Finite Group Databasebeta

← #4320116

Subgroups — #4320116

Order 512.

2338 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2735C212yesmaximal
2#17059C211yesmaximal, characteristic, complemented
3#16824C211yesmaximal, characteristic, complemented
4#16422C214yesmaximal, complemented
5#16391C212yesmaximal
6#15837C214yesmaximal, complemented
7#15706C214yesmaximal, complemented
8#15016C212yesmaximal, complemented
9#14998C214yesmaximal, complemented
10#14820C212yesmaximal
11#14422C212yesmaximal, complemented
12#14229C212yesmaximal, complemented
13#13775C211yesmaximal, characteristic
14C23×C22⋊C4C2211yescharacteristic
15#2735—216no—
16C2×#399C2212yes—
17C2×#397C2212yes—
18C2×#397C2214yes—
19C2×#397C2212yes—
20C2×#397C2212yes—
21C2×#397C2212yes—
22C2×#396C2214yes—
23C2×#389C2212yescomplemented
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#382C2212yescomplemented
28C2×#380C2212yes—
29C2×#379C2211yescharacteristic
30C2×#379C2214yes—
31C2×#379C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1445C2212yescomplemented
35#1445C2212yescomplemented
36#1445C2212yescomplemented
37#1445C2214yescomplemented
38#1444C2214yescomplemented
39#1442C2214yescomplemented
40#1442C2214yescomplemented
41#1442C2212yescomplemented
42#1441C2214yescomplemented
43#1441C2214yescomplemented
44#1441C2214yescomplemented
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1439C2212yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1434C2214yescomplemented
61#1434C2214yescomplemented
62#1434C2212yescomplemented
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2211yescharacteristic
67C4×#95C2212yescomplemented
68C26C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—42no—
81C22×C22⋊C4—42no—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2311yescharacteristic
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2311yescharacteristic
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2311yescharacteristic
96C22×C42C2312yes—
97C22×C42C2314yes—
98C22×C42C2312yes—
99#399—28no—
100#397—28no—
101#397—216no—
102#397—28no—
103#397—44no—
104#397—44no—
105#396—216no—
106#389—44no—
107#388—216no—
108#388—216no—
109#388—216no—
110#382—44no—
111#380—28no—
112#379—216no—
113#379—24no—
114#379D418yes—
115C4×C22⋊C4—42no—
116C4×C22⋊C4—42no—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C25—22no—
143C25—22no—
144C25—41no—
145C25—41no—
146C25—41no—
147C25—41no—
148C25—44no—
149C25—44no—
150C25C2×D414yes—
151C25C2412yes—
152C25C2411yescharacteristic
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4#4114yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—416no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4C2×D414yes—
234C2×C22⋊C4C2×D4116yes—
235C2×C22⋊C4C2×D414yes—
236C2×C22⋊C4#4114yes—
237C2×C42—24no—
238C2×C42—28no—
239C2×C42—24no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244#95—28no—
245#95—24no—
246#95—28no—
247#95—28no—
248#95—24no—
249#95—28no—
250#95—24no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—42no—
265#95—42no—
266#95—44no—
267#95—42no—
268#95—42no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—24no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—28no—
286C24—44no—
287C24—44no—
288C24—416no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—84no—
294C24—84no—
295C24—88no—
296C24—84no—
297C24—84no—
298C24—84no—
299C24—84no—
300C24—88no—
301C24—88no—
302C24C22×D414yes—
303C24C22×D412yes—
304C24C2×#4112yes—
305C24C2×#4112yes—
306C24C2511yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
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—44no—
364C22×C4—48no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4C22×D414yes—
371C22×C4C22×D414yes—
372C22×C4C2×#4114yes—
373C22×C4C2×#4114yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C4⋊C4—48no—
377C4⋊C4—88no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4—88no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C22⋊C4—416no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—44no—
391C22⋊C4—416no—
392C22⋊C4—416no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C42—48no—
399C42—88no—
400C42—84no—
401C23—22no—
402C23—22no—
403C23—28no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—28no—
411C23—24no—
412C23—28no—
413C23—28no—
414C23—24no—
415C23—24no—
416C23—416no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—816no—
442C23—816no—
443C23—816no—
444C23—816no—
445C23—816no—
446C23—88no—
447C23—816no—
448C23—88no—
449C23#53214yes—
450C23C23×D411yescharacteristic
451C23C22×#4111yescharacteristic
452C23C22×#4111yescharacteristic
453C23C2×H(5,𝔽2)11yescharacteristic
454C23C2×H(5,𝔽2)11yescharacteristic
455C23#58511yescharacteristic
456C23#58511yescharacteristic
457C23#58514yes—
458C23#58514yes—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
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—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—88no—
521C2×C4—88no—
522C22—28no—
523C22—24no—
524C22—44no—
525C22—44no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—44no—
531C22—48no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—48no—
536C22—832no—
537C22—82no—
538C22—82no—
539C22—84no—
540C22—82no—
541C22—84no—
542C22—82no—
543C22C2×#53211yescharacteristic
544C22#345911yescharacteristic
545C22#346011yescharacteristic
546C22#348514yes—
547C22#349011yescharacteristic
548C22#349112yes—
549C22#349311yescharacteristic
550C22#349611yescharacteristic
551C22#349611yescharacteristic
552C22#351012yes—
553C22#352114yes—
554C22#353312yes—
555C22#354512yes—
556C22#355112yes—
557C22#355112yes—
558C22#356614yes—
559C22#356614yes—
560C4—44no—
561C4—44no—
562C4—44no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—82no—
567C4—84no—
568C4—84no—
569C4—84no—
570C4—84no—
571C2—44no—
572C2—84no—
573C2#3500311yescharacteristic
574C2#3601212yes—
575C2#3717512yes—
576C2#4183712yes—
577C2#4681212yes—
578C2#4703511yescharacteristic
579C2#4719511yescharacteristic
580C2#4881112yes—
581C2#5018912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17059C2semidirect
3. #16824C2semidirect
4. #16422C2semidirect
6. #15837C2semidirect
7. #15706C2semidirect
8. #15016C2semidirect
9. #14998C2semidirect
11. #14422C2semidirect
12. #14229C2semidirect
23. C2×#389C22semidirect
27. C2×#382C22semidirect
34. #1445C22semidirect
35. #1445C22semidirect
36. #1445C22semidirect
37. #1445C22semidirect
38. #1444C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1434C22semidirect
61. #1434C22semidirect
62. #1434C22semidirect
67. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#27352
#170592
#168242
#164222
#163912
#158372
#157062
#150162
#149982
#148202
#144222
#142292
#137752