Finite Group Databasebeta

← #6508997

Subgroups — #6508997

Order 512.

1844 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27616C214yesmaximal, complemented
4#27601C212yesmaximal, complemented
5#27600C212yesmaximal, complemented
6#24954C218yesmaximal, complemented
7#24950C218yesmaximal, 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#2353—22no—
22#2353—22no—
23#2352C2214yes—
24#2352C2214yes—
25#2313—21no—
26#2312—21no—
27#2308—21no—
28C2×#389C2212yes—
29#2283C2212yes—
30#2280C2212yes—
31C2×#381—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36#2273—21no—
37#2101C2218yescomplemented
38#2085C2218yescomplemented
39#2077C2218yescomplemented
40#2075C2218yescomplemented
41#1921C2218yescomplemented
42#1920C2218yescomplemented
43#1870C2214yes—
44#1870C2214yes—
45#1854C2218yescomplemented
46#1853C2214yes—
47#1853C2214yescomplemented
48#1852C2214yes—
49#1852C2214yescomplemented
50#1824C2214yescomplemented
51#1824C2218yescomplemented
52#1823C2214yes—
53#1823C2214yes—
54#1823C2214yescomplemented
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×#117C2314yes—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4D411yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#107C2318yes—
91C2×#106C2318yes—
92#407—28no—
93C2×C22⋊C8C2314yes—
94C2×C22⋊C8C2314yes—
95C2×C22⋊C8C2314yes—
96C2×C22⋊C8C2314yes—
97#398—24no—
98#398—24no—
99#397—24no—
100#397—24no—
101#389—24no—
102#389C2318yes—
103#388—24no—
104#388—24no—
105#387—24no—
106#387—24no—
107#386—24no—
108#386—24no—
109#383—22no—
110#382C2314yes—
111#381—22no—
112#381—22no—
113#381—24no—
114#381—24no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380C2314yes—
123#380C2314yes—
124#380C2312yes—
125#380C2314yes—
126#380C2312yes—
127#380C2314yes—
128#380C2312yes—
129#380C2312yes—
130#379—22no—
131C4×C4⋊C4C2314yes—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137#376—22no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—22no—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2318yes—
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—22no—
162C23×C4—22no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4#4118yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C2×M16—28no—
181C2×M16—48no—
182C22×C8C2418yes—
183#123—416no—
184#123—416no—
185C4×D4—28no—
186C4×D4—28no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191#117—48no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
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—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4C2×D412yes—
230C2×C22⋊C4C2×D412yes—
231C2×C22⋊C4C2×D412yes—
232C2×C22⋊C4C2×D412yes—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42C2×D412yes—
237C2×C42C2412yes—
238#107—48no—
239#106—48no—
240C22⋊C8—28no—
241C22⋊C8—28no—
242C22⋊C8—44no—
243C22⋊C8—44no—
244#95—22no—
245#95—22no—
246#95—24no—
247#95—24no—
248#95—22no—
249#95—22no—
250#95—24no—
251#95—28no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24#12314yes—
265C24AGL(1,ℤ/8)14yes—
266C24C22×D412yes—
267C24C22×D411yescharacteristic
268C24C22×D411yescharacteristic
269C24C22×D411yescharacteristic
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—416no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—416no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—48no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—88no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—816no—
318C22×C4#12314yes—
319C22×C4#12318yes—
320C22×C4C22×D412yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D412yes—
324C22×C4C22×D412yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4114yes—
327C22×C4C2×#4114yes—
328C22×C4C2×#4114yes—
329C22×C4C2511yescharacteristic
330M16—88no—
331C2×C8—28no—
332C2×C8—28no—
333C2×C8—28no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—44no—
341C4⋊C4—416no—
342C4⋊C4—44no—
343C4⋊C4—48no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—44no—
351C22⋊C4—48no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—48no—
355C22⋊C4—48no—
356C22⋊C4—44no—
357C22⋊C4—84no—
358C22⋊C4—84no—
359C42—24no—
360C42—22no—
361C42—22no—
362C42—84no—
363C42—84no—
364C23—22no—
365C23—22no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—28no—
370C23—21no—
371C23—21no—
372C23—24no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—21no—
377C23—22no—
378C23—21no—
379C23—28no—
380C23—44no—
381C23—44no—
382C23—48no—
383C23—48no—
384C23—48no—
385C23—82no—
386C23—84no—
387C23—82no—
388C23C2×#12314yes—
389C23C2×#12312yes—
390C23C2×AGL(1,ℤ/8)12yes—
391C23C2×AGL(1,ℤ/8)12yes—
392C23C2×AGL(1,ℤ/8)12yes—
393C23#57512yes—
394C23C23×D411yescharacteristic
395D4—48no—
396D4—48no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—88no—
433C2×C4—816no—
434C2×C4—88no—
435C2×C4—164no—
436C2×C4—164no—
437C2×C4C2×#12312yes—
438C2×C4C2×#12314yes—
439C2×C4C22×#4112yes—
440C2×C4#58514yes—
441C8—48no—
442C22—24no—
443C22—24no—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—22no—
450C22—22no—
451C22—24no—
452C22—24no—
453C22—416no—
454C22—44no—
455C22—42no—
456C22—44no—
457C22—42no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—88no—
463C22#309414yes—
464C22C22×#12311yescharacteristic
465C22#348512yes—
466C22C22×AGL(1,ℤ/8)11yescharacteristic
467C22C2×#57511yescharacteristic
468C22#358612yes—
469C4—22no—
470C4—24no—
471C4—42no—
472C4—42no—
473C4—41no—
474C4—41no—
475C4—84no—
476C4—84no—
477C4—84no—
478C4—84no—
479C4#349114yes—
480C2—24no—
481C2—44no—
482C2—42no—
483C2—42no—
484C2#3544811yescharacteristic
485C2C2×#309411yescharacteristic
486C2#6051911yescharacteristic
487C2#6170612yes—
488C2#6171612yes—
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. #27616C2semidirect
4. #27601C2semidirect
5. #27600C2semidirect
6. #24954C2semidirect
7. #24950C2semidirect
8. #21944C2semidirect
10. #20700C2semidirect
37. #2101C22semidirect
38. #2085C22semidirect
39. #2077C22semidirect
40. #2075C22semidirect
41. #1921C22semidirect
42. #1920C22semidirect
45. #1854C22semidirect
47. #1853C22semidirect
49. #1852C22semidirect
50. #1824C22semidirect
51. #1824C22semidirect
54. #1823C22semidirect
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
#341852
#276162
#276012
#276002
#249542
#249502
#219442
#217422
#207002