Finite Group Databasebeta

← #4123544

Subgroups — #4123544

Order 512.

2922 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2459C214yesmaximal, complemented
2#17082C211yesmaximal, characteristic
3#15980C211yesmaximal, characteristic, complemented
4#15952C212yesmaximal, complemented
5#15832C214yesmaximal, complemented
6#15782C212yesmaximal, complemented
7#15761C212yesmaximal, complemented
8#15498C214yesmaximal, complemented
9#14227C212yesmaximal, complemented
10#14181C212yesmaximal, complemented
11#14176C212yesmaximal, complemented
12#14155C211yesmaximal, characteristic, complemented
13#14008C214yesmaximal, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2212yes—
16C22×C4×D4C2212yes—
17#2459—232no—
18C2×#394C2212yes—
19C2×#394C2212yes—
20C2×#394C2214yes—
21C2×#394C2212yes—
22C2×#394C2212yes—
23C2×#393C2214yes—
24C2×#392C2214yes—
25C2×#391C2214yes—
26C2×#390C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#385C2214yes—
34C2×#381C2214yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2214yes—
38#1446C2212yes—
39#1445C2211yescharacteristic
40#1445C2211yescharacteristic
41#1445C2211yescharacteristic
42#1445C2211yescharacteristic
43#1444C2212yes—
44#1443C2214yes—
45#1442C2212yes—
46#1441C2212yes—
47#1441C2214yes—
48#1441C2212yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2212yes—
56#1438C2212yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1436C2212yes—
61#1436C2212yes—
62#1436C2214yes—
63C4⋊#95C2214yes—
64#1434C2212yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C4×#95C2212yes—
68C23×Q8C2312yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C24×C4C2314yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—42no—
75C2×C4⋊Q8—42no—
76C2×#124—216no—
77C2×#124—44no—
78C2×C4×D4—216no—
79C2×C4×D4—44no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2314yes—
103C22×C42C2312yes—
104#394—28no—
105#394—28no—
106#394—216no—
107#394—44no—
108#394D4116yescomplemented
109#393—216no—
110#392—216no—
111#391—216no—
112#390—44no—
113#388—216no—
114#388—216no—
115#388—216no—
116#387—216no—
117#386—216no—
118#386—216no—
119#385—216no—
120#381—216no—
121C4×C4⋊C4—42no—
122C4×C4⋊C4—42no—
123C4×C22⋊C4—216no—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2311yescharacteristic
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C25C2414yes—
149C22×Q8—28no—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8—44no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—44no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4C2×D418yes—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—28no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C4⋊Q8—416no—
189#124—432no—
190C4×D4—432no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D418yes—
211C2×C4⋊C4#4114yes—
212C2×C4⋊C4#4118yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4C2×D4116yes—
245C2×C22⋊C4C2×D4116yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—28no—
250C2×C42—28no—
251C2×C42—24no—
252C2×C42—28no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—44no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—42no—
275#95—44no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—42no—
281#95C2×D418yes—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24C22×D418yes—
297C24C2511yescharacteristic
298C2×Q8—48no—
299C2×Q8—416no—
300C2×Q8—48no—
301C2×Q8—88no—
302C2×Q8—84no—
303C2×D4—416no—
304C2×D4—48no—
305C2×D4—416no—
306C2×D4—48no—
307C2×D4—416no—
308C2×D4—48no—
309C2×D4—416no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—84no—
313C2×D4—88no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—84no—
317C2×D4—88no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—416no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4C22×D414yes—
380C22×C4C22×D414yes—
381C22×C4C22×D418yes—
382C22×C4C22×D414yes—
383C22×C4C2×#4114yes—
384C22×C4C2×#4114yes—
385C22×C4C2×#4114yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4114yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—44no—
393C4⋊C4—416no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C4⋊C4—84no—
398C22⋊C4—232no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—416no—
402C22⋊C4—416no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C42—28no—
410C42—48no—
411C42—48no—
412C42—84no—
413C42—88no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23C23×D412yes—
450C23C22×#4112yes—
451C23C22×#4111yescharacteristic
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×H(5,𝔽2)11yescharacteristic
454C23#58514yes—
455C23#58512yes—
456C23#58512yes—
457Q8—816no—
458D4—432no—
459D4—816no—
460D4—816no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—832no—
525C2×C4D4214yes—
526C2×C4#55514yes—
527C22—48no—
528C22—48no—
529C22—48no—
530C22—48no—
531C22—48no—
532C22—48no—
533C22—48no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22C2×D4211yescharacteristic
543C22D4×#4112yes—
544C22D4×#4114yes—
545C22C2×#55511yescharacteristic
546C22#348012yes—
547C22#348514yes—
548C22#349011yescharacteristic
549C22#349612yes—
550C22#350512yes—
551C22#351012yes—
552C22#352414yes—
553C22#354012yes—
554C22#354511yescharacteristic
555C22#354511yescharacteristic
556C22#355112yes—
557C22#356614yes—
558C4—24no—
559C4—24no—
560C4—48no—
561C4—44no—
562C4—44no—
563C4—44no—
564C4—42no—
565C4—44no—
566C4—44no—
567C4—84no—
568C4—82no—
569C4—84no—
570C2—88no—
571C2#4681212yes—
572C2#4697512yes—
573C2#4703611yescharacteristic
574C2#5900012yes—
575C2#5917514yes—
576C2#5981411yescharacteristic
577C2#5984112yes—
578C2#5988511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17082C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2459C2semidirect
3. #15980C2semidirect
4. #15952C2semidirect
5. #15832C2semidirect
6. #15782C2semidirect
7. #15761C2semidirect
8. #15498C2semidirect
9. #14227C2semidirect
10. #14181C2semidirect
11. #14176C2semidirect
12. #14155C2semidirect
13. #14008C2semidirect
108. #394D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24592
#170822
#159802
#159522
#158322
#157822
#157612
#154982
#142272
#141812
#141762
#141552
#140082