Finite Group Databasebeta

← #146125

Subgroups — #146125

Order 512.

205 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 66 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#12033C211yesmaximal, characteristic
2#11637C211yesmaximal, characteristic
3#8649C211yesmaximal, characteristic
4#8617C211yesmaximal, characteristic
5#7288C211yesmaximal, characteristic
6#7217C211yesmaximal, characteristic
7#7217C211yesmaximal, characteristic
8C2×C8×Q8C2211yescharacteristic
9#2162C2211yescharacteristic
10#2162C2211yescharacteristic
11#1834C2211yescharacteristic
12#1803—21no—
13#1801—21no—
14#1567C2211yescharacteristic
15#1567C2211yescharacteristic
16C2×#335C2211yescharacteristic
17#1372—22no—
18Q8⋊C16—22no—
19#1296—21no—
20#1292—21no—
21C2×C4×Q8D411yescharacteristic
22C8×Q8—21no—
23C8×Q8—21no—
24C8×Q8—21no—
25C8×Q8—21no—
26#423—21no—
27#423—21no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8D411yescharacteristic
31C2×#103—41no—
32C2×#101—41no—
33C2×C4×C8—21no—
34C2×C4×C8C2311yescharacteristic
35#384—21no—
36C4⋊C16—21no—
37C4⋊C16—21no—
38C22⋊C16—21no—
39C22⋊C16—21no—
40C4×C16—21no—
41C4×C16D412yes—
42#343—41no—
43#340—21no—
44#340—21no—
45#340—41no—
46#335—21no—
47#335—21no—
48#335—21no—
49#335—21no—
50#335—21no—
51#335—21no—
52C22×Q8D811yescharacteristic
53C2×M16—41no—
54C2×M16—41no—
55C22×C8—21no—
56C22×C8—41no—
57C22×C8#4111yescharacteristic
58C22×C8#4111yescharacteristic
59C4×Q8—21no—
60C4×Q8—21no—
61C4×Q8—21no—
62C4×Q8—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4D811yescharacteristic
68C2×C42—21no—
69C2×C42C2×D411yescharacteristic
70C2×C16—22no—
71C2×C16—41no—
72C4⋊C8—21no—
73C4⋊C8—21no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76C4⋊C8—41no—
77C4⋊C8—41no—
78C4⋊C8—41no—
79C4⋊C8—41no—
80C4⋊C8—41no—
81C4⋊C8—41no—
82#103—82no—
83#101—82no—
84C4×C8—41no—
85C4×C8—41no—
86C4×C8C2×D411yescharacteristic
87C4×C8#4111yescharacteristic
88C4×C8#4111yescharacteristic
89C4×C8#4111yescharacteristic
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8SD3212yes—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—81no—
99C22×C4#12311yescharacteristic
100C22×C4C2×D811yescharacteristic
101C22×C4#13711yescharacteristic
102M16—81no—
103M16—81no—
104M16—81no—
105M16—81no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—81no—
110C2×C8—81no—
111C2×C8—81no—
112C2×C8—81no—
113C2×C8—81no—
114C2×C8—81no—
115C2×C8#12311yescharacteristic
116C2×C8#12411yescharacteristic
117C2×C8#12611yescharacteristic
118C2×C8#12611yescharacteristic
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—81no—
124C4⋊C4—81no—
125C4⋊C4—81no—
126C4⋊C4—81no—
127C4⋊C4—161no—
128C42—41no—
129C42—41no—
130C42#12111yescharacteristic
131C42#12311yescharacteristic
132C42#13511yescharacteristic
133C42#13711yescharacteristic
134C16—42no—
135C16—81no—
136C23#48011yescharacteristic
137Q8—22no—
138Q8—22no—
139Q8—41no—
140Q8—41no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—81no—
149C2×C4—161no—
150C2×C4—161no—
151C2×C4—161no—
152C2×C4#39711yescharacteristic
153C2×C4#46611yescharacteristic
154C2×C4#47011yescharacteristic
155C2×C4#48011yescharacteristic
156C2×C4#48511yescharacteristic
157C2×C4#48511yescharacteristic
158C2×C4C2×SD3211yescharacteristic
159C2×C4#50811yescharacteristic
160C2×C4#50911yescharacteristic
161C2×C4#51011yescharacteristic
162C8—21no—
163C8—21no—
164C8—21no—
165C8—21no—
166C8—81no—
167C8—161no—
168C8—161no—
169C8—161no—
170C22—21no—
171C22—21no—
172C22#207511yescharacteristic
173C22#223311yescharacteristic
174C22#223711yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—81no—
180C4—321no—
181C4#208011yescharacteristic
182C4#208211yescharacteristic
183C4#221711yescharacteristic
184C4#222011yescharacteristic
185C4#223411yescharacteristic
186C4#223511yescharacteristic
187C2—21no—
188C2—21no—
189C2#1063911yescharacteristic
190C2#1349411yescharacteristic
191C2#1349711yescharacteristic
subgroupG/H
center, socleC22#2075
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#480
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#120332
#116372
#86492
#86172
#72882
#72172