Finite Group Databasebeta

← #142073

Subgroups — #142073

Order 512.

248 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 76 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#33261C211yesmaximal, characteristic
2#12657C211yesmaximal, characteristic
3#12596C211yesmaximal, characteristic
4#7266C214yesmaximal
5#2647C412yes—
6#2150—21no—
7C2×#347C2212yes—
8C2×#347C2212yes—
9#1944—21no—
10#1943—21no—
11#1942—21no—
12#1942—21no—
13#1846C411yescharacteristic
14#1845C411yescharacteristic
15#1836C2212yes—
16#1726C2211yescharacteristic
17#1377—21no—
18#1376—21no—
19#1375—21no—
20C2×M32—22no—
21C2×M32—22no—
22C2×#107—21no—
23C2×#107—21no—
24C2×#107—21no—
25C2×#106—21no—
26C2×#97D411yescharacteristic
27C2×#97D411yescharacteristic
28C2×#97C2311yescharacteristic
29C2×C4×C8C2×C411yescharacteristic
30C2×C4×C8D411yescharacteristic
31C2×C4×C8D411yescharacteristic
32#400—42no—
33#398—42no—
34#389—22no—
35#389—22no—
36#384C2×C412yes—
37#382C2×C412yes—
38#376C2×C411yescharacteristic
39C4⋊C16—24no—
40#347—22no—
41#347—24no—
42#347Q814yes—
43#340—22no—
44#340—22no—
45C22×C8C22⋊C411yescharacteristic
46C22×C8C22⋊C411yescharacteristic
47C22×C8C2×D411yescharacteristic
48C22×C8C2×D411yescharacteristic
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C42C22⋊C411yescharacteristic
55C2×C42C22⋊C411yescharacteristic
56C2×C42C22×C411yescharacteristic
57M32—24no—
58M32—42no—
59M32—42no—
60M32—42no—
61C2×C16—24no—
62C2×C16—42no—
63#107—42no—
64#107—42no—
65#107—42no—
66#106—42no—
67#97—41no—
68#97—41no—
69#97C2×Q812yes—
70#97#4112yes—
71C4×C8—22no—
72C4×C8—22no—
73C4×C8—41no—
74#95—21no—
75#95—21no—
76#95—42no—
77#95—42no—
78#95—42no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4#10011yescharacteristic
84C22×C4#10011yescharacteristic
85C22×C4C2×C22⋊C411yescharacteristic
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—21no—
90C2×C8—21no—
91C2×C8—21no—
92C2×C8—21no—
93C2×C8—21no—
94C2×C8#10112yes—
95C2×C8#10412yes—
96C2×C8C22⋊Q812yes—
97C2×C8#12411yescharacteristic
98C2×C8C4⋊Q811yescharacteristic
99C4⋊C4—42no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4—42no—
103C4⋊C4—162no—
104C4⋊C4—162no—
105C4⋊C4—162no—
106C42—21no—
107C42—21no—
108C42—21no—
109C42—21no—
110C42—21no—
111C42C4×Q812yes—
112C16—44no—
113C16—44no—
114C23C2×#10011yescharacteristic
115C2×C4—22no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4#39111yescharacteristic
132C2×C4C2×#10111yescharacteristic
133C2×C4C2×#10411yescharacteristic
134C2×C4#42111yescharacteristic
135C2×C4#42111yescharacteristic
136C2×C4#42111yescharacteristic
137C8—21no—
138C8—21no—
139C8—22no—
140C8—22no—
141C8—22no—
142C22—21no—
143C22—21no—
144C22#153011yescharacteristic
145C22#153611yescharacteristic
146C22#199711yescharacteristic
147C4—42no—
148C4—41no—
149C4—41no—
150C4—81no—
151C4—162no—
152C4—162no—
153C4#152811yescharacteristic
154C4#153411yescharacteristic
155C4#198611yescharacteristic
156C4#199511yescharacteristic
157C2—21no—
158C2—21no—
159C2#989411yescharacteristic
160C2#1191311yescharacteristic
161C2#1191411yescharacteristic
subgroupG/H
center, socleC22#1530
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332612
#126572
#125962
#72662