Finite Group Databasebeta

← #128965

Subgroups — #128965

Order 512.

388 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 90 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#22213C211yesmaximal, characteristic
2#11728C212yesmaximal
3C8×#99C212yesmaximal
4#7287C211yesmaximal, characteristic
5#7283C211yesmaximal, characteristic
6C2×#446C411yescharacteristic, complemented
7C2×C8×D4C2211yescharacteristic
8#2398C2211yescharacteristic
9#1985C411yescharacteristic, complemented
10#1955C2211yescharacteristic
11#1924—22no—
12#1850—22no—
13#1843C411yescharacteristic, complemented
14C4×#99C2212yes—
15C8×C22⋊C4—22no—
16#1746C411yescharacteristic, complemented
17#1459C2212yes—
18#1351C414yescomplemented
19#1344C414yescomplemented
20#1282—22no—
21C23×C8D412yes—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×D4C2311yescharacteristic
24#446—42no—
25C8×D4—42no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×#99—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8D412yes—
34C2×#97—21no—
35C2×#97C2×C411yescharacteristic
36C2×C4×C8—42no—
37C2×C4×C8C2×C411yescharacteristic
38#400—21no—
39#397C2×C412yes—
40#394—21no—
41#393C2×C412yes—
42C4×C22⋊C4—22no—
43C4×C22⋊C4—22no—
44#376C2×C411yescharacteristic
45#342C2×C414yes—
46#336—22no—
47#336—21no—
48#336—21no—
49#336—42no—
50#324—22no—
51#323C2×C414yes—
52C22×Q8C4211yescharacteristic
53C22×D4#4111yescharacteristic
54C23×C4C2×D412yes—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8—42no—
60C22×C8—82no—
61C22×C8C22⋊C412yes—
62C22×C8C22⋊C411yescharacteristic
63C22×C8C22⋊C411yescharacteristic
64C4×Q8—41no—
65C4×Q8—41no—
66C4×D4—41no—
67C4×D4—41no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C4211yescharacteristic
70C2×C4⋊C4#4111yescharacteristic
71C2×C22⋊C4—24no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4C22×C412yes—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—42no—
78C2×C42C22×C411yescharacteristic
79C4⋊C8—41no—
80C4⋊C8—41no—
81C4⋊C8—42no—
82C4⋊C8—82no—
83#99—28no—
84C22⋊C8—24no—
85C22⋊C8—42no—
86C22⋊C8—42no—
87C22⋊C8—82no—
88#97—41no—
89#97—42no—
90C4×C8—22no—
91C4×C8—84no—
92#95—21no—
93#95—22no—
94#95—22no—
95#95—42no—
96#95C4212yes—
97C24C4×D412yes—
98C2×Q8—41no—
99C2×Q8—81no—
100C2×D4—24no—
101C2×D4—41no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—44no—
110C22×C4C2×C4211yescharacteristic
111C22×C4C2×C22⋊C411yescharacteristic
112C22×C4#11711yescharacteristic
113C22×C4C4×D412yes—
114C2×C8—21no—
115C2×C8—22no—
116C2×C8—41no—
117C2×C8—44no—
118C2×C8—42no—
119C2×C8—41no—
120C2×C8—44no—
121C2×C8—41no—
122C2×C8—42no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—82no—
127C2×C8—82no—
128C2×C8—82no—
129C2×C8#9912yes—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—42no—
133C22⋊C4—28no—
134C22⋊C4—42no—
135C22⋊C4—44no—
136C22⋊C4—44no—
137C22⋊C4—82no—
138C42—21no—
139C42—21no—
140C42—41no—
141C42—41no—
142C42—41no—
143C42—41no—
144C42—82no—
145C42—82no—
146C23—22no—
147C23—22no—
148C23—42no—
149C23—42no—
150C23C4×C22⋊C411yescharacteristic
151C23#44314yes—
152Q8—82no—
153D4—44no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—48no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—82no—
174C2×C4—84no—
175C2×C4—84no—
176C2×C4C2×#9911yescharacteristic
177C2×C4#41011yescharacteristic
178C8—22no—
179C8—42no—
180C8—41no—
181C8—81no—
182C8—84no—
183C8—81no—
184C22—21no—
185C22—21no—
186C22—24no—
187C22—44no—
188C22—42no—
189C22—42no—
190C22—82no—
191C22C4×#9911yescharacteristic
192C22#176412yes—
193C4—21no—
194C4—41no—
195C4—41no—
196C4—42no—
197C4—82no—
198C4—88no—
199C4—81no—
200C4—81no—
201C4#212711yescharacteristic
202C4#212911yescharacteristic
203C2—22no—
204C2—44no—
205C2#823811yescharacteristic
206C2#1262411yescharacteristic
207C2#1262711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC22×D4#41
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#446C4semidirect
9. #1985C4semidirect
13. #1843C4semidirect
16. #1746C4semidirect
18. #1351C4semidirect
19. #1344C4semidirect

Maximal subgroups

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

subgroupindex
#222132
#117282
C8×#992
#72872
#72832