Finite Group Databasebeta

← #142183

Subgroups — #142183

Order 512.

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

normal subgroups only

This group has 387 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#33207C211yesmaximal, characteristic
2#12788C211yesmaximal, characteristic
3#10818C211yesmaximal, characteristic
4#7263C212yesmaximal
5#7262C212yesmaximal
6C2×#498C411yescharacteristic, complemented
7C22×C4×C8C2211yescharacteristic
8#2857—21no—
9C2×#364—21no—
10#2112C2212yes—
11#1990—21no—
12#1941—21no—
13#1941C412yescomplemented
14#1844C411yescharacteristic, complemented
15#1807—21no—
16#1807C2212yes—
17C2×#341C2211yescharacteristic
18#1735C2211yescharacteristic
19C2×#339—21no—
20#1729—21no—
21#1376—21no—
22C22×M16—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—41no—
27C2×#117—21no—
28C22×C42D411yescharacteristic
29C2×M32—22no—
30#498—42no—
31C2×#106—21no—
32C2×#106—21no—
33C2×#106C2×C412yes—
34C2×#106C2×C412yes—
35C2×C4×C8—42no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#391—41no—
41#389—22no—
42#389—41no—
43#384—21no—
44#380—22no—
45#380—41no—
46#364—42no—
47C22⋊C16—22no—
48C22⋊C16—22no—
49#341—21no—
50#341Q812yescomplemented
51#340—22no—
52#340—22no—
53#340—21no—
54#340Q812yescomplemented
55#339—42no—
56#336—42no—
57C22×Q8—41no—
58C23×C4—21no—
59C23×C4C2×D411yescharacteristic
60C2×M16—22no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8SD1612yes—
74C22×C8Q1612yes—
75C22×C8C22×C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8C2×Q811yescharacteristic
78C22×C8C2×Q811yescharacteristic
79C22×C8#4111yescharacteristic
80C22×C8#4111yescharacteristic
81C4⋊Q8—42no—
82C4⋊Q8—42no—
83C22⋊Q8—84no—
84#117—41no—
85#117—41no—
86#117—41no—
87#117—41no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—42no—
97C2×C42—22no—
98C2×C42—21no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42C22⋊C411yescharacteristic
102C2×C42C22⋊C411yescharacteristic
103M32—42no—
104M32—42no—
105C2×C16—42no—
106#106—42no—
107#106—42no—
108#106—42no—
109#106—42no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114#95—42no—
115#95—42no—
116C24AGL(1,ℤ/8)11yescharacteristic
117C2×Q8—82no—
118C2×Q8—82no—
119C2×Q8—82no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C22⋊Q811yescharacteristic
134C22×C4C22⋊Q811yescharacteristic
135C22×C4C2×SD1611yescharacteristic
136C22×C4C2×Q1611yescharacteristic
137C22×C4AGL(1,ℤ/8)11yescharacteristic
138M16—42no—
139M16—42no—
140M16—81no—
141M16—81no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—81no—
158C2×C8—81no—
159C2×C8#10312yes—
160C2×C8#10312yes—
161C2×C8C4×Q811yescharacteristic
162C2×C8C4×Q811yescharacteristic
163C2×C8#12411yescharacteristic
164C2×C8C4⋊Q811yescharacteristic
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C4⋊C4—82no—
175C22⋊C4—81no—
176C22⋊C4—81no—
177C22⋊C4—84no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—42no—
182C42—81no—
183C42—81no—
184C42#9912yes—
185C16—82no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23—41no—
190C23#41811yescharacteristic
191C23#47611yescharacteristic
192C23#47711yescharacteristic
193Q8—164no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—48no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4#39111yescharacteristic
221C2×C4C2×#9911yescharacteristic
222C2×C4#41011yescharacteristic
223C2×C4C2×#10311yescharacteristic
224C2×C4#42111yescharacteristic
225C2×C4#42111yescharacteristic
226C2×C4#47611yescharacteristic
227C2×C4#47711yescharacteristic
228C8—21no—
229C8—21no—
230C8—21no—
231C8—21no—
232C8—22no—
233C8—22no—
234C8—82no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—42no—
243C22—42no—
244C22#151011yescharacteristic
245C22#151411yescharacteristic
246C22#198711yescharacteristic
247C22#223012yes—
248C4—24no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—162no—
253C4—162no—
254C4—162no—
255C4#151811yescharacteristic
256C4#152011yescharacteristic
257C4#199511yescharacteristic
258C4#199611yescharacteristic
259C2—21no—
260C2—21no—
261C2—22no—
262C2—41no—
263C2#991111yescharacteristic
264C2#1327111yescharacteristic
265C2#1327211yescharacteristic
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×#498C4semidirect
13. #1941C4semidirect
14. #1844C4semidirect
50. #341Q8semidirect
54. #340Q8semidirect

Maximal subgroups

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

subgroupindex
#332072
#127882
#108182
#72632
#72622