Finite Group Databasebeta

← #142015

Subgroups — #142015

Order 512.

401 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 102 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
2C4×#349C211yesmaximal, characteristic
3#10837C211yesmaximal, characteristic
4#7254C214yesmaximal
5C2×#499C411yescharacteristic, complemented
6C2×C4×M16C2211yescharacteristic
7#2851—21no—
8#2285—21no—
9C2×#349C2212yes—
10C4×M32—21no—
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—41no—
27#499—41no—
28C2×#107—21no—
29C2×#107C2×C412yes—
30C2×#106—21no—
31C2×#106C2×C412yes—
32C4×M16—41no—
33C4×M16—41no—
34C2×#97C2×C411yescharacteristic
35C2×#97D411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38#389—22no—
39#387—42no—
40#385—42no—
41#384—41no—
42#382—21no—
43#382—41no—
44#380—22no—
45#380—42no—
46C4×C4⋊C4—24no—
47#349—22no—
48#349Q814yes—
49#346—41no—
50C4×C16—41no—
51#340—22no—
52#340—22no—
53H(3,ℤ/4)—24no—
54H(3,ℤ/4)—42no—
55C23×C4D812yes—
56C23×C4C2×D411yescharacteristic
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16SD1612yes—
62C2×M16Q1612yes—
63C2×M16C2×Q812yes—
64C2×M16#4112yes—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69#125—41no—
70#125—41no—
71#125—42no—
72#117—42no—
73#117—42no—
74#117—42no—
75#117—42no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—22no—
87C2×C42—24no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42SD1612yes—
91M32—24no—
92M32—42no—
93C2×C16—42no—
94#107—42no—
95#107—42no—
96#106—42no—
97#106—42no—
98#97—41no—
99#97—41no—
100C4×C8—21no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104#95—44no—
105#95—42no—
106#95—42no—
107#95—42no—
108C24C2×D811yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—82no—
119C22×C4#10212yes—
120C22×C4#10212yes—
121C22×C4C2×C22⋊C411yescharacteristic
122C22×C4#12411yescharacteristic
123C22×C4C4⋊Q811yescharacteristic
124C22×C4C2×SD1611yescharacteristic
125C22×C4C2×SD1611yescharacteristic
126C22×C4C2×Q1611yescharacteristic
127M16—42no—
128M16—42no—
129M16—42no—
130M16—42no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8#10312yes—
140C2×C8#10312yes—
141C2×C8C4×Q812yes—
142C2×C8C22⋊Q812yes—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—82no—
148C4⋊C4—82no—
149C4⋊C4—88no—
150C4⋊C4—84no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C22⋊C4—82no—
154C22⋊C4—84no—
155C22⋊C4—82no—
156C42—22no—
157C42—22no—
158C42—21no—
159C42—21no—
160C42—44no—
161C42—44no—
162C42#9911yescharacteristic
163C42#9911yescharacteristic
164C42#10412yes—
165C16—44no—
166C23—21no—
167C23—21no—
168C23—42no—
169C23C2×#10211yescharacteristic
170C23#48611yescharacteristic
171C23#50011yescharacteristic
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—84no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—88no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#32712yes—
196C2×C4#32712yes—
197C2×C4#39111yescharacteristic
198C2×C4C2×#9911yescharacteristic
199C2×C4#41011yescharacteristic
200C2×C4C2×#10311yescharacteristic
201C2×C4C2×#10411yescharacteristic
202C2×C4#42111yescharacteristic
203C2×C4#48711yescharacteristic
204C2×C4#49811yescharacteristic
205C8—22no—
206C8—22no—
207C8—22no—
208C8—22no—
209C22—22no—
210C22—21no—
211C22—21no—
212C22—41no—
213C22—41no—
214C22—42no—
215C22—81no—
216C22C2×#32711yescharacteristic
217C22#151711yescharacteristic
218C22#199011yescharacteristic
219C4—21no—
220C4—21no—
221C4—22no—
222C4—22no—
223C4—42no—
224C4—162no—
225C4—164no—
226C4—162no—
227C4#150311yescharacteristic
228C4#152011yescharacteristic
229C4#198611yescharacteristic
230C4#199611yescharacteristic
231C2—21no—
232C2—21no—
233C2—41no—
234C2—41no—
235C2#988511yescharacteristic
236C2#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
C4×#3492
#108372
#72542