Finite Group Databasebeta

← #142167

Subgroups — #142167

Order 512.

420 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 96 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#33211C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic
3C4×H(3,ℤ/4)C211yesmaximal, characteristic
4#7254C214yesmaximal
5C2×#499C411yescharacteristic, complemented
6C2×C4×M16C2211yescharacteristic
7#2851—21no—
8C4×#117—21no—
9#2150—21no—
10C2×#349C2212yes—
11#1991—21no—
12#1944C412yescomplemented
13#1943—21no—
14#1846C411yescharacteristic, complemented
15#1809—21no—
16#1809C2212yes—
17C2×H(3,ℤ/4)—21no—
18C2×H(3,ℤ/4)C2212yes—
19#1377—21no—
20C22×M16D411yescharacteristic
21C22×M16C2311yescharacteristic
22C2×#125—21no—
23C2×#117—22no—
24C22×C42D411yescharacteristic
25C2×M32—22no—
26#499—42no—
27C2×#107—21no—
28C2×#107C2×C412yes—
29C2×#106—21no—
30C2×#106C2×C412yes—
31C4×M16—42no—
32C2×#97C2×C411yescharacteristic
33C2×#97D411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36#389—22no—
37#387—42no—
38#385—42no—
39#382—21no—
40#380—22no—
41C4×C4⋊C4—42no—
42C4×C22⋊C4—42no—
43#376—22no—
44C43—22no—
45C4⋊C16—42no—
46#349—24no—
47#340—22no—
48#340—22no—
49H(3,ℤ/4)—24no—
50H(3,ℤ/4)—22no—
51H(3,ℤ/4)Q814yes—
52C23×C4D812yes—
53C23×C4C2×D411yescharacteristic
54C2×M16—22no—
55C2×M16—22no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16C2×Q812yes—
60C2×M16#4112yes—
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22⋊C411yescharacteristic
63C22×C8C22×C411yescharacteristic
64C22×C8C2×D411yescharacteristic
65#125—42no—
66#125—41no—
67#125—41no—
68#117—24no—
69#117—24no—
70#117—42no—
71#117—42no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C42—22no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—24no—
83C2×C42—44no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42SD1612yes—
87M32—42no—
88M32—42no—
89C2×C16—42no—
90#107—42no—
91#107—42no—
92#106—42no—
93#106—42no—
94#97—41no—
95#97—41no—
96C4×C8—22no—
97C4×C8—22no—
98#95—44no—
99#95—42no—
100#95—42no—
101#95—42no—
102C24C2×D811yescharacteristic
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—44no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—82no—
113C22×C4#10212yes—
114C22×C4#10212yes—
115C22×C4C2×C22⋊C411yescharacteristic
116C22×C4C22⋊Q812yes—
117C22×C4C2×SD1611yescharacteristic
118C22×C4#13712yes—
119M16—42no—
120M16—42no—
121M16—44no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8#10412yes—
132C2×C8C4×Q812yes—
133C2×C8#12411yescharacteristic
134C2×C8C4⋊Q811yescharacteristic
135C4⋊C4—44no—
136C4⋊C4—44no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—84no—
142C4⋊C4—82no—
143C4⋊C4—82no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—84no—
147C42—22no—
148C42—22no—
149C42—21no—
150C42—21no—
151C42—28no—
152C42—44no—
153C42—88no—
154C42#9912yes—
155C42#10411yescharacteristic
156C42#10411yescharacteristic
157C16—82no—
158C23—21no—
159C23—21no—
160C23—42no—
161C23C2×#10211yescharacteristic
162C23#47412yes—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—48no—
174C2×C4—82no—
175C2×C4—88no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4#32712yes—
183C2×C4#32712yes—
184C2×C4#39111yescharacteristic
185C2×C4C2×#9911yescharacteristic
186C2×C4#41011yescharacteristic
187C2×C4#41911yescharacteristic
188C2×C4C2×#10411yescharacteristic
189C2×C4C2×#10411yescharacteristic
190C2×C4#47512yes—
191C8—21no—
192C8—22no—
193C8—21no—
194C8—22no—
195C8—41no—
196C22—22no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—81no—
201C22C2×#32711yescharacteristic
202C22#150211yescharacteristic
203C22#198811yescharacteristic
204C4—22no—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—88no—
209C4—162no—
210C4—162no—
211C4#150611yescharacteristic
212C4#150811yescharacteristic
213C4#199411yescharacteristic
214C4#199611yescharacteristic
215C2—21no—
216C2—21no—
217C2—42no—
218C2#990911yescharacteristic
219C2#1168512yes—
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×#499C4semidirect
12. #1944C4semidirect
14. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332112
#127852
C4×H(3,ℤ/4)2
#72542