Finite Group Databasebeta

← #116924

Subgroups — #116924

Order 512.

533 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 132 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#21821C211yesmaximal, characteristic
2#17814C211yesmaximal, characteristic
3#17749C212yesmaximal
4#8007C211yesmaximal, characteristic
5#7990C212yesmaximal
6C2×#533C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2212yes—
11#1985C414yes—
12#1984—22no—
13#1818—22no—
14#1571—24no—
15#1567—22no—
16#1559—24no—
17#1550—24no—
18#1548—22no—
19#1546—22no—
20#1293—22no—
21#1290—24no—
22C22×M16—22no—
23#533—22no—
24#533—22no—
25#533—22no—
26#533—22no—
27C2×C4⋊Q8D411yescharacteristic
28C2×C4⋊Q8D411yescharacteristic
29C2×#125—21no—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8Q812yes—
32C2×C4×Q8—21no—
33C2×C4×Q8C2×C412yes—
34C2×#117—21no—
35C22×C42C2311yescharacteristic
36#432—22no—
37#432—22no—
38#432—22no—
39#432D414yes—
40#431—22no—
41#431—22no—
42#431—22no—
43#431—22no—
44#423—22no—
45#423—22no—
46#423—22no—
47#423—22no—
48#423—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8C2×C412yes—
52C2×C4⋊C8D414yes—
53#407—22no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8D412yes—
56C4×M16—22no—
57C2×#97C2×C412yes—
58C2×C4×C8D412yes—
59#336—28no—
60#335—28no—
61#333—28no—
62#332—28no—
63C22×Q8—21no—
64C22×Q8C4⋊C412yes—
65C23×C4C2×D411yescharacteristic
66C23×C4C2×Q811yescharacteristic
67C23×C4#4111yescharacteristic
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—84no—
72C22×C8—22no—
73C22×C8—24no—
74C22×C8C22⋊C414yes—
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79#125—42no—
80#125—42no—
81C22⋊Q8—24no—
82C22⋊Q8—24no—
83C22⋊Q8—42no—
84C22⋊Q8—42no—
85C4×Q8—24no—
86C4×Q8—24no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×Q8—41no—
90C4×Q8—41no—
91#117—42no—
92#117—42no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C4⋊C412yes—
99C2×C22⋊C4—22no—
100C2×C42—22no—
101C2×C42C22×C411yescharacteristic
102C2×C42C2×D412yes—
103C2×C42C2×D412yes—
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42#4112yes—
107C2×C42#4112yes—
108C2×C42#4111yescharacteristic
109C4⋊C8—24no—
110C4⋊C8—24no—
111C4⋊C8—24no—
112C4⋊C8—24no—
113C4⋊C8—42no—
114C4⋊C8—41no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—41no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122#97—24no—
123#97—24no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—42no—
127C24C22⋊Q811yescharacteristic
128C2×Q8—24no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—82no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—81no—
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×C4⋊C411yescharacteristic
142C22×C4#11711yescharacteristic
143C22×C4#12112yes—
144C22×C4#12112yes—
145C22×C4#12112yes—
146C22×C4#12111yescharacteristic
147C22×C4#12111yescharacteristic
148C22×C4C22⋊Q812yes—
149C22×C4C22⋊Q812yes—
150C22×C4C22⋊Q811yescharacteristic
151C22×C4#12612yes—
152M16—84no—
153M16—84no—
154M16—84no—
155C2×C8—24no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—84no—
162C4⋊C4—24no—
163C4⋊C4—44no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C4⋊C4—42no—
170C22⋊C4—44no—
171C42—42no—
172C42—42no—
173C42C4×D412yes—
174C42C4×D412yes—
175C42#12012yes—
176C42#12011yescharacteristic
177C42#12312yes—
178C42#12311yescharacteristic
179C42#12312yes—
180C42#12412yes—
181C42#12411yescharacteristic
182C42#12711yescharacteristic
183C23—22no—
184C23—41no—
185C23—41no—
186C23#38011yescharacteristic
187C23#39612yes—
188Q8—84no—
189Q8—82no—
190Q8—81no—
191Q8—81no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4#38512yes—
209C2×C4#38611yescharacteristic
210C2×C4#38812yes—
211C2×C4#39011yescharacteristic
212C2×C4#39211yescharacteristic
213C2×C4#39311yescharacteristic
214C2×C4#39412yes—
215C2×C4#39711yescharacteristic
216C2×C4#39911yescharacteristic
217C2×C4#47212yes—
218C2×C4#47312yes—
219C8—44no—
220C8—82no—
221C8—84no—
222C8—84no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—82no—
229C22#143911yescharacteristic
230C22#181612yes—
231C4—21no—
232C4—22no—
233C4—22no—
234C4—21no—
235C4—21no—
236C4—22no—
237C4—82no—
238C4—82no—
239C4#190912yes—
240C4#201811yescharacteristic
241C4#202011yescharacteristic
242C4#207812yes—
243C2—22no—
244C2—42no—
245C2#1107411yescharacteristic
246C2#1110311yescharacteristic
247C2#1110711yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#218212
#178142
#177492
#80072
#79902