Finite Group Databasebeta

← #134249

Subgroups — #134249

Order 512.

420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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×#424C211yesmaximal, characteristic
2#12783C212yesmaximal
3C2×#1323C212yesmaximal
4#7186C212yesmaximal
5C2×#424C411yescharacteristic
6C2×#424C2211yescharacteristic
7C2×C4×M16C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C4×#117C411yescharacteristic
10C2×C4⋊C16—22no—
11C2×C22⋊C16C2214yes—
12#1768—22no—
13C4×C4⋊C8—22no—
14#1729C411yescharacteristic
15#1729C2211yescharacteristic
16#1374—22no—
17#1323—24no—
18C22×M16C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C23×C8C2×C411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#117C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C22×C16—24no—
26#424—22no—
27#424—22no—
28#424—22no—
29#424D412yes—
30#424Q812yes—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C4×M16—42no—
34C2×#97—21no—
35C2×C4×C8—21no—
36C2×C4×C8—42no—
37C2×C4×C8D412yes—
38C2×C4×C8D412yes—
39C4×C4⋊C4C812yes—
40C4×C4⋊C4C812yes—
41C4×C22⋊C4—22no—
42#376—21no—
43C43—21no—
44C4⋊C16—44no—
45C22⋊C16—24no—
46C22⋊C16—24no—
47#336—22no—
48#336—22no—
49C23×C4M1612yes—
50C23×C4C22×C411yescharacteristic
51C2×M16—21no—
52C2×M16—21no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16C4⋊C412yes—
58C2×M16M1612yes—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C22⋊C412yes—
68C22×C8C2×D412yes—
69C22×C8C2×D411yescharacteristic
70C22×C8C2×Q811yescharacteristic
71C22×C8#4112yes—
72#117—21no—
73#117—21no—
74#117—41no—
75#117—41no—
76#117C4⋊C412yes—
77#117M1612yes—
78C2×C4⋊C4C2×C812yes—
79C2×C4⋊C4C2×C811yescharacteristic
80C2×C4⋊C4C2×C811yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C42—21no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22⋊C411yescharacteristic
91C2×C42C2×C812yes—
92C2×C16—44no—
93C2×C16—44no—
94C2×C16—44no—
95C4⋊C8—28no—
96C4⋊C8—24no—
97C4⋊C8—24no—
98#97—42no—
99C4×C8—24no—
100C4×C8—42no—
101C4×C8—42no—
102#95—22no—
103#95—22no—
104C24C2×M1611yescharacteristic
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4C22⋊C812yes—
116C22×C4C22⋊C812yes—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4C2×C4⋊C411yescharacteristic
119C22×C4#11711yescharacteristic
120C22×C4C22×C811yescharacteristic
121C22×C4C2×M1611yescharacteristic
122C22×C4C2×M1611yescharacteristic
123M16—42no—
124M16—42no—
125M16—42no—
126M16—42no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8#12112yes—
144C2×C8C22⋊Q812yes—
145C4⋊C4—22no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4C4⋊C814yes—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4—42no—
152C42—24no—
153C42—22no—
154C42—22no—
155C42—21no—
156C42—21no—
157C42—44no—
158C42—44no—
159C42—41no—
160C42—41no—
161C42#9912yes—
162C42#10112yes—
163C16—84no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—21no—
168C23C2×C22⋊C811yescharacteristic
169C23#42311yescharacteristic
170C23#43211yescharacteristic
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4#32414yes—
194C2×C4#35014yes—
195C2×C4#38011yescharacteristic
196C2×C4#40711yescharacteristic
197C2×C4C2×#9911yescharacteristic
198C2×C4#41011yescharacteristic
199C2×C4C2×#10111yescharacteristic
200C2×C4#41311yescharacteristic
201C2×C4C2×C4⋊C811yescharacteristic
202C2×C4#43111yescharacteristic
203C8—22no—
204C8—22no—
205C8—42no—
206C8—42no—
207C8—42no—
208C22—21no—
209C22—21no—
210C22—22no—
211C22—22no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22C2×#32411yescharacteristic
216C22#145811yescharacteristic
217C22#179711yescharacteristic
218C22C2×#35012yes—
219C22#211712yes—
220C4—21no—
221C4—21no—
222C4—24no—
223C4—24no—
224C4—44no—
225C4—81no—
226C4—81no—
227C4#146511yescharacteristic
228C4#146711yescharacteristic
229C4#179911yescharacteristic
230C4#180111yescharacteristic
231C2—41no—
232C2—41no—
233C2#855811yescharacteristic
234C2#1144412yes—
235C2#1146211yescharacteristic
236C2#1146311yescharacteristic
237C2#1278712yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C4×#4242
#127832
C2×#13232
#71862