Finite Group Databasebeta

← #6508998

Subgroups — #6508998

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27617C214yesmaximal, complemented
4#27603C212yesmaximal, complemented
5#27601C212yesmaximal, complemented
6#24956C218yesmaximal, complemented
7#24952C218yesmaximal, complemented
8#21944C212yesmaximal, complemented
9#21742C211yesmaximal, characteristic
10#20700C212yesmaximal, complemented
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#407C2212yes—
14#2487—22no—
15#2487—22no—
16#2487C2214yes—
17#2487C2214yes—
18C2×#400—21no—
19C2×#399—21no—
20#2353—22no—
21#2353—22no—
22#2352C2214yes—
23#2352C2214yes—
24#2313—21no—
25#2312—21no—
26#2308—21no—
27C2×#389C2212yes—
28C2×#386—21no—
29C2×#385—21no—
30#2283C2212yes—
31#2280C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36#2273—21no—
37#2101C2218yescomplemented
38#2085C2218yescomplemented
39#2077C2218yescomplemented
40#2076C2218yescomplemented
41#1919C2218yescomplemented
42#1918C2218yescomplemented
43#1870C2214yes—
44#1870C2214yes—
45#1854C2218yescomplemented
46#1853C2214yes—
47#1853C2214yescomplemented
48#1852C2214yes—
49#1852C2214yescomplemented
50#1824C2214yes—
51#1824C2214yes—
52#1824C2214yescomplemented
53#1823C2218yescomplemented
54#1823C2214yescomplemented
55#1817C2212yes—
56#1809C2218yescomplemented
57#1779C2212yes—
58#1736C2218yescomplemented
59C23×D4D411yescharacteristic
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16C2314yes—
64C2×C4×D4—28no—
65C2×C4×D4D412yes—
66C2×C4×D4D412yes—
67C2×C4×D4C2312yes—
68C2×C4×D4C2312yes—
69C2×#117C2314yes—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107C2318yes—
88C2×#106C2318yes—
89#407—28no—
90C2×C22⋊C8C2314yes—
91C2×C22⋊C8C2314yes—
92C2×C22⋊C8C2314yes—
93C2×C22⋊C8C2314yes—
94#400—24no—
95#400—24no—
96#399—24no—
97#399—24no—
98#389—24no—
99#389C2318yes—
100#388—24no—
101#388—24no—
102#387—24no—
103#387—24no—
104#386—24no—
105#386—24no—
106#386—24no—
107#386—24no—
108#385—24no—
109#385—24no—
110#383—22no—
111#382C2314yes—
112#381—22no—
113#381—22no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380C2314yes—
122#380C2314yes—
123#380C2312yes—
124#380C2314yes—
125#380C2312yes—
126#380C2314yes—
127#380C2312yes—
128#380C2312yes—
129#379—22no—
130C4×C4⋊C4C2314yes—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—24no—
136#376—22no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95—22no—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2318yes—
149C25C2×D411yescharacteristic
150C25C2×D411yescharacteristic
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4C2×D412yes—
155C22×D4C2×D412yes—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—84no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4#4118yes—
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2412yes—
178C2×M16—28no—
179C2×M16—48no—
180C22×C8C2418yes—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187#117—48no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—84no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2414yes—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2414yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—84no—
225C2×C22⋊C4C2×D412yes—
226C2×C22⋊C4C2×D412yes—
227C2×C22⋊C4C2×D412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42C2×D412yes—
233C2×C42C2412yes—
234#107—48no—
235#106—48no—
236C22⋊C8—28no—
237C22⋊C8—28no—
238C22⋊C8—44no—
239C22⋊C8—44no—
240#95—22no—
241#95—22no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—24no—
247#95—28no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24#12314yes—
261C24#13714yes—
262C24C22×D412yes—
263C24C22×D411yescharacteristic
264C24C22×D411yescharacteristic
265C24C22×D411yescharacteristic
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—416no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—416no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—48no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—816no—
313C22×C4#12314yes—
314C22×C4#12318yes—
315C22×C4C22×D412yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D412yes—
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4114yes—
322C22×C4C2×#4114yes—
323C22×C4C2×#4114yes—
324C22×C4C2511yescharacteristic
325M16—88no—
326C2×C8—28no—
327C2×C8—28no—
328C2×C8—28no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—88no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—44no—
346C22⋊C4—48no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C42—24no—
355C42—22no—
356C42—22no—
357C42—84no—
358C42—84no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—28no—
365C23—21no—
366C23—21no—
367C23—24no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—21no—
373C23—21no—
374C23—28no—
375C23—44no—
376C23—44no—
377C23—48no—
378C23—48no—
379C23—48no—
380C23—82no—
381C23—84no—
382C23—82no—
383C23C2×#12314yes—
384C23C2×#12312yes—
385C23C2×#13712yes—
386C23C2×#13712yes—
387C23C2×#13712yes—
388C23#57512yes—
389C23C23×D411yescharacteristic
390D4—48no—
391D4—48no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—48no—
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—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—88no—
428C2×C4—88no—
429C2×C4—816no—
430C2×C4—164no—
431C2×C4—164no—
432C2×C4C2×#12312yes—
433C2×C4C2×#12314yes—
434C2×C4C22×#4112yes—
435C2×C4#58514yes—
436C8—48no—
437C22—24no—
438C22—24no—
439C22—22no—
440C22—22no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—24no—
448C22—416no—
449C22—44no—
450C22—42no—
451C22—44no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—88no—
458C22#309514yes—
459C22C22×#12311yescharacteristic
460C22#348512yes—
461C22C22×#13711yescharacteristic
462C22C2×#57511yescharacteristic
463C22#358512yes—
464C4—22no—
465C4—24no—
466C4—42no—
467C4—42no—
468C4—41no—
469C4—41no—
470C4—84no—
471C4—84no—
472C4—84no—
473C4—84no—
474C4#349114yes—
475C2—24no—
476C2—44no—
477C2—42no—
478C2—42no—
479C2#3544811yescharacteristic
480C2C2×#309511yescharacteristic
481C2#6051811yescharacteristic
482C2#6170912yes—
483C2#6171812yes—
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. #27617C2semidirect
4. #27603C2semidirect
5. #27601C2semidirect
6. #24956C2semidirect
7. #24952C2semidirect
8. #21944C2semidirect
10. #20700C2semidirect
37. #2101C22semidirect
38. #2085C22semidirect
39. #2077C22semidirect
40. #2076C22semidirect
41. #1919C22semidirect
42. #1918C22semidirect
45. #1854C22semidirect
47. #1853C22semidirect
49. #1852C22semidirect
52. #1824C22semidirect
53. #1823C22semidirect
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×#24872
#341852
#276172
#276032
#276012
#249562
#249522
#219442
#217422
#207002