Finite Group Databasebeta

← #142182

Subgroups — #142182

Order 512.

365 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 90 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#33208C211yesmaximal, characteristic
2#12788C211yesmaximal, characteristic
3#10818C211yesmaximal, characteristic
4#7263C212yesmaximal
5#7262C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C22×C4×C8C2211yescharacteristic
8#2851—21no—
9C2×#364—21no—
10#2112C2212yes—
11#1991—21no—
12#1942—21no—
13#1941C412yescomplemented
14#1846C411yescharacteristic, complemented
15#1808—21no—
16#1807C2212yes—
17C2×#341C2211yescharacteristic
18#1735C2211yescharacteristic
19C2×#339—21no—
20#1729—21no—
21#1376—21no—
22C22×M16—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#125—21no—
26C2×#117—21no—
27C22×C42D411yescharacteristic
28C2×M32—22no—
29#499—42no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#106C2×C412yes—
33C2×#106C2×C412yes—
34C2×C4×C8—42no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#389—22no—
40#387—42no—
41#385—42no—
42#382—21no—
43#380—22no—
44#364—42no—
45C22⋊C16—22no—
46C22⋊C16—22no—
47#341—21no—
48#341Q812yes—
49#340—22no—
50#340—22no—
51#340—21no—
52#340Q812yes—
53#339—42no—
54#336—42no—
55C23×C4—21no—
56C23×C4C2×D411yescharacteristic
57C2×M16—22no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8SD1612yes—
71C22×C8Q1612yes—
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×Q811yescharacteristic
75C22×C8C2×Q811yescharacteristic
76C22×C8#4111yescharacteristic
77C22×C8#4111yescharacteristic
78#125—42no—
79#125—42no—
80#117—41no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
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—
104C4×C8—22no—
105C4×C8—22no—
106C4×C8—41no—
107C4×C8—41no—
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—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C22⋊Q811yescharacteristic
126C22×C4C22⋊Q811yescharacteristic
127C22×C4C2×SD1611yescharacteristic
128C22×C4C2×Q1611yescharacteristic
129C22×C4AGL(1,ℤ/8)11yescharacteristic
130M16—42no—
131M16—42no—
132M16—81no—
133M16—81no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—42no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8#10312yes—
152C2×C8#10312yes—
153C2×C8C4×Q811yescharacteristic
154C2×C8C4×Q811yescharacteristic
155C2×C8#12411yescharacteristic
156C2×C8C4⋊Q811yescharacteristic
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C22⋊C4—84no—
166C22⋊C4—81no—
167C22⋊C4—81no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—42no—
172C42—81no—
173C42—81no—
174C42#9912yes—
175C16—82no—
176C23—22no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23#41811yescharacteristic
181C23#47611yescharacteristic
182C23#47711yescharacteristic
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—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—48no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4#39111yescharacteristic
210C2×C4C2×#9911yescharacteristic
211C2×C4#41011yescharacteristic
212C2×C4C2×#10311yescharacteristic
213C2×C4#42111yescharacteristic
214C2×C4#42111yescharacteristic
215C2×C4#47611yescharacteristic
216C2×C4#47711yescharacteristic
217C8—21no—
218C8—21no—
219C8—21no—
220C8—21no—
221C8—22no—
222C8—22no—
223C8—82no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—42no—
233C22#151011yescharacteristic
234C22#151411yescharacteristic
235C22#198711yescharacteristic
236C22#223112yes—
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—162no—
242C4—162no—
243C4—162no—
244C4#151811yescharacteristic
245C4#152011yescharacteristic
246C4#199511yescharacteristic
247C4#199611yescharacteristic
248C2—21no—
249C2—21no—
250C2—22no—
251C2—41no—
252C2#991111yescharacteristic
253C2#1327111yescharacteristic
254C2#1327311yescharacteristic
subgroupG/H
center, socleC22#1987
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#499C4semidirect
13. #1941C4semidirect
14. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332082
#127882
#108182
#72632
#72622