Finite Group Databasebeta

← #6482285

Subgroups — #6482285

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3#33147C214yesmaximal
4#29967C214yesmaximal
5#26508C218yesmaximal, complemented
6#26442C218yesmaximal, complemented
7#21951C211yesmaximal, characteristic
8#21834C212yesmaximal
9#21359C212yesmaximal
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2212yes—
13#2857C2214yes—
14#2659—22no—
15#2552C2214yes—
16#2551C2214yes—
17#2550—24no—
18#2369—24no—
19#2369C2218yes—
20#2316—21no—
21C2×#391—22no—
22C2×#389—21no—
23C2×#389C2212yes—
24#2285C2212yes—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2099C2218yes—
30#2088C22116yes—
31#2058C2218yes—
32#1990C2214yes—
33#1990C2214yes—
34#1988C2218yes—
35#1987C2218yes—
36#1948C2214yes—
37#1948C2214yes—
38#1948C2218yes—
39#1939C2218yes—
40#1919C22116yes—
41#1853C2214yes—
42#1853C2214yes—
43#1824C2218yes—
44#1797C2214yes—
45C4×#106C2218yes—
46C23×Q8—21no—
47#544—48no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8Q818yescomplemented
52C2×C4⋊Q8C2314yes—
53C2×C4⋊Q8C2314yes—
54C2×#125—24no—
55C2×C22⋊Q8—24no—
56C2×#117—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—22no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68#431—28no—
69C2×#106C2318yes—
70C2×#106C2318yes—
71C2×C4⋊C8C2314yes—
72C2×C4⋊C8C2318yes—
73C2×C22⋊C8C2314yes—
74C2×C4×C8C2314yes—
75#398—28no—
76#395—28no—
77#391—24no—
78#391—216no—
79#389—24no—
80#389—24no—
81#389—28no—
82#389—24no—
83#389—24no—
84#389C2318yes—
85#389C2318yes—
86#384C2314yes—
87#384C2312yes—
88#384C2312yes—
89#382—22no—
90#380—24no—
91#380—22no—
92#380C2314yes—
93C4×C4⋊C4—28no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4D414yes—
98C4×C4⋊C4C2318yes—
99C4×C4⋊C4C2314yes—
100C4×C4⋊C4C2314yes—
101C4×C4⋊C4C2314yes—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95D412yes—
106#340C23116yes—
107#340C23116yes—
108#340C2318yes—
109#340C2318yes—
110#336C2318yes—
111C22×Q8—28no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—82no—
115C22×Q8—82no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8C2418yes—
125C4⋊Q8—216no—
126C4⋊Q8—28no—
127C4⋊Q8—48no—
128C4⋊Q8—416no—
129C4⋊Q8C2×Q8116yes—
130#125—416no—
131C22⋊Q8—416no—
132#117—48no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—84no—
152C2×C4⋊C4SD1618yes—
153C2×C4⋊C4C2×D414yes—
154C2×C4⋊C4C2×D414yes—
155C2×C4⋊C4C2×Q818yes—
156C2×C4⋊C4C2412yes—
157C2×C4⋊C4C2414yes—
158C2×C4⋊C4C2414yes—
159C2×C4⋊C4C2414yes—
160C2×C4⋊C4C2412yes—
161C2×C22⋊C4—24no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—28no—
165C2×C42—44no—
166C2×C42—44no—
167C2×C42—84no—
168C2×C42C2×D414yes—
169C2×C42C2×D412yes—
170C2×C42C2×Q814yes—
171C2×C42#4114yes—
172C2×C42C2414yes—
173C2×C42C2412yes—
174#106—216no—
175#106—48no—
176C4⋊C8—28no—
177C4⋊C8—48no—
178C22⋊C8—28no—
179C4×C8—28no—
180#95—28no—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—28no—
186#95C2×D418yes—
187C24C22×D411yescharacteristic
188C2×Q8—216no—
189C2×Q8—48no—
190C2×Q8—84no—
191C2×Q8—84no—
192C2×Q8—84no—
193C2×Q8—84no—
194C2×Q8—816no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4C22⋊Q814yes—
219C22×C4C22⋊Q814yes—
220C22×C4C2×SD1618yes—
221C22×C4C2×SD1614yes—
222C22×C4#13714yes—
223C22×C4C22×D414yes—
224C22×C4C22×D412yes—
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D414yes—
227C22×C4C22×D412yes—
228C22×C4C22×Q812yes—
229C22×C4C2×#4112yes—
230C22×C4C2511yescharacteristic
231C2×C8—28no—
232C2×C8—28no—
233C2×C8—28no—
234C4⋊C4—24no—
235C4⋊C4—216no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—28no—
241C4⋊C4—24no—
242C4⋊C4—28no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—48no—
247C4⋊C4—416no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—84no—
251C4⋊C4—88no—
252C4⋊C4—84no—
253C4⋊C4C2×SD16116yes—
254C4⋊C4C22×Q818yes—
255C4⋊C42−1+414yes—
256C22⋊C4—48no—
257C42—28no—
258C42—44no—
259C42—48no—
260C42—48no—
261C42—88no—
262C42C22⋊Q818yes—
263C42C22×Q814yes—
264C42C2×#4114yes—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23C2×C22⋊Q812yes—
270C23C22×SD1612yes—
271C23C2×#13712yes—
272C23C23×D411yescharacteristic
273Q8—816no—
274Q8—168no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—416no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4#475116yes—
308C2×C4C2×C22⋊Q812yes—
309C2×C4C2×C22⋊Q814yes—
310C2×C4C2×C22⋊Q814yes—
311C2×C4C22×SD1618yes—
312C2×C4C22×SD1614yes—
313C2×C4C2×#13714yes—
314C2×C4C23×Q811yescharacteristic
315C2×C4C22×#4111yescharacteristic
316C2×C4C2×2−1+412yes—
317C2×C4#58512yes—
318C8—48no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22C2×#47514yes—
326C22C22×C22⋊Q811yescharacteristic
327C22#348511yescharacteristic
328C22#348611yescharacteristic
329C22C23×SD1611yescharacteristic
330C22C22×#13711yescharacteristic
331C22#358511yescharacteristic
332C22#358511yescharacteristic
333C4—28no—
334C4—24no—
335C4—22no—
336C4—44no—
337C4—48no—
338C4—44no—
339C4—84no—
340C4—82no—
341C4—84no—
342C4C2×#47518yes—
343C4#346112yes—
344C4#350212yes—
345C2—24no—
346C2#3515411yescharacteristic
347C2C22×#47511yescharacteristic
348C2#6049011yescharacteristic
349C2#6169711yescharacteristic
350C2#6172111yescharacteristic
351C2#6182011yescharacteristic
352C2#6182611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #26508C2semidirect
6. #26442C2semidirect
51. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23692
#331472
#299672
#265082
#264422
#219512
#218342
#213592