Finite Group Databasebeta

← #353072

Subgroups — #353072

Order 512.

622 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 234 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#33311C211yesmaximal, characteristic
2#33263C212yesmaximal
3#33256C211yesmaximal, characteristic
4#30307C212yesmaximal
5#8821C212yesmaximal
6#8820C212yesmaximal
7#8819C212yesmaximal
8#8819C212yesmaximal
9#7842C211yesmaximal, characteristic
10C2×#499—22no—
11#2859C2212yes—
12#2859C2212yes—
13#2085—24no—
14#2059—24no—
15#1944—24no—
16#1943—24no—
17#1941—22no—
18#1941—22no—
19#1846—22no—
20#1845C2212yes—
21#1845C2212yes—
22#1845C2212yes—
23#1844C2211yescharacteristic
24#1844C2211yescharacteristic
25#1844C2212yes—
26#1844C2211yescharacteristic
27#1844C2212yes—
28#1844C2211yescharacteristic
29#1834C2212yes—
30#1834C2212yes—
31#1834C2212yes—
32#1834C2212yes—
33#1834C2212yes—
34#1744C2211yescharacteristic
35C4×#97C2211yescharacteristic
36C42×C8C2211yescharacteristic
37C2×#322C2212yes—
38C2×#322C2212yes—
39C2×#125—22no—
40C2×#125—22no—
41#499—48no—
42C2×#107—24no—
43C2×#107—24no—
44C2×#107—24no—
45C2×#106—22no—
46C2×#106—22no—
47C2×#106—22no—
48C2×#106—22no—
49C2×#106—24no—
50C2×#106—24no—
51C2×#97D414yes—
52C2×C4×C8D412yes—
53C2×C4×C8D412yes—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2312yes—
56C2×C4×C8C2312yes—
57C2×C4×C8C2311yescharacteristic
58#389—22no—
59#389—22no—
60#389—22no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#384C2312yes—
67#384C2312yes—
68#384C2312yes—
69#384C2312yes—
70#382—22no—
71#382—22no—
72C43C2311yescharacteristic
73#340—24no—
74#340—24no—
75#340—24no—
76#340—24no—
77#340—24no—
78#336D414yes—
79#322Q818yes—
80#322Q818yes—
81C22×C8C2×D414yes—
82C22×C8C2×D412yes—
83C22×C8C2×D414yes—
84C22×C8C2×D412yes—
85#125—48no—
86#125—84no—
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—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D412yes—
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D412yes—
107C2×C42C2411yescharacteristic
108#107—48no—
109#107—48no—
110#107—84no—
111#106—48no—
112#106—44no—
113#106—44no—
114#106—44no—
115#106—44no—
116#106—84no—
117#97—44no—
118C4×C8—24no—
119C4×C8—24no—
120C4×C8C2×Q814yes—
121C4×C8C2×Q814yes—
122C4×C8C2×Q814yes—
123C4×C8C2×Q814yes—
124C4×C8C2×Q814yes—
125C4×C8C2×Q814yes—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4#12712yes—
135C22×C4#12712yes—
136C22×C4C22×D411yescharacteristic
137C22×C4C22×D411yescharacteristic
138C22×C4C22×D411yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—24no—
142C2×C8—24no—
143C2×C8—24no—
144C2×C8—24no—
145C2×C8C4⋊Q818yes—
146C2×C8C4⋊Q814yes—
147C2×C8C4⋊Q818yes—
148C2×C8C4⋊Q814yes—
149C2×C8AGL(1,ℤ/8)14yes—
150C2×C8#13714yes—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C42—22no—
168C42—22no—
169C42—24no—
170C42—24no—
171C42C22×Q812yes—
172C42C22×Q812yes—
173C23C2×#12711yescharacteristic
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4C2×C4⋊Q811yescharacteristic
193C2×C4C2×C4⋊Q812yes—
194C2×C4C2×C4⋊Q811yescharacteristic
195C2×C4C2×C4⋊Q812yes—
196C2×C4C2×AGL(1,ℤ/8)12yes—
197C2×C4C2×#13712yes—
198C2×C4#57514yes—
199C2×C4#57714yes—
200C8—28no—
201C8—24no—
202C8—28no—
203C8#50118yes—
204C22#286811yescharacteristic
205C22#315012yes—
206C22#315111yescharacteristic
207C22#315211yescharacteristic
208C22#315412yes—
209C4—42no—
210C4—42no—
211C4—162no—
212C4—162no—
213C4—162no—
214C4—162no—
215C4C2×#50112yes—
216C4#316314yes—
217C4#316512yes—
218C4#316614yes—
219C2#1969312yes—
220C2#1969412yes—
221C2#3331511yescharacteristic
222C2#3331611yescharacteristic
223C2#3331811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#333112
#332632
#332562
#303072
#88212
#88202
#88192
#78422