Finite Group Databasebeta

← #151613

Subgroups — #151613

Order 512.

183 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 50 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#33947C211yesmaximal, characteristic
2#13731C211yesmaximal, characteristic, complemented
3#13705C211yesmaximal, characteristic
4#12458C211yesmaximal, characteristic, complemented
5#7451C211yesmaximal, characteristic, complemented
6Q8⋊C32C211yesmaximal, characteristic, complemented
7#7383C211yesmaximal, characteristic
8#3380—21no—
9#2253—21no—
10#2252C2211yescharacteristic
11#2252C2211yescharacteristic
12#2228—22no—
13#2226—21no—
14Q8×C16C2211yescharacteristic
15C4×D16—21no—
16C4×D16C2211yescharacteristic
17#1676—21no—
18#1424—21no—
19#1423—22no—
20C4⋊C32C2211yescharacteristic
21#1415—21no—
22C4×C32C2211yescharacteristic
23#1371C2211yescharacteristic
24#1365—21no—
25#554—41no—
26C2×D16D411yescharacteristic
27#500—21no—
28#500—21no—
29#499—22no—
30#478—42no—
31#477—41no—
32#476—41no—
33C8×Q8D411yescharacteristic
34C4×D8—22no—
35C4×D8—21no—
36C2×C32—21no—
37C2×C32—21no—
38#367—22no—
39#366—21no—
40#366—21no—
41#366—21no—
42#366D411yescharacteristic
43#366D412yes—
44C4⋊C16—21no—
45C4⋊C16D411yescharacteristic
46#357—22no—
47#357—21no—
48C4×C16—21no—
49C4×C16C2311yescharacteristic
50#335—21no—
51#333—41no—
52C2×D8—21no—
53C4⋊Q8—41no—
54C4⋊Q8—41no—
55#125—42no—
56C22⋊Q8—82no—
57C22⋊Q8—81no—
58C4×Q8D811yescharacteristic
59C4×D4—42no—
60C4×D4—41no—
61C2×C4⋊C4—81no—
62D16—21no—
63D16Q1612yes—
64C2×C16—21no—
65C2×C16C2×D411yescharacteristic
66C2×C16C2×D411yescharacteristic
67#107—22no—
68#107—21no—
69#107—41no—
70#107—41no—
71#107—41no—
72#107—81no—
73#106—42no—
74C4⋊C8—21no—
75C4⋊C8—41no—
76C4⋊C8D811yescharacteristic
77#103—81no—
78#102—42no—
79#102—41no—
80C4×C8—21no—
81C4×C8C2×D411yescharacteristic
82C32—22no—
83C32—41no—
84C2×Q8—81no—
85C2×Q8—81no—
86C2×Q8D1611yescharacteristic
87C2×D4—41no—
88C22×C4—82no—
89C22×C4—81no—
90D8—22no—
91D8—41no—
92C2×C8—21no—
93C2×C8—81no—
94C2×C8#12011yescharacteristic
95C2×C8AGL(1,ℤ/8)11yescharacteristic
96C4⋊C4—21no—
97C4⋊C4—41no—
98C4⋊C4—42no—
99C4⋊C4—82no—
100C4⋊C4—82no—
101C4⋊C4—81no—
102C4⋊C4—81no—
103C4⋊C4—81no—
104C4⋊C4D1611yescharacteristic
105C22⋊C4—82no—
106C22⋊C4—81no—
107C42—21no—
108C42C2×D811yescharacteristic
109C16—21no—
110C16—41no—
111C16C2×Q1611yescharacteristic
112C16#13711yescharacteristic
113C23—81no—
114Q8—21no—
115Q8—21no—
116Q8—321no—
117Q8—321no—
118D4—42no—
119D4—81no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—81no—
123C2×C4—82no—
124C2×C4—161no—
125C2×C4—161no—
126C2×C4—161no—
127C2×C4—322no—
128C2×C4#44711yescharacteristic
129C2×C4C2×D1611yescharacteristic
130C2×C4#50911yescharacteristic
131C8—21no—
132C8—41no—
133C8—161no—
134C8#45111yescharacteristic
135C8#45611yescharacteristic
136C22—82no—
137C22—161no—
138C22—161no—
139C22#218411yescharacteristic
140C4—21no—
141C4—41no—
142C4—41no—
143C4—161no—
144C4—162no—
145C4—321no—
146C4#164911yescharacteristic
147C4#219511yescharacteristic
148C4#226211yescharacteristic
149C4#226411yescharacteristic
150C2—162no—
151C2#1240411yescharacteristic
152C2#1366611yescharacteristic
153C2#1366911yescharacteristic
subgroupG/H
center, socleC22#2184
commutator, Frattini, cosocleC4×C16C23
prime radicalC2×D16D4
LCSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ Q8×C16 ◁ G
LECSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13731C2semidirect
4. #12458C2semidirect
5. #7451C2semidirect
6. Q8⋊C32C2semidirect

Maximal subgroups

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

subgroupindex
#339472
#137312
#137052
#124582
#74512
Q8⋊C322
#73832