Finite Group Databasebeta

← #242835

Subgroups — #242835

Order 512.

717 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 186 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#21867C212yesmaximal
2#21375C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic, complemented
4#10626C214yesmaximal, complemented
5#10625C214yesmaximal, complemented
6#8040C212yesmaximal, complemented
7#7875C211yesmaximal, characteristic, complemented
8C2×#529C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2212yes—
11C22×C4×C8C2211yescharacteristic
12C2×#384C2212yes—
13#1992—22no—
14#1984C2212yes—
15#1959—22no—
16#1954C2212yes—
17#1874—22no—
18#1872—22no—
19#1869—22no—
20#1834—22no—
21#1834C2214yes—
22#1803C2214yes—
23#1802C2214yes—
24#1750—22no—
25C2×#340—22no—
26#1296C2218yes—
27#1277C2214yes—
28C23×C8D412yes—
29#529—24no—
30#529—24no—
31C2×#126—21no—
32C2×#125D411yescharacteristic
33C2×#124D411yescharacteristic
34C2×#123—21no—
35C2×C22⋊Q8D411yescharacteristic
36C2×#121D411yescharacteristic
37C2×C4×Q8C2311yescharacteristic
38C2×C4×D4—21no—
39C2×C4×D4C2311yescharacteristic
40C2×#117—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C42C2311yescharacteristic
45#432—24no—
46#432—24no—
47#432—24no—
48C2×C4⋊C8C2314yes—
49C2×C4⋊C8C2312yes—
50C2×#103—24no—
51C2×#102—24no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—24no—
55C2×#97—22no—
56C2×#97—21no—
57C2×C4×C8—24no—
58C2×C4×C8D412yes—
59C2×C4×C8C2312yes—
60#384—22no—
61#384—22no—
62#384—22no—
63#384C2314yes—
64#340—28no—
65#340—24no—
66#340—24no—
67#336—24no—
68C22×Q8C2×D411yescharacteristic
69C22×D4—21no—
70C22×D4C2×D411yescharacteristic
71C23×C4—21no—
72C23×C4—42no—
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4#4111yescharacteristic
76C22×C8—24no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8—44no—
80C22×C8C2×D414yes—
81#126—44no—
82#125—24no—
83#124—24no—
84#123—44no—
85C22⋊Q8—24no—
86#121—24no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—42no—
92C4×D4—42no—
93#117—42no—
94#117—42no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—48no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—84no—
107C2×C4⋊C4—84no—
108C2×C4⋊C4C2×D411yescharacteristic
109C2×C4⋊C4C2×D411yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C42—21no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42#4114yes—
119C2×C42#4114yes—
120C2×C42#4111yescharacteristic
121C2×C42C2411yescharacteristic
122C4⋊C8—24no—
123C4⋊C8—24no—
124C4⋊C8—24no—
125#103—84no—
126#102—84no—
127C22⋊C8—48no—
128C22⋊C8—44no—
129C22⋊C8—42no—
130#97—44no—
131#97—42no—
132C4×C8—24no—
133C4×C8—24no—
134C24—21no—
135C24#12111yescharacteristic
136C2×Q8—22no—
137C2×Q8—24no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—84no—
155C22×C4—84no—
156C22×C4—84no—
157C22×C4#12111yescharacteristic
158C22×C4#12111yescharacteristic
159C22×C4#12111yescharacteristic
160C22×C4#12314yes—
161C22×C4#12312yes—
162C22×C4#12412yes—
163C22×C4#12614yes—
164C22×C4C22×D411yescharacteristic
165C22×C4C22×D411yescharacteristic
166C22×C4C2×#4111yescharacteristic
167C2×C8—24no—
168C2×C8—24no—
169C2×C8—24no—
170C2×C8—48no—
171C2×C8—44no—
172C2×C8—42no—
173C2×C8—44no—
174C2×C8—44no—
175C2×C8—44no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—48no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—84no—
186C4⋊C4—84no—
187C4⋊C4—84no—
188C4⋊C4—84no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C42—41no—
195C42—41no—
196C42#12312yes—
197C42#12314yes—
198C42#12412yes—
199C42#12614yes—
200C42C2×#4112yes—
201C42C2×#4112yes—
202C23—22no—
203C23—22no—
204C23—41no—
205C23—42no—
206C23—42no—
207C23—41no—
208C23—41no—
209C23#39712yes—
210C23#47212yes—
211C23#47312yes—
212C23C2×#12111yescharacteristic
213Q8—44no—
214D4—44no—
215D4—84no—
216C2×C4—28no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—24no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4—168no—
239C2×C4#39712yes—
240C2×C4#39712yes—
241C2×C4#39712yes—
242C2×C4C2×#12311yescharacteristic
243C2×C4C2×#12312yes—
244C2×C4C2×#12411yescharacteristic
245C2×C4C2×#12612yes—
246C2×C4#57514yes—
247C2×C4#57714yes—
248C8—44no—
249C8—84no—
250C8—82no—
251C22—22no—
252C22—22no—
253C22—24no—
254C22—24no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—82no—
259C22—82no—
260C22#207814yes—
261C22C2×#39711yescharacteristic
262C22#304012yes—
263C22#305712yes—
264C22C2×#47211yescharacteristic
265C22C2×#47311yescharacteristic
266C4—22no—
267C4—22no—
268C4—22no—
269C4—22no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4—164no—
274C4#309312yes—
275C4#309914yes—
276C4#314212yes—
277C2—24no—
278C2—161no—
279C2#1936212yes—
280C2#1936312yes—
281C2#2363111yescharacteristic
282C2#2364011yescharacteristic
283C2C2×#207811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1834C2semidirect
4. #10626C2semidirect
5. #10625C2semidirect
6. #8040C2semidirect
7. #7875C2semidirect

Maximal subgroups

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

subgroupindex
#218672
#213752
C2×#18342
#106262
#106252
#80402
#78752