Finite Group Databasebeta

← #6468688

Subgroups — #6468688

Order 512.

2772 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C212yesmaximal, complemented
2#29408C214yesmaximal, complemented
3#29234C214yesmaximal, complemented
4#28114C214yesmaximal, complemented
5#27600C211yesmaximal, characteristic, complemented
6#24002C212yesmaximal, complemented
7#23918C211yesmaximal, characteristic
8#23696C214yesmaximal, complemented
9C4×#480C214yesmaximal, complemented
10C4×#447C214yesmaximal, complemented
11#20714C211yesmaximal, characteristic, complemented
12C22×#121C2212yes—
13C22×C4×D4—22no—
14C22×C4×D4C2212yescomplemented
15C23×C42C2211yescharacteristic
16C2×#480C2214yes—
17C2×#447C2214yes—
18C2×C4×D8C2214yescomplemented
19C2×#431C2212yescomplemented
20C4×#121—22no—
21C4×#121—22no—
22C4×#121—22no—
23C4×#121—22no—
24C4×#121C2214yes—
25C4×#121C2214yes—
26C4×#121C2214yescomplemented
27C4×#121C2214yescomplemented
28C2×#390C2212yescomplemented
29C2×#389C2212yescomplemented
30C2×#388—22no—
31C2×#386—22no—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yescomplemented
34C2×C4×C22⋊C4—22no—
35#2086C2214yescomplemented
36#2075C2214yescomplemented
37#2067C2218yescomplemented
38#2054C2214yescomplemented
39#2033C2218yescomplemented
40#1999C2214yes—
41#1925C2218yescomplemented
42#1918C2218yescomplemented
43#1879C2218yescomplemented
44#1874C2214yes—
45#1874C2214yes—
46#1862C2218yescomplemented
47#1852C2214yescomplemented
48#1852C2214yescomplemented
49#1823C2214yescomplemented
50#1817C2212yescomplemented
51#1797C2214yescomplemented
52#1786C2214yescomplemented
53#1779C2211yescharacteristic
54#1779C2211yescharacteristic
55C4×#107C2214yescomplemented
56C4×#102C2218yescomplemented
57C4×C22⋊C8C2214yescomplemented
58C23×D4—22no—
59C23×D4C2312yes—
60C24×C4—22no—
61C24×C4D411yescharacteristic
62C24×C4D411yescharacteristic
63C24×C4C2311yescharacteristic
64C22×D8C2314yes—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—216no—
68C2×#121C2314yes—
69C2×#121C2314yes—
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—48no—
77C2×C4×D4—416no—
78C2×C4×D4C2314yes—
79C2×C4×D4C2314yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C42—22no—
90C22×C42—44no—
91C22×C42D412yes—
92C22×C42D412yes—
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2312yes—
98C22×C42C2311yescharacteristic
99#480—216no—
100#447—216no—
101C4×D8—48no—
102#431—44no—
103C2×#107C2318yes—
104C2×#107C2314yes—
105C2×C4⋊C8C2314yes—
106C2×#102C2318yes—
107C2×#102C2318yes—
108C2×#102C2318yes—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×C4×C8C2314yes—
114#390—22no—
115#390—22no—
116#390C2314yes—
117#390C2314yes—
118#389—22no—
119#389—22no—
120#389C2314yes—
121#389C2314yes—
122#388—24no—
123#388—24no—
124#388—24no—
125#388—24no—
126#386—24no—
127#386—24no—
128#386—24no—
129#386—24no—
130#380—22no—
131#380—22no—
132#380C2314yes—
133#380C2314yes—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145#340C2314yes—
146#340C2318yes—
147#340C2318yes—
148#340C2314yes—
149#336C2318yes—
150C25—22no—
151C25C2×D411yescharacteristic
152C22×D4—216no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—28no—
156C22×D4—22no—
157C22×D4—28no—
158C22×D4—44no—
159C22×D4—48no—
160C22×D4—48no—
161C22×D4C2414yes—
162C23×C4—22no—
163C23×C4—28no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—48no—
174C23×C4—48no—
175C23×C4—48no—
176C23×C4D818yescomplemented
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C2×D8—216no—
190C2×D8—44no—
191C22×C8C2414yes—
192C22×C8C2414yes—
193#121—216no—
194#121—216no—
195#121—28no—
196#121—28no—
197#121—28no—
198#121#41116yes—
199C4×D4—216no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—416no—
204C4×D4—416no—
205C4×D4—416no—
206C4×D4—416no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2414yes—
219C2×C22⋊C4—216no—
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—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—48no—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—44no—
243C2×C42—42no—
244C2×C42—48no—
245C2×C42C2×D412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2412yes—
253C2×C42C2411yescharacteristic
254C2×C42C2411yescharacteristic
255#107—216no—
256#107—44no—
257C4⋊C8—44no—
258#102—216no—
259#102—216no—
260#102—48no—
261C22⋊C8—28no—
