Finite Group Databasebeta

← #124615

Subgroups — #124615

Order 512.

295 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 80 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#21603C211yesmaximal, characteristic
2#12847C211yesmaximal, characteristic
3#12786C211yesmaximal, characteristic
4#7133C212yesmaximal, complemented
5#7124C212yesmaximal, complemented
6C2×#446C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2396C411yescharacteristic
9#2150—21no—
10C2×#349—21no—
11#2112—21no—
12#1985C2211yescharacteristic
13#1940C2211yescharacteristic
14#1917—21no—
15#1851—21no—
16#1798C411yescharacteristic
17#1771—21no—
18#1770C411yescharacteristic
19#1752—21no—
20#1377—21no—
21#1377C2212yes—
22#1377C2212yes—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4×Q8C2×C411yescharacteristic
26C2×#117C2×C411yescharacteristic
27C2×M32—22no—
28C2×M32—22no—
29#446—42no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C22⋊C8—21no—
35#405—42no—
36C2×#97—21no—
37C2×#97—21no—
38C2×#97D411yescharacteristic
39C2×#97C2311yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#400C811yescharacteristic
43#399C811yescharacteristic
44#398—21no—
45#396—21no—
46#395C811yescharacteristic
47#393C811yescharacteristic
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2×C411yescharacteristic
50C4×C22⋊C4—21no—
51C4⋊C16—42no—
52#349—42no—
53C22⋊C16—42no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59C22×Q8C2×C811yescharacteristic
60C23×C4C22⋊C411yescharacteristic
61C2×M16—21no—
62C2×M16—21no—
63C2×M16—41no—
64C2×M16—41no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4×Q8—41no—
75C4×Q8—41no—
76#117—41no—
77#117—41no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4C2×C811yescharacteristic
83C2×C4⋊C4C2×C811yescharacteristic
84C2×C4⋊C4M1611yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4C2×C811yescharacteristic
88C2×C22⋊C4M1611yescharacteristic
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22×C411yescharacteristic
93M32—42no—
94M32—42no—
95M32—42no—
96M32—42no—
97C2×C16—42no—
98C2×C16—42no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—42no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C22⋊C8—42no—
105#97—41no—
106#97—41no—
107#97—42no—
108#97—42no—
109C4×C8—22no—
110C4×C8—22no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95C2×C811yescharacteristic
116#95C2×C811yescharacteristic
117C24C22⋊C811yescharacteristic
118C2×Q8—41no—
119C2×Q8—81no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22⋊C811yescharacteristic
131C22×C4C22⋊C811yescharacteristic
132C22×C4C22⋊C811yescharacteristic
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C22×C811yescharacteristic
135C22×C4C2×M1611yescharacteristic
136M16—42no—
137M16—42no—
138C2×C8—21no—
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—44no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8#9912yes—
155C2×C8AGL(1,ℤ/8)11yescharacteristic
156C2×C8#13711yescharacteristic
157C2×C8#13712yes—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C22⋊C4—41no—
166C22⋊C4—41no—
167C22⋊C4—81no—
168C22⋊C4—81no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—41no—
174C42—41no—
175C42—41no—
176C42—41no—
177C16—82no—
178C16—82no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23C2×C22⋊C811yescharacteristic
185Q8—82no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4C2×#9911yescharacteristic
214C2×C4#41311yescharacteristic
215C2×C4#41711yescharacteristic
216C2×C4#41911yescharacteristic
217C2×C4#42111yescharacteristic
218C2×C4#42111yescharacteristic
219C8—22no—
220C8—22no—
221C8—21no—
222C8—21no—
223C8—41no—
224C8—81no—
225C8—81no—
226C22—21no—
227C22—21no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—81no—
233C22#145911yescharacteristic
234C22#147711yescharacteristic
235C22#147911yescharacteristic
236C4—41no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—41no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4—81no—
245C4#151111yescharacteristic
246C4#151311yescharacteristic
247C4#153511yescharacteristic
248C4#155911yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2#760011yescharacteristic
253C2#1148411yescharacteristic
254C2#1148611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22#1459
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7133C2semidirect
5. #7124C2semidirect

Maximal subgroups

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

subgroupindex
#216032
#128472
#127862
#71332
#71242