Finite Group Databasebeta

← #94198

Subgroups — #94198

Order 512.

293 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 46 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#11511C211yesmaximal, characteristic
2#10892C211yesmaximal, characteristic
3#8718C211yesmaximal, characteristic
4C2×#382C2211yescharacteristic
5#2274C411yescharacteristic
6#1749—21no—
7C2×#343C411yescharacteristic
8#1736—21no—
9C2×#339—21no—
10#1463—21no—
11#1324—21no—
12#1324C412yescomplemented
13#1295—21no—
14#1295C412yescomplemented
15C22×M16—21no—
16C2×#117—21no—
17C22×C4⋊C4C2×C411yescharacteristic
18C22×C4⋊C4D411yescharacteristic
19C22×C42D411yescharacteristic
20#407—41no—
21C2×C22⋊C8—41no—
22C2×#97—22no—
23#382—41no—
24#382C2×C411yescharacteristic
25#382C2×C411yescharacteristic
26#382D411yescharacteristic
27#382Q811yescharacteristic
28C4×C4⋊C4—42no—
29C4×C22⋊C4—41no—
30C2×#95—41no—
31#343—41no—
32#340—42no—
33#339—24no—
34#336—41no—
35#324—22no—
36#324—41no—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4—21no—
40C23×C4C22⋊C411yescharacteristic
41C2×M16—22no—
42C2×M16—41no—
43C2×M16—41no—
44C2×M16—41no—
45C2×M16—41no—
46C2×M16—41no—
47C2×M16—81no—
48C22×C8—41no—
49C22×C8—42no—
50#117—22no—
51#117—22no—
52#117—41no—
53#117—41no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4C4211yescharacteristic
62C2×C4⋊C4C22⋊C411yescharacteristic
63C2×C4⋊C4C4⋊C411yescharacteristic
64C2×C4⋊C4C4⋊C411yescharacteristic
65C2×C22⋊C4—41no—
66C2×C42—22no—
67C2×C42—21no—
68C2×C42—41no—
69C2×C42—41no—
70C2×C42—42no—
71C2×C42C22⋊C411yescharacteristic
72C2×C42C4⋊C411yescharacteristic
73C22⋊C8—44no—
74C22⋊C8—42no—
75C22⋊C8—81no—
76#97—42no—
77#97—42no—
78#95—41no—
79#95—41no—
80#95—41no—
81#95—41no—
82#95—42no—
83C24#9911yescharacteristic
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—41no—
99C22×C4—42no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—81no—
104C22×C4—82no—
105C22×C4—82no—
106C22×C4#9511yescharacteristic
107C22×C4#9911yescharacteristic
108M16—42no—
109M16—42no—
110M16—81no—
111M16—81no—
112M16—162no—
113C2×C8—42no—
114C2×C8—44no—
115C2×C8—82no—
116C2×C8—81no—
117C2×C8—81no—
118C2×C8—82no—
119C2×C8—82no—
120C4⋊C4—24no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—41no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—81no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C42—21no—
131C42—21no—
132C42—21no—
133C42—42no—
134C42—42no—
135C42—42no—
136C42—162no—
137C42#10412yes—
138C23—21no—
139C23—21no—
140C23—22no—
141C23—41no—
142C23#34211yescharacteristic
143C23AΣL(1,𝔽16)11yescharacteristic
144C23#35211yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—44no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—84no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—161no—
178C2×C4—161no—
179C2×C4H(3,ℤ/4)11yescharacteristic
180C2×C4#33911yescharacteristic
181C2×C4#35511yescharacteristic
182C2×C4#35611yescharacteristic
183C8—82no—
184C8—84no—
185C8—162no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—42no—
194C22—42no—
195C22#128411yescharacteristic
196C22#139011yescharacteristic
197C22#139411yescharacteristic
198C22#140412yes—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—42no—
204C4—81no—
205C4—84no—
206C4—81no—
207C4—81no—
208C4—81no—
209C4—81no—
210C2—21no—
211C2—22no—
212C2—21no—
213C2—41no—
214C2#721511yescharacteristic
215C2#748911yescharacteristic
216C2#749511yescharacteristic
subgroupG/H
center, socleC22#1390
commutatorC2×C4⋊C4C42
Frattini, cosocleC2×#382C22
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ C2×#382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1324C4semidirect
14. #1295C4semidirect

Maximal subgroups

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

subgroupindex
#115112
#108922
#87182