Finite Group Databasebeta

← #142184

Subgroups — #142184

Order 512.

349 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 88 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#12791C211yesmaximal, characteristic
3#10813C211yesmaximal, characteristic
4#7255C212yesmaximal
5#7254C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C2×C4×M16C2211yescharacteristic
8#2851—21no—
9C2×#364—21no—
10C2×#349C2212yes—
11#1991—21no—
12#1944—21no—
13#1943C412yescomplemented
14#1846C411yescharacteristic, complemented
15#1809—21no—
16#1809C2212yes—
17C2×#344—21no—
18C2×#339—21no—
19C2×#338C2211yescharacteristic
20C2×H(3,ℤ/4)C2211yescharacteristic
21#1375—21no—
22C22×M16—21no—
23C22×M16D411yescharacteristic
24C22×M16C2311yescharacteristic
25C2×#125—21no—
26C2×#117—21no—
27C22×C42D411yescharacteristic
28C2×M32—22no—
29#499—42no—
30C2×#107—21no—
31C2×#107C2×C412yes—
32C2×#106—21no—
33C2×#106C2×C412yes—
34C4×M16—42no—
35C2×#97C2×C411yescharacteristic
36C2×#97D411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#389—22no—
40#387—42no—
41#385—42no—
42#382—21no—
43#380—22no—
44#364—42no—
45#349—22no—
46#349—22no—
47#344—42no—
48#340—22no—
49#340—22no—
50#339—42no—
51#338—21no—
52#338Q812yes—
53H(3,ℤ/4)—21no—
54H(3,ℤ/4)Q812yes—
55C23×C4—21no—
56C23×C4C2×D411yescharacteristic
57C2×M16—22no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16SD1612yes—
68C2×M16Q1612yes—
69C2×M16C2×Q811yescharacteristic
70C2×M16C2×Q811yescharacteristic
71C2×M16#4111yescharacteristic
72C2×M16#4111yescharacteristic
73C22×C8—41no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78#125—42no—
79#125—42no—
80#117—22no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—42no—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—41no—
94C2×C42—41no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97M32—42no—
98M32—42no—
99C2×C16—42no—
100#107—42no—
101#107—42no—
102#106—42no—
103#106—42no—
104#97—41no—
105#97—41no—
106C4×C8—22no—
107C4×C8—22no—
108#95—42no—
109#95—42no—
110#95—42no—
111C24AGL(1,ℤ/8)11yescharacteristic
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—82no—
122C22×C4—81no—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C22⋊Q811yescharacteristic
125C22×C4C22⋊Q811yescharacteristic
126C22×C4C2×SD1611yescharacteristic
127C22×C4C2×Q1611yescharacteristic
128C22×C4AGL(1,ℤ/8)11yescharacteristic
129M16—42no—
130M16—42no—
131M16—41no—
132M16—41no—
133M16—41no—
134M16—41no—
135M16—42no—
136M16—42no—
137M16—81no—
138M16—81no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8#10312yes—
151C2×C8#10312yes—
152C2×C8C4×Q811yescharacteristic
153C2×C8C4×Q811yescharacteristic
154C2×C8#12411yescharacteristic
155C2×C8C4⋊Q811yescharacteristic
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—81no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C22⋊C4—42no—
165C22⋊C4—81no—
166C22⋊C4—84no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—42no—
171C42—42no—
172C42—81no—
173C42#9912yes—
174C16—82no—
175C23—21no—
176C23—21no—
177C23—42no—
178C23#41811yescharacteristic
179C23#47611yescharacteristic
180C23#47711yescharacteristic
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—84no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—162no—
204C2×C4#39111yescharacteristic
205C2×C4C2×#9911yescharacteristic
206C2×C4#41011yescharacteristic
207C2×C4C2×#10311yescharacteristic
208C2×C4#42111yescharacteristic
209C2×C4#42111yescharacteristic
210C2×C4#47611yescharacteristic
211C2×C4#47711yescharacteristic
212C8—21no—
213C8—21no—
214C8—21no—
215C8—21no—
216C8—22no—
217C8—22no—
218C8—82no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—81no—
226C22#151011yescharacteristic
227C22#151411yescharacteristic
228C22#198711yescharacteristic
229C4—21no—
230C4—21no—
231C4—42no—
232C4—81no—
233C4—162no—
234C4—162no—
235C4—162no—
236C4#151811yescharacteristic
237C4#152011yescharacteristic
238C4#199511yescharacteristic
239C4#199611yescharacteristic
240C2—21no—
241C2—21no—
242C2—41no—
243C2—41no—
244C2#991111yescharacteristic
245C2#1183612yes—
subgroupG/H
center, socleC22#1514
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ 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
6. C2×#499C4semidirect
13. #1943C4semidirect
14. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332112
#127912
#108132
#72552
#72542