Finite Group Databasebeta

← #6508999

Subgroups — #6508999

Order 512.

1830 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#27613C212yesmaximal, complemented
4#27610C212yesmaximal, complemented
5#27602C214yesmaximal, complemented
6#24962C218yesmaximal, complemented
7#24958C218yesmaximal, complemented
8#21944C212yesmaximal, complemented
9#21742C211yesmaximal, characteristic
10#20700C212yesmaximal, complemented
11C22×#123—21no—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#407C2212yes—
15#2486—22no—
16#2486—22no—
17#2486C2214yes—
18#2486C2214yes—
19C2×#398—21no—
20C2×#397—21no—
21#2357—22no—
22#2357—22no—
23#2354C2214yes—
24#2354C2214yes—
25#2312—21no—
26#2312—21no—
27#2309—21no—
28C2×#389C2212yes—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#2280C2212yes—
32#2280C2212yes—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36#2274—21no—
37#2100C2214yescomplemented
38#2099C2214yescomplemented
39#2084C2214yescomplemented
40#2083C2214yescomplemented
41#2077C2218yescomplemented
42#2076C2218yescomplemented
43#1919C2218yescomplemented
44#1918C2218yescomplemented
45#1870C2214yes—
46#1870C2214yes—
47#1854C2218yescomplemented
48#1854C2218yescomplemented
49#1853C2214yes—
50#1852C2214yes—
51#1825C2218yescomplemented
52#1824C2218yescomplemented
53#1823C2214yes—
54#1823C2214yes—
55#1817C2212yes—
56#1809C2218yescomplemented
57#1779C2212yes—
58#1736C2218yescomplemented
59C23×D4D411yescharacteristic
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16C2314yes—
64C2×#123—24no—
65C2×#123—24no—
66C2×#123—48no—
67C2×C4×D4—28no—
68C2×C4×D4D412yes—
69C2×C4×D4D412yes—
70C2×C4×D4C2312yes—
71C2×C4×D4C2312yes—
72C2×#117—21no—
73C2×#117—21no—
74C2×#117C2314yes—
75C2×#117C2314yes—
76C22×C4⋊C4—28no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4D411yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#107C2318yes—
90C2×#106C2318yes—
91#407—28no—
92C2×C22⋊C8C2314yes—
93C2×C22⋊C8C2314yes—
94C2×C22⋊C8C2318yes—
95#398—24no—
96#398—24no—
97#397—24no—
98#397—24no—
99#390—24no—
100#390—24no—
101#389—22no—
102#389—22no—
103#389C2314yes—
104#389C2314yes—
105#388—24no—
106#388—24no—
107#388—24no—
108#388—24no—
109#384C2312yes—
110#384C2312yes—
111#382—21no—
112#382—21no—
113#381—22no—
114#381—22no—
115#381—24no—
116#381—24no—
117#380—21no—
118#380—21no—
119#380—24no—
120#380—22no—
121#380C2314yes—
122#380C2314yes—
123#380C2314yes—
124#380C2318yes—
125#380C2312yes—
126#380C2312yes—
127#379—22no—
128#379—22no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2312yes—
134C4×C4⋊C4C2312yes—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95—22no—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2314yes—
149#336C2314yes—
150C25C2×D411yescharacteristic
151C25C2×D411yescharacteristic
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—42no—
155C22×D4C2×D412yes—
156C22×D4C2×D412yes—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4#4118yes—
173C23×C4C2412yes—
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C2×M16—28no—
177C2×M16—48no—
178C22×C8C2418yes—
179#123—416no—
180#123—416no—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187#117—44no—
188#117—44no—
189#117—48no—
190#117—48no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2414yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—48no—
226C2×C22⋊C4C2×D412yes—
227C2×C22⋊C4C2×D412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C22⋊C4C2×D412yes—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2411yescharacteristic
238C2×C42C2411yescharacteristic
239#107—48no—
240#106—48no—
241C22⋊C8—28no—
242C22⋊C8—28no—
243C22⋊C8—48no—
244#95—28no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—22no—
249#95—22no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24#12314yes—
264C24C22×D412yes—
265C24C22×D412yes—
266C24C22×D411yescharacteristic
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—416no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—28no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—416no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—48no—
296C22×C4—44no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—88no—
304C22×C4—816no—
305C22×C4#12314yes—
306C22×C4#12318yes—
307C22×C4AGL(1,ℤ/8)14yes—
308C22×C4C22×D412yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D412yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4114yes—
316C22×C4C2×#4114yes—
317C22×C4C2×#4114yes—
318C22×C4C2511yescharacteristic
319M16—88no—
320C2×C8—28no—
321C2×C8—28no—
322C2×C8—28no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—28no—
328C4⋊C4—24no—
329C4⋊C4—28no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—44no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—48no—
344C22⋊C4—416no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—44no—
356C42—44no—
357C42—84no—
358C42—84no—
359C23—22no—
360C23—22no—
361C23—24no—
362C23—28no—
363C23—24no—
364C23—22no—
365C23—22no—
366C23—24no—
367C23—22no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—44no—
372C23—44no—
373C23—48no—
374C23—48no—
375C23—48no—
376C23—44no—
377C23—82no—
378C23—84no—
379C23—82no—
380C23C2×#12314yes—
381C23C2×#12312yes—
382C23C2×AGL(1,ℤ/8)12yes—
383C23#57512yes—
384C23C23×D411yescharacteristic
385D4—48no—
386D4—48no—
387C2×C4—28no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—88no—
423C2×C4—816no—
424C2×C4—88no—
425C2×C4C2×#12312yes—
426C2×C4C2×#12314yes—
427C2×C4C2×AGL(1,ℤ/8)12yes—
428C2×C4C2×AGL(1,ℤ/8)12yes—
429C2×C4C22×#4112yes—
430C2×C4#58514yes—
431C8—48no—
432C22—24no—
433C22—24no—
434C22—22no—
435C22—22no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—416no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—44no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—88no—
448C22#309414yes—
449C22C22×#12311yescharacteristic
450C22#348512yes—
451C22C22×AGL(1,ℤ/8)11yescharacteristic
452C22C2×#57511yescharacteristic
453C22#358612yes—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—24no—
458C4—41no—
459C4—41no—
460C4—42no—
461C4—84no—
462C4—84no—
463C4—84no—
464C4—84no—
465C4#349114yes—
466C2—24no—
467C2—44no—
468C2—82no—
469C2#3544811yescharacteristic
470C2C2×#309411yescharacteristic
471C2#6051911yescharacteristic
472C2#6171112yes—
473C2#6171712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27613C2semidirect
4. #27610C2semidirect
5. #27602C2semidirect
6. #24962C2semidirect
7. #24958C2semidirect
8. #21944C2semidirect
10. #20700C2semidirect
37. #2100C22semidirect
38. #2099C22semidirect
39. #2084C22semidirect
40. #2083C22semidirect
41. #2077C22semidirect
42. #2076C22semidirect
43. #1919C22semidirect
44. #1918C22semidirect
47. #1854C22semidirect
48. #1854C22semidirect
51. #1825C22semidirect
52. #1824C22semidirect
56. #1809C22semidirect
58. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#341862
#276132
#276102
#276022
#249622
#249582
#219442
#217422
#207002