Finite Group Databasebeta

← #153257

Subgroups — #153257

Order 512.

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

normal subgroups only

This group has 266 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#33611C211yesmaximal, characteristic
2C2×#1426C211yesmaximal, characteristic, complemented
3#13625C211yesmaximal, characteristic, complemented
4#13622C211yesmaximal, characteristic, complemented
5#13558C211yesmaximal, characteristic, complemented
6#12841C211yesmaximal, characteristic, complemented
7#7477C211yesmaximal, characteristic, complemented
8C2×#508C2211yescharacteristic
9C2×#504C2211yescharacteristic
10#3061—21no—
11C2×M64C2211yescharacteristic
12C2×M64C2211yescharacteristic
13#2219—21no—
14#2218—21no—
15#2217—22no—
16C2×#368C2211yescharacteristic
17#2158C2211yescharacteristic
18C2×#367C2211yescharacteristic
19#2139—21no—
20#1814—21no—
21#1426—22no—
22#1420—21no—
23#1419—21no—
24#1418—22no—
25#1401—22no—
26#1389—21no—
27C2×#135—21no—
28C2×#131D411yescharacteristic
29#537—41no—
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#468—42no—
46#467—22no—
47#466—22no—
48C2×#108—21no—
49#427—21no—
50C2×#107—21no—
51#421—41no—
52#417—41no—
53M64—21no—
54M64—21no—
55M64—21no—
56M64D412yescomplemented
57C2×C32—21no—
58C2×C32—21no—
59#368—21no—
60#368—21no—
61#367—21no—
62#367—21no—
63#367D412yes—
64#366—21no—
65#358—22no—
66#357—22no—
67#344—41no—
68C2×#41—41no—
69C2×#41D811yescharacteristic
70#135—41no—
71#135—41no—
72#135—41no—
73#135—41no—
74C2×Q16—21no—
75C2×SD16—41no—
76C2×D8—21no—
77#131—21no—
78#131—21no—
79#131—21no—
80#131—21no—
81C2×M16—21no—
82C2×M16—41no—
83C2×M16D811yescharacteristic
84C22×C8—21no—
85C22×C8C2×D411yescharacteristic
86C22⋊Q8—42no—
87C22⋊Q8—81no—
88#121—42no—
89#121—82no—
90#121—81no—
91#117—41no—
92C2×C4⋊C4—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—41no—
113#108—41no—
114#107—22no—
115#107—41no—
116#107—41no—
117#106—41no—
118#104—82no—
119#103—42no—
120#102—42no—
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×D4—82no—
137C2×D4SD3211yescharacteristic
138C22×C4—21no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4C2×D811yescharacteristic
142Q16—41no—
143Q16—41no—
144SD16—81no—
145SD16—81no—
146D8—41no—
147D8—41no—
148M16—21no—
149M16—21no—
150M16—41no—
151M16—41no—
152M16—81no—
153M16—81no—
154C2×C8—21no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—81no—
158C2×C8#12011yescharacteristic
159C2×C8#12111yescharacteristic
160C2×C8#12311yescharacteristic
161C2×C8#13511yescharacteristic
162C2×C8AGL(1,ℤ/8)11yescharacteristic
163C2×C8#13711yescharacteristic
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C22⋊C4—82no—
169C22⋊C4—82no—
170C22⋊C4—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—
192D4—322no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—161no—
199C2×C4—161no—
200C2×C4—161no—
201C2×C4—321no—
202C2×C4#44711yescharacteristic
203C2×C4#46611yescharacteristic
204C2×C4#48011yescharacteristic
205C2×C4C2×SD3211yescharacteristic
206C2×C4#50811yescharacteristic
207C2×C4#50911yescharacteristic
208C8—41no—
209C8—161no—
210C8#39411yescharacteristic
211C8#45211yescharacteristic
212C8#46811yescharacteristic
213C8#48211yescharacteristic
214C22—41no—
215C22—42no—
216C22—41no—
217C22—161no—
218C22—161no—
219C22—161no—
220C22#218711yescharacteristic
221C22#221811yescharacteristic
222C22#223411yescharacteristic
223C4—41no—
224C4—162no—
225C4—161no—
226C4—321no—
227C4#203311yescharacteristic
228C4#218611yescharacteristic
229C4#221711yescharacteristic
230C4#223311yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—321no—
236C2#1338111yescharacteristic
subgroupG/H
center, socleC2#13381
commutator, Frattini, cosocleC22×C16C23
prime radical#33611C2
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ 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. C2×#1426C2semidirect
3. #13625C2semidirect
4. #13622C2semidirect
5. #13558C2semidirect
6. #12841C2semidirect
7. #7477C2semidirect
56. M64D4semidirect

Maximal subgroups

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

subgroupindex
#336112
C2×#14262
#136252
#136222
#135582
#128412
#74772