Finite Group Databasebeta

← #6629636

Subgroups — #6629636

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2624C211yesmaximal, characteristic
2#33996C211yesmaximal, characteristic, complemented
3#33103C218yesmaximal, complemented
4#33102C218yesmaximal, complemented
5#21060C214yesmaximal, complemented
6#21034C214yesmaximal, complemented
7#20912C212yesmaximal
8#20846C212yesmaximal, complemented
9#20683C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C22×#117C2212yes—
12C22×#117C2211yescharacteristic
13C2×#428C2214yes—
14C2×#418C2214yes—
15C2×#407C2212yes—
16#2624—24no—
17#2624C2218yes—
18C2×#399—21no—
19C2×#397—21no—
20C2×#394—22no—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#380C2211yescharacteristic
24#2274—21no—
25#2274—21no—
26#2274C2214yescomplemented
27#2274C2214yescomplemented
28#2273—22no—
29C4×#117—24no—
30C4×#117C2214yes—
31#1926C2218yescomplemented
32#1926C2218yescomplemented
33#1925C2218yescomplemented
34#1925C2218yescomplemented
35#1921C2218yescomplemented
36#1920C2218yescomplemented
37#1919C2218yescomplemented
38#1918C2218yescomplemented
39#1875C2218yes—
40#1874C2214yes—
41#1874C2214yes—
42#1854C2218yescomplemented
43#1853C2214yescomplemented
44#1852C2214yescomplemented
45#1818C2218yescomplemented
46#1736C2214yes—
47#1736C2214yes—
48#1728C2214yes—
49C23×D4—21no—
50C23×D4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16C2314yes—
53C2×#121—22no—
54C2×#121—28no—
55C2×#121C2314yes—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—28no—
59C2×#117—216no—
60C2×#117D414yes—
61C2×#117C2314yes—
62C2×#117C2314yes—
63C2×#117C2314yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D412yes—
77C22×C42—22no—
78C22×C42C2312yes—
79#428—216no—
80C2×#107C2318yes—
81C2×#106C2318yes—
82#418—216no—
83C2×#102C2318yes—
84C2×#102C2318yes—
85#407—28no—
86C2×C22⋊C8C2314yes—
87C2×C22⋊C8C2314yes—
88#399—28no—
89#397—28no—
90#394—28no—
91#394—28no—
92#390—24no—
93#390C2318yes—
94#389—24no—
95#389C2318yes—
96#380—22no—
97#380C2314yes—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4C2314yes—
103C4×C4⋊C4C2314yes—
104C4×C4⋊C4C2314yes—
105C4×C4⋊C4C2314yes—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—24no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—24no—
112#376—24no—
113C2×#95—22no—
114C2×#95—21no—
115C43—24no—
116#340C2318yes—
117#340C2318yes—
118#340C2318yes—
119#340C2318yes—
120#336C2318yes—
121#336C2318yes—
122C25—21no—
123C25C2×D411yescharacteristic
124C22×D4—24no—
125C22×D4—21no—
126C22×D4—24no—
127C22×D4—28no—
128C22×D4—44no—
129C22×D4—84no—
130C22×D4C2412yes—
131C23×C4—28no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4#4118yes—
141C23×C4C2412yes—
142C23×C4C2411yescharacteristic
143C2×M16—28no—
144C2×M16—28no—
145C2×M16—28no—
146C22×C8C2418yes—
147#121—28no—
148#121—28no—
149#121—48no—
150#117—216no—
151#117—216no—
152#117—216no—
153#117—28no—
154#117—28no—
155#117—28no—
156#117—28no—
157#117—28no—
158#117#41116yes—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×D414yes—
173C2×C4⋊C4C2412yes—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2414yes—
176C2×C4⋊C4C2414yes—
177C2×C22⋊C4—216no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—84no—
187C2×C22⋊C4—84no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4C2×D414yes—
190C2×C22⋊C4C2×D414yes—
191C2×C42—216no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—28no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42C2×D414yes—
