Finite Group Databasebeta

← #318876

Subgroups — #318876

Order 512.

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

normal subgroups only

This group has 512 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#29289C212yesmaximal
2#22303C211yesmaximal, characteristic
3#22248C211yesmaximal, characteristic
4#10597C212yesmaximal
5#10589C212yesmaximal
6#9739C212yesmaximal
7#9622C212yesmaximal
8#8809C211yesmaximal, characteristic
9#8587C212yesmaximal
10C2×#500C2211yescharacteristic
11C4×C4⋊Q8C2212yes—
12C4×#125C2211yescharacteristic
13#2088—22no—
14#2086—22no—
15#1990C2212yes—
16#1989—22no—
17#1987C2212yes—
18#1987C2212yes—
19#1949—22no—
20#1948C2212yes—
21#1947C2212yes—
22#1940—22no—
23#1939C2212yes—
24#1921—22no—
25#1921—22no—
26#1917—22no—
27#1842C2211yescharacteristic
28#1842C2212yes—
29#1834C2211yescharacteristic
30#1770C2212yes—
31#1744C2211yescharacteristic
32C2×#334C2212yes—
33C2×#334C2211yescharacteristic
34C2×#334C2211yescharacteristic
35C2×#333C2212yes—
36C2×#333C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39C2×C4⋊Q8C2312yes—
40C2×#125D411yescharacteristic
41C2×C4×Q8—22no—
42#500—22no—
43#500Q814yes—
44C2×#107—22no—
45C2×#107—22no—
46C2×#106—22no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#97—22no—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#391—22no—
58#389—22no—
59#389—22no—
60#389D412yes—
61#387—22no—
62#384C2312yes—
63#382D411yescharacteristic
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4C2312yes—
68C43C2311yescharacteristic
69#340—22no—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#336—22no—
79#336—22no—
80#336D412yes—
81#336D412yes—
82#334—24no—
83#334—41no—
84#334—41no—
85#333—42no—
86#333—42no—
87C22×Q8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8C2×D411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D412yes—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8C2×Q814yes—
98#125—24no—
99C4×Q8—84no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C42—22no—
111C2×C42—42no—
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42#4112yes—
116C2×C42C2411yescharacteristic
117#107—28no—
118#107—82no—
119#106—82no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—44no—
127C4⋊C8—82no—
128C4⋊C8—82no—
129C4⋊C8—82no—
130#97—82no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8C2×Q812yes—
134C4×C8#4112yes—
135#95—22no—
136#95—22no—
137#95—22no—
138#95—42no—
139#95—42no—
140C2×Q8—24no—
141C2×Q8—42no—
142C2×Q8—82no—
143C2×Q8—82no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4#12112yes—
149C22×C4#12112yes—
150C22×C4C22×D411yescharacteristic
151C22×C4C22×D411yescharacteristic
152C22×C4C2×#4111yescharacteristic
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8C22⋊Q814yes—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C4⋊C4—84no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—42no—
188C42—42no—
189C42—42no—
190C42—84no—
191C42#13714yes—
192C42C22×Q811yescharacteristic
193C42C2×#4111yescharacteristic
194C42H(5,𝔽2)11yescharacteristic
195C422−1+411yescharacteristic
196C23C2×#12111yescharacteristic
197Q8—84no—
198Q8—84no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#47014yes—
218C2×C4C2×C22⋊Q811yescharacteristic
219C2×C4#53711yescharacteristic
220C2×C4D4×Q811yescharacteristic
221C2×C4#55011yescharacteristic
222C2×C4#55411yescharacteristic
223C2×C4#55611yescharacteristic
224C2×C4C2×#13712yes—
225C2×C4#57712yes—
226C2×C4#57712yes—
227C2×C4#57812yes—
228C8—24no—
229C8—41no—
230C8—41no—
231C8—82no—
232C8—82no—
233C8—82no—
234C22#253811yescharacteristic
235C22#304612yes—
236C22C2×#47011yescharacteristic
237C22#306011yescharacteristic
238C22#306812yes—
239C4—21no—
240C4—21no—
241C4—24no—
242C4—42no—
243C4—82no—
244C4—84no—
245C4—82no—
246C4—162no—
247C4#308212yes—
248C4#326011yescharacteristic
249C4#326111yescharacteristic
250C4#338712yes—
251C4#340012yes—
252C2#1945312yes—
253C2#1954912yes—
254C2#2827411yescharacteristic
255C2#2829411yescharacteristic
256C2#2829911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#292892
#223032
#222482
#105972
#105892
#97392
#96222
#88092
#85872