Finite Group Databasebeta

← #120308

Subgroups — #120308

Order 512.

510 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 138 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#20808C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic
3#8066C212yesmaximal
4#8006C211yesmaximal, characteristic
5#7867C212yesmaximal
6#2920C2212yes—
7#2920C2212yes—
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×C43C2211yescharacteristic
11#1842C412yes—
12#1831C412yes—
13#1797—22no—
14C4×C4⋊C8—22no—
15#1574—24no—
16#1568—22no—
17#1299—24no—
18#1293—22no—
19#1276—22no—
20#1276—22no—
21C22×C42—21no—
22C22×C42—21no—
23C22×C42Q811yescharacteristic
24C22×C42Q811yescharacteristic
25C22×C42C2311yescharacteristic
26#432—22no—
27#432—22no—
28#432—22no—
29#432Q814yes—
30#431—22no—
31#431—22no—
32#431—22no—
33#431—22no—
34#431—22no—
35#431—22no—
36#431—22no—
37#431—22no—
38#423—22no—
39#423D414yes—
40C2×C4⋊C8—24no—
41C2×C4⋊C8C2×C412yes—
42C2×C4⋊C8C2×C412yes—
43#407—22no—
44#407—22no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8D412yes—
47C4×M16—22no—
48C4×M16—22no—
49C2×C4×C8—21no—
50C2×C4×C8D412yes—
51C43—21no—
52C43—21no—
53C43C2×C411yescharacteristic
54C43C2×C411yescharacteristic
55C43D411yescharacteristic
56C43D411yescharacteristic
57#336—24no—
58#335—24no—
59#334—24no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×Q811yescharacteristic
64C23×C4#4111yescharacteristic
65C2×M16—24no—
66C2×M16—42no—
67C22×C8—22no—
68C22×C8C22⋊C414yes—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—42no—
82C2×C42—44no—
83C2×C42—44no—
84C2×C42C4⋊C411yescharacteristic
85C2×C42C4⋊C411yescharacteristic
86C2×C42C4⋊C411yescharacteristic
87C2×C42C4⋊C411yescharacteristic
88C2×C42C22×C411yescharacteristic
89C2×C42C2×D412yes—
90C2×C42C2×D411yescharacteristic
91C2×C42C2×D411yescharacteristic
92C2×C42C2×Q812yes—
93C2×C42#4112yes—
94C2×C42#4112yes—
95C2×C42#4111yescharacteristic
96C4⋊C8—24no—
97C4⋊C8—24no—
98C4⋊C8—24no—
99C4⋊C8—24no—
100C4⋊C8—24no—
101C4⋊C8—48no—
102C4⋊C8—42no—
103C22⋊C8—24no—
104C22⋊C8—24no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#97—24no—
110#97—42no—
111C4×C8—22no—
112C4×C8—22no—
113C4×C8—41no—
114C4×C8—42no—
115C4×C8—42no—
116C4×C8—41no—
117C24C22⋊Q811yescharacteristic
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C2×C4⋊C411yescharacteristic
135C22×C4#11711yescharacteristic
136C22×C4#12112yes—
137C22×C4#12111yescharacteristic
138C22×C4#12111yescharacteristic
139C22×C4C22⋊Q811yescharacteristic
140C22×C4C22⋊Q812yes—
141C22×C4#12312yes—
142C22×C4#12512yes—
143C22×C4#12512yes—
144C22×C4C4⋊Q812yes—
145M16—84no—
146M16—84no—
147C2×C8—24no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8—42no—
151C2×C8—44no—
152C2×C8—44no—
153C42—24no—
154C42—24no—
155C42—24no—
156C42—21no—
157C42—21no—
158C42—22no—
159C42—22no—
160C42—42no—
161C42—42no—
162C42—42no—
163C42—42no—
164C42—42no—
165C42—42no—
166C42—42no—
167C42#10812yes—
168C42#10812yes—
169C42C4×D412yes—
170C42C4×Q812yes—
171C42#12011yescharacteristic
172C42#12112yes—
173C42C22⋊Q812yes—
174C42C22⋊Q812yes—
175C42#12311yescharacteristic
176C42#12411yescharacteristic
177C42#12612yes—
178C42C4⋊Q811yescharacteristic
179C23—21no—
180C23—21no—
181C23—41no—
182C23—41no—
183C23#38011yescharacteristic
184C23#39812yes—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—48no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—48no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4#38611yescharacteristic
214C2×C4#38712yes—
215C2×C4#38912yes—
216C2×C4#39111yescharacteristic
217C2×C4#39411yescharacteristic
218C2×C4#39511yescharacteristic
219C2×C4#39612yes—
220C2×C4#39811yescharacteristic
221C2×C4#40011yescharacteristic
222C2×C4C2×#10811yescharacteristic
223C2×C4C2×#10811yescharacteristic
224C2×C4#43012yes—
225C8—44no—
226C8—84no—
227C8—84no—
228C8—82no—
229C22—21no—
230C22—21no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—42no—
235C22—42no—
236C22#144111yescharacteristic
237C22#181011yescharacteristic
238C22#181111yescharacteristic
239C4—24no—
240C4—22no—
241C4—21no—
242C4—21no—
243C4—21no—
244C4—22no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—42no—
249C4#188412yes—
250C4#195012yes—
251C4#195012yes—
252C4#205012yes—
253C2—21no—
254C2—21no—
255C2—42no—
256C2#1121211yescharacteristic
257C2#1123311yescharacteristic
258C2#1124311yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1441
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#208082
#178192
#80662
#80062
#78672