Finite Group Databasebeta

← #134302

Subgroups — #134302

Order 512.

418 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 122 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#20435C211yesmaximal, characteristic
2#12783C212yesmaximal
3C2×#1323C212yesmaximal
4#7186C212yesmaximal
5C2×#424C2211yescharacteristic
6C2×#405C411yescharacteristic
7C2×#405C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9#2296C411yescharacteristic
10C2×C4⋊C16—22no—
11C2×C22⋊C16C2214yes—
12#1771—22no—
13C8×C4⋊C4—22no—
14#1729C2211yescharacteristic
15#1728C411yescharacteristic
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#424D412yes—
28#424Q812yes—
29C2×C4⋊C8—22no—
30#405—22no—
31#405—22no—
32#405—42no—
33C2×#97—21no—
34C2×#97—21no—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8—42no—
38C2×C4×C8D412yes—
39C2×C4×C8D412yes—
40#389C812yes—
41#389C812yes—
42#385—21no—
43#385—21no—
44C4×C4⋊C4—22no—
45C4⋊C16—44no—
46C22⋊C16—24no—
47C22⋊C16—24no—
48#336—22no—
49#336—22no—
50C23×C4M1611yescharacteristic
51C23×C4M1611yescharacteristic
52C23×C4C22×C411yescharacteristic
53C2×M16—21no—
54C2×M16—21no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16C4⋊C412yes—
60C2×M16M1612yes—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C2×D412yes—
72C22×C8C2×D411yescharacteristic
73C22×C8C2×Q811yescharacteristic
74C22×C8#4112yes—
75#117—21no—
76#117—21no—
77#117—41no—
78#117—41no—
79#117C4⋊C412yes—
80#117M1612yes—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4C2×C812yes—
84C2×C4⋊C4C2×C811yescharacteristic
85C2×C4⋊C4C2×C811yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—41no—
91C2×C42—41no—
92C2×C42C22⋊C412yes—
93C2×C42C2×C812yes—
94C2×C16—44no—
95C2×C16—44no—
96C2×C16—44no—
97C4⋊C8—24no—
98C4⋊C8—24no—
99#97—24no—
100#97—42no—
101C4×C8—24no—
102C4×C8—24no—
103C4×C8—42no—
104C4×C8—42no—
105#95—22no—
106#95—22no—
107C24C2×M1611yescharacteristic
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4C22⋊C812yes—
120C22×C4C22⋊C812yes—
121C22×C4C2×C22⋊C411yescharacteristic
122C22×C4C2×C4⋊C411yescharacteristic
123C22×C4#11711yescharacteristic
124C22×C4C22×C811yescharacteristic
125C22×C4C2×M1611yescharacteristic
126C22×C4C2×M1611yescharacteristic
127M16—42no—
128M16—42no—
129M16—42no—
130M16—42no—
131C2×C8—24no—
132C2×C8—24no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—42no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8#9912yes—
149C2×C8#10112yes—
150C2×C8#12112yes—
151C2×C8C22⋊Q812yes—
152C4⋊C4—22no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4C4⋊C814yes—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—42no—
161C42—22no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—42no—
166C42—41no—
167C42—41no—
168C16—84no—
169C23—22no—
170C23—22no—
171C23—21no—
172C23—21no—
173C23C2×C22⋊C811yescharacteristic
174C23#42311yescharacteristic
175C23#43211yescharacteristic
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4#35014yes—
201C2×C4#38011yescharacteristic
202C2×C4#40711yescharacteristic
203C2×C4C2×#9911yescharacteristic
204C2×C4#41011yescharacteristic
205C2×C4C2×#10111yescharacteristic
206C2×C4#41311yescharacteristic
207C2×C4C2×C4⋊C811yescharacteristic
208C2×C4#43111yescharacteristic
209C8—22no—
210C8—22no—
211C8—22no—
212C8—22no—
213C8—42no—
214C8—42no—
215C22—21no—
216C22—21no—
217C22—22no—
218C22—22no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22#145911yescharacteristic
223C22#146111yescharacteristic
224C22#179711yescharacteristic
225C22C2×#35012yes—
226C22#211712yes—
227C4—21no—
228C4—21no—
229C4—24no—
230C4—42no—
231C4—44no—
232C4—81no—
233C4—81no—
234C4#146311yescharacteristic
235C4#146411yescharacteristic
236C4#179911yescharacteristic
237C4#180111yescharacteristic
238C2—41no—
239C2—41no—
240C2#856011yescharacteristic
241C2#1146811yescharacteristic
242C2#1146911yescharacteristic
243C2#1149812yes—
244C2#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
#204352
#127832
C2×#13232
#71862