Finite Group Databasebeta

← #150381

Subgroups — #150381

Order 512.

231 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 72 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#33735C211yesmaximal, characteristic
2#13640C211yesmaximal, characteristic, complemented
3C4×#366C211yesmaximal, characteristic, complemented
4#7419C212yesmaximal, complemented
5#7418C212yesmaximal
6#3168—21no—
7#2253C411yescharacteristic, complemented
8#2249C411yescharacteristic, complemented
9#2242—21no—
10#2217—21no—
11#2216C2211yescharacteristic
12#2215C2211yescharacteristic
13C4×Q32C411yescharacteristic, complemented
14C4×SD32—21no—
15C4×D16C411yescharacteristic, complemented
16C2×#366—21no—
17C2×C4×C16C2211yescharacteristic
18C4×#107—21no—
19C4⋊C32—21no—
20C22⋊C32C2212yes—
21#1397—21no—
22#1380C2212yes—
23#529—41no—
24C2×Q32C2×C411yescharacteristic
25C2×SD32—21no—
26C2×D16C2×C411yescharacteristic
27C22×C16D411yescharacteristic
28C22×C16C2311yescharacteristic
29#499—21no—
30#495—21no—
31#488—41no—
32#467—21no—
33#466—21no—
34#465—41no—
35C4×Q16—21no—
36C4×SD16—41no—
37C4×D8—21no—
38C2×#107—22no—
39C2×C4×C8D411yescharacteristic
40C4×C4⋊C4—41no—
41C2×C32—22no—
42#367—21no—
43#366—24no—
44#366C2×C411yescharacteristic
45#366C2×C411yescharacteristic
46#358—21no—
47#358—21no—
48#357—21no—
49#357—21no—
50C4×C16C2×C411yescharacteristic
51C4×C16C2×C411yescharacteristic
52C4×C16D411yescharacteristic
53C4×C16D411yescharacteristic
54#340—41no—
55C2×Q16—21no—
56C2×SD16—41no—
57C2×D8—21no—
58C22×C8D811yescharacteristic
59C22×C8C2×D411yescharacteristic
60#126—81no—
61#125—41no—
62#124—41no—
63#123—81no—
64C22⋊Q8—41no—
65#121—41no—
66C4×Q8—41no—
67C4×D4—41no—
68C4×D4—81no—
69#117—81no—
70C2×C4⋊C4—42no—
71C2×C42—81no—
72C2×C42D811yescharacteristic
73Q32—41no—
74SD32—42no—
75D16—41no—
76C2×C16—22no—
77C2×C16—41no—
78C2×C16C22⋊C411yescharacteristic
79C2×C16C22⋊C411yescharacteristic
80C2×C16C22×C411yescharacteristic
81C2×C16C2×D411yescharacteristic
82#107—24no—
83#107—21no—
84#107—21no—
85#107—42no—
86#106—41no—
87#103—41no—
88#103—41no—
89#102—41no—
90#102—41no—
91C4×C8C22⋊C411yescharacteristic
92C4×C8C22⋊C411yescharacteristic
93C4×C8SD1611yescharacteristic
94C4×C8SD1611yescharacteristic
95#95—81no—
96C32—42no—
97C2×Q8—41no—
98C2×D4—41no—
99C2×D4—81no—
100C22×C4—82no—
101C22×C4—81no—
102C22×C4D1611yescharacteristic
103C22×C4D1611yescharacteristic
104C22×C4C2×D811yescharacteristic
105Q16—41no—
106Q16—41no—
107SD16—82no—
108D8—41no—
109D8—41no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—41no—
113C2×C8#10211yescharacteristic
114C2×C8#10211yescharacteristic
115C2×C8C2×C22⋊C411yescharacteristic
116C2×C8C2×SD1611yescharacteristic
117C4⋊C4—44no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—82no—
121C4⋊C4—81no—
122C4⋊C4—81no—
123C4⋊C4—81no—
124C4⋊C4—81no—
125C22⋊C4—81no—
126C22⋊C4—81no—
127C22⋊C4—81no—
128C22⋊C4—81no—
129C22⋊C4—81no—
130C42—81no—
131C42—162no—
132C42#10211yescharacteristic
133C42#10211yescharacteristic
134C42SD3211yescharacteristic
135C42SD3211yescharacteristic
136C16—21no—
137C16—21no—
138C16—21no—
139C16#10412yes—
140C23—81no—
141C23C2×D1611yescharacteristic
142Q8—81no—
143Q8—81no—
144D4—81no—
145D4—81no—
146D4—161no—
147D4—161no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—84no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—162no—
158C2×C4—162no—
159C2×C4—162no—
160C2×C4#35711yescharacteristic
161C2×C4#35711yescharacteristic
162C2×C4#35711yescharacteristic
163C2×C4#35711yescharacteristic
164C2×C4C2×#10211yescharacteristic
165C2×C4C2×SD3211yescharacteristic
166C8—21no—
167C8—21no—
168C8—21no—
169C8#41011yescharacteristic
170C8C2×#10411yescharacteristic
171C22—21no—
172C22—21no—
173C22—161no—
174C22—161no—
175C22—161no—
176C22C2×#35711yescharacteristic
177C22#226311yescharacteristic
178C22#226411yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—164no—
185C4—161no—
186C4—161no—
187C4—161no—
188C4#150211yescharacteristic
189C4#214511yescharacteristic
190C2—21no—
191C2—21no—
192C2—321no—
193C2#1174211yescharacteristic
194C2#1361611yescharacteristic
195C2#1362211yescharacteristic
subgroupG/H
center, socleC22C2×#357
commutatorC2×C16C22×C4
Frattini, cosocleC22×C16C23
prime radical#2215C22
LCSG ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G
LECSG ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13640C2semidirect
3. C4×#366C2semidirect
4. #7419C2semidirect
7. #2253C4semidirect
8. #2249C4semidirect
13. C4×Q32C4semidirect
15. C4×D16C4semidirect

Maximal subgroups

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

subgroupindex
#337352
#136402
C4×#3662
#74192
#74182