Finite Group Databasebeta

← #325352

Subgroups — #325352

Order 512.

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

normal subgroups only

This group has 538 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#33291C211yesmaximal, characteristic
2#29123C212yesmaximal
3C8×C4⋊Q8C211yesmaximal, characteristic
4#10626C214yesmaximal
5#9900C212yesmaximal
6#8649C212yesmaximal
7#8649C212yesmaximal
8C4×#335C211yesmaximal, characteristic
9C2×C8×Q8C2212yes—
10#2859C2212yes—
11C4×C4⋊Q8C2211yescharacteristic
12#2100—22no—
13#2043—22no—
14#1991—22no—
15#1991—22no—
16#1984C2212yes—
17#1939—22no—
18#1933—24no—
19#1842C2211yescharacteristic
20#1834C2212yes—
21#1834C2211yescharacteristic
22#1834C2211yescharacteristic
23#1803C2214yes—
24#1803C2214yes—
25C8×C4⋊C4—22no—
26C4×C4⋊C8C2212yes—
27C42×C8C2211yescharacteristic
28C2×#335C2212yes—
29#1296C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32C2×C4⋊Q8D411yescharacteristic
33C2×#125—22no—
34C2×C4×Q8C2312yes—
35C8×Q8—24no—
36C8×Q8—24no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8C2314yes—
40C2×C4⋊C8C2312yes—
41C2×#103—24no—
42C2×#103—24no—
43C2×C4×C8—22no—
44C2×C4×C8—22no—
45C2×C4×C8D412yes—
46C2×C4×C8D412yes—
47C2×C4×C8C2311yescharacteristic
48C2×C4×C8C2311yescharacteristic
49#391D412yes—
50#389—22no—
51#389—22no—
52#389—22no—
53#389—22no—
54#389—22no—
55#384D411yescharacteristic
56#384C2312yes—
57#384C2312yes—
58#382—22no—
59C4×C4⋊C4—22no—
60C43C2311yescharacteristic
61#340—24no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—21no—
66#340—24no—
67#340—24no—
68#336—24no—
69#336—22no—
70#335—24no—
71C22×Q8C2×D412yes—
72C22×C8—22no—
73C22×C8—24no—
74C22×C8C2×D412yes—
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C4⋊Q8—24no—
78#125—44no—
79#125—44no—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4C2×D412yes—
96C2×C42—22no—
97C2×C42C2×D411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D412yes—
100C2×C42#4112yes—
101C2×C42C2411yescharacteristic
102C4⋊C8—24no—
103C4⋊C8—22no—
104C4⋊C8—22no—
105C4⋊C8—24no—
106C4⋊C8—44no—
107C4⋊C8—44no—
108#103—48no—
109#103—84no—
110C4×C8—24no—
111C4×C8—24no—
112C4×C8—22no—
113C4×C8—44no—
114C4×C8—82no—
115C4×C8#4114yes—
116#95—24no—
117#95—42no—
118#95—42no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8#13514yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4#12112yes—
130C22×C4#12112yes—
131C22×C4C22×D411yescharacteristic
132C22×C4C22×D411yescharacteristic
133C22×C4C2×#4111yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—42no—
140C2×C8—44no—
141C2×C8—44no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8#12412yes—
145C2×C8#12412yes—
146C4⋊C4—28no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—84no—
162C4⋊C4—84no—
163C42—22no—
164C42—21no—
165C42—21no—
166C42—42no—
167C42—42no—
168C42—42no—
169C42#13514yes—
170C42#13714yes—
171C42C2×#4112yes—
172C422−1+412yes—
173C23C2×#12111yescharacteristic
174Q8—24no—
175Q8—24no—
176Q8—42no—
177Q8—42no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4#46414yes—
195C2×C4#46414yes—
196C2×C4C2×#12411yescharacteristic
197C2×C4#53611yescharacteristic
198C2×C4#54712yes—
199C2×C4#55612yes—
200C2×C4C2×#13512yes—
201C2×C4C2×#13512yes—
202C2×C4C2×#13712yes—
203C2×C4#57611yescharacteristic
204C2×C4#57811yescharacteristic
205C8—22no—
206C8—22no—
207C8—42no—
208C8—84no—
209C8—82no—
210C22#254711yescharacteristic
211C22C2×#46412yes—
212C22#305011yescharacteristic
213C22#306111yescharacteristic
214C22#306712yes—
215C4—24no—
216C4—22no—
217C4—24no—
218C4—44no—
219C4—82no—
220C4—162no—
221C4—162no—
222C4#313111yescharacteristic
223C4#313911yescharacteristic
224C4#323112yes—
225C4#333612yes—
226C4#333912yes—
227C2#1912312yes—
228C2#1921712yes—
229C2#2870811yescharacteristic
230C2#2871911yescharacteristic
231C2#2873311yescharacteristic
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
#332912
#291232
C8×C4⋊Q82
#106262
#99002
#86492
C4×#3352