Finite Group Databasebeta

← #120230

Subgroups — #120230

Order 512.

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

normal subgroups only

This group has 438 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
1#21974C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic
3#8121C212yesmaximal
4#8071C212yesmaximal
5#7986C211yesmaximal, characteristic
6#2920C2212yes—
7C2×#432C2212yes—
8C2×#431C2211yescharacteristic
9C2×#423C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1939—21no—
12#1939—21no—
13#1851C412yes—
14#1851C412yes—
15#1842—21no—
16#1831—21no—
17#1572—24no—
18#1567—21no—
19#1566—21no—
20#1307—24no—
21#1300—24no—
22#1290—21no—
23#1290—21no—
24C22×M16C2×C412yes—
25C2×#117—21no—
26C22×C42C2311yescharacteristic
27#432—22no—
28#432—22no—
29#432—22no—
30#432—22no—
31#432D414yes—
32#432Q814yes—
33#431—22no—
34#431—22no—
35#431—22no—
36#431—22no—
37#423—22no—
38#423—22no—
39#423—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47#407—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8C2×C411yescharacteristic
50C2×C22⋊C8C2×C411yescharacteristic
51C4×M16—22no—
52C2×#97—22no—
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C4×C4⋊C4—21no—
56C4×C4⋊C4Q811yescharacteristic
57C4×C4⋊C4Q811yescharacteristic
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2×C412yes—
60#376D412yes—
61C43—21no—
62#336—22no—
63#336—22no—
64#335—24no—
65#334—24no—
66C23×C4C22×C411yescharacteristic
67C23×C4#4111yescharacteristic
68C23×C4#4111yescharacteristic
69C2×M16—24no—
70C2×M16—42no—
71C2×M16—82no—
72C22×C8—22no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78#117—44no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C22⋊C4—22no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2×D412yes—
87C2×C42C2×D411yescharacteristic
88C2×C42C2×D411yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C42C2×Q812yes—
91C2×C42C2×Q811yescharacteristic
92C2×C42#4112yes—
93C2×C42#4112yes—
94C4⋊C8—24no—
95C4⋊C8—22no—
96C4⋊C8—22no—
97C4⋊C8—24no—
98C4⋊C8—41no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—41no—
102C4⋊C8—42no—
103C4⋊C8—82no—
104C4⋊C8—82no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107C22⋊C8—24no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111#97—24no—
112#97—24no—
113#97—42no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—42no—
119#95—22no—
120#95C4⋊C412yes—
121#95C4⋊C412yes—
122C24#11711yescharacteristic
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—42no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C2×C4⋊C411yescharacteristic
132C22×C4C4×D412yes—
133C22×C4C4×Q812yes—
134C22×C4#12111yescharacteristic
135C22×C4#12111yescharacteristic
136C22×C4C22⋊Q811yescharacteristic
137C22×C4C22⋊Q811yescharacteristic
138C22×C4C22⋊Q812yes—
139C22×C4#12312yes—
140C22×C4#12512yes—
141C22×C4#12612yes—
142M16—82no—
143M16—82no—
144M16—84no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—84no—
159C4⋊C4—24no—
160C4⋊C4—42no—
161C22⋊C4—44no—
162C42—42no—
163C42—42no—
164C42—42no—
165C42—42no—
166C42—44no—
167C42#12011yescharacteristic
168C42#12112yes—
169C42#12112yes—
170C42C22⋊Q812yes—
171C42C22⋊Q812yes—
172C42#12311yescharacteristic
173C42#12411yescharacteristic
174C42#12512yes—
175C42C4⋊Q812yes—
176C42C4⋊Q811yescharacteristic
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23#38011yescharacteristic
182C23#38712yes—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4#38611yescharacteristic
206C2×C4#38912yes—
207C2×C4#39111yescharacteristic
208C2×C4#39411yescharacteristic
209C2×C4#39511yescharacteristic
210C2×C4#39612yes—
211C2×C4#39812yes—
212C2×C4#39811yescharacteristic
213C2×C4#40011yescharacteristic
214C8—41no—
215C8—41no—
216C8—41no—
217C8—41no—
218C8—84no—
219C8—84no—
220C8—82no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—81no—
227C22—81no—
228C22#144111yescharacteristic
229C22#181411yescharacteristic
230C22#181411yescharacteristic
231C4—22no—
232C4—21no—
233C4—21no—
234C4—22no—
235C4—21no—
236C4—22no—
237C4—21no—
238C4—44no—
239C4—82no—
240C4#191411yescharacteristic
241C4#191511yescharacteristic
242C4#210212yes—
243C2—21no—
244C2—21no—
245C2—42no—
246C2#1120611yescharacteristic
247C2#1124111yescharacteristic
248C2#1124411yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#117
LCSG ▷ C23×C4 ▷ 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
#219742
#178152
#81212
#80712
#79862