Finite Group Databasebeta

← #312155

Subgroups — #312155

Order 512.

562 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 158 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#26446C212yesmaximal
2#23158C212yesmaximal
3#10543C212yesmaximal
4#10349C211yesmaximal, characteristic
5#10309C211yesmaximal, characteristic
6#9568C212yesmaximal
7#9536C212yesmaximal
8#9066C212yesmaximal
9#8929C211yesmaximal, characteristic
10C2×#500C2212yes—
11C2×#494C2211yescharacteristic
12#2551C2211yescharacteristic
13#2369C2212yes—
14#2369C2211yescharacteristic
15#2087—22no—
16#2057—22no—
17#1990C2212yes—
18#1987—22no—
19#1987C2212yes—
20#1987C2212yes—
21#1987C2212yes—
22#1956C2211yescharacteristic
23#1947C2212yes—
24#1942—22no—
25#1939C2212yes—
26#1921—22no—
27#1921—22no—
28#1921—22no—
29C4×#107—22no—
30C4×#107C2212yes—
31C8×C4⋊C4C2212yes—
32C2×#333C2211yescharacteristic
33C2×#333C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C22×Q16—41no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2311yescharacteristic
43#500—24no—
44#500—24no—
45#494—24no—
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—22no—
49C2×#107D412yes—
50C2×#107D412yes—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—41no—
55C2×C4×C8—22no—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2312yes—
58#395—41no—
59#391—41no—
60#391—42no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#384C2311yescharacteristic
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4C2312yes—
70C4×C4⋊C4C2312yes—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340D412yes—
81#340D412yes—
82#336—22no—
83#336—22no—
84#333—24no—
85#333—22no—
86C22×Q8—22no—
87C22×Q8—21no—
88C22×Q8—41no—
89C2×Q16—44no—
90C2×Q16—82no—
91C2×Q16—82no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8C2×D412yes—
96C22×C8C2×D412yes—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—21no—
102C4⋊Q8—44no—
103C4⋊Q8—82no—
104C4⋊Q8#4114yes—
105C4⋊Q8#4112yes—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D412yes—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—41no—
121C2×C42—42no—
122C2×C42C2411yescharacteristic
123#107—28no—
124#107—28no—
125#107—42no—
126#107—82no—
127#107Q1618yescomplemented
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—82no—
133#103—82no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—44no—
138#95—22no—
139#95—22no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—41no—
144C2×Q8—22no—
145C2×Q8—24no—
146C2×Q8—42no—
147C2×Q8—41no—
148C2×Q8—81no—
149C2×Q8—82no—
150C2×Q8—82no—
151C2×Q8—81no—
152C2×Q8—84no—
153C2×Q8—161no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4C22×D412yes—
162C22×C4H(5,𝔽2)11yescharacteristic
163Q16—88no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8C2×Q1614yes—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4—84no—
195C4⋊C4—161no—
196C4⋊C4—162no—
197C4⋊C4C2×Q1614yes—
198C4⋊C4C2×Q1614yes—
199C42—42no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—161no—
204C42—162no—
205C42C2×#4111yescharacteristic
206C42C2×#4112yes—
207C422−1+411yescharacteristic
208C23D4211yescharacteristic
209Q8—84no—
210Q8—82no—
211Q8—161no—
212Q8—162no—
213Q8—164no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4#52512yes—
237C2×C4#53911yescharacteristic
238C2×C4#54812yes—
239C2×C4#55111yescharacteristic
240C2×C4C22×Q1612yes—
241C2×C4#57512yes—
242C2×C4#57712yes—
243C2×C4#57812yes—
244C8—24no—
245C8—42no—
246C8—42no—
247C8—82no—
248C8—82no—
249C22#246311yescharacteristic
250C22#326712yes—
251C22D4×Q1612yes—
252C22#328811yescharacteristic
253C22#329211yescharacteristic
254C4—21no—
255C4—21no—
256C4—44no—
257C4—44no—
258C4—84no—
259C4—82no—
260C4—82no—
261C4—82no—
262C4—161no—
263C4#310412yes—
264C4#312112yes—
265C4#336312yes—
266C4#337712yes—
267C2#1981412yes—
268C2Q16211yescharacteristic
269C2#2013111yescharacteristic
270C2#2689412yes—
271C2#2692011yescharacteristic
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
127. #107Q16semidirect

Maximal subgroups

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

subgroupindex
#264462
#231582
#105432
#103492
#103092
#95682
#95362
#90662
#89292