Finite Group Databasebeta

← #305071

Subgroups — #305071

Order 512.

823 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 823 conjugacy classes of subgroups — too many to draw. Only its 192 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#59924C211yesmaximal, characteristic
2#29218C212yesmaximal
3#28440C211yesmaximal, characteristic
4#10349C214yesmaximal
5#9864C212yesmaximal
6#9066C214yesmaximal
7C4×#333C211yesmaximal, characteristic
8C2×#494C2212yes—
9C2×#462—22no—
10#2868C2211yescharacteristic
11#2811—22no—
12#2567—24no—
13#2567—22no—
14#2551—22no—
15#2551C2212yes—
16#2369—22no—
17#2369C2212yes—
18C4×C4⋊Q8C2212yes—
19C4×C4⋊Q8C2211yescharacteristic
20#2100—24no—
21#2016—22no—
22#1990—22no—
23#1987C2218yes—
24#1956C2212yes—
25#1834—22no—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#333C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C22×Q16—24no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8D412yes—
37C2×C4⋊Q8C2312yes—
38C2×C4⋊Q8C2311yescharacteristic
39C2×C4×Q8—22no—
40C2×C4×Q8D412yes—
41#494—28no—
42#462—48no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8C2314yes—
45C2×#103—24no—
46C2×C4×C8—22no—
47C2×C4×C8C2312yes—
48#395—24no—
49#395—28no—
50#395—24no—
51#391—28no—
52#391—24no—
53#391—22no—
54#391D412yes—
55#389—24no—
56#389—24no—
57#389—24no—
58#389—22no—
59#389—22no—
60#389D414yes—
61#384—24no—
62#384—22no—
63#384—22no—
64#384D412yes—
65#384C2311yescharacteristic
66C4×C4⋊C4—22no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2314yes—
71C43C2311yescharacteristic
72#340—28no—
73#340—28no—
74#336—22no—
75#336—21no—
76#333—24no—
77C22×Q8—24no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8C2×D412yes—
81C2×Q16—48no—
82C2×Q16—48no—
83C2×Q16—84no—
84C22×C8—22no—
85C22×C8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—24no—
88C4⋊Q8—22no—
89C4⋊Q8—24no—
90C4⋊Q8—44no—
91C4⋊Q8—88no—
92C4⋊Q8—84no—
93C4⋊Q8—82no—
94C4⋊Q8Q1618yescomplemented
95C4⋊Q8#4114yes—
96C4×Q8—44no—
97C4×Q8—84no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C2×D414yes—
112C2×C4⋊C4C2×D414yes—
113C2×C4⋊C4C2×D412yes—
114C2×C42—22no—
115C2×C42—24no—
116C2×C42—24no—
117C2×C42C2×D412yes—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D412yes—
121C2×C42C2411yescharacteristic
122C4⋊C8—24no—
123C4⋊C8—24no—
124C4⋊C8—24no—
125#103—48no—
126C4×C8—24no—
127C4×C8—44no—
128#95—28no—
129#95—24no—
130#95—24no—
131#95—24no—
132#95—24no—
133C2×Q8—22no—
134C2×Q8—28no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—82no—
142C2×Q8—82no—
143C2×Q8—88no—
144C2×Q8—84no—
145C2×Q8#12414yes—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—42no—
151C22×C4#12011yescharacteristic
152C22×C4#12012yes—
153C22×C4#12011yescharacteristic
154C22×C4C22×D411yescharacteristic
155C22×C4C22×D412yes—
156Q16—816no—
157C2×C8—24no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—44no—
163C4⋊C4—28no—
164C4⋊C4—24no—
165C4⋊C4—28no—
166C4⋊C4—28no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C4⋊C4—82no—
175C4⋊C4—84no—
176C4⋊C4—88no—
177C4⋊C4—84no—
178C4⋊C4—82no—
179C4⋊C4C2×Q1618yes—
180C42—24no—
181C42—22no—
182C42—41no—
183C42—48no—
184C42—44no—
185C42—84no—
186C42C2×Q1614yes—
187C42#13714yes—
188C42C2×#4112yes—
189C422−1+412yes—
190C23C2×#12011yescharacteristic
191Q8—44no—
192Q8—82no—
193Q8—88no—
194Q8—84no—
195Q8—84no—
196Q8—168no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—28no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#45114yes—
218C2×C4#45114yes—
219C2×C4#45312yes—
220C2×C4C2×#12411yescharacteristic
221C2×C4#53611yescharacteristic
222C2×C4#54814yes—
223C2×C4C22×Q1612yes—
224C2×C4C2×#13712yes—
225C2×C4#57714yes—
226C8—44no—
227C8—84no—
228C22#243611yescharacteristic
229C22C2×#45112yes—
230C22#300912yes—
231C22C2×#45311yescharacteristic
232C22#302111yescharacteristic
233C4—21no—
234C4—21no—
235C4—24no—
236C4—41no—
237C4—44no—
238C4—48no—
239C4—84no—
240C4—82no—
241C4—162no—
242C4#163314yes—
243C4#336314yes—
244C4#336914yes—
245C2C2×#163311yescharacteristic
246C2#1871511yescharacteristic
247C2#1889512yes—
248C2#2591811yescharacteristic
249C2#2593612yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
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
94. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#599242
#292182
#284402
#103492
#98642
#90662
C4×#3332