Finite Group Databasebeta

← #135609

Subgroups — #135609

Order 512.

356 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 92 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#22705C211yesmaximal, characteristic
2#11487C211yesmaximal, characteristic
3#8616C211yesmaximal, characteristic
4#7296C212yesmaximal, complemented
5#7217C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7D4×C4⋊C4—21no—
8#2162C2212yes—
9#1947—21no—
10#1852—22no—
11#1834C2211yescharacteristic
12#1808C412yes—
13#1800—21no—
14C4×#107C2211yescharacteristic
15C2×#340C412yes—
16#1567C2212yes—
17#1314—22no—
18#1292—21no—
19C23×C8C2×C412yes—
20C2×C4×D4—41no—
21C2×C4×D4D411yescharacteristic
22C22×C4⋊C4—22no—
23C8×D4—22no—
24C8×D4—22no—
25C2×#107—21no—
26C2×#107—22no—
27#423—22no—
28C2×C4⋊C8D411yescharacteristic
29C2×#100—41no—
30C2×C22⋊C8D412yes—
31C2×C4×C8C2311yescharacteristic
32#389—41no—
33#385—42no—
34#384—21no—
35#380—22no—
36C4×C4⋊C4—21no—
37C4⋊C16—22no—
38C22⋊C16—22no—
39C4×C16—22no—
40#343—41no—
41#340—22no—
42#340—21no—
43#340C2×C411yescharacteristic
44#340C2×C412yes—
45#340C2×C411yescharacteristic
46#335—22no—
47#335—22no—
48C22×D4D811yescharacteristic
49C23×C4—42no—
50C23×C4C22⋊C412yes—
51C2×M16—42no—
52C22×C8—22no—
53C22×C8—22no—
54C22×C8—82no—
55C22×C8C22⋊C412yes—
56C22×C8C22×C411yescharacteristic
57C22×C8C2×D411yescharacteristic
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—84no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—82no—
69C2×C4⋊C4D811yescharacteristic
70C2×C22⋊C4—42no—
71C2×C22⋊C4SD1612yes—
72C2×C42—41no—
73C2×C42C2×D411yescharacteristic
74C2×C16—42no—
75#107—42no—
76#107—82no—
77C4⋊C8—22no—
78C4⋊C8—42no—
79C4⋊C8—42no—
80#100—82no—
81C22⋊C8—24no—
82C4×C8#4112yes—
83C4×C8#4112yes—
84#95—42no—
85#95—41no—
86C24#10212yes—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4SD3212yes—
90C2×D4Q3212yes—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4—84no—
99C22×C4—84no—
100C22×C4—162no—
101C22×C4#10212yes—
102C22×C4C2×C22⋊C411yescharacteristic
103C22×C4C2×D811yescharacteristic
104C22×C4C2×SD1611yescharacteristic
105M16—82no—
106M16—82no—
107C2×C8—22no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—42no—
111C2×C8—82no—
112C2×C8—82no—
113C2×C8—82no—
114C2×C8#11711yescharacteristic
115C2×C8#11711yescharacteristic
116C2×C8#12312yes—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—84no—
125C4⋊C4—82no—
126C4⋊C4—161no—
127C22⋊C4—24no—
128C22⋊C4—84no—
129C42—82no—
130C42#12312yes—
131C42#13712yes—
132C16—82no—
133C23—22no—
134C23—22no—
135C23—42no—
136C23—42no—
137C23#35814yes—
138C23C2×#10211yescharacteristic
139D4—24no—
140D4—24no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—44no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—82no—
153C2×C4—81no—
154C2×C4—82no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—82no—
158C2×C4—168no—
159C2×C4#38111yescharacteristic
160C2×C4#41911yescharacteristic
161C2×C4#48012yes—
162C2×C4#48312yes—
163C2×C4C2×SD3211yescharacteristic
164C2×C4C2×Q3211yescharacteristic
165C2×C4#50912yes—
166C8—22no—
167C8—21no—
168C8—21no—
169C8—42no—
170C8—162no—
171C22—21no—
172C22—21no—
173C22—24no—
174C22—44no—
175C22—44no—
176C22—42no—
177C22—42no—
178C22#182311yescharacteristic
179C22C2×#35811yescharacteristic
180C22#214011yescharacteristic
181C4—21no—
182C4—22no—
183C4—42no—
184C4—82no—
185C4—161no—
186C4—162no—
187C4#182711yescharacteristic
188C4#182911yescharacteristic
189C4#223312yes—
190C4#223612yes—
191C2—21no—
192C2—21no—
193C2—44no—
194C2#876411yescharacteristic
195C2#1286711yescharacteristic
196C2#1287411yescharacteristic
subgroupG/H
center, socleC22#1823
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7296C2semidirect
5. #7217C2semidirect

Maximal subgroups

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

subgroupindex
#227052
#114872
#86162
#72962
#72172