Finite Group Databasebeta

← #343565

Subgroups — #343565

Order 512.

624 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 142 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#26044C212yesmaximal, complemented
2#24965C212yesmaximal
3#10656C212yesmaximal, complemented
4#10619C212yesmaximal, complemented
5#10573C211yesmaximal, characteristic, complemented
6#10430C211yesmaximal, characteristic
7#9560C212yesmaximal, complemented
8#8886C212yesmaximal, complemented
9#8819C211yesmaximal, characteristic, complemented
10C2×#476—22no—
11#2551C2211yescharacteristic
12#2537C2211yescharacteristic
13D4×C4⋊C4C2212yes—
14#2088—22no—
15#2075—22no—
16#2058—22no—
17#2033—22no—
18#1954C2212yes—
19#1948C2212yes—
20#1947C2212yes—
21#1939C2212yes—
22#1874—22no—
23#1864—22no—
24#1845C2211yescharacteristic
25#1844C2212yes—
26#1844C2211yescharacteristic
27#1834C2212yes—
28#1823—22no—
29#1802C2212yes—
30#1802C2212yes—
31C4×#102C2212yes—
32C2×D4⋊C8C2212yes—
33C2×#322C2211yescharacteristic
34#1296C2211yescharacteristic
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1277C2212yes—
39C2×C4⋊Q8—22no—
40C2×#121—22no—
41C2×C4×D4—22no—
42C2×C4×D4C2312yes—
43C22×C4⋊C4—22no—
44#476—48no—
45C2×#107—22no—
46C2×#107—41no—
47C2×#106—22no—
48C2×#106—22no—
49C2×#106—22no—
50C2×#106—41no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#102—22no—
55C2×#102—22no—
56C2×#102D412yes—
57C2×#102D412yes—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C4×C8C2312yes—
61C2×C4×C8C2311yescharacteristic
62#395—41no—
63#394—41no—
64#389—22no—
65#389—22no—
66#385—42no—
67#384C2311yescharacteristic
68#384C2312yes—
69#384C2311yescharacteristic
70#380—22no—
71#380—22no—
72C4×C4⋊C4C2312yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340D412yes—
80#340D412yes—
81#336—22no—
82D4⋊C8—24no—
83#322—22no—
84C22×Q8—41no—
85C22×D4—41no—
86C22×D4C2×D412yes—
87C23×C4—22no—
88C23×C4—42no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8C2×D412yes—
93C22×C8C2×D412yes—
94C4⋊Q8—44no—
95C4⋊Q8—82no—
96#121—44no—
97#121—82no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—44no—
103C4×D4—82no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—82no—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D412yes—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C42—22no—
122C2×C42C2411yescharacteristic
123#107—82no—
124#107—82no—
125#106—44no—
126#106—44no—
127#106—82no—
128#106—82no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—44no—
134#102—24no—
135#102—24no—
136#102—44no—
137#102—82no—
138C22⋊C8—82no—
139C22⋊C8—82no—
140C4×C8—22no—
141C4×C8—21no—
142C4×C8—22no—
143C4×C8#4112yes—
144#95—22no—
145#95—41no—
146#95—42no—
147#95—41no—
148C24—22no—
149C2×Q8—84no—
150C2×Q8—161no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—84no—
157C2×D4—161no—
158C2×D4AGL(1,ℤ/8)14yes—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—84no—
172C22×C4—84no—
173C22×C4—162no—
174C22×C4C22×D412yes—
175C22×C4H(5,𝔽2)11yescharacteristic
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—84no—
205C4⋊C4—82no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—84no—
209C4⋊C4#13714yes—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—84no—
213C22⋊C4—84no—
214C42—42no—
215C42—42no—
216C42C2×#4111yescharacteristic
217C422−1+411yescharacteristic
218C422−1+412yes—
219C23—24no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23D4211yescharacteristic
227Q8—164no—
228D4—24no—
229D4—24no—
230D4—42no—
231D4—42no—
232D4—164no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—168no—
256C2×C4#53612yes—
257C2×C4#53911yescharacteristic
258C2×C4#54712yes—
259C2×C4#56311yescharacteristic
260C2×C4C2×AGL(1,ℤ/8)12yes—
261C2×C4C2×#13712yes—
262C2×C4#57612yes—
263C2×C4#57712yes—
264C8—24no—
265C8—42no—
266C8—42no—
267C8—82no—
268C8—82no—
269C22—44no—
270C22—44no—
271C22—44no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22#268011yescharacteristic
277C22#326412yes—
278C22#326612yes—
279C22#329111yescharacteristic
280C22#329111yescharacteristic
281C4—21no—
282C4—21no—
283C4—44no—
284C4—82no—
285C4—82no—
286C4—161no—
287C4—161no—
288C4—162no—
289C4#323812yes—
290C4#324112yes—
291C4#334512yes—
292C4#335712yes—
293C2—84no—
294C2#2002311yescharacteristic
295C2#2003812yes—
296C2#2010711yescharacteristic
297C2#3110412yes—
298C2#3111211yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#2537C22
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
1. #26044C2semidirect
3. #10656C2semidirect
4. #10619C2semidirect
5. #10573C2semidirect
7. #9560C2semidirect
8. #8886C2semidirect
9. #8819C2semidirect

Maximal subgroups

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

subgroupindex
#260442
#249652
#106562
#106192
#105732
#104302
#95602
#88862
#88192