Finite Group Databasebeta

← #153242

Subgroups — #153242

Order 512.

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

normal subgroups only

This group has 269 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#33610C211yesmaximal, characteristic
2C2×#1424C211yesmaximal, characteristic, complemented
3C2×#1417C211yesmaximal, characteristic, complemented
4#13616C211yesmaximal, characteristic, complemented
5C2×#1401C211yesmaximal, characteristic, complemented
6#12842C211yesmaximal, characteristic, complemented
7#7478C211yesmaximal, characteristic, complemented
8C2×#508C2211yescharacteristic
9C2×#504C2211yescharacteristic
10#3061—21no—
11C2×M64C2211yescharacteristic
12C22×C32C2211yescharacteristic
13#2220—22no—
14#2216—21no—
15#2215—21no—
16C2×#368C2211yescharacteristic
17C2×#366C2211yescharacteristic
18C2×#366C2211yescharacteristic
19#2139—21no—
20#1816—21no—
21#1424—22no—
22#1417—22no—
23#1416—21no—
24#1415—21no—
25#1401—22no—
26#1387—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—
49C2×#107—21no—
50C2×#107—21no—
51#417—41no—
52#417—41no—
53M64—21no—
54M64—21no—
55C2×C32—21no—
56C2×C32—21no—
57C2×C32—21no—
58C2×C32D412yescomplemented
59#368—21no—
60#368—21no—
61#366—21no—
62#366—21no—
63#366—21no—
64#366D412yes—
65#358—22no—
66#357—22no—
67#343—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—
91C2×C4⋊C4—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#107—41no—
118#103—42no—
119#103—81no—
120#102—42no—
121#102—81no—
122C32—22no—
123C32—41no—
124#41—21no—
125#41—21no—
126#41—21no—
127#41—21no—
128#41—81no—
129#41—81no—
130#41—81no—
131#41—81no—
132C2×Q8—41no—
133C2×Q8D1611yescharacteristic
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—81no—
137C2×D4—82no—
138C2×D4D1611yescharacteristic
139C22×C4—21no—
140C22×C4—81no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4C2×D811yescharacteristic
144Q16—41no—
145Q16—41no—
146SD16—81no—
147SD16—81no—
148D8—41no—
149D8—41no—
150M16—21no—
151M16—21no—
152M16—41no—
153M16—41no—
154M16—81no—
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—42no—
167C4⋊C4—82no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C22⋊C4—82no—
172C22⋊C4—82no—
173C16—41no—
174C16#12111yescharacteristic
175C16#12111yescharacteristic
176C16#12311yescharacteristic
177C16#12411yescharacteristic
178C23—21no—
179C23—81no—
180C23#50911yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—81no—
184Q8—81no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—81no—
190D4—81no—
191D4—161no—
192D4—161no—
193D4—322no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—161no—
200C2×C4—161no—
201C2×C4—161no—
202C2×C4—321no—
203C2×C4—321no—
204C2×C4#44711yescharacteristic
205C2×C4#46611yescharacteristic
206C2×C4#48011yescharacteristic
207C2×C4C2×D1611yescharacteristic
208C2×C4#50811yescharacteristic
209C2×C4#51011yescharacteristic
210C8—41no—
211C8—161no—
212C8#39411yescharacteristic
213C8#45211yescharacteristic
214C8#46811yescharacteristic
215C8#48211yescharacteristic
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—161no—
222C22—161no—
223C22—161no—
224C22#218411yescharacteristic
225C22#222011yescharacteristic
226C22#223411yescharacteristic
227C4—41no—
228C4—161no—
229C4—162no—
230C4—321no—
231C4#203311yescharacteristic
232C4#218911yescharacteristic
233C4#221511yescharacteristic
234C4#223211yescharacteristic
235C2—21no—
236C2—21no—
237C2—81no—
238C2—321no—
239C2#1338011yescharacteristic
240C2#1367011yescharacteristic
241C2#1367111yescharacteristic
subgroupG/H
center, socleC22#2184
commutator, Frattini, cosocleC22×C16C23
prime radical#33610C2
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. C2×#1424C2semidirect
3. C2×#1417C2semidirect
4. #13616C2semidirect
5. C2×#1401C2semidirect
6. #12842C2semidirect
7. #7478C2semidirect
58. C2×C32D4semidirect

Maximal subgroups

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

subgroupindex
#336102
C2×#14242
C2×#14172
#136162
C2×#14012
#128422
#74782