Finite Group Databasebeta

← #345009

Subgroups — #345009

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26283C212yesmaximal
2#24632C212yesmaximal
3#10600C212yesmaximal
4#10478C212yesmaximal
5#10067C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9656C211yesmaximal, characteristic
8#9594C211yesmaximal, characteristic
9#8883C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#462C2212yes—
12C2×C4×Q16C2211yescharacteristic
13#2546C2212yes—
14#2355C2212yes—
15#2101—22no—
16#2085—22no—
17#2056—22no—
18#2016—22no—
19#1990C2212yes—
20#1988C2212yes—
21#1984C2212yes—
22#1981—22no—
23#1956C2212yes—
24#1948C2212yes—
25#1939C2212yes—
26#1928—22no—
27#1921—22no—
28#1920—22no—
29#1845C2211yescharacteristic
30#1803C2212yes—
31C4×#107C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#332C2212yes—
35C2×Q8⋊C8C2212yes—
36C2×C82C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×Q16—22no—
40C22×Q16D411yescharacteristic
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2312yes—
43C2×C4×Q8C2312yes—
44#500—22no—
45#500—41no—
46#462—42no—
47#462—42no—
48C4×Q16—41no—
49C4×Q16—41no—
50C2×#107—41no—
51C2×#107—41no—
52C2×#107D411yescharacteristic
53C2×#107D411yescharacteristic
54C2×#107D411yescharacteristic
55C2×#106—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103D412yes—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#398—42no—
68#391—22no—
69#391—22no—
70#389—22no—
71#389—22no—
72#387—42no—
73#384C2312yes—
74#382—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4C2312yes—
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#332—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—44no—
113C4⋊Q8—82no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—42no—
132C2×C42C2411yescharacteristic
133#107—41no—
134#107—41no—
135#107—44no—
136#107—82no—
137#107Q1614yescomplemented
138#106—44no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—82no—
144#103—42no—
145#103—44no—
146#103—44no—
147#103—82no—
148#103—82no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8—21no—
154C4×C8#4112yes—
155#95—22no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—42no—
160#95—42no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D411yescharacteristic
181C22×C4H(5,𝔽2)11yescharacteristic
182Q16—44no—
183Q16—88no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—22no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8C2×Q1612yes—
204C2×C8#13512yes—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C4⋊C4C2×Q1614yes—
224C42—42no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—162no—
229C42C2×#4111yescharacteristic
230C42H(5,𝔽2)11yescharacteristic
231C42H(5,𝔽2)11yescharacteristic
232C422−1+411yescharacteristic
233C23D4211yescharacteristic
234Q8—42no—
235Q8—42no—
236Q8—42no—
237Q8—42no—
238Q8—82no—
239Q8—82no—
240Q8—82no—
241Q8—82no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4#53411yescharacteristic
263C2×C4#53711yescharacteristic
264C2×C4#53811yescharacteristic
265C2×C4#54711yescharacteristic
266C2×C4#55011yescharacteristic
267C2×C4#56211yescharacteristic
268C2×C4C22×Q1611yescharacteristic
269C2×C4C2×#13511yescharacteristic
270C2×C4#57511yescharacteristic
271C2×C4#57511yescharacteristic
272C2×C4#57611yescharacteristic
273C2×C4#57611yescharacteristic
274C2×C4#57811yescharacteristic
275C2×C4#57811yescharacteristic
276C8—24no—
277C8—22no—
278C8—22no—
279C8—41no—
280C8—41no—
281C8—82no—
282C8—82no—
283C22#268211yescharacteristic
284C22#327311yescharacteristic
285C22#327711yescharacteristic
286C22D4×Q1611yescharacteristic
287C22#328711yescharacteristic
288C22#329111yescharacteristic
289C22#329311yescharacteristic
290C4—21no—
291C4—21no—
292C4—44no—
293C4—82no—
294C4—82no—
295C4—82no—
296C4—162no—
297C4—162no—
298C4#319111yescharacteristic
299C4#319911yescharacteristic
300C4#324911yescharacteristic
301C4#326111yescharacteristic
302C4#334111yescharacteristic
303C4#335511yescharacteristic
304C4#339511yescharacteristic
305C4#340211yescharacteristic
306C2#1992711yescharacteristic
307C2#1996511yescharacteristic
308C2#2001011yescharacteristic
309C2#2030611yescharacteristic
310C2#3119211yescharacteristic
311C2#3121311yescharacteristic
312C2#3123011yescharacteristic
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
137. #107Q16semidirect

Maximal subgroups

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

subgroupindex
#262832
#246322
#106002
#104782
#100672
#98232
#96562
#95942
#88832