Finite Group Databasebeta

← #508559

Subgroups — #508559

Order 512.

417 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 132 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#33215C211yesmaximal, characteristic
2#20829C211yesmaximal, characteristic
3#13541C212yesmaximal, complemented
4#13510C212yesmaximal, complemented
5#13501C214yesmaximal
6#13279C212yesmaximal, complemented
7#13184C212yesmaximal, complemented
8#12786C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10#2921—21no—
11C2×#405C2211yescharacteristic
12#2628—21no—
13C2×#365C2214yes—
14#2150C2212yes—
15C2×#349C2212yes—
16#1993—21no—
17#1944C2212yes—
18#1942—21no—
19#1849C2211yescharacteristic
20#1849C2211yescharacteristic
21#1846C2211yescharacteristic
22#1837—21no—
23#1811C2212yes—
24#1810—21no—
25#1809C2212yes—
26#1808—21no—
27#1778—21no—
28C4×#108C2212yes—
29#1387C2214yes—
30#1387C2214yes—
31#1383C2214yes—
32#1377C2212yes—
33C22×M16C2311yescharacteristic
34C23×C8D411yescharacteristic
35C2×C4⋊Q8—21no—
36C2×#117D411yescharacteristic
37C2×M32C2314yes—
38#500—24no—
39#433—44no—
40C2×#108—21no—
41C2×#108—21no—
42C2×#108C2312yes—
43C2×#108C2312yes—
44C2×#107—21no—
45C2×#107—21no—
46C2×#107C2312yes—
47C2×#106C2312yes—
48#423—21no—
49#423—21no—
50#407—22no—
51#405—24no—
52C4×M16—22no—
53C2×#97D411yescharacteristic
54C2×#97C2311yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#400—41no—
58#399—41no—
59#395—41no—
60#393—41no—
61#389—22no—
62#382—21no—
63#380—22no—
64#365—44no—
65C4⋊C16—24no—
66#349—42no—
67#341—22no—
68#341—22no—
69#340—22no—
70#340—22no—
71C22×Q8—41no—
72C23×C4C2×D411yescharacteristic
73C2×M16—22no—
74C2×M16—21no—
75C2×M16—22no—
76C2×M16—21no—
77C2×M16—22no—
78C2×M16—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—41no—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2411yescharacteristic
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88#117—22no—
89#117—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4SD1611yescharacteristic
95C2×C4⋊C4SD1611yescharacteristic
96C2×C22⋊C4—42no—
97C2×C22⋊C4SD1611yescharacteristic
98C2×C22⋊C4SD1611yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42C2×D411yescharacteristic
101M32—24no—
102M32—24no—
103C2×C16—24no—
104#108—22no—
105#108—22no—
106#108—24no—
107#108—42no—
108#108—42no—
109#107—24no—
110#107—42no—
111#107—42no—
112#106—22no—
113#106—22no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C22⋊C8—44no—
117#97—22no—
118#97—22no—
119#97—42no—
120C4×C8—22no—
121C4×C8—42no—
122C4×C8#4114yes—
123#95—42no—
124#95—41no—
125#95—41no—
126#95—42no—
127C24C2×SD1611yescharacteristic
128C2×Q8—82no—
129C2×Q8—162no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4C2×SD1611yescharacteristic
139C22×C4C2×SD1611yescharacteristic
140C22×C4C2×SD1611yescharacteristic
141C22×C4C2×SD1611yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143C22×C4C22×D411yescharacteristic
144M16—42no—
145M16—42no—
146M16—42no—
147M16—42no—
148M16—42no—
149M16—42no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—44no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8#12312yes—
161C2×C8#13712yes—
162C2×C8C2×#4112yes—
163C2×C8H(5,𝔽2)11yescharacteristic
164C2×C82−1+411yescharacteristic
165C2×C82−1+412yes—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—82no—
173C4⋊C4—162no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—162no—
177C42—21no—
178C42—21no—
179C42#12312yes—
180C42AGL(1,ℤ/8)12yes—
181C16—44no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—41no—
186C23—41no—
187C23C22×SD1611yescharacteristic
188Q8—164no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
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—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4#48112yes—
209C2×C4#48112yes—
210C2×C4C2×#12311yescharacteristic
211C2×C4#53611yescharacteristic
212C2×C4#53711yescharacteristic
213C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C4C2×#13711yescharacteristic
215C2×C4#57511yescharacteristic
216C8—22no—
217C8—22no—
218C8—41no—
219C8—82no—
220C8—82no—
221C8#54112yes—
222C8#54212yes—
223C22—21no—
224C22—21no—
225C22—42no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—81no—
230C22C2×#48111yescharacteristic
231C22#322911yescharacteristic
232C22#324611yescharacteristic
233C4—22no—
234C4—22no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—162no—
239C4—162no—
240C4#262511yescharacteristic
241C4#309511yescharacteristic
242C4#323511yescharacteristic
243C4#325411yescharacteristic
244C2—21no—
245C2—21no—
246C2—81no—
247C2#2993711yescharacteristic
248C2#3365711yescharacteristic
249C2#3365811yescharacteristic
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13541C2semidirect
4. #13510C2semidirect
6. #13279C2semidirect
7. #13184C2semidirect

Maximal subgroups

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

subgroupindex
#332152
#208292
#135412
#135102
#135012
#132792
#131842
#127862