Finite Group Databasebeta

← #806542

Subgroups — #806542

Order 512.

1994 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16473C212yesmaximal
2#16236C212yesmaximal, complemented
3#16181C214yesmaximal, complemented
4#15981C212yesmaximal, complemented
5#15233C212yesmaximal
6#15195C211yesmaximal, characteristic
7#15054C214yesmaximal
8#14143C212yesmaximal
9#14130C211yesmaximal, characteristic
10#14042C212yesmaximal, complemented
11#14030C212yesmaximal, complemented
12#13955C212yesmaximal, complemented
13#13884C212yesmaximal, complemented
14C4×#399C211yesmaximal, characteristic
15C22⋊C4×C4⋊C4C212yesmaximal
16C23×C4⋊C4C2211yescharacteristic
17C2×#399C2211yescharacteristic
18C2×#389C2212yes—
19C2×#387C2214yes—
20C2×#387C2214yes—
21C2×#385C2212yes—
22C2×#384C2211yescharacteristic
23C2×#382C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2214yes—
33#1445C2212yes—
34#1445C2211yescharacteristic
35#1444C2211yescharacteristic
36#1444C2212yes—
37#1443C2214yes—
38#1443C2214yes—
39#1443C2214yes—
40#1443C2214yes—
41#1442C2212yes—
42#1442C2212yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2212yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2211yescharacteristic
53#1439C2212yes—
54#1439C2212yes—
55#1439C2212yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2211yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2314yes—
102C22×C42C2314yes—
103#399—24no—
104#389—28no—
105#387—48no—
106#387—48no—
107#385—28no—
108#384—42no—
109#382—28no—
110#380—24no—
111#380—28no—
112C4×C4⋊C4—28no—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—42no—
115C4×C4⋊C4—44no—
116C4×C22⋊C4—28no—
117C4×C22⋊C4—48no—
118C4×C22⋊C4—42no—
119C2×#95C2311yescharacteristic
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2311yescharacteristic
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C25C2411yescharacteristic
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—28no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4#4114yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2411yescharacteristic
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—416no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4#4118yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4#4114yes—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—28no—
238C2×C42—28no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42#4118yes—
244#95—24no—
245#95—24no—
246#95—28no—
247#95—24no—
248#95—24no—
249#95—28no—
250#95—24no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—22no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—42no—
260#95—44no—
261#95—42no—
262#95—41no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—41no—
269#95—41no—
270#95—42no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—21no—
285C24C2×#4112yes—
286C24C2×#4112yes—
287C242−1+412yes—
288C24C2511yescharacteristic
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—216no—
330C22×C4—28no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—48no—
365C22×C4—416no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—44no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4C2×#4112yes—
373C22×C4C2×#4114yes—
374C22×C4C2×#4112yes—
375C22×C4C2×#4114yes—
376C22×C4C2×#4118yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4114yes—
380C22×C4C2×#4114yes—
381C22×C4H(5,𝔽2)14yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+412yes—
384C22×C42−1+412yes—
385C4⋊C4—216no—
386C4⋊C4—216no—
387C4⋊C4—48no—
388C4⋊C4—416no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C22⋊C4—216no—
398C22⋊C4—44no—
399C22⋊C4—48no—
400C22⋊C4—44no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C42—216no—
404C42—44no—
405C42—44no—
406C42—48no—
407C42—88no—
408C42—88no—
409C23—24no—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—41no—
422C23—42no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23#55614yes—
432C23C22×#4111yescharacteristic
433C23C22×#4112yes—
434C23C22×#4112yes—
435C23C22×#4111yescharacteristic
436C23C2×H(5,𝔽2)11yescharacteristic
437C23C2×H(5,𝔽2)12yes—
438C23C2×2−1+411yescharacteristic
439C23C2×2−1+411yescharacteristic
440C23C2×2−1+411yescharacteristic
441C23#58512yes—
442C23#58511yescharacteristic
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—416no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—816no—
526C2×C4#52918yes—
527C2×C4#53914yes—
528C2×C4#55114yes—
529C2×C4#55218yes—
530C2×C4#55914yes—
531C22—24no—
532C22—28no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22C2×#52912yes—
543C22C2×#53911yescharacteristic
544C22C2×#55111yescharacteristic
545C22C2×#55212yes—
546C22C2×#55611yescharacteristic
547C22#347512yes—
548C22C2×#55911yescharacteristic
549C22#348112yes—
550C22#349212yes—
551C22#349212yes—
552C22#350511yescharacteristic
553C22#350612yes—
554C22#350711yescharacteristic
555C22#350911yescharacteristic
556C22#351212yes—
557C22#351412yes—
558C22#351612yes—
559C22#351812yes—
560C22#353711yescharacteristic
561C22#353811yescharacteristic
562C22#354712yes—
563C22#355111yescharacteristic
564C22#356311yescharacteristic
565C4—44no—
566C4—42no—
567C4—44no—
568C4—44no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—48no—
573C4—84no—
574C4—84no—
575C4—84no—
576C4—82no—
577C4—81no—
578C4#251718yes—
579C2—44no—
580C2C2×#251711yescharacteristic
581C2#3571012yes—
582C2#3625512yes—
583C2#3951211yescharacteristic
584C2#4252211yescharacteristic
585C2#4491912yes—
586C2#4530811yescharacteristic
587C2#4537511yescharacteristic
588C2#5929412yes—
589C2#5937811yescharacteristic
590C2#5986511yescharacteristic
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
2. #16236C2semidirect
3. #16181C2semidirect
4. #15981C2semidirect
10. #14042C2semidirect
11. #14030C2semidirect
12. #13955C2semidirect
13. #13884C2semidirect

Maximal subgroups

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

subgroupindex
#164732
#162362
#161812
#159812
#152332
#151952
#150542
#141432
#141302
#140422
#140302
#139552
#138842
C4×#3992
C22⋊C4×C4⋊C42