Finite Group Databasebeta

← #128976

Subgroups — #128976

Order 512.

389 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 92 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#21635C211yesmaximal, characteristic
2#11728C212yesmaximal
3#8160C212yesmaximal
4#7287C211yesmaximal, characteristic
5#7283C211yesmaximal, characteristic
6C2×C8×Q8C411yescharacteristic, complemented
7C2×C8×D4C2211yescharacteristic
8#2398C2211yescharacteristic
9#1985C411yescharacteristic, complemented
10#1955C2211yescharacteristic
11#1924—22no—
12#1832C411yescharacteristic, complemented
13C4×#99C2212yes—
14#1752—22no—
15C8×C22⋊C4—22no—
16#1726C411yescharacteristic, complemented
17#1459C2212yes—
18#1351C414yescomplemented
19#1344C414yescomplemented
20#1282—22no—
21C23×C8D412yes—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×D4C2311yescharacteristic
24C8×Q8—41no—
25C8×Q8—41no—
26C8×D4—41no—
27C8×D4—41no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×#99—22no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8D412yes—
35C2×#97—21no—
36C2×#97—42no—
37C2×#97C2×C411yescharacteristic
38C2×C4×C8—21no—
39C2×C4×C8—21no—
40C2×C4×C8C2×C411yescharacteristic
41#400—21no—
42#397C2×C412yes—
43#394—21no—
44#393C2×C412yes—
45C4×C22⋊C4—22no—
46C4×C22⋊C4—22no—
47#376C2×C411yescharacteristic
48#342C2×C414yes—
49#336—22no—
50#336—22no—
51#336—42no—
52#324—22no—
53#323C2×C414yes—
54C22×Q8C4211yescharacteristic
55C22×D4#4111yescharacteristic
56C23×C4C2×D412yes—
57C22×C8—21no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—21no—
61C22×C8—44no—
62C22×C8—42no—
63C22×C8C22⋊C412yes—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C4×Q8—41no—
67C4×Q8—41no—
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C4211yescharacteristic
72C2×C4⋊C4#4111yescharacteristic
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4C22×C412yes—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—42no—
80C2×C42C22×C411yescharacteristic
81C4⋊C8—41no—
82C4⋊C8—41no—
83C4⋊C8—42no—
84#99—44no—
85C22⋊C8—24no—
86C22⋊C8—42no—
87C22⋊C8—44no—
88C22⋊C8—42no—
89#97—41no—
90#97—42no—
91#97—84no—
92C4×C8—21no—
93C4×C8—21no—
94C4×C8—42no—
95C4×C8—41no—
96C4×C8—41no—
97#95—21no—
98#95—22no—
99#95—22no—
100#95—42no—
101#95C4212yes—
102C24C4×D412yes—
103C2×Q8—42no—
104C2×Q8—41no—
105C2×D4—24no—
106C2×D4—41no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4C2×C4211yescharacteristic
116C22×C4C2×C22⋊C411yescharacteristic
117C22×C4#11711yescharacteristic
118C22×C4C4×D412yes—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—44no—
126C2×C8—41no—
127C2×C8—42no—
128C2×C8—44no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—82no—
132C2×C8—82no—
133C2×C8—82no—
134C2×C8#9911yescharacteristic
135C2×C8#9911yescharacteristic
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—44no—
141C22⋊C4—44no—
142C22⋊C4—44no—
143C22⋊C4—44no—
144C42—21no—
145C42—21no—
146C42—41no—
147C42—41no—
148C42—41no—
149C42—41no—
150C42—82no—
151C42—82no—
152C23—22no—
153C23—22no—
154C23—42no—
155C23—42no—
156C23C4×C22⋊C411yescharacteristic
157C23#44414yes—
158Q8—42no—
159Q8—42no—
160D4—44no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—82no—
180C2×C4—84no—
181C2×C4—84no—
182C2×C4—84no—
183C2×C4C2×#9911yescharacteristic
184C2×C4#41011yescharacteristic
185C8—21no—
186C8—42no—
187C8—42no—
188C8—41no—
189C8—84no—
190C8—81no—
191C8#35412yes—
192C22—21no—
193C22—21no—
194C22—24no—
195C22—44no—
196C22—42no—
197C22—42no—
198C22—82no—
199C22C4×#9911yescharacteristic
200C22#176512yes—
201C4—21no—
202C4—41no—
203C4—42no—
204C4—41no—
205C4—42no—
206C4—82no—
207C4—81no—
208C4—164no—
209C4C2×#35411yescharacteristic
210C4#212611yescharacteristic
211C2—22no—
212C2—44no—
213C2#823911yescharacteristic
214C2#1262311yescharacteristic
215C2C4×#35411yescharacteristic
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×C8×Q8C4semidirect
9. #1985C4semidirect
12. #1832C4semidirect
16. #1726C4semidirect
18. #1351C4semidirect
19. #1344C4semidirect

Maximal subgroups

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

subgroupindex
#216352
#117282
#81602
#72872
#72832