Finite Group Databasebeta

← #318362

Subgroups — #318362

Order 512.

596 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 158 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#28206C212yesmaximal
2#22248C211yesmaximal, characteristic
3#22238C211yesmaximal, characteristic
4C2×#1676C212yesmaximal
5#10482C214yesmaximal
6#8811C211yesmaximal, characteristic
7#8649C212yesmaximal
8#8594C212yesmaximal
9C2×#500C2211yescharacteristic
10C2×#477C2214yes—
11C2×C8×Q8C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×C4⋊Q8C2211yescharacteristic
14#2083—22no—
15#2043—22no—
16#1990C2212yes—
17#1984C2212yes—
18#1928—24no—
19#1921—24no—
20#1916—24no—
21#1834C2211yescharacteristic
22#1831C2212yes—
23#1831C2211yescharacteristic
24#1803C2214yes—
25#1744C2211yescharacteristic
26#1676—28no—
27C2×#335C2212yes—
28C2×#334C2212yes—
29C2×#333C2214yes—
30#1296C2214yes—
31C2×C4⋊Q8D411yescharacteristic
32C2×C4⋊Q8C2312yes—
33C2×C4×Q8—22no—
34C2×C4×Q8C2312yes—
35#500—22no—
36#500D414yes—
37#477—28no—
38#477—44no—
39C8×Q8—24no—
40C8×Q8—24no—
41C2×#107—22no—
42C2×#107—24no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8C2314yes—
45C2×C4⋊C8C2312yes—
46C2×#103—24no—
47C2×#103—24no—
48C2×C4×C8—22no—
49C2×C4×C8C2311yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#391—22no—
52#391D412yes—
53#389—22no—
54#389—22no—
55#389—22no—
56#384D411yescharacteristic
57#384C2312yes—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C43C2311yescharacteristic
62#340—22no—
63#340—24no—
64#340—24no—
65#336—24no—
66#336—22no—
67#336—22no—
68#336—22no—
69#336D414yes—
70#335—24no—
71#334—42no—
72#333—28no—
73C22×Q8—22no—
74C22×Q8C2×D412yes—
75C22×C8—22no—
76C22×C8—24no—
77C22×C8C2×D412yes—
78C22×C8C2×D412yes—
79C4⋊Q8—24no—
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82C4⋊Q8—22no—
83C4⋊Q8C2×D414yes—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—84no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4C2×D412yes—
99C2×C42—22no—
100C2×C42—42no—
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D412yes—
104C2×C42#4112yes—
105C2×C42C2411yescharacteristic
106#107—28no—
107#107—48no—
108C4⋊C8—22no—
109C4⋊C8—24no—
110C4⋊C8—22no—
111C4⋊C8—24no—
112C4⋊C8—44no—
113#103—48no—
114#103—48no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—44no—
118C4×C8C2×D412yes—
119C4×C8#4112yes—
120#95—24no—
121#95—42no—
122#95—42no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—24no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—42no—
130C2×Q8—82no—
131C2×Q8—82no—
132C2×Q8#13514yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4#12112yes—
138C22×C4#12112yes—
139C22×C4C22×D411yescharacteristic
140C22×C4C22×D411yescharacteristic
141C22×C4C2×#4111yescharacteristic
142C2×C8—24no—
143C2×C8—22no—
144C2×C8—24no—
145C2×C8—42no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8#12114yes—
151C4⋊C4—28no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C4⋊C4—84no—
165C42—22no—
166C42—24no—
167C42—22no—
168C42—42no—
169C42—42no—
170C42—42no—
171C42—84no—
172C42C22×D411yescharacteristic
173C42C2×#4111yescharacteristic
174C42H(5,𝔽2)11yescharacteristic
175C422−1+411yescharacteristic
176C23C2×#12111yescharacteristic
177Q8—24no—
178Q8—24no—
179Q8—42no—
180Q8—42no—
181Q8—84no—
182Q8—84no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4C2×#12111yescharacteristic
202C2×C4#53711yescharacteristic
203C2×C4D4211yescharacteristic
204C2×C4#54711yescharacteristic
205C2×C4#54711yescharacteristic
206C2×C4#55511yescharacteristic
207C2×C4C2×#13512yes—
208C2×C4#57812yes—
209C2×C4#57814yes—
210C8—24no—
211C8—42no—
212C8—84no—
213C8—82no—
214C22#253711yescharacteristic
215C22#304412yes—
216C22#305011yescharacteristic
217C22#305911yescharacteristic
218C22#306812yes—
219C4—21no—
220C4—21no—
221C4—44no—
222C4—42no—
223C4—42no—
224C4—82no—
225C4—84no—
226C4—162no—
227C4#304412yes—
228C4#326112yes—
229C4#328512yes—
230C4#333612yes—
231C2#1940612yes—
232C2#1953112yes—
233C2#2822611yescharacteristic
234C2#2824311yescharacteristic
235C2#2826111yescharacteristic
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
#282062
#222482
#222382
C2×#16762
#104822
#88112
#86492
#85942