262C22⋊C8—28no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265C4×C8—28no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272C24—24no—
273C24—24no—
274C24—28no—
275C24—24no—
276C24—41no—
277C24—41no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24—44no—
282C24—44no—
283C24—44no—
284C24C2×D814yes—
285C24C22×D411yescharacteristic
286C24C22×D412yes—
287C2×D4—28no—
288C2×D4—216no—
289C2×D4—216no—
290C2×D4—28no—
291C2×D4—24no—
292C2×D4—28no—
293C2×D4—432no—
294C2×D4—44no—
295C2×D4—464no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—44no—
301C2×D4C2×#4118yes—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—416no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—416no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—44no—
338C22×C4—416no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—832no—
351C22×C4C2×D818yes—
352C22×C4C2×D814yes—
353C22×C4C2×D818yes—
354C22×C4C2×D814yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4118yes—
364C22×C4C2511yescharacteristic
365D8—416no—
366C2×C8—28no—
367C2×C8—28no—
368C2×C8—24no—
369C2×C8—24no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4C2×#4118yes—
379C22⋊C4—216no—
380C22⋊C4—216no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—48no—
388C22⋊C4—48no—
389C42—22no—
390C42—22no—
391C42—24no—
392C42—24no—
393C42—22no—
394C42—24no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—48no—
399C42—48no—
400C42H(5,𝔽2)14yes—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—216no—
406C23—22no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—28no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—48no—
415C23—416no—
416C23—416no—
417C23—416no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—88no—
425C23—88no—
426C23—88no—
427C23—88no—
428C23#52514yes—
429C23C22×D812yes—
430C23C22×D814yes—
431C23#57514yes—
432C23C23×D411yescharacteristic
433D4—216no—
434D4—416no—
435D4—432no—
436D4—432no—
437D4—48no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—216no—
446C2×C4—28no—
447C2×C4—216no—
448C2×C4—22no—
449C2×C4—28no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—44no—
457C2×C4—48no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—416no—
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—816no—
475C2×C4—88no—
476C2×C4—88no—
477C2×C4—816no—
478C2×C4#52514yes—
479C2×C4#53412yes—
480C2×C4#53412yes—
481C2×C4C22×D814yes—
482C2×C4C22×D812yes—
483C2×C4C22×D812yes—
484C2×C4#57512yes—
485C2×C4C22×#4114yes—
486C2×C4C2×H(5,𝔽2)12yes—
487C8—44no—
488C8—44no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—22no—
496C22—24no—
497C22—416no—
498C22—41no—
499C22—41no—
500C22—42no—
501C22—48no—
502C22—48no—
503C22—48no—
504C22—832no—
505C22—84no—
506C22—84no—
507C22—84no—
508C22—84no—
509C22#310018yes—
510C22C2×#52512yes—
511C22C2×#53411yescharacteristic
512C22C23×D811yescharacteristic
513C22C2×#57511yescharacteristic
514C22C2×#57512yes—
515C4—24no—
516C4—22no—
517C4—22no—
518C4—24no—
519C4—42no—
520C4—41no—
521C4—48no—
522C4—41no—
523C4—84no—
524C4—84no—
525C4#318614yes—
526C4#348114yes—
527C2—28no—
528C2—42no—
529C2—88no—
530C2#3495611yescharacteristic
531C2C2×#310012yes—
532C2#6056312yes—
533C2C2×#318611yescharacteristic
534C2#6074111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radical#23918C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×#121C2semidirect
2. #29408C2semidirect
3. #29234C2semidirect
4. #28114C2semidirect
5. #27600C2semidirect
6. #24002C2semidirect
8. #23696C2semidirect
9. C4×#480C2semidirect
10. C4×#447C2semidirect
11. #20714C2semidirect
14. C22×C4×D4C22semidirect
18. C2×C4×D8C22semidirect
19. C2×#431C22semidirect
26. C4×#121C22semidirect
27. C4×#121C22semidirect
28. C2×#390C22semidirect
29. C2×#389C22semidirect
33. C2×C4×C4⋊C4C22semidirect
35. #2086C22semidirect
36. #2075C22semidirect
37. #2067C22semidirect
38. #2054C22semidirect
39. #2033C22semidirect
41. #1925C22semidirect
42. #1918C22semidirect
43. #1879C22semidirect
46. #1862C22semidirect
47. #1852C22semidirect
48. #1852C22semidirect
49. #1823C22semidirect
50. #1817C22semidirect
51. #1797C22semidirect
52. #1786C22semidirect
55. C4×#107C22semidirect
56. C4×#102C22semidirect
57. C4×C22⋊C8C22semidirect
176. C23×C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#294082
#292342
#281142
#276002
#240022
#239182
#236962
C4×#4802
C4×#4472
#207142