Finite Group Databasebeta

← #325419

Subgroups — #325419

Order 512.

549 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33256C211yesmaximal, characteristic, complemented
2#29122C212yesmaximal
3#21631C211yesmaximal, characteristic
4#10620C212yesmaximal, complemented
5#10619C212yesmaximal
6#9901C212yesmaximal, complemented
7#8657C211yesmaximal, characteristic, complemented
8#8657C211yesmaximal, characteristic, complemented
9#8656C211yesmaximal, characteristic
10#8656C211yesmaximal, characteristic
11C4×#334C211yesmaximal, characteristic, complemented
12C2×#446C2211yescharacteristic
13C2×#436C2211yescharacteristic
14#2859C2212yes—
15C4×#124C2211yescharacteristic
16#2101—22no—
17#2042—22no—
18#1984C2211yescharacteristic
19#1954C2211yescharacteristic
20#1941—22no—
21#1941—22no—
22#1934—22no—
23#1933—22no—
24#1923—22no—
25#1844C2212yes—
26#1834C2211yescharacteristic
27#1834C2211yescharacteristic
28#1831C2211yescharacteristic
29#1803C2212yes—
30#1803C2212yes—
31#1802C2212yes—
32#1802C2212yes—
33C4×C4⋊C8C2212yes—
34#1753—22no—
35C42×C8C2211yescharacteristic
36C2×#334C2211yescharacteristic
37C2×#334C2211yescharacteristic
38#1296C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1277C2212yes—
43C22×M16—21no—
44C2×#125—22no—
45C2×#124D411yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×C4×D4C2311yescharacteristic
48#446—22no—
49#446—22no—
50#436—22no—
51#436—22no—
52C2×#106—22no—
53C2×#106—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2311yescharacteristic
58C2×C4⋊C8C2311yescharacteristic
59C2×#103—22no—
60C2×#103—22no—
61C2×#102—22no—
62C2×#102—22no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—22no—
65C2×#97—21no—
66C2×C4×C8—22no—
67C2×C4×C8D412yes—
68C2×C4×C8D412yes—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#391D411yescharacteristic
72#390D411yescharacteristic
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#388—22no—
78#384C2312yes—
79#384C2312yes—
80#382—22no—
81#382D411yescharacteristic
82C4×C22⋊C4—22no—
83C43C2311yescharacteristic
84#340—21no—
85#340—21no—
86#340—21no—
87#340—21no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#336—22no—
93#336—22no—
94#336—22no—
95#334—22no—
96#334—22no—
97C22×Q8C2×D411yescharacteristic
98C22×D4C2×D411yescharacteristic
99C23×C4—21no—
100C2×M16—44no—
101C2×M16—42no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D412yes—
109#125—84no—
110#124—42no—
111C4×Q8—21no—
112C4×Q8—21no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×D4—21no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—21no—
133C2×C42—21no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42#4112yes—
138C2×C42C2411yescharacteristic
139#106—44no—
140#106—44no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—42no—
150#103—44no—
151#103—82no—
152#102—44no—
153#102—82no—
154C22⋊C8—42no—
155C22⋊C8—44no—
156#97—42no—
157C4×C8—24no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—42no—
161C4×C8—82no—
162C4×C8#4112yes—
163C4×C8#4112yes—
164#95—22no—
165#95—22no—
166#95—42no—
167#95—42no—
168C24—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#12112yes—
187C22×C4#12112yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C2×#4111yescharacteristic
191M16—84no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8#12412yes—
207C2×C8#12412yes—
208C2×C8#13514yes—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—84no—
226C4⋊C4—84no—
227C22⋊C4—44no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—44no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—24no—
236C42—41no—
237C42—41no—
238C42C2×#4111yescharacteristic
239C42C2×#4111yescharacteristic
240C422−1+411yescharacteristic
241C422−1+411yescharacteristic
242C23—42no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23C2×#12111yescharacteristic
248Q8—42no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4#47114yes—
275C2×C4C2×#12411yescharacteristic
276C2×C4#53611yescharacteristic
277C2×C4#54711yescharacteristic
278C2×C4#54711yescharacteristic
279C2×C4#55611yescharacteristic
280C2×C4#55611yescharacteristic
281C2×C4C2×#13512yes—
282C2×C4#57512yes—
283C2×C4#57512yes—
284C2×C4#57712yes—
285C8—22no—
286C8—22no—
287C8—24no—
288C8—82no—
289C8—82no—
290C8—81no—
291C8—81no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—82no—
297C22#254711yescharacteristic
298C22#303512yes—
299C22#305411yescharacteristic
300C22#306011yescharacteristic
301C22C2×#47112yes—
302C4—24no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—162no—
310C4—162no—
311C4#313111yescharacteristic
312C4#314011yescharacteristic
313C4#323212yes—
314C4#335112yes—
315C4#335112yes—
316C2—161no—
317C2#1915912yes—
318C2#1922112yes—
319C2#2871011yescharacteristic
320C2#2872011yescharacteristic
321C2#2873211yescharacteristic
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
1. #33256C2semidirect
4. #10620C2semidirect
6. #9901C2semidirect
7. #8657C2semidirect
8. #8657C2semidirect
11. C4×#334C2semidirect

Maximal subgroups

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

subgroupindex
#332562
#291222
#216312
#106202
#106192
#99012
#86572
#86562
C4×#3342