Finite Group Databasebeta

← #153245

Subgroups — #153245

Order 512.

263 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 68 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#33608C211yesmaximal, characteristic
2#13634C211yesmaximal, characteristic, complemented
3C2×#1420C211yesmaximal, characteristic, complemented
4C2×#1416C211yesmaximal, characteristic, complemented
5C2×#1401C211yesmaximal, characteristic
6#12842C211yesmaximal, characteristic
7#7473C211yesmaximal, characteristic, complemented
8C22×Q32C2211yescharacteristic
9C2×#504C2211yescharacteristic
10#3059—21no—
11C2×M64C2211yescharacteristic
12C22×C32C2211yescharacteristic
13#2219—22no—
14#2216—21no—
15#2216—21no—
16C2×#368C2211yescharacteristic
17#2158C2211yescharacteristic
18C2×#366C2211yescharacteristic
19#2141—21no—
20#1816—21no—
21#1424—21no—
22#1423—21no—
23#1420—22no—
24#1416—22no—
25#1401—22no—
26#1387—21no—
27C22×Q16—21no—
28C2×#131D411yescharacteristic
29#536—41no—
30C2×Q32—21no—
31C2×Q32—21no—
32C2×Q32—21no—
33C2×Q32—21no—
34C2×Q32—21no—
35C2×Q32D412yes—
36#504—21no—
37#504—21no—
38#504—21no—
39#504—21no—
40C2×M32—21no—
41C2×M32D411yescharacteristic
42C22×C16—21no—
43C22×C16C2311yescharacteristic
44#470—42no—
45#467—22no—
46#467—22no—
47C2×#108—21no—
48#427—21no—
49C2×#107—21no—
50#419—41no—
51#417—41no—
52M64—21no—
53M64D412yes—
54C2×C32—21no—
55C2×C32—21no—
56C2×C32—21no—
57C2×C32—21no—
58#368—21no—
59#368—21no—
60#367—21no—
61#366—21no—
62#366—21no—
63#366D411yescharacteristic
64#366D411yescharacteristic
65#358—22no—
66#358—22no—
67#343—41no—
68C2×#41D811yescharacteristic
69C22×Q8—41no—
70C2×Q16—22no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×Q16—41no—
76#131—21no—
77#131—21no—
78#131—21no—
79#131—21no—
80C2×M16—21no—
81C2×M16—41no—
82C2×M16D811yescharacteristic
83C22×C8—21no—
84C22×C8C2×D411yescharacteristic
85C4⋊Q8—81no—
86#124—81no—
87#123—82no—
88C22⋊Q8—42no—
89C22⋊Q8—42no—
90#117—41no—
91C2×C4⋊C4—41no—
92Q32—22no—
93Q32—22no—
94Q32—41no—
95Q32—41no—
96M32—21no—
97M32—21no—
98M32—41no—
99M32—41no—
100C2×C16—21no—
101C2×C16—41no—
102C2×C16—41no—
103C2×C16C2×D411yescharacteristic
104C2×C16C2×D411yescharacteristic
105C2×C16C2×D411yescharacteristic
106C2×C16#4111yescharacteristic
107C2×C16#4111yescharacteristic
108C2×C16#4111yescharacteristic
109#108—41no—
110#108—41no—
111#107—21no—
112#107—21no—
113#107—41no—
114#107—41no—
115#106—41no—
116#103—42no—
117#103—42no—
118#103—81no—
119#102—81no—
120C32—22no—
121C32—41no—
122#41—21no—
123#41—21no—
124#41—21no—
125#41—21no—
126C2×Q8—42no—
127C2×Q8—81no—
128C2×Q8—81no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×Q8—81no—
132C2×Q8D1611yescharacteristic
133C2×D4—21no—
134C2×D4D1611yescharacteristic
135C22×C4—21no—
136C22×C4—81no—
137C22×C4—81no—
138C22×C4C2×D811yescharacteristic
139Q16—42no—
140Q16—42no—
141Q16—81no—
142Q16—81no—
143M16—21no—
144M16—21no—
145M16—41no—
146M16—41no—
147M16—81no—
148M16—81no—
149C2×C8—21no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—81no—
153C2×C8#12011yescharacteristic
154C2×C8#12111yescharacteristic
155C2×C8#12311yescharacteristic
156C2×C8#13511yescharacteristic
157C2×C8AGL(1,ℤ/8)11yescharacteristic
158C2×C8#13711yescharacteristic
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C22⋊C4—82no—
166C22⋊C4—82no—
167C22⋊C4—81no—
168C42—81no—
169C16—41no—
170C16#12111yescharacteristic
171C16#12111yescharacteristic
172C16#12311yescharacteristic
173C16#12411yescharacteristic
174C23—21no—
175C23#50911yescharacteristic
176Q8—21no—
177Q8—21no—
178Q8—82no—
179Q8—82no—
180Q8—161no—
181Q8—161no—
182D4—21no—
183D4—21no—
184D4—41no—
185D4—41no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4—161no—
193C2×C4—161no—
194C2×C4—161no—
195C2×C4—321no—
196C2×C4#44711yescharacteristic
197C2×C4#46611yescharacteristic
198C2×C4#48011yescharacteristic
199C2×C4C2×D1611yescharacteristic
200C2×C4#50811yescharacteristic
201C2×C4#51011yescharacteristic
202C8—41no—
203C8—161no—
204C8#39411yescharacteristic
205C8#45211yescharacteristic
206C8#46811yescharacteristic
207C8#48211yescharacteristic
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22#218411yescharacteristic
214C22#222011yescharacteristic
215C22#223411yescharacteristic
216C4—41no—
217C4—162no—
218C4—161no—
219C4—161no—
220C4—321no—
221C4#203311yescharacteristic
222C4#218911yescharacteristic
223C4#221511yescharacteristic
224C4#223211yescharacteristic
225C2—21no—
226C2—21no—
227C2—81no—
228C2#1338011yescharacteristic
229C2#1367011yescharacteristic
230C2#1367211yescharacteristic
subgroupG/H
center, socleC22#2184
commutator, Frattini, cosocleC22×C16C23
prime radicalC2×#41D8
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13634C2semidirect
3. C2×#1420C2semidirect
4. C2×#1416C2semidirect
7. #7473C2semidirect

Maximal subgroups

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

subgroupindex
#336082
#136342
C2×#14202
C2×#14162
C2×#14012
#128422
#74732