Finite Group Databasebeta

← #120259

Subgroups — #120259

Order 512.

562 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 162 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#21349C211yesmaximal, characteristic
2C2×#1566C211yesmaximal, characteristic
3C4×#341C211yesmaximal, characteristic
4C2×#1299C212yesmaximal
5#7903C212yesmaximal
6C2×#431C2211yescharacteristic
7C2×#423C2212yes—
8C2×C4×M16C2212yes—
9C22×C4×C8C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1939C412yes—
12#1939C412yes—
13#1850—22no—
14C4×C4⋊C8—22no—
15C2×#341—22no—
16#1566—24no—
17C2×#334—22no—
18#1299—24no—
19#1279—24no—
20C22×M16—22no—
21C22×M16—22no—
22C23×C8D412yes—
23C2×#117Q811yescharacteristic
24C2×#117Q811yescharacteristic
25C22×C42C2311yescharacteristic
26#431—22no—
27#431—22no—
28#423—22no—
29#423—22no—
30#423—22no—
31#423D414yes—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8C2×C412yes—
37C2×C22⋊C8—21no—
38C4×M16—22no—
39C4×M16—22no—
40C4×M16—22no—
41C4×M16Q814yes—
42C2×#97—22no—
43C2×C4×C8—22no—
44C2×C4×C8—22no—
45C2×C4×C8—22no—
46C2×C4×C8—21no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D412yes—
50C4×C4⋊C4C2×C412yes—
51C4×C4⋊C4D412yes—
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54#376—21no—
55C43—21no—
56#341—28no—
57#336—22no—
58#336—22no—
59#334—28no—
60C23×C4C2×D411yescharacteristic
61C23×C4C2×Q811yescharacteristic
62C23×C4#4111yescharacteristic
63C2×M16—28no—
64C2×M16—24no—
65C2×M16—24no—
66C2×M16—44no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—84no—
70C22×C8—24no—
71C22×C8—24no—
72C22×C8—22no—
73C22×C8—42no—
74C22×C8—42no—
75C22×C8—42no—
76C22×C8C22⋊C412yes—
77C22×C8C22⋊C412yes—
78#117—24no—
79#117—42no—
80C2×C4⋊C4C4⋊C412yes—
81C2×C4⋊C4C4⋊C412yes—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C42—21no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—21no—
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—44no—
100C4⋊C8—44no—
101C4⋊C8—42no—
102C4⋊C8—42no—
103C22⋊C8—42no—
104#97—24no—
105#97—42no—
106C4×C8—22no—
107C4×C8—22no—
108C4×C8—22no—
109C4×C8—24no—
110C4×C8—22no—
111C4×C8—41no—
112C4×C8—42no—
113C4×C8—41no—
114#95—22no—
115#95—22no—
116C24C22⋊Q811yescharacteristic
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C2×C4⋊C411yescharacteristic
126C22×C4#11711yescharacteristic
127C22×C4#12112yes—
128C22×C4#12111yescharacteristic
129C22×C4#12111yescharacteristic
130C22×C4C22⋊Q812yes—
131C22×C4C22⋊Q811yescharacteristic
132C22×C4#12312yes—
133C22×C4#12512yes—
134C22×C4#12512yes—
135C22×C4C4⋊Q812yes—
136M16—28no—
137M16—48no—
138M16—44no—
139M16—84no—
140M16—84no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—48no—
147C2×C8—48no—
148C2×C8—44no—
149C2×C8—42no—
150C2×C8—44no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—84no—
154C2×C8#10414yes—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4#10814yes—
158C22⋊C4—24no—
159C22⋊C4—42no—
160C42—24no—
161C42—42no—
162C42—42no—
163C42—42no—
164C42—44no—
165C42C4×D412yes—
166C42C4×Q812yes—
167C42#12011yescharacteristic
168C42#12112yes—
169C42C22⋊Q812yes—
170C42C22⋊Q812yes—
171C42#12311yescharacteristic
172C42#12411yescharacteristic
173C42#12612yes—
174C42C4⋊Q811yescharacteristic
175C23—22no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23#38011yescharacteristic
180C23#39812yes—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4#38611yescharacteristic
200C2×C4#38712yes—
201C2×C4#38912yes—
202C2×C4#39111yescharacteristic
203C2×C4#39411yescharacteristic
204C2×C4#39511yescharacteristic
205C2×C4#39612yes—
206C2×C4#39811yescharacteristic
207C2×C4#40011yescharacteristic
208C2×C4C2×#10412yes—
209C2×C4#42112yes—
210C2×C4C2×#10812yes—
211C2×C4#43012yes—
212C2×C4#47114yes—
213C8—24no—
214C8—48no—
215C8—42no—
216C8—82no—
217C8—84no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—42no—
222C22—42no—
223C22#144111yescharacteristic
224C22#180612yes—
225C22#181011yescharacteristic
226C22#181111yescharacteristic
227C22#181512yes—
228C4—22no—
229C4—21no—
230C4—21no—
231C4—22no—
232C4—44no—
233C4—82no—
234C4#187112yes—
235C4#195014yes—
236C4#199411yescharacteristic
237C4#199511yescharacteristic
238C4#204612yes—
239C4#209012yes—
240C2—42no—
241C2#867012yes—
242C2#867212yes—
243C2#1120911yescharacteristic
244C2#1123311yescharacteristic
245C2#1123811yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#380
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×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
#213492
C2×#15662
C4×#3412
C2×#12992
#79032