Finite Group Databasebeta

← #4453474

Subgroups — #4453474

Order 512.

2368 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#400C211yesmaximal, characteristic, complemented
2C2×#2322C211yesmaximal, characteristic
3#16641C214yesmaximal, complemented
4#16496C214yesmaximal, complemented
5#16418C214yesmaximal, complemented
6#16414C214yesmaximal, complemented
7#15092C211yesmaximal, characteristic, complemented
8#15091C211yesmaximal, characteristic
9#14955C211yesmaximal, characteristic, complemented
10#14945C211yesmaximal, characteristic
11#14919C212yesmaximal, complemented
12#14912C212yesmaximal
13#14342C211yesmaximal, characteristic, complemented
14#14040C212yesmaximal, complemented
15#14016C212yesmaximal
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#400—28no—
21C2×#400C2214yes—
22C2×#400C2214yes—
23#2322—28no—
24C2×#389C2212yes—
25C2×#388C2212yes—
26C2×#387C2212yes—
27C2×#386C2211yescharacteristic
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yes—
31C2×#385C2211yescharacteristic
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#381C2211yescharacteristic
35C2×#381C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C22×#95C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2212yes—
40#1445C2214yes—
41#1445C2214yes—
42#1445C2214yes—
43#1445C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1434C2214yes—
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×#123—24no—
81C2×#123—24no—
82C2×#123—24no—
83C2×C4×D4—24no—
84C2×C4×D4—44no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109#400—216no—
110#400—216no—
111#389—28no—
112#388—28no—
113#387—28no—
114#386—28no—
115#386—24no—
116#386—44no—
117#385—24no—
118#385—28no—
119#385—28no—
120#385—44no—
121#381—24no—
122#381—44no—
123C4×C22⋊C4—24no—
124C2×#95—28no—
125C2×#95—28no—
126C2×#95—24no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2411yescharacteristic
156C25C2411yescharacteristic
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4#4114yes—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—44no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4C2×D414yes—
184C23×C4C2×D414yes—
185C23×C4C2×D414yes—
186C23×C4C2411yescharacteristic
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199#123—232no—
200C4×D4—416no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—416no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—416no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×D418yes—
224C2×C4⋊C4C2×D418yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4#4114yes—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—44no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—28no—
266#95—28no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283C24—24no—
284C24—24no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24—24no—
290C24—24no—
291C24C22×D412yes—
292C24C2×#4112yes—
293C24C2×#4112yes—
294C24C2×#4112yes—
295C24H(5,𝔽2)14yes—
296C242−1+414yes—
297C24C2511yescharacteristic
298C2×D4—28no—
299C2×D4—216no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—48no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
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—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—416no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—84no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4C22×D412yes—
389C22×C4C22×D414yes—
390C22×C4C22×D412yes—
391C22×C4C22×D414yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4114yes—
394C4⋊C4—216no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—416no—
398C4⋊C4—44no—
399C4⋊C4—84no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C22⋊C4—216no—
403C22⋊C4—28no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—84no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C42—44no—
413C42—84no—
414C23—24no—
415C23—24no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—28no—
422C23—22no—
423C23—24no—
424C23—28no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23#52514yes—
436C23#53414yes—
437C23#53614yes—
438C23#53918yes—
439C23#54218yes—
440C23C23×D411yescharacteristic
441C23C22×#4111yescharacteristic
442C23C22×#4111yescharacteristic
443C23C2×H(5,𝔽2)12yes—
444C23C2×2−1+412yes—
445C23#58514yes—
446C23#58514yes—
447D4—416no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—416no—
463C2×C4—416no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
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—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
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—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—88no—
518C2×C4—88no—
519C2×C4—816no—
520C2×C4—816no—
521C22—28no—
522C22—24no—
523C22—216no—
524C22—216no—
525C22—24no—
526C22—28no—
527C22—42no—
528C22—48no—
529C22—44no—
530C22—82no—
531C22—81no—
532C22—84no—
533C22—81no—
534C22#263818yes—
535C22C2×#52511yescharacteristic
536C22C2×#53411yescharacteristic
537C22C2×#53611yescharacteristic
538C22C2×#53912yes—
539C22C2×#54212yes—
540C22#348512yes—
541C22#348712yes—
542C22#352114yes—
543C22#352114yes—
544C22#353314yes—
545C22#354514yes—
546C22#355314yes—
547C22#355314yes—
548C4—44no—
549C4—42no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—44no—
554C4—82no—
555C4—81no—
556C4—84no—
557C4—84no—
558C4—84no—
559C4—81no—
560C4—84no—
561C2—28no—
562C2—82no—
563C2C2×#263811yescharacteristic
564C2#3545412yes—
565C2#3696511yescharacteristic
566C2#3712012yes—
567C2#3718711yescharacteristic
568C2#4155214yes—
569C2#4155314yes—
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
1. C22×#400C2semidirect
3. #16641C2semidirect
4. #16496C2semidirect
5. #16418C2semidirect
6. #16414C2semidirect
7. #15092C2semidirect
9. #14955C2semidirect
11. #14919C2semidirect
13. #14342C2semidirect
14. #14040C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4002
C2×#23222
#166412
#164962
#164182
#164142
#150922
#150912
#149552
#149452
#149192
#149122
#143422
#140402
#140162