Finite Group Databasebeta

← #316969

Subgroups — #316969

Order 512.

641 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28254C212yesmaximal
2#22247C211yesmaximal, characteristic, complemented
3#22214C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10622C211yesmaximal, characteristic
6#10265C211yesmaximal, characteristic, complemented
7#10264C211yesmaximal, characteristic
8#9226C212yesmaximal, complemented
9#9225C212yesmaximal
10#8787C211yesmaximal, characteristic, complemented
11#8586C212yesmaximal, complemented
12C2×#500C2211yescharacteristic
13C2×#464—22no—
14C2×#462C2212yes—
15C2×#461C2212yes—
16C2×#446C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C4×C4⋊Q8C2212yes—
19C4×#124C2211yescharacteristic
20#2056—22no—
21#2055—22no—
22#1990C2212yes—
23#1984C2211yescharacteristic
24#1954C2211yescharacteristic
25#1924—22no—
26#1918—22no—
27#1918—22no—
28#1851—22no—
29#1842C2212yes—
30#1842C2211yescharacteristic
31#1834C2211yescharacteristic
32#1805—22no—
33#1803C2212yes—
34#1802C2212yes—
35#1744C2211yescharacteristic
36C2×#332C2212yes—
37C2×#332C2212yes—
38#1296C2212yes—
39#1296C2212yes—
40#1277C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C22×Q16—22no—
44C22×SD16—22no—
45C22×M16—21no—
46C2×C4⋊Q8C2312yes—
47C2×#124D411yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×D4C2311yescharacteristic
51#500—24no—
52#464—48no—
53#462—24no—
54#462—42no—
55#461—24no—
56#461—42no—
57#446—42no—
58#436—42no—
59C2×#107—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#102—22no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—22no—
74C2×#97—21no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—22no—
78#391D411yescharacteristic
79#390D411yescharacteristic
80#389—22no—
81#389—22no—
82#388—22no—
83#384C2312yes—
84#382D411yescharacteristic
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C22⋊C4—22no—
88C43C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#336—22no—
95#336D412yes—
96#336D412yes—
97#332—42no—
98#332—42no—
99C22×Q8—22no—
100C22×Q8C2×D411yescharacteristic
101C22×D4C2×D411yescharacteristic
102C23×C4—21no—
103C2×Q16—44no—
104C2×Q16—44no—
105C2×Q16—82no—
106C2×SD16—44no—
107C2×SD16—44no—
108C2×SD16—82no—
109C2×M16—42no—
110C2×M16—82no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D412yes—
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8#4114yes—
122#124—24no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—44no—
126C4×Q8—44no—
127C4×D4—22no—
128C4×D4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C22⋊C4—22no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C42—21no—
142C2×C42—42no—
143C2×C42C2×D412yes—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4112yes—
147C2×C42C2411yescharacteristic
148#107—28no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—42no—
156C4⋊C8—44no—
157C4⋊C8—82no—
158C4⋊C8—82no—
159#103—44no—
160#103—44no—
161#103—44no—
162#102—44no—
163C22⋊C8—42no—
164C22⋊C8—82no—
165#97—42no—
166C4×C8—22no—
167C4×C8—22no—
168#95—22no—
169#95—22no—
170#95—42no—
171#95—42no—
172C24—21no—
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4#12112yes—
197C22×C4#12112yes—
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C2×#4111yescharacteristic
201Q16—88no—
202SD16—88no—
203M16—84no—
204C2×C8—22no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C22⋊C4—42no—
235C22⋊C4—44no—
236C22⋊C4—82no—
237C42—21no—
238C42—22no—
239C42—22no—
240C42—21no—
241C42—42no—
242C42—41no—
243C42—41no—
244C42—84no—
245C42AGL(1,ℤ/8)14yes—
246C42C2×#4112yes—
247C42H(5,𝔽2)12yes—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23C2×#12111yescharacteristic
254Q8—42no—
255Q8—44no—
256Q8—44no—
257Q8—42no—
258Q8—82no—
259Q8—82no—
260D4—42no—
261D4—42no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4#46914yes—
285C2×C4#52511yescharacteristic
286C2×C4#53511yescharacteristic
287C2×C4#54612yes—
288C2×C4#55312yes—
289C2×C4C2×AGL(1,ℤ/8)12yes—
290C2×C4#57611yescharacteristic
291C2×C4#57712yes—
292C2×C4#57811yescharacteristic
293C2×C4#57812yes—
294C8—42no—
295C8—41no—
296C8—41no—
297C8—81no—
298C8—81no—
299C8—82no—
300C8—82no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22—81no—
305C22—82no—
306C22#253511yescharacteristic
307C22#304612yes—
308C22C2×#46911yescharacteristic
309C22#306111yescharacteristic
310C22#306912yes—
311C4—22no—
312C4—24no—
313C4—42no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—84no—
318C4—162no—
319C4#312112yes—
320C4#322211yescharacteristic
321C4#322311yescharacteristic
322C4#332512yes—
323C4#333012yes—
324C2—161no—
325C2#1945212yes—
326C2#1955012yes—
327C2#2814911yescharacteristic
328C2#2817311yescharacteristic
329C2#2817911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22247C2semidirect
4. #10624C2semidirect
6. #10265C2semidirect
8. #9226C2semidirect
10. #8787C2semidirect
11. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#282542
#222472
#222142
#106242
#106222
#102652
#102642
#92262
#92252
#87872
#85862