Finite Group Databasebeta

← #6519029

Subgroups — #6519029

Order 512.

2346 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#27601C214yesmaximal, complemented
4#25012C218yesmaximal, complemented
5#24967C218yesmaximal, complemented
6#24005C214yesmaximal, complemented
7#21981C212yesmaximal, complemented
8#21786C211yesmaximal, characteristic, complemented
9#20732C212yesmaximal, complemented
10C22×#121—21no—
11C22×#121C2212yes—
12C2×#518—21no—
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#481C2218yescomplemented
16C2×#431C2212yes—
17#2537—24no—
18#2537C2218yes—
19C2×#394—22no—
20#2349—24no—
21D4×C4⋊C4C2214yescomplemented
22D4×C4⋊C4C2214yescomplemented
23#2316—21no—
24#2315—22no—
25#2285C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×#380C2212yes—
29C2×#380C2214yes—
30#2077C22116yescomplemented
31#2034C22116yescomplemented
32#1954C2218yescomplemented
33#1948C2218yescomplemented
34#1874C2218yescomplemented
35#1853C2218yes—
36#1853C2218yescomplemented
37#1844C2214yescomplemented
38#1834C2214yescomplemented
39#1824C2218yescomplemented
40#1823C2214yes—
41#1802C2214yescomplemented
42#1802C2214yescomplemented
43#1786C2214yes—
44#1779C2214yes—
45C4×#102C2218yescomplemented
46C23×D4—22no—
47C23×D4D411yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2312yes—
51C2×#121—28no—
52C2×#121—24no—
53C2×#121—48no—
54C2×#121—48no—
55C2×#121C2318yes—
56#518—48no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—22no—
60C2×C4×D4—28no—
61C2×C4×D4—28no—
62C2×C4×D4D414yes—
63C2×C4×D4C2314yes—
64C2×C4×D4C2314yes—
65C2×#117—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—24no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42C2311yescharacteristic
79#481—416no—
80#431—28no—
81C2×#106C23116yes—
82C2×C4⋊C8C2314yes—
83C2×#102C2318yes—
84C2×#102C2318yes—
85C2×C22⋊C8C2314yes—
86C2×C22⋊C8C2318yes—
87C2×C22⋊C8C2318yes—
88C2×C4×C8C2314yes—
89#394—216no—
90#389—24no—
91#389—24no—
92#385—28no—
93#385—28no—
94#384—22no—
95#384C2312yes—
96#384C2312yes—
97#384C2314yes—
98#382—22no—
99#380—22no—
100#380—24no—
101#380—24no—
102#380—24no—
103#380—28no—
104#380C2318yes—
105#380C2318yes—
106#380C2314yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2314yes—
109C2×#95—24no—
110C2×#95—22no—
111#340C2318yes—
112#340C2318yes—
113#340C23116yes—
114C25—21no—
115C25C2×D412yes—
116C22×D4—24no—
117C22×D4—28no—
118C22×D4—24no—
119C22×D4—24no—
120C22×D4—21no—
121C22×D4—24no—
122C22×D4—42no—
123C22×D4—48no—
124C22×D4—48no—
125C22×D4—84no—
126C22×D4SD1618yes—
127C22×D4C2×D414yes—
128C22×D4C2412yes—
129C23×C4—28no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4C2×D412yes—
141C23×C4C2×D414yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2414yes—
145C23×C4C2411yescharacteristic
146C22×C8C2418yes—
147#121—216no—
148#121—48no—
149#121—416no—
150#121—432no—
151C4×D4—216no—
152C4×D4—28no—
153C4×D4—28no—
154C4×D4—28no—
155C4×D4—28no—
156C4×D4—416no—
157C4×D4—416no—
158C4×D4—48no—
159#117—48no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4C2×D414yes—
178C2×C4⋊C4C2414yes—
179C2×C4⋊C4C2414yes—
180C2×C4⋊C4C2412yes—
181C2×C4⋊C4C2414yes—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—416no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4C2×D418yes—
193C2×C42—24no—
194C2×C42C2×D412yes—
195C2×C42#4118yes—
196C2×C42C2412yes—
197#106—416no—
198C4⋊C8—28no—
199#102—216no—
200#102—48no—
201C22⋊C8—28no—
202C22⋊C8—48no—
203C22⋊C8—48no—
204C4×C8—28no—
205#95—24no—
206#95—28no—
207#95—28no—
208#95—24no—
209C24—24no—
210C24—24no—
211C24—24no—
212C24—22no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24C2×SD1618yes—
224C24C22×D414yes—
