Finite Group Databasebeta

← #628565

Subgroups — #628565

Order 512.

2040 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16214C212yesmaximal
2#16167C212yesmaximal, complemented
3#16058C214yesmaximal, complemented
4#15968C212yesmaximal, complemented
5#15616C212yesmaximal
6#15531C214yesmaximal
7#15131C212yesmaximal
8#15131C212yesmaximal
9#14712C212yesmaximal, complemented
10#14343C211yesmaximal, characteristic
11#14266C212yesmaximal, complemented
12#14005C212yesmaximal, complemented
13#13981C212yesmaximal, complemented
14#13777C211yesmaximal, characteristic
15#13761C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C2×#396C2214yes—
18C2×#396C2214yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2214yes—
23C2×#389C2214yes—
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#387C2214yes—
27C2×#384C2212yes—
28C2×#384C2212yes—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×#376C2211yescharacteristic
36C2×C43C2211yescharacteristic
37#1445C2212yes—
38#1445C2212yes—
39#1445C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2212yes—
52#1440C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1437C2211yescharacteristic
56#1437C2212yes—
57#1437C2211yescharacteristic
58#1437C2212yes—
59#1437C2212yes—
60#1436C2211yescharacteristic
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2211yescharacteristic
65#1436C2212yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74C4×#95C2212yes—
75C24×C4C2311yescharacteristic
76C2×#117—42no—
77C2×#117—42no—
78C2×#117Q818yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109#396—48no—
110#396—48no—
111#389—28no—
112#389—28no—
113#389—28no—
114#389—48no—
115#389—44no—
116#389—48no—
117#389—48no—
118#387—48no—
119#384—28no—
120#384—44no—
121#380—28no—
122#380—42no—
123#380—42no—
124C4×C4⋊C4—42no—
125C4×C4⋊C4—42no—
126C4×C22⋊C4—28no—
127#376—42no—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C43—42no—
157C25C2411yescharacteristic
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C2×Q814yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177#117—416no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C2×Q814yes—
218C2×C4⋊C4C2×Q814yes—
219C2×C4⋊C4#4118yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
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⋊C4C2×Q814yes—
238C2×C22⋊C4C2×Q814yes—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—41no—
253C2×C42—42no—
254C2×C42—41no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42C2×Q814yes—
260C2×C42C2×Q814yes—
261C2×C42#4118yes—
262#95—24no—
263#95—24no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—41no—
273#95—42no—
274#95—41no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—44no—
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#95—44no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—22no—
298C24C22×Q812yes—
299C24C2511yescharacteristic
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
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—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4C22×Q812yes—
375C22×C4C22×Q812yes—
376C22×C4C22×Q812yes—
377C22×C4C22×Q812yes—
378C22×C4C22×Q812yes—
379C22×C4C22×Q812yes—
380C22×C4C2×#4118yes—
381C22×C4C2×#4114yes—
382C22×C4C2×#4114yes—
383C22×C4C2×#4114yes—
384C22×C4C2×#4114yes—
385C22×C4H(5,𝔽2)14yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+414yes—
390C22×C42−1+414yes—
391C4⋊C4—216no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—416no—
395C4⋊C4—416no—
396C4⋊C4—416no—
397C4⋊C4—416no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—88no—
402C4⋊C4—88no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—88no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—28no—
413C42—28no—
414C42—216no—
415C42—48no—
416C42—44no—
417C42—44no—
418C42—48no—
419C42—84no—
420C42—84no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23C23×Q811yescharacteristic
443C23C22×#4112yes—
444C23C22×#4112yes—
445C23C2×H(5,𝔽2)12yes—
446C23C2×H(5,𝔽2)12yes—
447C23C2×H(5,𝔽2)11yescharacteristic
448C23C2×H(5,𝔽2)11yescharacteristic
449C23C2×2−1+412yes—
450C23C2×2−1+412yes—
451C2×C4—24no—
452C2×C4—24no—
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—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—416no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4#53314yes—
532C2×C4#53314yes—
533C2×C4#54018yes—
534C2×C4#54118yes—
535C2×C4#54314yes—
536C2×C4#55318yes—
537C2×C4#55318yes—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22C2×#53311yescharacteristic
548C22C2×#53311yescharacteristic
549C22C2×#54012yes—
550C22C2×#54112yes—
551C22C2×#54311yescharacteristic
552C22C2×#55312yes—
553C22C2×#55312yes—
554C22#350312yes—
555C22#350312yes—
556C22#351112yes—
557C22#351112yes—
558C22#351912yes—
559C22#352012yes—
560C22#352212yes—
561C22#352312yes—
562C22#353612yes—
563C22#353711yescharacteristic
564C22#353811yescharacteristic
565C22#354812yes—
566C22#354911yescharacteristic
567C22#355011yescharacteristic
568C4—28no—
569C4—28no—
570C4—48no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—84no—
578C4—82no—
579C4—84no—
580C4—84no—
581C4#260818yes—
582C2—82no—
583C2C2×#260811yescharacteristic
584C2#3528412yes—
585C2#3557812yes—
586C2#3730611yescharacteristic
587C2#3730711yescharacteristic
588C2#3812212yes—
589C2#3812312yes—
590C2#3919112yes—
591C2#3919212yes—
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. #16167C2semidirect
3. #16058C2semidirect
4. #15968C2semidirect
9. #14712C2semidirect
11. #14266C2semidirect
12. #14005C2semidirect
13. #13981C2semidirect

Maximal subgroups

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

subgroupindex
#162142
#161672
#160582
#159682
#156162
#155312
#151312
#147122
#143432
#142662
#140052
#139812
#137772
#137612