Finite Group Databasebeta

← #135394

Subgroups — #135394

Order 512.

327 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 84 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#20829C211yesmaximal, characteristic
2#12845C211yesmaximal, characteristic
3#10814C211yesmaximal, characteristic
4#7260C214yesmaximal
5#2921—21no—
6C2×#405C2211yescharacteristic
7C2×C22⋊C16C2212yes—
8#1849C411yescharacteristic, complemented
9#1849C411yescharacteristic, complemented
10#1837—21no—
11#1811—21no—
12#1810C2212yes—
13#1778C412yescomplemented
14C4×#108—21no—
15#1740—21no—
16#1735C2212yes—
17C2×H(3,ℤ/4)—21no—
18#1374—21no—
19#1374—21no—
20C22×M16D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#117—22no—
23C2×#117D411yescharacteristic
24C22×C16—22no—
25#433—44no—
26C2×#108—21no—
27C2×#108—21no—
28C2×#108C2×C412yes—
29C2×#108C2×C412yes—
30#423—21no—
31#423—21no—
32#407—22no—
33#405—24no—
34C4×M16—22no—
35C2×#97C2×C411yescharacteristic
36C2×#97C2×C411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C22⋊C16—24no—
40#342—42no—
41#341—22no—
42#341—22no—
43#340—24no—
44H(3,ℤ/4)—24no—
45C23×C4C2×D411yescharacteristic
46C2×M16—22no—
47C2×M16—22no—
48C2×M16—22no—
49C2×M16—22no—
50C2×M16—41no—
51C22×C8—41no—
52C22×C8—41no—
53C22×C8C22⋊C411yescharacteristic
54C22×C8C22⋊C411yescharacteristic
55C22×C8C22×C411yescharacteristic
56C22×C8C2×D411yescharacteristic
57C22×C8#4114yes—
58#117—22no—
59#117—22no—
60#117—24no—
61#117—42no—
62#117—42no—
63#117—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4Q1612yes—
66C2×C22⋊C4—42no—
67C2×C22⋊C4SD1612yes—
68C2×C42—42no—
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22⋊C411yescharacteristic
71C2×C16—42no—
72C2×C16—42no—
73C2×C16—42no—
74#108—42no—
75#108—42no—
76#108—42no—
77#108—42no—
78C4⋊C8—42no—
79C4⋊C8—42no—
80C22⋊C8—44no—
81#97—22no—
82#97—22no—
83#97—42no—
84C4×C8—22no—
85C4×C8—22no—
86C4×C8—42no—
87C24C2×SD1611yescharacteristic
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—82no—
93C22×C4#10312yes—
94C22×C4#10312yes—
95C22×C4C2×C22⋊C411yescharacteristic
96C22×C4#12312yes—
97C22×C4C2×Q1611yescharacteristic
98C22×C4AGL(1,ℤ/8)12yes—
99M16—44no—
100M16—42no—
101M16—42no—
102M16—42no—
103M16—42no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—44no—
114C2×C8—42no—
115C2×C8—42no—
116C2×C8#10412yes—
117C2×C8#11712yes—
118C2×C8#12312yes—
119C4⋊C4—24no—
120C4⋊C4—44no—
121C4⋊C4—82no—
122C22⋊C4—24no—
123C22⋊C4—44no—
124C22⋊C4—82no—
125C42—21no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—44no—
130C42—82no—
131C16—82no—
132C23—21no—
133C23—21no—
134C23—42no—
135C23C2×#10311yescharacteristic
136C23#48112yes—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—84no—
148C2×C4—164no—
149C2×C4#38111yescharacteristic
150C2×C4#41011yescharacteristic
151C2×C4#41011yescharacteristic
152C2×C4#41811yescharacteristic
153C2×C4C2×#10411yescharacteristic
154C2×C4#42111yescharacteristic
155C2×C4#48412yes—
156C8—22no—
157C8—22no—
158C8—22no—
159C8—41no—
160C8—82no—
161C8—82no—
162C22—22no—
163C22—22no—
164C22—24no—
165C22—81no—
166C22—81no—
167C22#150311yescharacteristic
168C22#151811yescharacteristic
169C22#182411yescharacteristic
170C4—22no—
171C4—42no—
172C4—42no—
173C4—164no—
174C4#150511yescharacteristic
175C4#151411yescharacteristic
176C4#182011yescharacteristic
177C4#182611yescharacteristic
178C2—21no—
179C2—21no—
180C2—42no—
181C2#871711yescharacteristic
182C2#1287712yes—
subgroupG/H
centerC2×C4#418
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
socleC22#1824
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #1849C4semidirect
9. #1849C4semidirect
13. #1778C4semidirect

Maximal subgroups

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

subgroupindex
#208292
#128452
#108142
#72602