Finite Group Databasebeta

← #136059

Subgroups — #136059

Order 512.

376 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 86 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#21952C211yesmaximal, characteristic
2#11447C211yesmaximal, characteristic
3#11212C211yesmaximal, characteristic
4#8862C212yesmaximal
5#7252C212yesmaximal
6C2×#431—21no—
7#2905C2211yescharacteristic
8#2285C2211yescharacteristic
9#1939—22no—
10#1939—22no—
11#1798—21no—
12#1798C2212yes—
13#1740C2211yescharacteristic
14#1733—21no—
15C2×#339—21no—
16#1570—22no—
17#1458C2212yes—
18#1325C412yescomplemented
19#1324C412yescomplemented
20#1301—22no—
21C22×M16—21no—
22C2×#117—21no—
23C2×#117D411yescharacteristic
24C2×#117C2311yescharacteristic
25C22×C42D411yescharacteristic
26#431—22no—
27#431—42no—
28#431—42no—
29#424—42no—
30#423—22no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8D412yes—
36C2×C4⋊C8D412yes—
37C2×C22⋊C8—22no—
38C2×C4×C8—22no—
39#384C2×C411yescharacteristic
40#384Q811yescharacteristic
41#382C2×C411yescharacteristic
42#382Q811yescharacteristic
43#380—22no—
44C4×C4⋊C4—22no—
45#342—22no—
46#339—42no—
47#339—42no—
48#336—22no—
49#336—22no—
50#334—44no—
51#324C2×C414yes—
52#323—22no—
53C23×C4—21no—
54C23×C4C2×D411yescharacteristic
55C2×M16—22no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—41no—
64#117—22no—
65#117—41no—
66#117—41no—
67#117—41no—
68#117—41no—
69#117—41no—
70#117—41no—
71#117—41no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4C4⋊C411yescharacteristic
76C2×C4⋊C4C4⋊C411yescharacteristic
77C2×C4⋊C4C22×C411yescharacteristic
78C2×C4⋊C4C2×Q811yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4#4112yes—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—41no—
87C2×C42C4⋊C411yescharacteristic
88C2×C42C4⋊C411yescharacteristic
89C2×C42SD1612yes—
90C2×C42SD1612yes—
91C2×C42C2×D412yes—
92C4⋊C8—24no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4⋊C8—81no—
100C4⋊C8—81no—
101C22⋊C8—24no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104C4×C8—42no—
105C4×C8—44no—
106#95—22no—
107#95—22no—
108C24#12311yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—81no—
122C22×C4C2×C4⋊C411yescharacteristic
123C22×C4C4×D412yes—
124C22×C4#12011yescharacteristic
125C22×C4C22⋊Q812yes—
126C22×C4C2×SD1611yescharacteristic
127C22×C4C2×SD1611yescharacteristic
128C22×C4AGL(1,ℤ/8)11yescharacteristic
129C22×C4#13711yescharacteristic
130M16—42no—
131M16—42no—
132M16—81no—
133M16—81no—
134C2×C8—44no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—81no—
143C2×C8—81no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—81no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—81no—
155C22⋊C4—81no—
156C42—21no—
157C42—21no—
158C42—22no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42—42no—
163C42—41no—
164C42—41no—
165C42—81no—
166C42#10612yes—
167C42#10612yes—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23#38511yescharacteristic
173C23#48111yescharacteristic
174C23#48311yescharacteristic
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—82no—
202C2×C4—161no—
203C2×C4—161no—
204C2×C4C2×#10611yescharacteristic
205C2×C4#42811yescharacteristic
206C2×C4#44811yescharacteristic
207C2×C4#45011yescharacteristic
208C2×C4#45411yescharacteristic
209C2×C4#45611yescharacteristic
210C2×C4B(4,𝔽2)11yescharacteristic
211C2×C4#45811yescharacteristic
212C8—84no—
213C8—82no—
214C8—82no—
215C8—82no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22#185311yescharacteristic
226C22#185711yescharacteristic
227C22#186711yescharacteristic
228C4—24no—
229C4—21no—
230C4—21no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4—81no—
235C4—81no—
236C4—162no—
237C4#159911yescharacteristic
238C4#160411yescharacteristic
239C4#161111yescharacteristic
240C4#161611yescharacteristic
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2#885711yescharacteristic
246C2#1296811yescharacteristic
247C2#1297511yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1867
prime radicalC24#123
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1325C4semidirect
19. #1324C4semidirect

Maximal subgroups

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

subgroupindex
#219522
#114472
#112122
#88622
#72522