Finite Group Databasebeta

← #4930511

Subgroups — #4930511

Order 512.

3373 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2622C211yesmaximal, characteristic
2C2×#2273C211yesmaximal, characteristic, complemented
3#14828C214yesmaximal, complemented
4#14803C214yesmaximal, complemented
5#14749C214yesmaximal, complemented
6#14739C214yesmaximal, complemented
7#14134C212yesmaximal
8#14047C214yesmaximal, complemented
9#13999C214yesmaximal, complemented
10#13877C212yesmaximal, complemented
11C2×#1440C211yesmaximal, characteristic, complemented
12C22×#120C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16#2622—28no—
17C2×#397C2214yes—
18C2×#394C2214yes—
19C2×#388C2214yescomplemented
20C2×#387C2214yescomplemented
21C2×#386C2214yes—
22C2×#386C2212yes—
23C2×#386C2214yes—
24C2×#385C2212yes—
25C2×#385C2214yes—
26C2×#380C2214yes—
27C2×#379C2211yescharacteristic
28#2273—28no—
29C2×C4×C22⋊C4C2214yes—
30C2×C4×C22⋊C4C2214yes—
31C2×C4×C22⋊C4C2214yes—
32C22×#95C2214yes—
33#1445C2214yescomplemented
34#1443C2214yescomplemented
35#1443C2214yescomplemented
36#1442C2214yescomplemented
37#1442C2214yescomplemented
38#1441C2214yescomplemented
39#1440C2214yes—
40#1440C2214yes—
41#1440C2212yes—
42#1440C2212yes—
43#1440C2214yescomplemented
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yescomplemented
47#1438C2212yes—
48#1438C2212yes—
49#1437C2214yescomplemented
50#1436C2214yescomplemented
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53#1434C2214yescomplemented
54#1434C2214yescomplemented
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C26C2311yescharacteristic
60C23×D4C2311yescharacteristic
61C23×D4C2312yes—
62C24×C4C2314yes—
63C24×C4C2314yes—
64C24×C4C2312yes—
65C2×#120—24no—
66C2×#120—28no—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C22⋊C4—28no—
71C22×C22⋊C4—28no—
72C22×C22⋊C4—216no—
73C22×C22⋊C4—28no—
74C22×C22⋊C4—48no—
75C22×C22⋊C4D418yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2311yescharacteristic
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C42C2314yes—
91C22×C42C2314yes—
92#397—216no—
93#394—216no—
94#388—48no—
95#387—48no—
96#386—28no—
97#386—216no—
98#386—48no—
99#385—28no—
100#385—48no—
101#380—216no—
102#379—24no—
103C4×C22⋊C4—216no—
104C4×C22⋊C4—216no—
105C4×C22⋊C4—216no—
106C2×#95—48no—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C25—28no—
128C25—44no—
129C25—44no—
130C25C2×D414yes—
131C25#4118yes—
132C25C2411yescharacteristic
133C25C2412yes—
134C25C2411yescharacteristic
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—28no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—44no—
142C22×D4—44no—
143C23×C4—28no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—28no—
147C23×C4—28no—
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—28no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4C2×D418yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2411yescharacteristic
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170#120—416no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C22⋊C4—264no—
181C2×C22⋊C4—232no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—432no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4C2×D4116yes—
218C2×C22⋊C4#41116yes—
219C2×C42—28no—
220C2×C42—28no—
221C2×C42—44no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—44no—
225#95—28no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—216no—
254C24—232no—
255C24—21no—
256C24—24no—
257C24—22no—
258C24—24no—
259C24—24no—
260C24—48no—
261C24—416no—
262C24—48no—
263C24—416no—
264C24—88no—
265C24—88no—
266C24—88no—
267C24—88no—
268C24#123116yes—
269C24C22×D414yes—
270C24C22×D412yes—
271C24C2×#4114yes—
272C24C2×#4114yes—
273C24C2×#4114yes—
274C24C2×#4118yes—
275C24C2511yescharacteristic
276C2×D4—416no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—416no—
281C2×D4—84no—
282C2×D4—84no—
283C2×D4—88no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—232no—
309C22×C4—216no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—416no—
333C22×C4—416no—
334C22×C4—416no—
335C22×C4—416no—
336C22×C4—416no—
337C22×C4—88no—
338C22×C4—88no—
339C22×C4—88no—
340C22×C4—88no—
341C22×C4—84no—
342C22×C4—84no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4C22×D418yes—
350C22×C4C2×#4118yes—
351C22×C4C2×#4118yes—
352C22×C4H(5,𝔽2)18yes—
353C4⋊C4—416no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C22⋊C4—216no—
357C22⋊C4—216no—
358C22⋊C4—216no—
359C22⋊C4—232no—
360C22⋊C4—232no—
361C22⋊C4—416no—
362C22⋊C4—416no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—44no—
366C22⋊C4—48no—
367C22⋊C4—88no—
368C22⋊C4—88no—
369C22⋊C4—88no—
370C42—88no—
371C42—88no—
372C23—24no—
373C23—24no—
374C23—28no—
375C23—28no—
376C23—28no—
377C23—28no—
378C23—28no—
379C23—28no—
380C23—24no—
381C23—24no—
382C23—28no—
383C23—28no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—232no—
393C23—432no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—41no—
403C23—42no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—832no—
414C23—832no—
415C23—832no—
416C23C2×#12318yes—
417C23C2×#12314yes—
418C23#52518yes—
419C23#529116yes—
420C23C23×D411yescharacteristic
421C23C22×#4112yes—
422C23C22×#4114yes—
423C23C2×H(5,𝔽2)14yes—
424C23#58514yes—
425D4—816no—
426D4—88no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—816no—
478C2×C4—832no—
479C2×C4—832no—
480C22—28no—
481C22—216no—
482C22—216no—
483C22—216no—
484C22—28no—
485C22—28no—
486C22—28no—
487C22—48no—
488C22—48no—
489C22—48no—
490C22—48no—
491C22—48no—
492C22—48no—
493C22—832no—
494C22—81no—
495C22—82no—
496C22—84no—
497C22—84no—
498C22—84no—
499C22#285318yes—
500C22C22×#12311yescharacteristic
501C22C2×#52512yes—
502C22C2×#52914yes—
503C22#349012yes—
504C22#349112yes—
505C22#349614yes—
506C22#350514yes—
507C22#351314yes—
508C22#351414yes—
509C22#351914yes—
510C22#352114yes—
511C4—48no—
512C4—48no—
513C4—48no—
514C4—48no—
515C4—81no—
516C4—82no—
517C4—84no—
518C4—84no—
519C2—28no—
520C2—48no—
521C2—161no—
522C2#3536012yes—
523C2#3536514yes—
524C2#3571714yes—
525C2#3579414yes—
526C2C2×#285311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2273C2semidirect
3. #14828C2semidirect
4. #14803C2semidirect
5. #14749C2semidirect
6. #14739C2semidirect
8. #14047C2semidirect
9. #13999C2semidirect
10. #13877C2semidirect
11. C2×#1440C2semidirect
19. C2×#388C22semidirect
20. C2×#387C22semidirect
33. #1445C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
43. #1440C22semidirect
46. #1439C22semidirect
49. #1437C22semidirect
50. #1436C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect
53. #1434C22semidirect
54. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26222
C2×#22732
#148282
#148032
#147492
#147392
#141342
#140472
#139992
#138772
C2×#14402