Finite Group Databasebeta

← #6477782

Subgroups — #6477782

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic
3#33147C214yesmaximal, complemented
4#29967C214yesmaximal, complemented
5#25992C2116yesmaximal, complemented
6#21938C212yesmaximal
7#21930C211yesmaximal, characteristic
8C2×#1824C212yesmaximal
9C2×#543—21no—
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2857C2214yes—
13#2659—22no—
14#2549—24no—
15#2440C2218yes—
16#2345—24no—
17#2345C2218yes—
18#2315—21no—
19C2×#389C2212yes—
20C2×#387—22no—
21C2×#380—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C22⋊C4—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2099C2218yescomplemented
28#2088C22116yescomplemented
29#2058C2218yescomplemented
30#1988C2218yescomplemented
31#1948C2218yescomplemented
32#1919C22116yescomplemented
33#1853C2218yes—
34#1853C2218yes—
35#1850C2218yes—
36#1824C2214yes—
37#1824C2214yes—
38#1824C2218yes—
39#1807C2218yes—
40#1797C2214yes—
41C2×#340C2218yes—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2314yes—
45#543—48no—
46C2×C4⋊Q8C2314yes—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—28no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4Q818yescomplemented
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2314yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2314yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4D411yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C42—22no—
64C22×C42C2312yes—
65C2×#106C23116yes—
66C2×C4⋊C8C2318yes—
67C2×C22⋊C8—28no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70#398—28no—
71#395—28no—
72#391—24no—
73#389—24no—
74#389—24no—
75#389C2318yes—
76#389C2318yes—
77#387—216no—
78#385—28no—
79#384C2314yes—
80#380—24no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—28no—
85#380C2314yes—
86#380C2314yes—
87#380C2314yes—
88C4×C4⋊C4—24no—
89C4×C4⋊C4C2318yes—
90C4×C22⋊C4—28no—
91C4×C22⋊C4D414yes—
92C4×C22⋊C4C2314yes—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95D412yes—
98#340C23116yes—
99#340C23116yes—
100#340C23116yes—
101#336C2318yes—
102C25C2×D411yescharacteristic
103C22×Q8—24no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—28no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—44no—
111C23×C4—84no—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×Q814yes—
117C23×C4#4114yes—
118C23×C4C2412yes—
119C23×C4C2411yescharacteristic
120C22×C8—28no—
121C22×C8—48no—
122C22×C8C2418yes—
123C4⋊Q8—48no—
124C22⋊Q8—416no—
125C22⋊Q8—432no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—216no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—84no—
144C2×C4⋊C4C2×Q814yes—
145C2×C4⋊C4C2×Q814yes—
146C2×C4⋊C4C2×Q8116yes—
147C2×C4⋊C4C2418yes—
148C2×C4⋊C4C2414yes—
149C2×C4⋊C4C2412yes—
150C2×C4⋊C4C2412yes—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—48no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4SD1618yes—
158C2×C22⋊C4C2×D414yes—
159C2×C22⋊C4C2×D414yes—
160C2×C42—28no—
161C2×C42—44no—
162C2×C42—84no—
163C2×C42C2×D414yes—
164C2×C42C2414yes—
165#106—416no—
166C4⋊C8—48no—
167C22⋊C8—28no—
168C22⋊C8—28no—
169#95—28no—
170#95—28no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—28no—
175#95C2×D418yes—
176C24—42no—
177C24—42no—
178C24C22⋊Q814yes—
179C24C2×SD1614yes—
180C24#13714yes—
181C24C22×D412yes—
182C24C22×D411yescharacteristic
183C2×Q8—48no—
184C2×Q8—84no—
185C2×Q8—84no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—216no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—48no—
210C22×C4—88no—
211C22×C4—816no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4C22⋊Q814yes—
215C22×C4C22⋊Q818yes—
216C22×C4C2×SD1618yes—
217C22×C4C22×D414yes—
218C22×C4C22×D412yes—
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D414yes—
221C22×C4C22×Q814yes—
222C22×C4C22×Q812yes—
223C22×C4C2×#4112yes—
224C22×C4C2×#4114yes—
225C22×C4C2511yescharacteristic
226C2×C8—28no—
227C2×C8—216no—
228C2×C8—88no—
229C4⋊C4—24no—
230C4⋊C4—216no—
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—28no—
235C4⋊C4—216no—
236C4⋊C4—44no—
237C4⋊C4—48no—
238C4⋊C4—416no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C4⋊C4—88no—
242C4⋊C4—84no—
243C4⋊C4C22×Q814yes—
244C4⋊C4C22×Q814yes—
245C22⋊C4—28no—
246C22⋊C4—48no—
247C22⋊C4—48no—
248C22⋊C4—48no—
249C22⋊C4C2×SD16116yes—
250C42—28no—
251C42—44no—
252C42—88no—
253C23—22no—
254C23—24no—
255C23—22no—
256C23—24no—
257C23—44no—
258C23—44no—
259C23—48no—
260C23—48no—
261C23#475116yes—
262C23C2×C22⋊Q812yes—
263C23C2×C22⋊Q814yes—
264C23C22×SD1612yes—
265C23C22×SD1614yes—
266C23C2×#13714yes—
267C23C2×#13712yes—
268C23C23×D411yescharacteristic
269Q8—168no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—48no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—44no—
296C2×C4—84no—
297C2×C4—816no—
298C2×C4—88no—
299C2×C4—168no—
300C2×C4C2×C22⋊Q814yes—
301C2×C4C2×C22⋊Q812yes—
302C2×C4C22×SD1618yes—
303C2×C4C23×Q811yescharacteristic
304C2×C4C22×#4111yescharacteristic
305C2×C4#58514yes—
306C8—48no—
307C22—24no—
308C22—28no—
309C22—24no—
310C22—28no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—24no—
315C22—416no—
316C22C2×#47514yes—
317C22C2×#47518yes—
318C22C22×C22⋊Q811yescharacteristic
319C22#348512yes—
320C22C23×SD1611yescharacteristic
321C22C22×#13711yescharacteristic
322C22#358512yes—
323C4—28no—
324C4—22no—
325C4—24no—
326C4—44no—
327C4—44no—
328C4—44no—
329C4—88no—
330C4—84no—
331C4#345812yes—
332C4#349112yes—
333C2—24no—
334C2—28no—
335C2#3509011yescharacteristic
336C2C22×#47511yescharacteristic
337C2#6049011yescharacteristic
338C2#6169712yes—
339C2#6170812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33147C2semidirect
4. #29967C2semidirect
5. #25992C2semidirect
27. #2099C22semidirect
28. #2088C22semidirect
29. #2058C22semidirect
30. #1988C22semidirect
31. #1948C22semidirect
32. #1919C22semidirect
53. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23452
#331472
#299672
#259922
#219382
#219302
C2×#18242