Finite Group Databasebeta

← #125047

Subgroups — #125047

Order 512.

372 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 102 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#12879C211yesmaximal, characteristic
3#8785C211yesmaximal, characteristic
4#7113C214yesmaximal
5C2×#494C411yescharacteristic
6C2×C4×Q16—21no—
7C2×C4×Q16C412yes—
8C4×C4⋊Q8—21no—
9C2×C4⋊C16—21no—
10#1969—21no—
11#1969C2212yes—
12#1956—21no—
13#1845C411yescharacteristic
14#1831—21no—
15C42×C8C2211yescharacteristic
16#1374—21no—
17#1374C2212yes—
18#1277—21no—
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—42no—
28C4×Q16—42no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×C4⋊C8—22no—
32C2×#103—22no—
33C2×#103—22no—
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
47C4⋊C16—42no—
48#336—41no—
49#336—41no—
50C22×Q8—22no—
51C2×Q16—24no—
52C2×Q16—22no—
53C2×Q16—42no—
54C2×Q16—42no—
55C2×Q16—42no—
56C2×Q16—42no—
57C2×Q16M1614yes—
58C22×C8—42no—
59C22×C8C22⋊C411yescharacteristic
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22×C411yescharacteristic
62C22×C8C2×D411yescharacteristic
63C4⋊Q8—24no—
64C4⋊Q8—42no—
65C4×Q8—44no—
66C4×Q8—44no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C42—42no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42D811yescharacteristic
77C2×C42D811yescharacteristic
78C2×C42SD1611yescharacteristic
79C2×C42SD1611yescharacteristic
80C2×C42C2×D411yescharacteristic
81C2×C16—42no—
82C2×C16—42no—
83C2×C16—42no—
84#107—42no—
85#107—42no—
86C4⋊C8—42no—
87C4⋊C8—42no—
88#103—44no—
89#103—42no—
90#103—42no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—41no—
94C4×C8—41no—
95C4×C8—41no—
96C4×C8D812yes—
97C4×C8SD1612yes—
98#95—42no—
99#95—42no—
100C2×Q8—24no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4#10211yescharacteristic
109C22×C4#10211yescharacteristic
110C22×C4#10211yescharacteristic
111C22×C4#10211yescharacteristic
112C22×C4C2×C22⋊C411yescharacteristic
113C22×C4C2×D811yescharacteristic
114C22×C4C2×SD1611yescharacteristic
115Q16—44no—
116Q16—44no—
117Q16—48no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—82no—
128C2×C8—82no—
129C2×C8—82no—
130C2×C8#10212yes—
131C2×C8C2×M1612yes—
132C2×C8C2×D811yescharacteristic
133C2×C8C2×SD1611yescharacteristic
134C4⋊C4—44no—
135C4⋊C4—44no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—82no—
139C4⋊C4—84no—
140C4⋊C4—82no—
141C42—22no—
142C42—22no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—22no—
148C42—41no—
149C42—84no—
150C42#10012yes—
151C42#10212yes—
152C16—82no—
153C23C2×#10211yescharacteristic
154Q8—44no—
155Q8—44no—
156Q8—82no—
157Q8—82no—
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—21no—
171C2×C4—44no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4#33112yes—
178C2×C4#33112yes—
179C2×C4#35912yes—
180C2×C4#36212yes—
181C2×C4#40711yescharacteristic
182C2×C4#41011yescharacteristic
183C2×C4C2×#10011yescharacteristic
184C2×C4C2×#10211yescharacteristic
185C2×C4#42111yescharacteristic
186C2×C4#42111yescharacteristic
187C2×C4#51012yes—
188C8—22no—
189C8—22no—
190C8—21no—
191C8—21no—
192C8—41no—
193C8—162no—
194C22#148611yescharacteristic
195C22#151711yescharacteristic
196C22C2×#33111yescharacteristic
197C22#214111yescharacteristic
198C22C2×#35911yescharacteristic
199C22#214511yescharacteristic
200C22C2×#36211yescharacteristic
201C4—22no—
202C4—22no—
203C4—22no—
204C4—22no—
205C4—22no—
206C4—41no—
207C4—84no—
208C4—162no—
209C4#146611yescharacteristic
210C4#148611yescharacteristic
211C4#151711yescharacteristic
212C4#153611yescharacteristic
213C2#765111yescharacteristic
214C2#1166211yescharacteristic
215C2#1166811yescharacteristic
216C2#1185911yescharacteristic
217C2#1186311yescharacteristic
218C2#1192411yescharacteristic
219C2#1193211yescharacteristic
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
#128792
#87852
#71132