Finite Group Databasebeta

← #345015

Subgroups — #345015

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26277C212yesmaximal
2#24682C212yesmaximal
3#10601C212yesmaximal
4#10483C212yesmaximal
5#10057C212yesmaximal
6#9825C211yesmaximal, characteristic
7#9656C211yesmaximal, characteristic
8#9594C211yesmaximal, characteristic
9#8883C212yesmaximal
10C2×#494C2211yescharacteristic
11C2×#477C2212yes—
12C2×C4×Q16C2211yescharacteristic
13#2567C2212yes—
14#2355C2212yes—
15#2101—22no—
16#2085—22no—
17#2072—22no—
18#2056—22no—
19#1990C2212yes—
20#1990C2212yes—
21#1987C2212yes—
22#1984C2212yes—
23#1948C2212yes—
24#1947—22no—
25#1939C2212yes—
26#1928—22no—
27#1919—22no—
28#1918—22no—
29#1845C2211yescharacteristic
30#1803C2212yes—
31C4×#107C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#333C2212yes—
35C2×Q8⋊C8C2212yes—
36C2×C82C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C22×Q16D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×C4×Q8C2312yes—
43#494—22no—
44#494—41no—
45#477—42no—
46#477—42no—
47C4×Q16—41no—
48C4×Q16—41no—
49C2×#107—22no—
50C2×#107—41no—
51C2×#107D411yescharacteristic
52C2×#107D411yescharacteristic
53C2×#107D411yescharacteristic
54C2×#106—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#103—22no—
59C2×#103—22no—
60C2×#103D412yes—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#395—42no—
65#391—22no—
66#391—22no—
67#389—22no—
68#389—22no—
69#387—42no—
70#384—22no—
71#384C2312yes—
72#382—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4C2312yes—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340D412yes—
85#336—22no—
86#336—22no—
87#336—22no—
88#335—41no—
89#335—41no—
90#333—42no—
91Q8⋊C8—24no—
92C82—41no—
93C22×Q8—22no—
94C22×Q8C2×D412yes—
95C2×Q16—22no—
96C2×Q16—41no—
97C2×Q16—41no—
98C2×Q16—44no—
99C2×Q16—82no—
100C2×Q16—82no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4C2×D412yes—
125C2×C4⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—42no—
130C2×C42C2411yescharacteristic
131#107—41no—
132#107—41no—
133#107—44no—
134#107—82no—
135#107Q1614yescomplemented
136#106—44no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—44no—
142#103—42no—
143#103—44no—
144#103—44no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8#4112yes—
151#95—22no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—42no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—82no—
166C2×Q8—82no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4C22×D411yescharacteristic
174C22×C4C22×D411yescharacteristic
175C22×C4H(5,𝔽2)11yescharacteristic
176Q16—44no—
177Q16—88no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8C2×Q1612yes—
198C2×C8#13512yes—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—162no—
216C4⋊C4—162no—
217C4⋊C4C2×Q1614yes—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—162no—
223C42C2×#4111yescharacteristic
224C42H(5,𝔽2)11yescharacteristic
225C42H(5,𝔽2)11yescharacteristic
226C422−1+411yescharacteristic
227C23D4211yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—42no—
231Q8—42no—
232Q8—84no—
233Q8—162no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4#53411yescharacteristic
255C2×C4#53711yescharacteristic
256C2×C4#53811yescharacteristic
257C2×C4#54711yescharacteristic
258C2×C4#55011yescharacteristic
259C2×C4#56211yescharacteristic
260C2×C4C22×Q1611yescharacteristic
261C2×C4C2×#13511yescharacteristic
262C2×C4#57511yescharacteristic
263C2×C4#57511yescharacteristic
264C2×C4#57811yescharacteristic
265C2×C4#57811yescharacteristic
266C2×C4#57811yescharacteristic
267C2×C4#57811yescharacteristic
268C8—24no—
269C8—22no—
270C8—22no—
271C8—41no—
272C8—41no—
273C8—82no—
274C8—82no—
275C22#268211yescharacteristic
276C22#327711yescharacteristic
277C22#327711yescharacteristic
278C22#328311yescharacteristic
279C22D4×Q1611yescharacteristic
280C22#329111yescharacteristic
281C22#329211yescharacteristic
282C4—21no—
283C4—21no—
284C4—44no—
285C4—82no—
286C4—82no—
287C4—82no—
288C4—162no—
289C4—162no—
290C4#319111yescharacteristic
291C4#320211yescharacteristic
292C4#324911yescharacteristic
293C4#326111yescharacteristic
294C4#334211yescharacteristic
295C4#335211yescharacteristic
296C4#339511yescharacteristic
297C4#340211yescharacteristic
298C2#1993111yescharacteristic
299C2#1995811yescharacteristic
300C2#2000611yescharacteristic
301C2#2031911yescharacteristic
302C2#3119311yescharacteristic
303C2#3121311yescharacteristic
304C2#3122811yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
135. #107Q16semidirect

Maximal subgroups

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

subgroupindex
#262772
#246822
#106012
#104832
#100572
#98252
#96562
#95942
#88832