Finite Group Databasebeta

← #6509000

Subgroups — #6509000

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#27614C212yesmaximal, complemented
4#27611C212yesmaximal, complemented
5#27604C214yesmaximal, complemented
6#24964C218yesmaximal, complemented
7#24960C218yesmaximal, 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#2357—22no—
21#2357—22no—
22#2354C2214yes—
23#2354C2214yes—
24#2312—21no—
25#2312—21no—
26#2309—21no—
27C2×#389C2212yes—
28C2×#386—21no—
29C2×#385—21no—
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32#2280C2212yes—
33#2280C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36#2274—21no—
37#2100C2214yescomplemented
38#2099C2214yescomplemented
39#2084C2214yescomplemented
40#2083C2214yescomplemented
41#2077C2218yescomplemented
42#2075C2218yescomplemented
43#1921C2218yescomplemented
44#1920C2218yescomplemented
45#1870C2214yes—
46#1870C2214yes—
47#1854C2218yescomplemented
48#1854C2218yescomplemented
49#1853C2214yes—
50#1852C2214yes—
51#1825C2218yescomplemented
52#1824C2214yes—
53#1824C2214yes—
54#1823C2218yescomplemented
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×#117—21no—
70C2×#117—21no—
71C2×#117C2314yes—
72C2×#117C2314yes—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107C2318yes—
87C2×#106C2318yes—
88#407—28no—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91C2×C22⋊C8C2318yes—
92#400—24no—
93#400—24no—
94#399—24no—
95#399—24no—
96#390—24no—
97#390—24no—
98#389—22no—
99#389—22no—
100#389C2314yes—
101#389C2314yes—
102#388—24no—
103#388—24no—
104#388—24no—
105#388—24no—
106#386—24no—
107#386—24no—
108#385—24no—
109#385—24no—
110#384C2312yes—
111#384C2312yes—
112#382—21no—
113#382—21no—
114#381—22no—
115#381—22no—
116#380—21no—
117#380—21no—
118#380—24no—
119#380—22no—
120#380C2314yes—
121#380C2314yes—
122#380C2314yes—
123#380C2318yes—
124#380C2312yes—
125#380C2312yes—
126#379—22no—
127#379—22no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—22no—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
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—24no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—84no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4#4118yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C2×M16—28no—
175C2×M16—48no—
176C22×C8C2418yes—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183#117—44no—
184#117—44no—
185#117—48no—
186#117—48no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
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—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—84no—
222C2×C22⋊C4C2×D412yes—
223C2×C22⋊C4C2×D412yes—
224C2×C22⋊C4C2×D412yes—
225C2×C22⋊C4C2×D412yes—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2411yescharacteristic
234C2×C42C2411yescharacteristic
235#107—48no—
236#106—48no—
237C22⋊C8—28no—
238C22⋊C8—28no—
239C22⋊C8—48no—
240#95—28no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—24no—
247#95—24no—
248#95—22no—
249#95—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24#12314yes—
260C24C22×D412yes—
261C24C22×D412yes—
262C24C22×D411yescharacteristic
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—416no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—416no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—48no—
296C22×C4—44no—
297C22×C4—88no—
298C22×C4—88no—
299C22×C4—816no—
300C22×C4#12314yes—
301C22×C4#12318yes—
302C22×C4#13714yes—
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D412yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4114yes—
311C22×C4C2×#4114yes—
312C22×C4C2×#4114yes—
313C22×C4C2511yescharacteristic
314M16—88no—
315C2×C8—28no—
316C2×C8—28no—
317C2×C8—28no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—28no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—48no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C22⋊C4—88no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—44no—
351C42—44no—
352C42—84no—
353C42—84no—
354C23—22no—
355C23—22no—
356C23—24no—
357C23—28no—
358C23—24no—
359C23—22no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—24no—
364C23—22no—
365C23—22no—
366C23—44no—
367C23—44no—
368C23—48no—
369C23—48no—
370C23—48no—
371C23—44no—
372C23—82no—
373C23—84no—
374C23—82no—
375C23C2×#12314yes—
376C23C2×#12312yes—
377C23C2×#13712yes—
378C23#57512yes—
379C23C23×D411yescharacteristic
380D4—48no—
381D4—48no—
382C2×C4—28no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
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—84no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—88no—
418C2×C4—88no—
419C2×C4—816no—
420C2×C4C2×#12312yes—
421C2×C4C2×#12314yes—
422C2×C4C2×#13712yes—
423C2×C4C2×#13712yes—
424C2×C4C22×#4112yes—
425C2×C4#58514yes—
426C8—48no—
427C22—24no—
428C22—24no—
429C22—22no—
430C22—22no—
431C22—24no—
432C22—24no—
433C22—24no—
434C22—416no—
435C22—44no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—88no—
443C22#309514yes—
444C22C22×#12311yescharacteristic
445C22#348512yes—
446C22C22×#13711yescharacteristic
447C22C2×#57511yescharacteristic
448C22#358512yes—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—24no—
453C4—41no—
454C4—41no—
455C4—42no—
456C4—84no—
457C4—84no—
458C4—84no—
459C4—84no—
460C4#349114yes—
461C2—24no—
462C2—44no—
463C2—82no—
464C2#3544811yescharacteristic
465C2C2×#309511yescharacteristic
466C2#6051811yescharacteristic
467C2#6171312yes—
468C2#6171912yes—
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. #27614C2semidirect
4. #27611C2semidirect
5. #27604C2semidirect
6. #24964C2semidirect
7. #24960C2semidirect
8. #21944C2semidirect
10. #20700C2semidirect
37. #2100C22semidirect
38. #2099C22semidirect
39. #2084C22semidirect
40. #2083C22semidirect
41. #2077C22semidirect
42. #2075C22semidirect
43. #1921C22semidirect
44. #1920C22semidirect
47. #1854C22semidirect
48. #1854C22semidirect
51. #1825C22semidirect
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
#341862
#276142
#276112
#276042
#249642
#249602
#219442
#217422
#207002