Finite Group Databasebeta

← #770969

Subgroups — #770969

Order 512.

1912 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16242C212yesmaximal, complemented
2#16186C212yesmaximal, complemented
3#16167C212yesmaximal, complemented
4#15968C212yesmaximal, complemented
5#15869C214yesmaximal
6#15810C214yesmaximal
7#15220C212yesmaximal
8#15131C212yesmaximal
9#14030C212yesmaximal, complemented
10#14029C212yesmaximal, complemented
11#14006C212yesmaximal, complemented
12#14005C212yesmaximal, complemented
13#13853C211yesmaximal, characteristic
14#13777C211yesmaximal, characteristic
15#13777C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C2×#396C2214yes—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2214yes—
22C2×#389C2212yes—
23C2×#387C2214yes—
24C2×#384C2212yes—
25C2×#382C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1445C2211yescharacteristic
35#1445C2211yescharacteristic
36#1445C2211yescharacteristic
37#1445C2211yescharacteristic
38#1445C2214yes—
39#1444C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2212yes—
51#1440C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1437C2212yes—
56#1437C2212yes—
57#1437C2212yes—
58#1437C2212yes—
59#1436C2212yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C24×C4C2311yescharacteristic
76C2×#117—42no—
77C2×#117—42no—
78C2×#117Q818yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2311yescharacteristic
97C22×C42C2312yes—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2312yes—
105#396—48no—
106#389—28no—
107#389—28no—
108#389—48no—
109#389—44no—
110#389—44no—
111#387—48no—
112#384—28no—
113#382—28no—
114#380—28no—
115#380—28no—
116C4×C4⋊C4—42no—
117C4×C4⋊C4—42no—
118C4×C4⋊C4—42no—
119C4×C4⋊C4—42no—
120C4×C22⋊C4—42no—
121C4×C22⋊C4—42no—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154C25C2411yescharacteristic
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4C2×Q814yes—
160C23×C4C2412yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165C23×C4C2411yescharacteristic
166C23×C4C2411yescharacteristic
167C23×C4C2411yescharacteristic
168C23×C4C2411yescharacteristic
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174#117—416no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
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—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4#4118yes—
210C2×C4⋊C4#4118yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4C2×Q814yes—
228C2×C22⋊C4C2×Q814yes—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42C2×Q814yes—
248C2×C42C2×Q814yes—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—24no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—42no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24C22×Q812yes—
290C24C2511yescharacteristic
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4C22×Q812yes—
369C22×C4C22×Q812yes—
370C22×C4C22×Q812yes—
371C22×C4C22×Q812yes—
372C22×C4C22×Q812yes—
373C22×C4C22×Q812yes—
374C22×C4C2×#4118yes—
375C22×C4C2×#4118yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4114yes—
378C22×C4H(5,𝔽2)14yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+414yes—
382C4⋊C4—216no—
383C4⋊C4—216no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—88no—
391C4⋊C4—88no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C42—28no—
401C42—28no—
402C42—48no—
403C42—48no—
404C42—84no—
405C42—84no—
406C42—84no—
407C42—84no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23C23×Q811yescharacteristic
430C23C22×#4112yes—
431C23C22×#4112yes—
432C23C2×H(5,𝔽2)11yescharacteristic
433C23C2×H(5,𝔽2)11yescharacteristic
434C23C2×H(5,𝔽2)12yes—
435C23C2×2−1+412yes—
436C23#58512yes—
437C23#58512yes—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—416no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
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—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4#53314yes—
521C2×C4#53314yes—
522C2×C4#54314yes—
523C2×C4#55318yes—
524C2×C4#55318yes—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—44no—
529C22—44no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22C2×#53311yescharacteristic
535C22C2×#53311yescharacteristic
536C22C2×#54311yescharacteristic
537C22C2×#55312yes—
538C22C2×#55312yes—
539C22#350312yes—
540C22#350412yes—
541C22#351312yes—
542C22#351312yes—
543C22#351412yes—
544C22#351512yes—
545C22#351612yes—
546C22#351612yes—
547C22#351712yes—
548C22#351812yes—
549C22#353612yes—
550C22#353912yes—
551C22#354911yescharacteristic
552C22#355011yescharacteristic
553C22#355111yescharacteristic
554C22#355111yescharacteristic
555C4—48no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—48no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—84no—
566C4—82no—
567C4—84no—
568C4#260818yes—
569C2—82no—
570C2C2×#260811yescharacteristic
571C2#3557712yes—
572C2#3557912yes—
573C2#3730611yescharacteristic
574C2#3742511yescharacteristic
575C2#4506312yes—
576C2#4521612yes—
577C2#4522412yes—
578C2#4534812yes—
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
1. #16242C2semidirect
2. #16186C2semidirect
3. #16167C2semidirect
4. #15968C2semidirect
9. #14030C2semidirect
10. #14029C2semidirect
11. #14006C2semidirect
12. #14005C2semidirect

Maximal subgroups

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

subgroupindex
#162422
#161862
#161672
#159682
#158692
#158102
#152202
#151312
#140302
#140292
#140062
#140052
#138532
#137772