Finite Group Databasebeta

← #6508964

Subgroups — #6508964

Order 512.

2032 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#27610C214yesmaximal, complemented
4#27600C214yesmaximal, complemented
5#24933C2116yesmaximal, complemented
6#21940C212yesmaximal, complemented
7#21738C211yesmaximal, characteristic
8C2×#1823C212yesmaximal, complemented
9C22×#123—21no—
10C2×#518—21no—
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14#2486—24no—
15#2486C2218yes—
16C2×#398—21no—
17C2×#397—21no—
18#2349—24no—
19D4×C4⋊C4C2218yes—
20#2316—21no—
21#2315—21no—
22#2315—21no—
23C2×#389C2212yes—
24#2285C2212yes—
25C2×#381—21no—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30#2099C2218yescomplemented
31#2083C2218yescomplemented
32#2075C22116yescomplemented
33#1918C22116yescomplemented
34#1869C2218yes—
35#1852C2218yes—
36#1852C2218yescomplemented
37#1852C2218yescomplemented
38#1823C2218yes—
39#1823C2218yescomplemented
40#1823C2218yescomplemented
41#1817C2212yes—
42#1808C2218yescomplemented
43#1779C2212yes—
44C2×#340C2218yescomplemented
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2314yes—
50C2×#123—28no—
51C2×#123—48no—
52#518—48no—
53C2×C4×D4—28no—
54C2×C4×D4—28no—
55C2×C4×D4D414yes—
56C2×C4×D4C2314yes—
57C2×#117—22no—
58C22×C4⋊C4—28no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107C23116yes—
76C2×C22⋊C8—28no—
77C2×C22⋊C8C2314yes—
78C2×C22⋊C8C2314yes—
79C2×C22⋊C8C2318yes—
80#398—28no—
81#397—28no—
82#389—28no—
83#389—24no—
84#389C2318yes—
85#385—28no—
86#385—28no—
87#384C2314yes—
88#382—22no—
89#381—28no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—24no—
96#380—22no—
97#380C2314yes—
98#380C2314yes—
99#380C2314yes—
100#380C2318yes—
101#380C2314yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2314yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109#340C23116yes—
110#340C23116yes—
111#340C23116yes—
112#336C2318yes—
113C25C2×D411yescharacteristic
114C25C2×D411yescharacteristic
115C22×D4—24no—
116C22×D4—24no—
117C22×D4D818yes—
118C22×D4C2×D414yes—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4#4118yes—
137C23×C4C2412yes—
138C23×C4C2412yes—
139C23×C4C2411yescharacteristic
140C22×C8—28no—
141C22×C8—48no—
142C22×C8C2418yes—
143#123—432no—
144C4×D4—216no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—432no—
148#117—48no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2418yes—
170C2×C4⋊C4C2414yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—48no—
182C2×C22⋊C4C2×D414yes—
183C2×C22⋊C4C2×D414yes—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42C2×D412yes—
187C2×C42C2412yes—
188#107—416no—
189C22⋊C8—28no—
190C22⋊C8—28no—
191C22⋊C8—48no—
192#95—24no—
193#95—28no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—28no—
198#95—24no—
199C24—22no—
200C24—22no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24#12314yes—
206C24C2×D814yes—
207C24C2×D814yes—
208C24#13714yes—
209C24C22×D412yes—
210C24C22×D412yes—
211C24C22×D411yescharacteristic
212C2×D4—28no—
213C2×D4—28no—
214C2×D4—232no—
215C2×D4C2×D8116yes—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—28no—
227C22×C4—24no—
228C22×C4—416no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—88no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4—84no—
250C22×C4—816no—
251C22×C4#12314yes—
252C22×C4#12318yes—
253C22×C4C2×D814yes—
254C22×C4C22×D412yes—
255C22×C4C22×D412yes—
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D412yes—
258C22×C4C22×D412yes—
259C22×C4C2×#4114yes—
260C22×C4C2×#4114yes—
261C22×C4C2×#4114yes—
262C22×C4C2511yescharacteristic
263C2×C8—28no—
264C2×C8—216no—
265C2×C8—88no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—216no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—48no—
273C4⋊C4—416no—
274C4⋊C4—48no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—84no—
278C22⋊C4—28no—
279C22⋊C4—28no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C22⋊C4—48no—
284C22⋊C4—416no—
285C42—24no—
286C42—24no—
287C42—48no—
288C23—22no—
289C23—24no—
290C23—22no—
291C23—24no—
292C23—24no—
293C23—22no—
294C23—24no—
295C23—22no—
296C23—28no—
297C23—44no—
298C23—44no—
299C23—48no—
300C23—48no—
301C23—48no—
302C23—82no—
303C23—84no—
304C23—82no—
305C23#480116yes—
306C23C2×#12314yes—
307C23C2×#12312yes—
308C23C22×D814yes—
309C23C22×D812yes—
310C23C22×D814yes—
311C23C2×#13712yes—
312C23C2×#13714yes—
313C23C23×D411yescharacteristic
314D4—232no—
315C2×C4—28no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—48no—
328C2×C4—44no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—84no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—88no—
343C2×C4—88no—
344C2×C4—816no—
345C2×C4—168no—
346C2×C4—168no—
347C2×C4C2×#12312yes—
348C2×C4C2×#12314yes—
349C2×C4C22×D814yes—
350C2×C4C22×#4112yes—
351C2×C4#58514yes—
352C8—48no—
353C22—24no—
354C22—28no—
355C22—22no—
356C22—24no—
357C22—22no—
358C22—24no—
359C22—24no—
360C22—28no—
361C22—24no—
362C22—28no—
363C22—416no—
364C22—44no—
365C22—42no—
366C22—42no—
367C22—88no—
368C22C2×#48014yes—
369C22C2×#48018yes—
370C22C22×#12311yescharacteristic
371C22#348512yes—
372C22C23×D811yescharacteristic
373C22C22×#13711yescharacteristic
374C22#358612yes—
375C4—24no—
376C4—22no—
377C4—24no—
378C4—42no—
379C4—42no—
380C4—84no—
381C4—84no—
382C4—88no—
383C4#349114yes—
384C2—24no—
385C2—28no—
386C2—44no—
387C2#3544811yescharacteristic
388C2C22×#48011yescharacteristic
389C2#6051911yescharacteristic
390C2#6169412yes—
391C2#6170512yes—
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. #27610C2semidirect
4. #27600C2semidirect
5. #24933C2semidirect
6. #21940C2semidirect
8. C2×#1823C2semidirect
30. #2099C22semidirect
31. #2083C22semidirect
32. #2075C22semidirect
33. #1918C22semidirect
36. #1852C22semidirect
37. #1852C22semidirect
39. #1823C22semidirect
40. #1823C22semidirect
42. #1808C22semidirect
44. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#341812
#276102
#276002
#249332
#219402
#217382
C2×#18232