Finite Group Databasebeta

← #318089

Subgroups — #318089

Order 512.

616 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 166 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#28265C212yesmaximal
2#28196C212yesmaximal
3C2×#1591C211yesmaximal, characteristic
4C2×C8×Q16C211yesmaximal, characteristic
5#10437C214yesmaximal
6#8814C211yesmaximal, characteristic
7#8655C212yesmaximal
8#8645C212yesmaximal
9C2×#446C2212yes—
10C2×C8×Q8C2212yes—
11C2×C4×Q16C2211yescharacteristic
12C2×C4×Q16C2211yescharacteristic
13#2582C2214yes—
14#2086—22no—
15#2068—24no—
16#2056—22no—
17#1969—22no—
18#1942—22no—
19#1935—24no—
20#1845C2212yes—
21#1844C2212yes—
22#1834C2212yes—
23#1803C2214yes—
24#1803C2214yes—
25#1591—24no—
26C8×Q16—24no—
27C2×#335C2211yescharacteristic
28C2×#335C2211yescharacteristic
29C2×Q8⋊C8C2212yes—
30C2×Q8⋊C8C2212yes—
31C2×C82C2211yescharacteristic
32#1296C2214yes—
33C22×Q16D411yescharacteristic
34C22×Q16D411yescharacteristic
35C2×C4×Q8C2312yes—
36C2×C4×Q8C2312yes—
37#446—24no—
38#446—24no—
39C8×Q8—24no—
40C8×Q8—24no—
41C4×Q16—22no—
42C4×Q16—22no—
43C4×Q16—22no—
44C4×Q16—22no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#107D411yescharacteristic
48C2×#107D411yescharacteristic
49C2×#106—42no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—24no—
55C2×#103—24no—
56C2×#103D412yes—
57C2×#103D412yes—
58C2×#97—22no—
59C2×C4×C8—22no—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#398—44no—
64#391—24no—
65#391—24no—
66#389—24no—
67#389—24no—
68#384C2314yes—
69#340—22no—
70#340—22no—
71#340—24no—
72#340—24no—
73#335—22no—
74#335—22no—
75Q8⋊C8—24no—
76Q8⋊C8—24no—
77C82D414yes—
78C22×Q8C2×D412yes—
79C22×Q8C2×D412yes—
80C2×Q16—24no—
81C2×Q16—22no—
82C2×Q16—22no—
83C2×Q16—24no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4C2×D412yes—
106C2×C4⋊C4C2×D412yes—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42C2411yescharacteristic
110#107—22no—
111#107—22no—
112#107—44no—
113#107—82no—
114#106—84no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—44no—
120C4⋊C8—44no—
121#103—24no—
122#103—24no—
123#103—48no—
124#103—84no—
125#97—44no—
126C4×C8—21no—
127C4×C8—44no—
128C4×C8C2×D412yes—
129C4×C8C2×D412yes—
130C4×C8C2×D412yes—
131C4×C8#4112yes—
132C4×C8#4112yes—
133#95—44no—
134#95—44no—
135#95—44no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8#13514yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—44no—
151C22×C4C22×D411yescharacteristic
152C22×C4C22×D411yescharacteristic
153C22×C4H(5,𝔽2)11yescharacteristic
154Q16—28no—
155Q16—44no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—44no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8#12112yes—
168C2×C8#12112yes—
169C2×C8#12112yes—
170C2×C8#12112yes—
171C2×C8#13714yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—84no—
185C4⋊C4—164no—
186C42—42no—
187C42—42no—
188C42—42no—
189C42—42no—
190C42C22×D411yescharacteristic
191C42C2×#4111yescharacteristic
192C42C2×#4111yescharacteristic
193C422−1+411yescharacteristic
194C23D4211yescharacteristic
195Q8—24no—
196Q8—24no—
197Q8—42no—
198Q8—42no—
199Q8—44no—
200Q8—42no—
201Q8—42no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—84no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4C2×#12111yescharacteristic
219C2×C4C2×#12111yescharacteristic
220C2×C4#53711yescharacteristic
221C2×C4#54711yescharacteristic
222C2×C4#54711yescharacteristic
223C2×C4#55511yescharacteristic
224C2×C4C2×#13512yes—
225C2×C4C2×#13712yes—
226C2×C4#57712yes—
227C2×C4#57812yes—
228C8—24no—
229C8—22no—
230C8—22no—
231C8—22no—
232C8—82no—
233C8—82no—
234C8#47014yes—
235C22#253711yescharacteristic
236C22#327812yes—
237C22#328512yes—
238C22#329211yescharacteristic
239C22#329311yescharacteristic
240C4—21no—
241C4—21no—
242C4—44no—
243C4—82no—
244C4—82no—
245C4—82no—
246C4—164no—
247C4#305011yescharacteristic
248C4C2×#47011yescharacteristic
249C4#305911yescharacteristic
250C4#306011yescharacteristic
251C4#333612yes—
252C4#335612yes—
253C2#1994212yes—
254C2#1994412yes—
255C2#2822611yescharacteristic
256C2#2823211yescharacteristic
257C2#2824711yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#282652
#281962
C2×#15912
C2×C8×Q162
#104372
#88142
#86552
#86452