225C24C22×D411yescharacteristic
226C2×D4—24no—
227C2×D4—28no—
228C2×D4—28no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—232no—
233C2×D4—416no—
234C2×D4—42no—
235C2×D4—416no—
236C2×D4—416no—
237C2×D4—42no—
238C2×D4—88no—
239C2×D4—832no—
240C2×D4C2×SD16116yes—
241C2×D4H(5,𝔽2)14yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—816no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4—816no—
276C22×C4#12314yes—
277C22×C4#12314yes—
278C22×C4C2×SD1614yes—
279C22×C4AGL(1,ℤ/8)14yes—
280C22×C4C22×D414yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D412yes—
283C22×C4C22×D412yes—
284C22×C4C2×#4114yes—
285C22×C4C2511yescharacteristic
286C2×C8—28no—
287C2×C8—28no—
288C2×C8—28no—
289C4⋊C4—28no—
290C4⋊C4—24no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—48no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C22⋊C4—216no—
305C22⋊C4—416no—
306C22⋊C4—416no—
307C22⋊C4—416no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C42—48no—
311C42#12318yes—
312C42C2×#4114yes—
313C42C2×#4114yes—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—24no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—24no—
324C23—216no—
325C23—24no—
326C23—48no—
327C23—48no—
328C23—48no—
329C23—48no—
330C23—48no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—84no—
338C23—84no—
339C23—84no—
340C23—84no—
341C23—84no—
342C23—84no—
343C23—84no—
344C23—84no—
345C23C2×#12312yes—
346C23C22×SD1618yes—
347C23C22×SD1612yes—
348C23C2×AGL(1,ℤ/8)12yes—
349C23C23×D411yescharacteristic
350D4—232no—
351D4—28no—
352D4—48no—
353D4—44no—
354D4—832no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—44no—
364C2×C4—416no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4—88no—
380C2×C4—84no—
381C2×C4—816no—
382C2×C4—168no—
383C2×C4—168no—
384C2×C4#481116yescomplemented
385C2×C4C2×#12312yes—
386C2×C4C2×#12314yes—
387C2×C4C2×#12314yes—
388C2×C4C22×SD1614yes—
389C2×C4C2×AGL(1,ℤ/8)14yes—
390C2×C4C22×#4112yes—
391C2×C4C2×H(5,𝔽2)12yes—
392C2×C4#58512yes—
393C8—48no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—28no—
401C22—28no—
402C22—28no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—44no—
409C22—44no—
410C22—816no—
411C22—816no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22C2×#48114yes—
417C22C22×#12311yescharacteristic
418C22#348411yescharacteristic
419C22#348511yescharacteristic
420C22C23×SD1611yescharacteristic
421C22C22×AGL(1,ℤ/8)11yescharacteristic
422C22#358511yescharacteristic
423C22#358611yescharacteristic
424C4—24no—
425C4—22no—
426C4—44no—
427C4—84no—
428C4—82no—
429C4—84no—
430C4—84no—
431C4C2×#48118yes—
432C4#349414yes—
433C2—24no—
434C2—48no—
435C2—84no—
436C2#3554011yescharacteristic
437C2C22×#48111yescharacteristic
438C2#6052011yescharacteristic
439C2#6156011yescharacteristic
440C2#6158511yescharacteristic
441C2#6169511yescharacteristic
442C2#6172011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2537C2
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
2. #34181C2semidirect
3. #27601C2semidirect
4. #25012C2semidirect
5. #24967C2semidirect
6. #24005C2semidirect
7. #21981C2semidirect
8. #21786C2semidirect
9. #20732C2semidirect
15. C2×#481C22semidirect
21. D4×C4⋊C4C22semidirect
22. D4×C4⋊C4C22semidirect
30. #2077C22semidirect
31. #2034C22semidirect
32. #1954C22semidirect
33. #1948C22semidirect
34. #1874C22semidirect
36. #1853C22semidirect
37. #1844C22semidirect
38. #1834C22semidirect
39. #1824C22semidirect
41. #1802C22semidirect
42. #1802C22semidirect
45. C4×#102C22semidirect
384. C2×C4#481semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
#341812
#276012
#250122
#249672
#240052
#219812
#217862
#207322