Finite Group Databasebeta

← #351071

Subgroups — #351071

Order 512.

626 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 170 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#33294C211yesmaximal, characteristic
2#33163C211yesmaximal, characteristic
3#33069C211yesmaximal, characteristic
4#20520C211yesmaximal, characteristic
5#9902C211yesmaximal, characteristic
6#9703C211yesmaximal, characteristic
7#9566C214yesmaximal
8#9226C214yesmaximal
9#8786C211yesmaximal, characteristic
10C2×#462—22no—
11C2×#462C2212yes—
12C2×#446C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C42×Q8C2211yescharacteristic
16#1992—22no—
17#1990C2214yes—
18#1981—22no—
19#1981C2212yes—
20#1947C2212yes—
21#1921—22no—
22#1863—22no—
23#1842C2211yescharacteristic
24#1836—21no—
25#1836—21no—
26#1831C2211yescharacteristic
27#1803C2212yes—
28#1771—22no—
29C8×C4⋊C4C2212yes—
30#1767—21no—
31#1767C2211yescharacteristic
32C4×#97—21no—
33C4×#97C2211yescharacteristic
34C2×#332C2214yes—
35#1296C2214yes—
36#1277C2214yes—
37C22×Q16—22no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×C4×Q8D412yes—
42C2×C4×Q8C2311yescharacteristic
43#462—28no—
44#462—48no—
45#446—24no—
46#446—24no—
47C2×#107—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—22no—
54C2×#103—22no—
55C2×#103—24no—
56C2×#97—22no—
57C2×#97—21no—
58C2×#97D411yescharacteristic
59C2×#97D412yes—
60C2×#97D411yescharacteristic
61C2×C4×C8—22no—
62C2×C4×C8C2312yes—
63#389—22no—
64#389—22no—
65#384—21no—
66#384—21no—
67#384—22no—
68#384C2312yes—
69#382—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4D412yes—
73C4×C4⋊C4C2311yescharacteristic
74C43—21no—
75C43C2311yescharacteristic
76#340—24no—
77#340—24no—
78#340—22no—
79#340—22no—
80#336—22no—
81#336—22no—
82#336—21no—
83#332—28no—
84C22×Q8—22no—
85C22×Q8C2×D411yescharacteristic
86C2×Q16—48no—
87C2×Q16—44no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93C4⋊Q8—24no—
94C4⋊Q8—24no—
95C4⋊Q8—44no—
96C4⋊Q8—44no—
97#125—44no—
98C4×Q8—22no—
99C4×Q8—24no—
100C4×Q8—42no—
101C4×Q8#4114yes—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4C2×D411yescharacteristic
113C2×C4⋊C4C2×D412yes—
114C2×C42—21no—
115C2×C42—22no—
116C2×C42—21no—
117C2×C42—22no—
118C2×C42C2×D412yes—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42#4112yes—
122C2×C42C2411yescharacteristic
123#107—44no—
124C4⋊C8—28no—
125C4⋊C8—22no—
126C4⋊C8—24no—
127C4⋊C8—24no—
128C4⋊C8—44no—
129C4⋊C8#4114yes—
130#103—44no—
131#103—44no—
132#103—48no—
133#97—28no—
134#97—22no—
135#97—42no—
136#97—42no—
137#97Q1614yescomplemented
138C4×C8—22no—
139C4×C8—22no—
140C4×C8—44no—
141#95—22no—
142#95—21no—
143#95—42no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8#12412yes—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4#12112yes—
158C22×C4#12112yes—
159C22×C4C22×D411yescharacteristic
160C22×C4C22×D411yescharacteristic
161C22×C4C2×#4111yescharacteristic
162Q16—88no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—24no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—44no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8C2×Q1614yes—
177C4⋊C4—24no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—82no—
187C4⋊C4—84no—
188C4⋊C4—84no—
189C4⋊C4—84no—
190C4⋊C4#12412yes—
191C4⋊C4AGL(1,ℤ/8)14yes—
192C42—24no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—42no—
197C42—44no—
198C42—42no—
199C42—82no—
200C42C2×Q1612yes—
201C42#13712yes—
202C42C2×#4112yes—
203C42C2×#4112yes—
204C23C2×#12111yescharacteristic
205Q8—22no—
206Q8—22no—
207Q8—42no—
208Q8—82no—
209Q8—82no—
210Q8—84no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—84no—
228C2×C4—82no—
229C2×C4#46214yes—
230C2×C4#52512yes—
231C2×C4C2×#12411yescharacteristic
232C2×C4#52912yes—
233C2×C4#53211yescharacteristic
234C2×C4C22×Q1611yescharacteristic
235C2×C4C2×AGL(1,ℤ/8)12yes—
236C2×C4C2×#13711yescharacteristic
237C2×C4#57512yes—
238C2×C4#57512yes—
239C8—24no—
240C8—42no—
241C8—44no—
242C8—82no—
243C8—82no—
244C22#285011yescharacteristic
245C22C2×#46211yescharacteristic
246C22#304011yescharacteristic
247C22#305512yes—
248C22#306912yes—
249C4—22no—
250C4—24no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—44no—
255C4—162no—
256C4—162no—
257C4#310412yes—
258C4#311612yes—
259C4#313811yescharacteristic
260C4#314211yescharacteristic
261C4#317112yes—
262C2#1917212yes—
263C2#1931912yes—
264C2#3302711yescharacteristic
265C2#3305811yescharacteristic
266C2#3306511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
137. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#332942
#331632
#330692
#205202
#99022
#97032
#95662
#92262
#87862