Finite Group Databasebeta

← #151612

Subgroups — #151612

Order 512.

179 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 52 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
1Q8×Q32C211yesmaximal, characteristic
2#13731C211yesmaximal, characteristic
3#13706C211yesmaximal, characteristic
4#12458C211yesmaximal, characteristic
5#7451C211yesmaximal, characteristic
6Q8⋊C32C211yesmaximal, characteristic
7#7384C211yesmaximal, characteristic
8Q8×Q16—21no—
9#2252—21no—
10#2252C2211yescharacteristic
11#2252C2211yescharacteristic
12#2227—22no—
13#2227—21no—
14Q8×C16C2211yescharacteristic
15C4×Q32—21no—
16C4×Q32C2211yescharacteristic
17#1676—21no—
18#1423—21no—
19#1423—22no—
20C4⋊C32C2211yescharacteristic
21#1416—21no—
22C4×C32C2211yescharacteristic
23#1371C2211yescharacteristic
24#1365—21no—
25Q82—41no—
26C2×Q32D411yescharacteristic
27#500—22no—
28#500—21no—
29#500—21no—
30#477—42no—
31#477—41no—
32#477—41no—
33C8×Q8D411yescharacteristic
34C4×Q16—22no—
35C4×Q16—21no—
36C2×C32—21no—
37C2×C32—21no—
38#366—22no—
39#366—21no—
40#366—21no—
41#366—21no—
42#366D411yescharacteristic
43#366D412yes—
44C4⋊C16—21no—
45C4⋊C16D411yescharacteristic
46#358—22no—
47#358—21no—
48C4×C16—21no—
49C4×C16C2311yescharacteristic
50#335—21no—
51#333—41no—
52C2×Q16—21no—
53C4⋊Q8—42no—
54C4⋊Q8—41no—
55C4⋊Q8—41no—
56C4⋊Q8—82no—
57C4⋊Q8—81no—
58C4×Q8—42no—
59C4×Q8—41no—
60C4×Q8—81no—
61C4×Q8D811yescharacteristic
62Q32—21no—
63Q32Q1612yes—
64C2×C16—21no—
65C2×C16C2×D411yescharacteristic
66C2×C16C2×D411yescharacteristic
67#107—22no—
68#107—21no—
69#107—42no—
70#107—41no—
71#107—41no—
72#107—41no—
73#107—81no—
74C4⋊C8—21no—
75C4⋊C8—41no—
76C4⋊C8D811yescharacteristic
77#103—42no—
78#103—41no—
79#103—81no—
80C4×C8—21no—
81C4×C8C2×D411yescharacteristic
82C32—22no—
83C32—41no—
84C2×Q8—41no—
85C2×Q8—82no—
86C2×Q8—81no—
87C2×Q8—81no—
88C2×Q8D1611yescharacteristic
89Q16—22no—
90Q16—41no—
91C2×C8—21no—
92C2×C8—81no—
93C2×C8#12011yescharacteristic
94C2×C8AGL(1,ℤ/8)11yescharacteristic
95C4⋊C4—21no—
96C4⋊C4—42no—
97C4⋊C4—41no—
98C4⋊C4—82no—
99C4⋊C4—82no—
100C4⋊C4—81no—
101C4⋊C4—81no—
102C4⋊C4—81no—
103C4⋊C4—81no—
104C4⋊C4D1611yescharacteristic
105C42—21no—
106C42—82no—
107C42—81no—
108C42C2×D811yescharacteristic
109C16—21no—
110C16—41no—
111C16C2×Q1611yescharacteristic
112C16#13711yescharacteristic
113Q8—21no—
114Q8—42no—
115Q8—81no—
116Q8—321no—
117Q8—321no—
118Q8—322no—
119Q8Q6412yes—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—81no—
123C2×C4—82no—
124C2×C4—81no—
125C2×C4—161no—
126C2×C4#44711yescharacteristic
127C2×C4C2×D1611yescharacteristic
128C2×C4#50911yescharacteristic
129C8—21no—
130C8—41no—
131C8—161no—
132C8#45111yescharacteristic
133C8#45611yescharacteristic
134C22#218411yescharacteristic
135C4—22no—
136C4—21no—
137C4—41no—
138C4—82no—
139C4—162no—
140C4—161no—
141C4—321no—
142C4#164911yescharacteristic
143C4#219511yescharacteristic
144C4C2×Q6411yescharacteristic
145C4#226411yescharacteristic
146C2#1240411yescharacteristic
147C2#1366511yescharacteristic
148C2#1366911yescharacteristic
subgroupG/H
center, socle, prime radicalC22#2184
commutator, Frattini, cosocleC4×C16C23
LCSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ Q8×C16 ◁ G
LECSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
Q8×Q322
#137312
#137062
#124582
#74512
Q8⋊C322
#73842