Finite Group Databasebeta

← #124642

Subgroups — #124642

Order 512.

291 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 80 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#21727C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic
3#8800C211yesmaximal, characteristic
4#7127C212yesmaximal
5#7126C212yesmaximal
6C2×#498C411yescharacteristic
7C4×C4⋊Q8—21no—
8#1990—21no—
9#1944—21no—
10#1944C2212yes—
11#1845C411yescharacteristic
12#1776—21no—
13#1776C412yes—
14C4×#97C2211yescharacteristic
15#1377C2212yes—
16#1376—21no—
17#1375—21no—
18#1281—21no—
19#1276—21no—
20#1275C2211yescharacteristic
21#1275C2211yescharacteristic
22C2×C4⋊Q8—21no—
23C2×C4×Q8—41no—
24C2×M32—22no—
25#498—22no—
26#498C814yescomplemented
27#431—21no—
28#431—21no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#106—21no—
32C2×#106C2×C412yes—
33C2×#97C2×C411yescharacteristic
34C2×#97D411yescharacteristic
35C2×#97D411yescharacteristic
36C2×#97C2311yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39#391—41no—
40#389—22no—
41#389—41no—
42#384—21no—
43C4×C4⋊C4—22no—
44C4×C4⋊C4—41no—
45C43D411yescharacteristic
46#340—22no—
47#340—22no—
48C22×Q8—41no—
49C22×C8C22⋊C411yescharacteristic
50C22×C8C22⋊C411yescharacteristic
51C22×C8C22×C411yescharacteristic
52C22×C8C2×D411yescharacteristic
53C4⋊Q8—24no—
54C4⋊Q8—42no—
55C4×Q8—84no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—42no—
62C2×C42—21no—
63C2×C42—21no—
64C2×C42—42no—
65C2×C42C22⋊C411yescharacteristic
66C2×C42C22⋊C411yescharacteristic
67C2×C42C2×D411yescharacteristic
68M32—42no—
69M32—42no—
70C2×C16—42no—
71#107—42no—
72#107—42no—
73#106—24no—
74#106—22no—
75#106C2×C814yes—
76C4⋊C8—41no—
77C4⋊C8—41no—
78C22⋊C8—41no—
79C22⋊C8—41no—
80#97—22no—
81#97—22no—
82#97—41no—
83#97—41no—
84C4×C8—41no—
85C4×C8—41no—
86C4×C8—41no—
87C4×C8C2×C812yes—
88C4×C8M1612yes—
89#95—42no—
90#95—42no—
91C2×Q8—44no—
92C2×Q8—82no—
93C2×Q8—82no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4C2×C22⋊C411yescharacteristic
99C22×C4AGL(1,ℤ/8)11yescharacteristic
100C22×C4AGL(1,ℤ/8)11yescharacteristic
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—81no—
111C2×C8—81no—
112C2×C8C22⋊C812yes—
113C2×C8C22×C811yescharacteristic
114C2×C8C2×M1611yescharacteristic
115C2×C8#13712yes—
116C4⋊C4—24no—
117C4⋊C4—44no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—82no—
122C4⋊C4—82no—
123C4⋊C4—84no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42—21no—
128C42—42no—
129C42—42no—
130C42—84no—
131C42C22⋊C812yes—
132C42#10112yes—
133C16—82no—
134C23#41811yescharacteristic
135Q8—88no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4C2×C22⋊C811yescharacteristic
153C2×C4#41011yescharacteristic
154C2×C4C2×#10111yescharacteristic
155C2×C4#41911yescharacteristic
156C2×C4#42111yescharacteristic
157C2×C4#42111yescharacteristic
158C8—21no—
159C8—21no—
160C8—22no—
161C8—22no—
162C8—41no—
163C8—161no—
164C8—161no—
165C22—21no—
166C22—21no—
167C22#147811yescharacteristic
168C22#151411yescharacteristic
169C22#155611yescharacteristic
170C4—24no—
171C4—42no—
172C4—41no—
173C4—41no—
174C4—84no—
175C4—162no—
176C4#146111yescharacteristic
177C4#147911yescharacteristic
178C4#151511yescharacteristic
179C4#153711yescharacteristic
180C2—21no—
181C2—21no—
182C2#760411yescharacteristic
183C2#1183711yescharacteristic
184C2#1183811yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC2×C8C22×C8
Frattini, cosocleC2×#97C23
prime radicalC23#418
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #498C8semidirect

Maximal subgroups

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

subgroupindex
#217272
#128962
#88002
#71272
#71262