
242C4×D4—232no—
243C4×D4—232no—
244C4×D4C22⋊C4116yes—
245C4×D4C22⋊C4116yes—
246C4×D4C22⋊C4116yes—
247C4×D4C22⋊C4116yes—
248#117—28no—
249#117—28no—
250#117—84no—
251#117—84no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—84no—
265C2×C4⋊C4C22×C418yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C414yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C418yes—
274C2×C4⋊C4C22×C414yes—
275C2×C4⋊C4C22×C414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—216no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4C22×C418yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C22×C414yes—
293C2×C22⋊C4C22×C414yes—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2418yes—
299C2×C22⋊C4C2418yes—
300C2×C22⋊C4C2418yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—84no—
306C2×C42C22×C418yes—
307C2×C42C22×C412yes—
308C2×C42C22×C412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325C24—24no—
326C24—81no—
327C24—81no—
328C24—81no—
329C24—81no—
330C24C23×C412yes—
331C24C23×C412yes—
332C24C22×D412yes—
333C24C22×D412yes—
334C24C22×D412yes—
335C24C22×D412yes—
336C242−1+418yes—
337C24C2511yescharacteristic
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—48no—
344C2×Q8—88no—
345C2×Q8—88no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×Q8—88no—
350C2×Q8—88no—
351C2×D4—216no—
352C2×D4—28no—
353C2×D4—216no—
354C2×D4—28no—
355C2×D4C2×C22⋊C4116yes—
356C22×C4—24no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C414yes—
391C22×C4C2×C22⋊C414yes—
392C22×C4C2×C22⋊C414yes—
393C22×C4C2×C22⋊C414yes—
394C22×C4C2×C22⋊C414yes—
395C22×C4C2×C22⋊C414yes—
396C22×C4C2×C22⋊C414yes—
397C22×C4C23×C412yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C23×C418yes—
400C22×C4C23×C418yes—
401C22×C4C23×C412yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C23×C412yes—
404C22×C4C23×C412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—216no—
424C4⋊C4—216no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—216no—
428C4⋊C4—28no—
429C4⋊C4—48no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4C2×C22⋊C4116yes—
434C22⋊C4—216no—
435C22⋊C4—216no—
436C22⋊C4—28no—
437C22⋊C4—28no—
438C22⋊C4—216no—
439C22⋊C4—216no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4C2×C22⋊C4116yes—
447C22⋊C4C2×C22⋊C4116yes—
448C42—216no—
449C42—48no—
450C42—84no—
451C42—84no—
452C42—84no—
453C42C2×C22⋊C414yes—
454C42C2×C22⋊C414yes—
455C42C2×C22⋊C414yes—
456C42C2×C22⋊C414yes—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—88no—
475C23—88no—
476C23C22×C22⋊C414yes—
477C23C22×C22⋊C414yes—
478C23#51918yes—
479C23#53614yes—
480C23#53618yes—
481C23#53614yes—
482C23C24×C411yescharacteristic
483C23C23×D411yescharacteristic
484C23C23×D411yescharacteristic
485C23C2×2−1+414yes—
486Q8—48no—
487Q8—48no—
488Q8—88no—
489Q8—88no—
490Q8—88no—
491D4—232no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—88no—
529C2×C4—88no—
530C2×C4—88no—
531C2×C4—88no—
532C2×C4—816no—
533C2×C4—168no—
534C2×C4C22×C22⋊C414yes—
535C2×C4C22×C22⋊C414yes—
536C2×C4C22×C22⋊C411yescharacteristic
537C2×C4C22×C22⋊C411yescharacteristic
538C2×C4C22×C22⋊C411yescharacteristic
539C2×C4C22×C22⋊C414yes—
540C2×C4C22×C22⋊C414yes—
541C2×C4C22×C22⋊C411yescharacteristic
542C22—21no—
543C22—21no—
544C22—21no—
545C22—24no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—48no—
554C22—44no—
555C22—44no—
556C22—48no—
557C22—44no—
558C22—48no—
559C22—84no—
560C22—84no—
561C22#231018yes—
562C22C23×C22⋊C411yescharacteristic
563C22C2×#51912yes—
564C22C2×#53612yes—
565C22C2×#53612yes—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—24no—
570C4—24no—
571C4—24no—
572C4—24no—
573C4—48no—
574C4—48no—
575C4—88no—
576C4—88no—
577C4#288312yes—
578C2—22no—
579C2—22no—
580C2—22no—
581C2—28no—
582C2—44no—
583C2C2×#231011yescharacteristic
584C2C2×#288311yescharacteristic
585C2#6007011yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×D4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21828C2semidirect
5. #21827C2semidirect
6. #21826C2semidirect
7. #21825C2semidirect
8. #21333C2semidirect
9. C2×#1996C2semidirect
15. C2×#554C4semidirect
16. C2×#554C4semidirect
17. C2×D4×Q8C4semidirect
52. C23×Q8C2×C4semidirect
72. C2×C4⋊Q8C2×C4semidirect
73. C2×C4⋊Q8C2×C4semidirect
74. C2×C4⋊Q8C2×C4semidirect
75. C2×#125C2×C4semidirect
76. C2×#125C2×C4semidirect
89. C2×C22⋊Q8C2×C4semidirect
90. C2×C22⋊Q8C2×C4semidirect
91. C2×C22⋊Q8C2×C4semidirect
92. C2×C22⋊Q8C2×C4semidirect
99. C2×C4×Q8C2×C4semidirect
100. C2×C4×Q8C2×C4semidirect
115. C22×C4⋊C4C2×C4semidirect
116. C22×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
#341852
#340572
#218282
#218272
#218262
#218252
#213332
C2×#19962
#206892