Finite Group Databasebeta

← #4314278

Subgroups — #4314278

Order 512.

2618 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2717C212yesmaximal
3C2×#2326C211yesmaximal, characteristic
4#16993C212yesmaximal, complemented
5#16563C212yesmaximal, complemented
6#16470C214yesmaximal, complemented
7#15485C214yesmaximal, complemented
8#15398C214yesmaximal, complemented
9#15122C212yesmaximal, complemented
10#14880C212yesmaximal, complemented
11#14736C214yesmaximal, complemented
12#14486C211yesmaximal, characteristic
13#13968C212yesmaximal
14C22×#124C2211yescharacteristic
15C22×#123C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2724—28no—
19#2717—216no—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#397C2214yes—
23C2×#397C2214yes—
24C2×#397C2214yes—
25C2×#395C2214yes—
26#2326—28no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#388C2214yes—
32C2×#387C2214yes—
33C2×#386C2212yes—
34C2×#385C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#384C2211yescharacteristic
38C2×#381C2212yes—
39C2×#380C2214yes—
40C2×#376C2211yescharacteristic
41C22×#95C2212yes—
42#1445C2212yes—
43#1445C2212yes—
44#1444C2212yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1434C2212yes—
65#1434C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×#124—24no—
76C2×#124—24no—
77C2×#124D418yes—
78C2×#123—28no—
79C2×#123—28no—
80C2×#123—28no—
81C2×C4×Q8—24no—
82C2×C4×Q8—42no—
83C2×C4×Q8—42no—
84C2×C4×D4—24no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#398—28no—
111#398—44no—
112#397—216no—
113#397—216no—
114#397—216no—
115#395—216no—
116#391—24no—
117#390D418yes—
118#389—28no—
119#389—44no—
120#388—216no—
121#387—216no—
122#386—28no—
123#385—216no—
124#385—216no—
125#385—216no—
126#384—24no—
127#381—44no—
128#380—216no—
129#376—24no—
130C2×#95—44no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C25C2412yes—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8C2×D414yes—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4C2×D414yes—
169C22×D4C2×D414yes—
170C22×D4#4114yes—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200#124—232no—
201#123—432no—
202C4×Q8—416no—
203C4×D4—416no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C2×D414yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4C2×D4116yes—
258C2×C22⋊C4C2×D4116yes—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—42no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42C2×D414yes—
270C2×C42C2×D414yes—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—42no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24C22×D414yes—
305C24C22×D414yes—
306C24C2×#4114yes—
307C24C2511yescharacteristic
308C2×Q8—28no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4#121116yes—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—416no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—84no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—88no—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D414yes—
402C22×C4C22×D414yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+418yes—
409C4⋊C4—44no—
410C4⋊C4—416no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—416no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4—88no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C22⋊C4—232no—
422C22⋊C4—416no—
423C22⋊C4—416no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—416no—
427C22⋊C4—84no—
428C42—28no—
429C42—28no—
430C42—44no—
431C42—48no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—28no—
436C23—24no—
437C23—28no—
438C23—28no—
439C23—24no—
440C23—24no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23C2×#12118yes—
455C23C23×D411yescharacteristic
456C23C23×D411yescharacteristic
457C23C22×#4111yescharacteristic
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×H(5,𝔽2)11yescharacteristic
460C23C2×2−1+414yes—
461C23#58511yescharacteristic
462C23#58511yescharacteristic
463C23#58514yes—
464Q8—416no—
465D4—416no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
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—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—816no—
536C2×C4—816no—
537C2×C4—816no—
538C2×C4C2×#12114yes—
539C22—28no—
540C22—216no—
541C22—44no—
542C22—44no—
543C22—48no—
544C22—44no—
545C22—44no—
546C22—48no—
547C22—44no—
548C22—44no—
549C22—48no—
550C22C22×#12111yescharacteristic
551C22#345911yescharacteristic
552C22#345911yescharacteristic
553C22#348411yescharacteristic
554C22#348512yes—
555C22#348511yescharacteristic
556C22#348914yes—
557C22#349611yescharacteristic
558C22#349611yescharacteristic
559C22#349912yes—
560C22#351514yes—
561C22#353312yes—
562C22#353812yes—
563C22#354012yes—
564C22#354412yes—
565C22#355714yes—
566C22#356414yes—
567C4—24no—
568C4—42no—
569C4—44no—
570C4—42no—
571C4—48no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—82no—
576C4—84no—
577C4—84no—
578C4—84no—
579C2—48no—
580C2#3498411yescharacteristic
581C2#3602012yes—
582C2#3705712yes—
583C2#4236912yes—
584C2#4681812yes—
585C2#4778812yes—
586C2#5154211yescharacteristic
587C2#5158912yes—
588C2#5205411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #16993C2semidirect
5. #16563C2semidirect
6. #16470C2semidirect
7. #15485C2semidirect
8. #15398C2semidirect
9. #15122C2semidirect
10. #14880C2semidirect
11. #14736C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#27172
C2×#23262
#169932
#165632
#164702
#154852
#153982
#151222
#148802
#147362
#144862
#139682