Finite Group Databasebeta

← #150390

Subgroups — #150390

Order 512.

213 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 62 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#33750C211yesmaximal, characteristic
2#13642C211yesmaximal, characteristic, complemented
3#12668C211yesmaximal, characteristic, complemented
4#7425C211yesmaximal, characteristic, complemented
5#7424C211yesmaximal, characteristic
6#7423C211yesmaximal, characteristic, complemented
7#7422C211yesmaximal, characteristic
8#3168—21no—
9#2253C411yescharacteristic, complemented
10#2249C411yescharacteristic, complemented
11#2242—21no—
12#2220—21no—
13#2219C2211yescharacteristic
14#2218C2211yescharacteristic
15#2177C411yescharacteristic, complemented
16#2176C411yescharacteristic, complemented
17#2175—21no—
18#2155—21no—
19C4×M32C2211yescharacteristic
20#1772—21no—
21#1422—21no—
22#1400C2211yescharacteristic
23#1400C2211yescharacteristic
24#1398—21no—
25#1385C2211yescharacteristic
26#1384C2211yescharacteristic
27#529—41no—
28C2×Q32C2×C411yescharacteristic
29C2×SD32—21no—
30C2×D16C2×C411yescharacteristic
31C2×M32D411yescharacteristic
32C2×M32C2311yescharacteristic
33#499—21no—
34#495—21no—
35#488—41no—
36#467—21no—
37#466—21no—
38#465—41no—
39C4×Q16—21no—
40C4×SD16—41no—
41C4×D8—21no—
42C2×#108—21no—
43#427—21no—
44#405—41no—
45C2×C4×C8D411yescharacteristic
46M64—21no—
47M64—21no—
48#367C2×C411yescharacteristic
49#367C2×C411yescharacteristic
50#366—21no—
51#365—24no—
52#358—21no—
53#358—21no—
54#357—21no—
55#357—21no—
56#346C2×C411yescharacteristic
57#346D411yescharacteristic
58C4×C16C2×C411yescharacteristic
59C4×C16D411yescharacteristic
60#339—41no—
61C2×Q16—21no—
62C2×SD16—41no—
63C2×D8—21no—
64C2×M16—41no—
65C22×C8D811yescharacteristic
66C22×C8C2×D411yescharacteristic
67#126—81no—
68#125—41no—
69#124—41no—
70#123—81no—
71C22⋊Q8—41no—
72#121—41no—
73C4×Q8—41no—
74C4×D4—41no—
75C4×D4—81no—
76#117—41no—
77#117—81no—
78C2×C42D811yescharacteristic
79Q32—41no—
80SD32—42no—
81D16—41no—
82M32—22no—
83M32—41no—
84C2×C16C22⋊C411yescharacteristic
85C2×C16C22⋊C411yescharacteristic
86C2×C16C22×C411yescharacteristic
87C2×C16C2×D411yescharacteristic
88#108—22no—
89#108—41no—
90#107—21no—
91#107—21no—
92#107—41no—
93#106—22no—
94#106—41no—
95#103—41no—
96#103—41no—
97#102—41no—
98#102—41no—
99#97—81no—
100C4×C8—81no—
101C4×C8C22⋊C411yescharacteristic
102C4×C8C22⋊C411yescharacteristic
103C4×C8SD1611yescharacteristic
104C4×C8SD1611yescharacteristic
105C32—41no—
106C32—41no—
107C2×Q8—41no—
108C2×D4—41no—
109C2×D4—81no—
110C22×C4—21no—
111C22×C4—81no—
112C22×C4C2×D811yescharacteristic
113Q16—41no—
114Q16—41no—
115SD16—81no—
116SD16—81no—
117D8—41no—
118D8—41no—
119M16—42no—
120M16—81no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—81no—
126C2×C8#10211yescharacteristic
127C2×C8#10211yescharacteristic
128C2×C8C2×C22⋊C411yescharacteristic
129C2×C8C2×SD1611yescharacteristic
130C4⋊C4—42no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—81no—
134C4⋊C4—81no—
135C4⋊C4—81no—
136C4⋊C4—81no—
137C22⋊C4—81no—
138C22⋊C4—81no—
139C22⋊C4—81no—
140C22⋊C4—81no—
141C22⋊C4—81no—
142C22⋊C4—81no—
143C42—21no—
144C42—81no—
145C42—81no—
146C42#10211yescharacteristic
147C42#10211yescharacteristic
148C16—21no—
149C16—21no—
150C16—21no—
151C16#10412yes—
152C23—81no—
153C23#50911yescharacteristic
154Q8—81no—
155Q8—81no—
156D4—81no—
157D4—81no—
158D4—162no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—81no—
164C2×C4—82no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—162no—
169C2×C4C2×#10211yescharacteristic
170C2×C4#50911yescharacteristic
171C8—21no—
172C8—21no—
173C8—82no—
174C8#36011yescharacteristic
175C8#36011yescharacteristic
176C8#41011yescharacteristic
177C8C2×#10411yescharacteristic
178C22—21no—
179C22—41no—
180C22—161no—
181C22—161no—
182C22—161no—
183C22#214011yescharacteristic
184C4—21no—
185C4—41no—
186C4—41no—
187C4—161no—
188C4—161no—
189C4—162no—
190C4—161no—
191C4#150211yescharacteristic
192C4C2×#36011yescharacteristic
193C2—21no—
194C2—41no—
195C2—321no—
196C2#1174411yescharacteristic
subgroupG/H
center, socleC2#11744
commutatorC2×C16C22×C4
Frattini, cosocleC2×M32C23
prime radical#2218C22
LCSG ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G
LECSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13642C2semidirect
3. #12668C2semidirect
4. #7425C2semidirect
6. #7423C2semidirect
9. #2253C4semidirect
10. #2249C4semidirect
15. #2177C4semidirect
16. #2176C4semidirect

Maximal subgroups

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

subgroupindex
#337502
#136422
#126682
#74252
#74242
#74232
#74222