Finite Group Databasebeta

← #6477706

Subgroups — #6477706

Order 512.

2492 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3192C211yesmaximal, characteristic
2C2×#2920C211yesmaximal, characteristic, complemented
3C2×#2854C212yesmaximal, complemented
4#33143C214yesmaximal, complemented
5#29782C214yesmaximal, complemented
6#24607C214yesmaximal, complemented
7#24603C214yesmaximal, complemented
8#24506C214yesmaximal, complemented
9#24500C214yesmaximal, complemented
10#21942C211yesmaximal, characteristic, complemented
11#21764C212yesmaximal, complemented
12C22×#123—22no—
13C22×#121C2212yes—
14C23×C42C2211yescharacteristic
15#3192—28no—
16C2×#481C2214yes—
17C2×#469C2214yes—
18C2×#447C2214yes—
19#2920—28no—
20C2×#432C2214yescomplemented
21C2×#431C2214yescomplemented
22C2×#423C2212yescomplemented
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25C2×C4×M16C2212yescomplemented
26#2854—22no—
27#2854—22no—
28#2854—22no—
29#2854—22no—
30#2854C2214yes—
31#2854C2214yes—
32#2854C2214yescomplemented
33#2854C2214yescomplemented
34C2×#390C2212yescomplemented
35C2×#390C2212yescomplemented
36C2×#389C2212yescomplemented
37C2×#389C2212yescomplemented
38C2×#388—22no—
39C2×#388—22no—
40C2×#386—22no—
41C2×#380C2212yes—
42#2088C2218yescomplemented
43#2067C2218yescomplemented
44#2065C2218yescomplemented
45#2054C2218yescomplemented
46#1972C2214yescomplemented
47#1970C2214yescomplemented
48#1943C2214yescomplemented
49#1941C2214yescomplemented
50#1936C2218yescomplemented
51#1934C2218yescomplemented
52#1895C2218yescomplemented
53#1874C2214yes—
54#1874C2214yes—
55#1853C2214yescomplemented
56#1852C2214yescomplemented
57#1824C2214yescomplemented
58#1809C2214yescomplemented
59#1786C2214yescomplemented
60C23×D4—22no—
61C23×D4C2312yes—
62C24×C4D412yes—
63C24×C4C2311yescharacteristic
64C22×D8C2314yes—
65C22×M16C2312yes—
66C22×M16C2312yes—
67C2×#123—24no—
68C2×#123—24no—
69C2×#123—24no—
70C2×#123—24no—
71C2×#123—416no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121—28no—
75C2×#121—28no—
76C2×#121C2314yes—
77C2×#121C2314yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C42—24no—
90C22×C42—24no—
91C22×C42D412yes—
92C22×C42D412yes—
93C22×C42D412yes—
94C22×C42C2312yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#481—216no—
99#469—216no—
100#447—216no—
101#432—216no—
102#431—216no—
103C2×#107C2314yes—
104C2×#106C2314yes—
105C2×#106C2314yes—
106#423—28no—
107C2×C4⋊C8C2318yes—
108C2×#102C2318yes—
109C2×#102C2318yes—
110C2×#102C2318yes—
111#407—24no—
112#407D418yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C4×M16—28no—
116C2×#97C2314yes—
117C2×C4×C8C2314yes—
118#390—22no—
119#390—22no—
120#390—22no—
121#390—22no—
122#390C2314yes—
123#390C2314yes—
124#390C2314yes—
125#390C2314yes—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389C2314yes—
131#389C2314yes—
132#389C2314yes—
133#389C2314yes—
134#388—24no—
135#388—24no—
136#388—24no—
137#388—24no—
138#388—24no—
139#388—24no—
140#388—24no—
141#388—24no—
142#386—24no—
143#386—24no—
144#386—24no—
145#386—24no—
146#380—22no—
147#380—22no—
148#380C2314yes—
149#380C2314yes—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—22no—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156C25—22no—
157C25C2×D411yescharacteristic
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—22no—
162C22×D4—24no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—48no—
166C22×D4—48no—
167C22×D4—48no—
168C22×D4C2414yes—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D414yes—
186C23×C4#4114yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C2×D8—28no—
190C2×D8—44no—
191C2×D8—44no—
192C2×M16—24no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—42no—
197C2×M16C2×D418yes—
198C22×C8C2414yes—
199C22×C8C2414yes—
200#123—416no—
201#123—416no—
202#123—416no—
203#123—416no—
204#121—28no—
205#121—28no—
206#121—28no—
207#121—28no—
208#121—48no—
209#121—48no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2414yes—
225C2×C4⋊C4C2414yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—48no—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—28no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2411yescharacteristic
