Finite Group Databasebeta

← #151600

Subgroups — #151600

Order 512.

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

normal subgroups only

This group has 185 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#33905C211yesmaximal, characteristic
2#13731C211yesmaximal, characteristic, complemented
3#13706C211yesmaximal, characteristic, complemented
4#12452C211yesmaximal, characteristic
5#7451C211yesmaximal, characteristic, complemented
6#7384C211yesmaximal, characteristic, complemented
7D4⋊C32C211yesmaximal, characteristic
8#3338—21no—
9#2252C2211yescharacteristic
10#2252C2211yescharacteristic
11#2236—22no—
12#2227—21no—
13#2216—21no—
14C4×Q32C2211yescharacteristic
15D4×C16C2211yescharacteristic
16C2×#366—21no—
17#1675—21no—
18#1423—21no—
19#1423—22no—
20C4⋊C32C2211yescharacteristic
21#1416—21no—
22C4×C32C2211yescharacteristic
23#1371C2211yescharacteristic
24#1365—21no—
25#554—41no—
26C2×Q32D411yescharacteristic
27C22×C16—21no—
28#500—21no—
29#500—21no—
30#484—42no—
31#477—41no—
32#476—41no—
33#467—22no—
34C4×Q16—21no—
35C8×D4D411yescharacteristic
36C2×#107—22no—
37C2×C32—21no—
38C2×C32—21no—
39#366—22no—
40#366—21no—
41#366—21no—
42#366—21no—
43#366D411yescharacteristic
44#366D412yes—
45C4⋊C16D411yescharacteristic
46#358—22no—
47#358—21no—
48C22⋊C16—21no—
49C4×C16C2311yescharacteristic
50#335—21no—
51#333—41no—
52C2×Q16—21no—
53C22×C8—21no—
54C4⋊Q8—41no—
55C4⋊Q8—41no—
56#125—81no—
57C22⋊Q8—42no—
58C22⋊Q8—82no—
59C4×Q8—41no—
60C4×D4—81no—
61C4×D4D811yescharacteristic
62C2×C4⋊C4—42no—
63Q32—21no—
64Q32—21no—
65C2×C16—21no—
66C2×C16—41no—
67C2×C16—41no—
68C2×C16C2×D411yescharacteristic
69C2×C16C2×D411yescharacteristic
70#107—21no—
71#107—22no—
72#107—42no—
73#107—41no—
74#107—41no—
75#107—41no—
76#106—81no—
77C4⋊C8—41no—
78C4⋊C8D811yescharacteristic
79#103—42no—
80#103—41no—
81#102—81no—
82C22⋊C8—21no—
83C4×C8C2×D411yescharacteristic
84C32—22no—
85C32—41no—
86C2×Q8—41no—
87C2×Q8—81no—
88C2×Q8—81no—
89C2×D4D1611yescharacteristic
90C22×C4—21no—
91C22×C4—82no—
92Q16—41no—
93Q16—41no—
94C2×C8—21no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—81no—
98C2×C8#12011yescharacteristic
99C2×C8AGL(1,ℤ/8)11yescharacteristic
100C4⋊C4—41no—
101C4⋊C4—42no—
102C4⋊C4—82no—
103C4⋊C4—82no—
104C4⋊C4—81no—
105C4⋊C4—81no—
106C4⋊C4—81no—
107C4⋊C4—81no—
108C4⋊C4D1611yescharacteristic
109C22⋊C4—21no—
110C22⋊C4—82no—
111C42—81no—
112C42C2×D811yescharacteristic
113C16—21no—
114C16—41no—
115C16#13511yescharacteristic
116C16#13711yescharacteristic
117C23—21no—
118Q8—81no—
119Q8—81no—
120Q8—321no—
121Q8—321no—
122D4—21no—
123D4Q6412yes—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—81no—
128C2×C4—81no—
129C2×C4—82no—
130C2×C4—161no—
131C2×C4—322no—
132C2×C4#44711yescharacteristic
133C2×C4C2×D1611yescharacteristic
134C2×C4#50911yescharacteristic
135C8—21no—
136C8—41no—
137C8—161no—
138C8#45211yescharacteristic
139C8#45611yescharacteristic
140C22—22no—
141C22—41no—
142C22—41no—
143C22#218411yescharacteristic
144C4—21no—
145C4—41no—
146C4—161no—
147C4—162no—
148C4—161no—
149C4—321no—
150C4#164711yescharacteristic
151C4#219511yescharacteristic
152C4C2×Q6411yescharacteristic
153C4#226411yescharacteristic
154C2—42no—
155C2#1239211yescharacteristic
156C2#1366511yescharacteristic
157C2#1366911yescharacteristic
subgroupG/H
center, socleC22#2184
commutator, Frattini, cosocleC4×C16C23
prime radicalC2×D4D16
LCSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ D4×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
3. #13706C2semidirect
5. #7451C2semidirect
6. #7384C2semidirect

Maximal subgroups

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

subgroupindex
#339052
#137312
#137062
#124522
#74512
#73842
D4⋊C322