200C2×C42C2414yes—
201#107—216no—
202#106—216no—
203#102—216no—
204#102—216no—
205C22⋊C8—28no—
206C22⋊C8—28no—
207#95—24no—
208#95—28no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213C24—24no—
214C24—22no—
215C24—41no—
216C24—42no—
217C24—41no—
218C24—42no—
219C24—44no—
220C24—42no—
221C24—42no—
222C24—44no—
223C24#12314yes—
224C24C22×D412yes—
225C24C22×D411yescharacteristic
226C2×D4—28no—
227C2×D4—24no—
228C2×D4—48no—
229C2×D4—42no—
230C2×D4—44no—
231C2×D4—416no—
232C2×D4—84no—
233C2×D4—84no—
234C2×D4—816no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—232no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4#12314yes—
263C22×C4#12318yes—
264C22×C4C22×D414yes—
265C22×C4C22×D414yes—
266C22×C4C22×D412yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4114yes—
269C22×C4C2×#4114yes—
270C22×C4C2×#4114yes—
271C22×C4C2×#4118yes—
272C22×C4C2511yescharacteristic
273M16—416no—
274C2×C8—28no—
275C2×C8—28no—
276C2×C8—28no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—28no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—88no—
288C4⋊C4C2×#4118yes—
289C4⋊C4C2×#4118yes—
290C22⋊C4—216no—
291C22⋊C4—28no—
292C22⋊C4—28no—
293C22⋊C4—216no—
294C22⋊C4—48no—
295C22⋊C4—88no—
296C22⋊C4—88no—
297C22⋊C4—84no—
298C42—216no—
299C42—28no—
300C42—216no—
301C42—28no—
302C42—84no—
303C42—84no—
304C42—84no—
305C42—84no—
306C23—22no—
307C23—21no—
308C23—21no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—28no—
316C23—44no—
317C23—44no—
318C23—42no—
319C23—44no—
320C23—42no—
321C23—42no—
322C23—44no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—88no—
328C23—88no—
329C23—84no—
330C23C2×#12314yes—
331C23C2×#12312yes—
332C23#52514yes—
333C23C23×D411yescharacteristic
334D4—48no—
335D4—816no—
336D4—84no—
337C2×C4—28no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—216no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—88no—
363C2×C4—88no—
364C2×C4—88no—
365C2×C4—84no—
366C2×C4C2×#12314yes—
367C2×C4C2×#12312yes—
368C2×C4#52514yes—
369C2×C4#529116yes—
370C2×C4C22×#4112yes—
371C2×C4C22×#4114yes—
372C8—48no—
373C22—22no—
374C22—22no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—88no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—84no—
389C22—84no—
390C22—168no—
391C22#285314yes—
392C22C22×#12311yescharacteristic
393C22C2×#52512yes—
394C22#358512yes—
395C22#358511yescharacteristic
396C22#358611yescharacteristic
397C4—24no—
398C4—44no—
399C4—44no—
400C4—48no—
401C4—48no—
402C4—82no—
403C4—84no—
404C4#285314yes—
405C4C2×#52914yes—
406C2—24no—
407C2—44no—
408C2—162no—
409C2C2×#285311yescharacteristic
410C2#6051811yescharacteristic
411C2#6052011yescharacteristic
412C2#6056712yes—
413C2#6056812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33996C2semidirect
3. #33103C2semidirect
4. #33102C2semidirect
5. #21060C2semidirect
6. #21034C2semidirect
8. #20846C2semidirect
9. #20683C2semidirect
26. #2274C22semidirect
27. #2274C22semidirect
31. #1926C22semidirect
32. #1926C22semidirect
33. #1925C22semidirect
34. #1925C22semidirect
35. #1921C22semidirect
36. #1920C22semidirect
37. #1919C22semidirect
38. #1918C22semidirect
42. #1854C22semidirect
43. #1853C22semidirect
44. #1852C22semidirect
45. #1818C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26242
#339962
#331032
#331022
#210602
#210342
#209122
#208462
#206832