263C2×C42C2411yescharacteristic
264C2×C42C2412yes—
265#107—44no—
266#106—28no—
267#106—44no—
268C4⋊C8—216no—
269#102—216no—
270#102—48no—
271#102—48no—
272C22⋊C8—28no—
273C22⋊C8C2×D4116yes—
274#97—28no—
275C4×C8—28no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—41no—
286C24—41no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—44no—
294C24#12114yes—
295C24C22×D411yescharacteristic
296C24C22×D412yes—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—44no—
310C2×D4—88no—
311C2×D4—88no—
312C2×D4—832no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
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—24no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—432no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4#12118yes—
354C22×C4#12114yes—
355C22×C4AGL(1,ℤ/8)18yes—
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D412yes—
360C22×C4C22×D414yes—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C2×#4114yes—
365C22×C4C2×#4112yes—
366C22×C4C2511yescharacteristic
367D8—88no—
368M16—48no—
369M16—48no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8C22×D418yes—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C22⋊C4—48no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C42—22no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—24no—
404C42—22no—
405C42—24no—
406C42—21no—
407C42—21no—
408C42—22no—
409C42—21no—
410C42—22no—
411C42—21no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—21no—
417C23—21no—
418C23—216no—
419C23—22no—
420C23—22no—
421C23—44no—
422C23—48no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—88no—
433C23—88no—
434C23—88no—
435C23—88no—
436C23—88no—
437C23—88no—
438C23—88no—
439C23C2×#12112yes—
440C23C2×#12114yes—
441C23C2×AGL(1,ℤ/8)14yes—
442C23#57514yes—
443C23C23×D411yescharacteristic
444D4—48no—
445D4—48no—
446D4—88no—
447D4—816no—
448D4—816no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—216no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—88no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4C2×#12112yes—
488C2×C4C2×#12114yes—
489C2×C4C2×AGL(1,ℤ/8)14yes—
490C2×C4C2×AGL(1,ℤ/8)14yes—
491C2×C4C23×D411yescharacteristic
492C2×C4C22×#4111yescharacteristic
493C2×C4#58512yes—
494C2×C4#58512yes—
495C8—28no—
496C8—44no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—22no—
504C22—24no—
505C22—416no—
506C22—42no—
507C22—41no—
508C22—41no—
509C22—84no—
510C22—84no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—84no—
515C22—84no—
516C22—1616no—
517C22#305518yes—
518C22C22×#12111yescharacteristic
519C22#344911yescharacteristic
520C22#344911yescharacteristic
521C22C22×AGL(1,ℤ/8)12yes—
522C22C2×#57512yes—
523C4—22no—
524C4—24no—
525C4—24no—
526C4—24no—
527C4—42no—
528C4—42no—
529C4—84no—
530C4—84no—
531C4—84no—
532C4#348512yes—
533C4#349112yes—
534C2—28no—
535C2—42no—
536C2—164no—
537C2#3508911yescharacteristic
538C2C2×#305512yes—
539C2#6094912yes—
540C2#6096411yescharacteristic
541C2#6096811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3192C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2920C2semidirect
3. C2×#2854C2semidirect
4. #33143C2semidirect
5. #29782C2semidirect
6. #24607C2semidirect
7. #24603C2semidirect
8. #24506C2semidirect
9. #24500C2semidirect
10. #21942C2semidirect
11. #21764C2semidirect
20. C2×#432C22semidirect
21. C2×#431C22semidirect
22. C2×#423C22semidirect
25. C2×C4×M16C22semidirect
32. #2854C22semidirect
33. #2854C22semidirect
34. C2×#390C22semidirect
35. C2×#390C22semidirect
36. C2×#389C22semidirect
37. C2×#389C22semidirect
42. #2088C22semidirect
43. #2067C22semidirect
44. #2065C22semidirect
45. #2054C22semidirect
46. #1972C22semidirect
47. #1970C22semidirect
48. #1943C22semidirect
49. #1941C22semidirect
50. #1936C22semidirect
51. #1934C22semidirect
52. #1895C22semidirect
55. #1853C22semidirect
56. #1852C22semidirect
57. #1824C22semidirect
58. #1809C22semidirect
59. #1786C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31922
C2×#29202
C2×#28542
#331432
#297822
#246072
#246032
#245062
#245002
#219422
#217642