Finite Group Databasebeta

← #141127

Subgroups — #141127

Order 512.

502 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 100 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#33176C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic
3C2×#1340C214yesmaximal
4C4×H(3,ℤ/4)C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#470—21no—
7C2×#470C2212yes—
8C2×C4×M16C2211yescharacteristic
9#2857—21no—
10C4×#117—21no—
11#2150—21no—
12C2×#349C2212yes—
13#1971—21no—
14#1971C412yescomplemented
15#1956—21no—
16#1844C411yescharacteristic, complemented
17C2×H(3,ℤ/4)—21no—
18C2×H(3,ℤ/4)C2212yes—
19#1377—21no—
20#1340—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C22×M16D411yescharacteristic
24C22×M16C2311yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C2×#117—22no—
28C22×C42D411yescharacteristic
29C2×M32—22no—
30#494—42no—
31#470—24no—
32#470—24no—
33#470—44no—
34C2×#106—21no—
35C2×#106C2×C412yes—
36C2×#103—22no—
37C2×#103—22no—
38C4×M16—42no—
39C2×#97C2×C411yescharacteristic
40C2×#97D411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#391—22no—
44#389—42no—
45#384—21no—
46#380—42no—
47C4×C4⋊C4—42no—
48C4×C22⋊C4—42no—
49#376—22no—
50C43—22no—
51C4⋊C16—42no—
52#349—24no—
53H(3,ℤ/4)—24no—
54H(3,ℤ/4)—22no—
55H(3,ℤ/4)D414yes—
56C22×Q8—22no—
57C23×C4D812yes—
58C23×C4C2×D411yescharacteristic
59C2×Q16—24no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16C2×D412yes—
71C2×M16#4112yes—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4⋊Q8—42no—
77C4⋊Q8—41no—
78C4⋊Q8—41no—
79C22⋊Q8—44no—
80C22⋊Q8—44no—
81#117—24no—
82#117—24no—
83#117—42no—
84#117—42no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—24no—
96C2×C42—44no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42SD1612yes—
100M32—42no—
101M32—42no—
102C2×C16—42no—
103#106—24no—
104#106—42no—
105#103—44no—
106#103—42no—
107#103—42no—
108#97—41no—
109#97—41no—
110C4×C8—22no—
111C4×C8—22no—
112#95—44no—
113#95—44no—
114C24C2×D811yescharacteristic
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4#10212yes—
132C22×C4#10212yes—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4#12112yes—
135C22×C4C2×SD1611yescharacteristic
136C22×C4AGL(1,ℤ/8)12yes—
137Q16—84no—
138Q16—84no—
139M16—42no—
140M16—42no—
141M16—44no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8#10412yes—
152C2×C8C4×D412yes—
153C2×C8#12411yescharacteristic
154C2×C8#12711yescharacteristic
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—84no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—84no—
163C22⋊C4—44no—
164C22⋊C4—44no—
165C22⋊C4—84no—
166C42—22no—
167C42—22no—
168C42—21no—
169C42—21no—
170C42—28no—
171C42—44no—
172C42—88no—
173C42#9912yes—
174C42#10411yescharacteristic
175C42#10411yescharacteristic
176C16—82no—
177C23—21no—
178C23—21no—
179C23—42no—
180C23C2×#10211yescharacteristic
181C23#45912yes—
182Q8—82no—
183Q8—82no—
184Q8—82no—
185Q8—82no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—48no—
197C2×C4—82no—
198C2×C4—88no—
199C2×C4—84no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4#32712yes—
205C2×C4#32712yes—
206C2×C4#39011yescharacteristic
207C2×C4C2×#9911yescharacteristic
208C2×C4#41011yescharacteristic
209C2×C4#41811yescharacteristic
210C2×C4C2×#10411yescharacteristic
211C2×C4C2×#10411yescharacteristic
212C2×C4#46012yes—
213C8—21no—
214C8—22no—
215C8—21no—
216C8—22no—
217C8—41no—
218C22—22no—
219C22—22no—
220C22—42no—
221C22—42no—
222C22—81no—
223C22C2×#32711yescharacteristic
224C22#150211yescharacteristic
225C22#198211yescharacteristic
226C4—22no—
227C4—22no—
228C4—42no—
229C4—88no—
230C4—162no—
231C4—162no—
232C4#134014yescomplemented
233C4#150511yescharacteristic
234C4#150711yescharacteristic
235C4#196211yescharacteristic
236C4#196411yescharacteristic
237C2—21no—
238C2—21no—
239C2—42no—
240C2#971211yescharacteristic
241C2C2×#134012yes—
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#494C4semidirect
14. #1971C4semidirect
16. #1844C4semidirect
232. C4#1340semidirect

Maximal subgroups

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

subgroupindex
#331762
#127852
C2×#13402
C4×H(3,ℤ/4)2