Finite Group Databasebeta

← #150501

Subgroups — #150501

Order 512.

306 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 58 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#28721C211yesmaximal, characteristic
2C4×#354C211yesmaximal, characteristic
3#11530C211yesmaximal, characteristic
4#7412C214yesmaximal
5C2×#488—21no—
6#2859C411yescharacteristic, complemented
7C4×#124C2211yescharacteristic
8C2×#354—21no—
9#2074C412yescomplemented
10#2073C2212yes—
11#1991—21no—
12#1803—21no—
13#1802—21no—
14C4×C4⋊C8C411yescharacteristic, complemented
15C4×#99—21no—
16#1467—21no—
17#1393C2212yes—
18#1324—21no—
19#1318C2212yes—
20C2×#125—21no—
21C2×#124D411yescharacteristic
22C2×C4×Q8D411yescharacteristic
23C2×C4×D4D411yescharacteristic
24#488—44no—
25#432—41no—
26C2×C4⋊C8—21no—
27C2×#103—22no—
28C2×#102—22no—
29C2×#99—22no—
30C2×C4×C8—21no—
31#391C2×C411yescharacteristic
32#390C2311yescharacteristic
33#389—41no—
34#389—41no—
35#388—41no—
36#384—21no—
37#382C2×C412yes—
38#382D411yescharacteristic
39C4×C22⋊C4—41no—
40C4×C22⋊C4—41no—
41C43C2×C411yescharacteristic
42#354—24no—
43#342—41no—
44#340—22no—
45#340—22no—
46#336C2×C412yes—
47#324—22no—
48#324—41no—
49C22×Q8C22⋊C411yescharacteristic
50C22×D4C2×D411yescharacteristic
51C23×C4—21no—
52C22×C8—22no—
53#125—42no—
54#125—42no—
55#124—22no—
56#124—22no—
57C4×Q8—41no—
58C4×Q8—41no—
59C4×D4—41no—
60C4×D4—41no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4C22⋊C411yescharacteristic
67C2×C4⋊C4C2×D411yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—41no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—41no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—81no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C22⋊C411yescharacteristic
78C2×C42C22×C411yescharacteristic
79C4⋊C8—41no—
80C4⋊C8—41no—
81C4⋊C8—81no—
82#103—44no—
83#102—44no—
84#99—24no—
85#99—42no—
86C22⋊C8—42no—
87#97—81no—
88C4×C8—42no—
89#95—41no—
90#95—41no—
91#95—42no—
92#95—81no—
93C24—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—41no—
97C2×Q8—41no—
98C2×D4—21no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4AGL(1,ℤ/8)12yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—42no—
110C22×C4—82no—
111C22×C4—82no—
112C22×C4#9911yescharacteristic
113C22×C4#9911yescharacteristic
114C22×C4C2×C22⋊C411yescharacteristic
115C2×C8—42no—
116C2×C8—42no—
117C2×C8—42no—
118C2×C8—82no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—81no—
124C4⋊C4—82no—
125C4⋊C4—84no—
126C4⋊C4—82no—
127C22⋊C4—44no—
128C22⋊C4—44no—
129C22⋊C4—81no—
130C22⋊C4—82no—
131C22⋊C4—82no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—41no—
136C42—42no—
137C42—41no—
138C42—41no—
139C42—81no—
140C42—162no—
141C42#10412yes—
142C23—22no—
143C23—41no—
144C23—41no—
145C23—81no—
146C23—81no—
147C23C2×#9911yescharacteristic
148Q8—42no—
149Q8—81no—
150D4—42no—
151D4—81no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—82no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—84no—
174C2×C4—82no—
175C2×C4—161no—
176C2×C4—162no—
177C2×C4—162no—
178C2×C4#35411yescharacteristic
179C2×C4#35411yescharacteristic
180C2×C4#41811yescharacteristic
181C2×C4C2×#10411yescharacteristic
182C8—82no—
183C8—162no—
184C22—21no—
185C22—21no—
186C22—42no—
187C22—82no—
188C22—81no—
189C22—81no—
190C22—161no—
191C22#150711yescharacteristic
192C22#212311yescharacteristic
193C22C2×#35411yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—41no—
199C4—81no—
200C4—81no—
201C4—81no—
202C4—81no—
203C4—162no—
204C4—164no—
205C4#141412yes—
206C2—21no—
207C2—21no—
208C2—82no—
209C2#1177711yescharacteristic
210C2#1359611yescharacteristic
211C2C2×#141411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #2859C4semidirect
9. #2074C4semidirect
14. C4×C4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#287212
C4×#3542
#115302
#74122