Finite Group Databasebeta

← #153258

Subgroups — #153258

Order 512.

260 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 66 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#33617C211yesmaximal, characteristic
2#13638C211yesmaximal, characteristic, complemented
3#13622C211yesmaximal, characteristic, complemented
4#13615C211yesmaximal, characteristic, complemented
5C2×#1401C211yesmaximal, characteristic, complemented
6#12839C211yesmaximal, characteristic, complemented
7#7473C211yesmaximal, characteristic, complemented
8C2×#508C2211yescharacteristic
9C2×#504C2211yescharacteristic
10#3067—21no—
11C2×M64C2211yescharacteristic
12C22×C32C2211yescharacteristic
13#2223—22no—
14#2222—21no—
15#2221—21no—
16C2×#368C2211yescharacteristic
17#2158C2211yescharacteristic
18#2158C2211yescharacteristic
19#2148—21no—
20#1812—21no—
21#1426—22no—
22#1418—22no—
23#1416—21no—
24#1415—21no—
25#1401—22no—
26#1384—21no—
27#575—41no—
28C2×#135—21no—
29C2×#131D411yescharacteristic
30#508—21no—
31#508—21no—
32#508—21no—
33#508—21no—
34C2×Q32D411yescharacteristic
35C2×SD32—21no—
36C2×D16D411yescharacteristic
37#504—21no—
38#504—21no—
39#504—21no—
40#504—21no—
41C2×M32—21no—
42C2×M32D411yescharacteristic
43C22×C16—21no—
44C22×C16C2311yescharacteristic
45#473—22no—
46#472—22no—
47#471—42no—
48C2×#108—21no—
49#427—21no—
50#427—21no—
51#413—41no—
52#410—41no—
53M64—21no—
54M64D412yes—
55C2×C32—21no—
56C2×C32—21no—
57C2×C32—21no—
58C2×C32—21no—
59#368—21no—
60#368D412yes—
61#367—21no—
62#367—21no—
63#366—21no—
64#366—21no—
65#362—22no—
66#361—22no—
67H(3,ℤ/4)—41no—
68C2×#41—41no—
69C2×#41D811yescharacteristic
70#137—42no—
71#137—81no—
72AGL(1,ℤ/8)—42no—
73AGL(1,ℤ/8)—81no—
74#135—41no—
75#135—41no—
76#135—41no—
77#135—41no—
78#135—82no—
79C2×Q16—21no—
80C2×SD16—41no—
81C2×D8—21no—
82#131—21no—
83#131—21no—
84#131—21no—
85#131—21no—
86C2×M16—21no—
87C2×M16—41no—
88C2×M16D811yescharacteristic
89C22×C8—21no—
90C22×C8C2×D411yescharacteristic
91#117—41no—
92#117—41no—
93Q32—21no—
94Q32—21no—
95SD32—41no—
96SD32—41no—
97D16—21no—
98D16—21no—
99M32—21no—
100M32—21no—
101M32—41no—
102M32—41no—
103C2×C16—21no—
104C2×C16—41no—
105C2×C16—41no—
106C2×C16C2×D411yescharacteristic
107C2×C16C2×D411yescharacteristic
108C2×C16C2×D411yescharacteristic
109C2×C16#4111yescharacteristic
110C2×C16#4111yescharacteristic
111C2×C16#4111yescharacteristic
112#108—22no—
113#108—41no—
114#107—41no—
115#107—41no—
116#106—41no—
117#106—41no—
118#101—42no—
119#100—42no—
120#99—82no—
121C32—22no—
122C32—41no—
123#41—21no—
124#41—21no—
125#41—21no—
126#41—21no—
127#41—81no—
128#41—81no—
129#41—81no—
130#41—81no—
131C2×Q8—41no—
132C2×Q8SD3211yescharacteristic
133C2×D4—21no—
134C2×D4—41no—
135C2×D4—81no—
136C2×D4SD3211yescharacteristic
137C22×C4—21no—
138C22×C4—81no—
139C22×C4C2×D811yescharacteristic
140Q16—41no—
141Q16—41no—
142Q16—82no—
143SD16—82no—
144SD16—82no—
145SD16—81no—
146SD16—81no—
147D8—41no—
148D8—41no—
149D8—82no—
150M16—21no—
151M16—21no—
152M16—41no—
153M16—41no—
154M16—42no—
155M16—81no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—81no—
160C2×C8#12011yescharacteristic
161C2×C8#12111yescharacteristic
162C2×C8#12311yescharacteristic
163C2×C8#13511yescharacteristic
164C2×C8AGL(1,ℤ/8)11yescharacteristic
165C2×C8#13711yescharacteristic
166C4⋊C4—81no—
167C4⋊C4—81no—
168C22⋊C4—81no—
169C22⋊C4—81no—
170C42—81no—
171C42—81no—
172C16—41no—
173C16#12111yescharacteristic
174C16#12111yescharacteristic
175C16#12311yescharacteristic
176C16#12411yescharacteristic
177C23—21no—
178C23—81no—
179C23#50911yescharacteristic
180Q8—21no—
181Q8—21no—
182Q8—81no—
183Q8—81no—
184D4—21no—
185D4—21no—
186D4—41no—
187D4—41no—
188D4—81no—
189D4—81no—
190D4—161no—
191D4—161no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—81no—
196C2×C4—161no—
197C2×C4—161no—
198C2×C4—161no—
199C2×C4—161no—
200C2×C4#44711yescharacteristic
201C2×C4#46611yescharacteristic
202C2×C4#48011yescharacteristic
203C2×C4C2×SD3211yescharacteristic
204C2×C4#50811yescharacteristic
205C2×C4#50911yescharacteristic
206C8—41no—
207C8—82no—
208C8#39411yescharacteristic
209C8#45211yescharacteristic
210C8#46811yescharacteristic
211C8#48211yescharacteristic
212C22—41no—
213C22—41no—
214C22—81no—
215C22—161no—
216C22—321no—
217C22#218611yescharacteristic
218C22#221811yescharacteristic
219C22#223311yescharacteristic
220C4—41no—
221C4—161no—
222C4—321no—
223C4—321no—
224C4#203311yescharacteristic
225C4#218711yescharacteristic
226C4#221711yescharacteristic
227C4#223411yescharacteristic
228C2—21no—
229C2—21no—
230C2—21no—
231C2—81no—
232C2—321no—
233C2#1338111yescharacteristic
subgroupG/H
centerC4#2234
commutator, Frattini, cosocleC22×C16C23
socleC2#13381
prime radical#33617C2
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ 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. #13638C2semidirect
3. #13622C2semidirect
4. #13615C2semidirect
5. C2×#1401C2semidirect
6. #12839C2semidirect
7. #7473C2semidirect

Maximal subgroups

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

subgroupindex
#336172
#136382
#136222
#136152
C2×#14012
#128392
#74732