Finite Group Databasebeta

← #4607224

Subgroups — #4607224

Order 512.

2256 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic
3#17157C214yesmaximal, complemented
4#16963C212yesmaximal, complemented
5#16563C212yesmaximal, complemented
6#16454C214yesmaximal, complemented
7#16267C214yesmaximal, complemented
8#15983C212yesmaximal, complemented
9#15972C214yesmaximal, complemented
10#15950C212yesmaximal
11#14969C212yesmaximal, complemented
12#14907C212yesmaximal, complemented
13#14667C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#2724—28no—
18#2500—28no—
19C2×#398C2212yes—
20C2×#397C2214yes—
21C2×#397C2212yes—
22C2×#394C2214yes—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#387C2214yes—
29C2×#386C2212yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1445C2214yes—
37#1445C2214yes—
38#1444C2212yes—
39#1444C2212yes—
40#1443C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2212yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2212yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2212yes—
65#1436C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C2×#123—28no—
74C2×#123—48no—
75C2×#121—24no—
76C2×#121—42no—
77C2×#121—42no—
78C2×#117—24no—
79C2×#117—42no—
80C2×#117—42no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107#398—28no—
108#397—216no—
109#397—44no—
110#394—216no—
111#390—24no—
112#390D418yes—
113#389—28no—
114#389—44no—
115#388—216no—
116#387—48no—
117#386—44no—
118#385—216no—
119#385—48no—
120#385—44no—
121#384—24no—
122#382—24no—
123C4×C22⋊C4—24no—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C25C2412yes—
148C25C2411yescharacteristic
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4C2×D414yes—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—42no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180#123—432no—
181#121—416no—
182#117—416no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C2×D414yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
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⋊C4—42no—
245C2×C22⋊C4C2×D4116yes—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C2×D414yes—
253#95—24no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—24no—
258#95—28no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—44no—
276C24—21no—
277C24—22no—
278C24—22no—
279C24—24no—
280C24—21no—
281C24—22no—
282C24—24no—
283C24—22no—
284C24—24no—
285C24—22no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24C22×D414yes—
292C24H(5,𝔽2)12yes—
293C24C2511yescharacteristic
294C2×D4—28no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—84no—
300C2×D4—84no—
301C2×D4—88no—
302C2×D4—84no—
303C2×D4—88no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—28no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—48no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4C22×D414yes—
372C22×C4C22×D414yes—
373C22×C4C22×D412yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)14yes—
376C22×C42−1+414yes—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—416no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—416no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—88no—
387C4⋊C4—88no—
388C22⋊C4—416no—
389C22⋊C4—48no—
390C22⋊C4—416no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—416no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—88no—
398C22⋊C4—84no—
399C22⋊C4—88no—
400C42—44no—
401C42—48no—
402C42—44no—
403C23—24no—
404C23—22no—
405C23—24no—
406C23—28no—
407C23—24no—
408C23—28no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—41no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23C23×D411yescharacteristic
441C23C2×H(5,𝔽2)12yes—
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×H(5,𝔽2)11yescharacteristic
444C23C2×2−1+412yes—
445C23#58514yes—
446C23#58512yes—
447C23#58512yes—
448D4—88no—
449D4—816no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—416no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
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—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—816no—
514C2×C4#53514yes—
515C22—48no—
516C22—42no—
517C22—44no—
518C22—42no—
519C22—44no—
520C22—48no—
521C22—42no—
522C22—44no—
523C22—48no—
524C22—84no—
525C22—84no—
526C22—82no—
527C22—82no—
528C22—84no—
529C22—82no—
530C22—82no—
531C22—84no—
532C22C2×#53511yescharacteristic
533C22#348411yescharacteristic
534C22#348512yes—
535C22#348511yescharacteristic
536C22#348912yes—
537C22#352911yescharacteristic
538C22#353312yes—
539C22#353412yes—
540C22#353412yes—
541C22#353914yes—
542C22#354012yes—
543C22#354114yes—
544C22#355011yescharacteristic
545C22#355912yes—
546C22#356514yes—
547C22#356614yes—
548C4—24no—
549C4—44no—
550C4—42no—
551C4—44no—
552C4—44no—
553C4—84no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—84no—
558C4—84no—
559C2—82no—
560C2—84no—
561C2#3672111yescharacteristic
562C2#4267912yes—
563C2#4622811yescharacteristic
564C2#4878112yes—
565C2#5205411yescharacteristic
566C2#5268712yes—
567C2#5653012yes—
568C2#5813014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2724C2semidirect
3. #17157C2semidirect
4. #16963C2semidirect
5. #16563C2semidirect
6. #16454C2semidirect
7. #16267C2semidirect
8. #15983C2semidirect
9. #15972C2semidirect
11. #14969C2semidirect
12. #14907C2semidirect
13. #14667C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#25002
#171572
#169632
#165632
#164542
#162672
#159832
#159722
#159502
#149692
#149072
#146672