Finite Group Databasebeta

← #345163

Subgroups — #345163

Order 512.

549 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26322C212yesmaximal
2#24818C212yesmaximal
3#10590C212yesmaximal
4#10487C212yesmaximal
5#10077C212yesmaximal
6#9834C211yesmaximal, characteristic
7#9658C211yesmaximal, characteristic
8#9598C211yesmaximal, characteristic
9#8892C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#475C2212yes—
12C2×C4×Q16C2211yescharacteristic
13#2568C2212yes—
14Q8×C4⋊C4C2212yes—
15#2095—22no—
16#2085—22no—
17#2072—22no—
18#2066—22no—
19#1990C2212yes—
20#1984C2212yes—
21#1981C2212yes—
22#1956C2212yes—
23#1947C2212yes—
24#1939C2212yes—
25#1928—22no—
26#1928—22no—
27#1878—22no—
28#1834C2211yescharacteristic
29#1805—22no—
30#1803C2212yes—
31C4×#107C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#332C2212yes—
35C2×Q8⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×Q16D411yescharacteristic
40C2×C4⋊Q8C2312yes—
41C2×C4×Q8—22no—
42C2×C4×Q8C2312yes—
43#500—22no—
44#500—41no—
45#475—42no—
46#475—42no—
47C4×Q16—41no—
48C4×Q16—41no—
49C2×#107—22no—
50C2×#107—41no—
51C2×#107D411yescharacteristic
52C2×#107D412yes—
53C2×#106—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#103—22no—
58C2×#103—22no—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103D412yes—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#400—42no—
68#391—22no—
69#391—22no—
70#391—22no—
71#389—22no—
72#389—42no—
73#384—22no—
74#384C2312yes—
75#382—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4C2312yes—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—41no—
83#340D411yescharacteristic
84#340D411yescharacteristic
85#336—22no—
86#336—22no—
87#336—22no—
88#335—41no—
89#334—41no—
90#332—42no—
91Q8⋊C8—24no—
92#322—41no—
93C22×Q8—22no—
94C22×Q8C2×D412yes—
95C2×Q16—41no—
96C2×Q16—41no—
97C2×Q16Q1614yes—
98C22×C8—21no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4×Q8—22no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8—44no—
114C4×Q8—82no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D412yes—
125C2×C4⋊C4C2×D412yes—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—42no—
129C2×C42C2411yescharacteristic
130#107—42no—
131#107—41no—
132#107—44no—
133#107—82no—
134#106—44no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—82no—
140#103—42no—
141#103—44no—
142#103—44no—
143#103—44no—
144#103—82no—
145#103—82no—
146#103—82no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8#4112yes—
153#95—22no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—42no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8C2×Q1614yes—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4H(5,𝔽2)11yescharacteristic
180Q16—44no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—21no—
189C2×C8—22no—
190C2×C8—21no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8C2×Q1612yes—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
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—42no—
215C4⋊C4—82no—
216C4⋊C4—84no—
217C4⋊C4—162no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—84no—
223C42C2×#4111yescharacteristic
224C42H(5,𝔽2)11yescharacteristic
225C42H(5,𝔽2)11yescharacteristic
226C422−1+411yescharacteristic
227C23D4211yescharacteristic
228Q8—24no—
229Q8—24no—
230Q8—42no—
231Q8—42no—
232Q8—82no—
233Q8—82no—
234Q8—82no—
235Q8—82no—
236C2×C4—22no—
237C2×C4—24no—
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—42no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4#53411yescharacteristic
256C2×C4#53711yescharacteristic
257C2×C4#53811yescharacteristic
258C2×C4#54711yescharacteristic
259C2×C4#55011yescharacteristic
260C2×C4#56211yescharacteristic
261C2×C4C22×Q1611yescharacteristic
262C2×C4#57511yescharacteristic
263C2×C4#57512yes—
264C2×C4#57611yescharacteristic
265C2×C4#57712yes—
266C2×C4#57811yescharacteristic
267C8—22no—
268C8—42no—
269C8—41no—
270C8—41no—
271C8—41no—
272C8—82no—
273C8—82no—
274C22#268211yescharacteristic
275C22#327112yes—
276C22#327311yescharacteristic
277C22D4×Q1611yescharacteristic
278C22#328912yes—
279C4—21no—
280C4—21no—
281C4—44no—
282C4—82no—
283C4—82no—
284C4—82no—
285C4—162no—
286C4—162no—
287C4#318911yescharacteristic
288C4#320311yescharacteristic
289C4#325611yescharacteristic
290C4#326311yescharacteristic
291C4#333811yescharacteristic
292C4#335211yescharacteristic
293C4#340112yes—
294C2#1981812yes—
295C2#2027112yes—
296C2#3119011yescharacteristic
297C2#3122711yescharacteristic
298C2#3123311yescharacteristic
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

Maximal subgroups

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

subgroupindex
#263222
#248182
#105902
#104872
#100772
#98342
#96582
#95982
#88922