Finite Group Databasebeta

← #124797

Subgroups — #124797

Order 512.

380 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C4×#494C211yesmaximal, characteristic
2#12894C211yesmaximal, characteristic
3#8274C211yesmaximal, characteristic
4#7113C214yesmaximal
5C2×#494C411yescharacteristic
6C2×C4×Q16—21no—
7C2×C4×Q16C412yes—
8C4×C4⋊Q8—21no—
9#1969—21no—
10#1969C2212yes—
11#1956—21no—
12#1845C411yescharacteristic
13#1767—21no—
14C42×C8C2211yescharacteristic
15C2×#335—21no—
16#1374—21no—
17#1374—21no—
18#1374C2212yes—
19#1277C2212yes—
20C22×Q16—21no—
21C22×Q16C2×C412yes—
22C2×C4⋊Q8—21no—
23C2×C4×Q8—22no—
24C22×C16—22no—
25#494—24no—
26C4×Q16—44no—
27C4×Q16—44no—
28C2×#107—21no—
29C2×#107C2×C412yes—
30C2×C4⋊C8—22no—
31C2×#103—22no—
32C2×#103—22no—
33C2×#97—41no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8C2311yescharacteristic
40#391—22no—
41#389—41no—
42#389—41no—
43#384—21no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
46C43D411yescharacteristic
47#336—41no—
48#335—24no—
49C22×Q8—22no—
50C2×Q16—24no—
51C2×Q16—22no—
52C2×Q16—42no—
53C2×Q16—42no—
54C2×Q16—42no—
55C2×Q16—42no—
56C2×Q16M1614yes—
57C22×C8—42no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8C22×C411yescharacteristic
61C22×C8C2×D411yescharacteristic
62C4⋊Q8—24no—
63C4⋊Q8—42no—
64C4×Q8—44no—
65C4×Q8—44no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C42—42no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22⋊C411yescharacteristic
75C2×C42D811yescharacteristic
76C2×C42D811yescharacteristic
77C2×C42SD1611yescharacteristic
78C2×C42SD1611yescharacteristic
79C2×C42C2×D411yescharacteristic
80C2×C16—42no—
81C2×C16—42no—
82C2×C16—42no—
83#107—42no—
84#107—42no—
85C4⋊C8—24no—
86C4⋊C8—42no—
87#103—44no—
88#103—44no—
89#97—82no—
90C4×C8—22no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8—41no—
95C4×C8—41no—
96#95—42no—
97#95—42no—
98C2×Q8—24no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4#10211yescharacteristic
107C22×C4#10211yescharacteristic
108C22×C4#10211yescharacteristic
109C22×C4#10211yescharacteristic
110C22×C4C2×C22⋊C411yescharacteristic
111C22×C4C2×D811yescharacteristic
112C22×C4C2×SD1611yescharacteristic
113Q16—44no—
114Q16—44no—
115Q16—48no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—44no—
125C2×C8—82no—
126C2×C8—82no—
127C2×C8#10012yes—
128C2×C8#10412yes—
129C2×C8C2×M1612yes—
130C2×C8AGL(1,ℤ/8)12yes—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C4⋊C4—82no—
137C4⋊C4—84no—
138C42—22no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—41no—
146C42—41no—
147C42—84no—
148C42SD3212yes—
149C42Q3212yes—
150C16—82no—
151C23C2×#10211yescharacteristic
152Q8—44no—
153Q8—44no—
154Q8—82no—
155Q8—82no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—44no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4#35812yes—
177C2×C4#35812yes—
178C2×C4#36212yes—
179C2×C4#40711yescharacteristic
180C2×C4#41011yescharacteristic
181C2×C4C2×#10011yescharacteristic
182C2×C4#41811yescharacteristic
183C2×C4C2×#10411yescharacteristic
184C2×C4#42111yescharacteristic
185C2×C4C2×SD3211yescharacteristic
186C2×C4C2×Q3211yescharacteristic
187C8—22no—
188C8—22no—
189C8—22no—
190C8—22no—
191C8—84no—
192C22#148611yescharacteristic
193C22#150211yescharacteristic
194C22#154511yescharacteristic
195C22C2×#35811yescharacteristic
196C22#214311yescharacteristic
197C22#214511yescharacteristic
198C22C2×#36211yescharacteristic
199C4—22no—
200C4—22no—
201C4—22no—
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—84no—
207C4—162no—
208C4#146611yescharacteristic
209C4#149011yescharacteristic
210C4#151411yescharacteristic
211C4#153011yescharacteristic
212C2#762111yescharacteristic
213C2#1165811yescharacteristic
214C2#1166811yescharacteristic
215C2#1174311yescharacteristic
216C2#1175211yescharacteristic
217C2#1196011yescharacteristic
218C2#1196211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C4×#4942
#128942
#82742
#71132