Finite Group Databasebeta

← #4452881

Subgroups — #4452881

Order 512.

2214 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic
3#16853C214yesmaximal, complemented
4#16843C214yesmaximal, complemented
5#16529C214yesmaximal, complemented
6#16409C214yesmaximal, complemented
7#15019C211yesmaximal, characteristic
8#15015C211yesmaximal, characteristic
9#15014C211yesmaximal, characteristic
10#14936C211yesmaximal, characteristic
11#14935C212yesmaximal
12#14728C212yesmaximal
13#14043C212yesmaximal
14#13972C212yesmaximal
15#13778C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2646—28no—
19C2×#400—28no—
20C2×#400C2212yes—
21C2×#400C2212yes—
22C2×#400C2214yes—
23C2×#400C2214yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#389C2214yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1445C2212yes—
44#1445C2214yes—
45#1445C2212yes—
46#1445C2212yes—
47#1445C2212yes—
48#1444C2212yes—
49#1444C2214yes—
50#1444C2212yes—
51#1444C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2211yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C42C2314yes—
106#400—28no—
107#400—28no—
108#400—216no—
109#400—216no—
110#398—28no—
111#398—28no—
112#398—48no—
113#389—48no—
114#385—28no—
115#385—28no—
116#385—28no—
117#385—44no—
118#381—28no—
119#381—44no—
120#380—24no—
121#380—24no—
122#380—28no—
123C4×C22⋊C4—44no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95—24no—
128C2×#95D418yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C25C2411yescharacteristic
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—416no—
228C2×C4⋊C4—416no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4#4118yes—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95C2×D418yes—
290#95C2×D418yes—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24C22×D412yes—
296C24C2×#4114yes—
297C24H(5,𝔽2)12yes—
298C242−1+414yes—
299C242−1+412yes—
300C24C2511yescharacteristic
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
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—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—88no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—88no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C2×#4118yes—
402C22×C4H(5,𝔽2)14yes—
403C22×C42−1+414yes—
404C4⋊C4—416no—
405C4⋊C4—416no—
406C4⋊C4—416no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C22⋊C4—216no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C42—88no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—28no—
427C23—28no—
428C23—28no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—28no—
434C23#53214yes—
435C23#53614yes—
436C23#53714yes—
437C23#53918yes—
438C23#54118yes—
439C23C23×D411yescharacteristic
440C23C22×#4112yes—
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)12yes—
443C23C2×2−1+412yes—
444C23C2×2−1+411yescharacteristic
445C23C2×2−1+412yes—
446C23#58512yes—
447C23#58512yes—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—416no—
468C2×C4—416no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
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—44no—
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—88no—
523C2×C4—88no—
524C2×C4—88no—
525C2×C4—88no—
526C2×C4—88no—
527C22—216no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—216no—
535C22#266418yes—
536C22C2×#53211yescharacteristic
537C22C2×#53611yescharacteristic
538C22C2×#53711yescharacteristic
539C22C2×#53912yes—
540C22C2×#54112yes—
541C22#348411yescharacteristic
542C22#348511yescharacteristic
543C22#348511yescharacteristic
544C22#348611yescharacteristic
545C22#351314yes—
546C22#351514yes—
547C22#353814yes—
548C22#355812yes—
549C22#355912yes—
550C22#356112yes—
551C22#356312yes—
552C22#356414yes—
553C4—48no—
554C4—44no—
555C4—44no—
556C4—44no—
557C4—44no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—82no—
567C2—28no—
568C2C2×#266411yescharacteristic
569C2#3539411yescharacteristic
570C2#3546811yescharacteristic
571C2#3670811yescharacteristic
572C2#3695711yescharacteristic
573C2#3696811yescharacteristic
574C2#3697011yescharacteristic
575C2#4070314yes—
576C2#4076314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16853C2semidirect
4. #16843C2semidirect
5. #16529C2semidirect
6. #16409C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
C22×#4002
#168532
#168432
#165292
#164092
#150192
#150152
#150142
#149362
#149352
#147282
#140432
#139722
#137782