Finite Group Databasebeta

← #150313

Subgroups — #150313

Order 512.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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#23560C211yesmaximal, characteristic
2#11447C211yesmaximal, characteristic
3#8031C211yesmaximal, characteristic
4#7442C214yesmaximal, complemented
5#3458C411yescharacteristic
6C2×#432—21no—
7#2911—21no—
8#2285C2211yescharacteristic
9#2061—21no—
10#2061C2212yes—
11#2060—21no—
12#2060C412yes—
13#1742C411yescharacteristic
14C2×#339—21no—
15#1458—21no—
16#1325—21no—
17#1324C2212yes—
18#1295—21no—
19#1295C2212yes—
20C22×M16—21no—
21#544—21no—
22#544—41no—
23#544—41no—
24#544C2×C412yes—
25#543—41no—
26#533—41no—
27#533—42no—
28C2×#117—21no—
29C2×#117C2×C411yescharacteristic
30C2×#117D411yescharacteristic
31C22×C42D411yescharacteristic
32#432—24no—
33#432—42no—
34#431—41no—
35#428—22no—
36#428—22no—
37#428—41no—
38#428—41no—
39#427—42no—
40C2×C4⋊C8—41no—
41C2×C22⋊C8—41no—
42C2×#97—22no—
43#384C2×C411yescharacteristic
44#384D411yescharacteristic
45#382D411yescharacteristic
46#382C2311yescharacteristic
47#380—41no—
48C4×C4⋊C4—41no—
49#343—21no—
50#343C2×C412yes—
51#339—22no—
52#339—22no—
53#324—22no—
54#323—41no—
55C23×C4—21no—
56C23×C4C22⋊C411yescharacteristic
57C2×M16—24no—
58C2×M16—22no—
59C2×M16—41no—
60C2×M16—42no—
61C22×C8—41no—
62C22×C8—42no—
63C4⋊Q8—42no—
64C4⋊Q8—42no—
65C4⋊Q8—41no—
66C4⋊Q8—41no—
67#125—42no—
68#125—42no—
69#125—42no—
70C22⋊Q8—42no—
71C22⋊Q8—44no—
72C22⋊Q8—44no—
73C22⋊Q8—42no—
74C4×Q8—42no—
75C4×Q8—42no—
76#117—22no—
77#117—22no—
78#117—41no—
79#117—41no—
80#117—41no—
81#117—41no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4C22⋊C411yescharacteristic
89C2×C4⋊C4C22⋊C411yescharacteristic
90C2×C4⋊C4C22×C411yescharacteristic
91C2×C4⋊C4C2×D411yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—41no—
95C2×C42—21no—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—41no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42SD1612yes—
102C2×C42Q1612yes—
103C2×C42C2×D411yescharacteristic
104#107—42no—
105#107—42no—
106#106—42no—
107#106—42no—
108C4⋊C8—44no—
109C4⋊C8—81no—
110C22⋊C8—44no—
111C22⋊C8—42no—
112#97—24no—
113#97—42no—
114C4×C8—81no—
115#95—41no—
116#95—41no—
117C24#9911yescharacteristic
118C2×Q8—42no—
119C2×Q8—42no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4#9911yescharacteristic
136C22×C4#10312yes—
137C22×C4#10412yes—
138C22×C4C2×C22⋊C411yescharacteristic
139M16—24no—
140M16—44no—
141M16—42no—
142C2×C8—42no—
143C2×C8—44no—
144C2×C8—44no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—82no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—81no—
166C42—22no—
167C42—22no—
168C42—21no—
169C42—21no—
170C42—42no—
171C42—41no—
172C42—41no—
173C42—41no—
174C42—41no—
175C42—42no—
176C42#10312yes—
177C42C2×SD1611yescharacteristic
178C42C2×Q1611yescharacteristic
179C23—21no—
180C23—21no—
181C23—41no—
182C23—41no—
183C23#32812yes—
184C23C2×#9911yescharacteristic
185Q8—162no—
186Q8—162no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—161no—
213C2×C4—161no—
214C2×C4#32812yes—
215C2×C4AΣL(1,𝔽16)11yescharacteristic
216C2×C4AΣL(1,𝔽16)11yescharacteristic
217C2×C4#35612yes—
218C2×C4C2×#10311yescharacteristic
219C2×C4C2×#10411yescharacteristic
220C8—44no—
221C8—84no—
222C8—162no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22—81no—
230C22C2×#32811yescharacteristic
231C22C2×AΣL(1,𝔽16)11yescharacteristic
232C22C2×#35611yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—42no—
237C4—41no—
238C4—81no—
239C4—81no—
240C4—82no—
241C4—81no—
242C4—81no—
243C4—82no—
244C4#135412yes—
245C4#135412yes—
246C2—21no—
247C2—21no—
248C2—42no—
249C2C2×#135411yescharacteristic
250C2#1356611yescharacteristic
251C2#1357011yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7442C2semidirect

Maximal subgroups

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

subgroupindex
#235602
#114472
#80312
#74422