Finite Group Databasebeta

← #123144

Subgroups — #123144

Order 512.

292 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 46 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#29286C211yesmaximal, characteristic
2#12289C212yesmaximal
3#10522C211yesmaximal, characteristic
4#7091C212yesmaximal
5#7063C211yesmaximal, characteristic
6#3249C2211yescharacteristic
7C4×C4⋊Q8—21no—
8#2235—22no—
9#2086—21no—
10#2086C2212yes—
11#1994—21no—
12#1988—22no—
13#1947—22no—
14#1842C2211yescharacteristic
15#1550—21no—
16#1348—22no—
17#1322C2212yes—
18#1292C2211yescharacteristic
19#533—21no—
20C2×C4⋊Q8—21no—
21C2×C4×Q8—41no—
22#477—41no—
23#467—22no—
24#462—41no—
25C2×#107—21no—
26C2×#107D412yes—
27#423—21no—
28C2×C4⋊C8—22no—
29C2×C4⋊C8D411yescharacteristic
30C2×C4⋊C8C2311yescharacteristic
31#391—41no—
32#389—22no—
33#389—41no—
34#384—21no—
35C4×C4⋊C4—22no—
36C4×C4⋊C4—41no—
37C43D411yescharacteristic
38#367—42no—
39#358—42no—
40C22⋊C16—22no—
41#340—22no—
42#340—21no—
43#340—22no—
44#340D412yes—
45#339—41no—
46#339—41no—
47#333—41no—
48#332—41no—
49C22×Q8—41no—
50C2×Q16—22no—
51C2×M16—41no—
52C22×C8—21no—
53C22×C8C2×D412yes—
54C4⋊Q8—24no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—42no—
58#125—41no—
59C22⋊Q8—41no—
60C22⋊Q8—41no—
61C4×Q8—41no—
62C4×Q8—41no—
63C4×Q8—82no—
64C4×Q8—82no—
65#117—41no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—42no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—42no—
75C2×C42D811yescharacteristic
76C2×C42D811yescharacteristic
77C2×C42C2×D411yescharacteristic
78C2×C16—42no—
79#107—22no—
80#107—42no—
81#107—81no—
82C4⋊C8—41no—
83C4⋊C8—41no—
84C4⋊C8—44no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87#103—42no—
88#95—42no—
89#95—42no—
90C2×Q8—44no—
91C2×Q8—41no—
92C2×Q8—41no—
93C2×Q8—82no—
94C2×Q8—82no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4#12011yescharacteristic
100C22×C4C2×D811yescharacteristic
101C22×C4AGL(1,ℤ/8)11yescharacteristic
102Q16—82no—
103M16—81no—
104M16—81no—
105C2×C8—22no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—41no—
109C2×C8—81no—
110C4⋊C4—24no—
111C4⋊C4—42no—
112C4⋊C4—44no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—42no—
117C4⋊C4—41no—
118C4⋊C4—42no—
119C4⋊C4—82no—
120C4⋊C4—81no—
121C4⋊C4—82no—
122C4⋊C4—81no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C22⋊C4—81no—
126C42—22no—
127C42—22no—
128C42—21no—
129C42—21no—
130C42—21no—
131C42—41no—
132C42—42no—
133C42—84no—
134C42—81no—
135C42SD3212yes—
136C16—82no—
137C23#44711yescharacteristic
138Q8—88no—
139Q8—81no—
140Q8—81no—
141Q8—161no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—44no—
155C2×C4—82no—
156C2×C4—82no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4#45411yescharacteristic
163C2×C4B(4,𝔽2)11yescharacteristic
164C2×C4C2×SD3211yescharacteristic
165C2×C4#50911yescharacteristic
166C8—42no—
167C8—81no—
168C8—161no—
169C22—21no—
170C22—21no—
171C22#159811yescharacteristic
172C22#218711yescharacteristic
173C22#219111yescharacteristic
174C4—24no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—41no—
180C4—84no—
181C4—161no—
182C4—162no—
183C4—161no—
184C4#219811yescharacteristic
185C4#220511yescharacteristic
186C2—21no—
187C2—21no—
188C2#1215311yescharacteristic
189C2#1216111yescharacteristic
190C2#1217411yescharacteristic
subgroupG/H
center, socleC22#1598
commutator, Frattini, cosocleC2×C4⋊C8C23
prime radicalC23#447
LCSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#292862
#122892
#105222
#70912
#70632