Finite Group Databasebeta

← #150902

Subgroups — #150902

Order 512.

241 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 60 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#27693C211yesmaximal, characteristic
2#12809C211yesmaximal, characteristic
3#11491C211yesmaximal, characteristic
4#7439C214yesmaximal, complemented
5#2931C2211yescharacteristic
6#2916C411yescharacteristic
7#2163—21no—
8#2132—21no—
9#2102C412yes—
10#2098C2212yes—
11#1945—21no—
12#1884—21no—
13#1868—21no—
14#1814—21no—
15#1813—21no—
16#1812C411yescharacteristic
17#1392C2212yes—
18#1315—21no—
19#1315C2212yes—
20#568—21no—
21C2×#131C2×C411yescharacteristic
22C2×#131D411yescharacteristic
23#532D411yescharacteristic
24#436—41no—
25C8×D4—41no—
26#430—41no—
27#430C2×C412yes—
28C2×#108—21no—
29C2×#108—41no—
30C2×#108—41no—
31#427—21no—
32#424—22no—
33#423C2×C411yescharacteristic
34#421—22no—
35#413—22no—
36#410—21no—
37#408D411yescharacteristic
38#408C2311yescharacteristic
39C2×C4×C8D411yescharacteristic
40C4⋊C16—41no—
41#356—41no—
42#355—41no—
43C22⋊C16—22no—
44#346—41no—
45#344—22no—
46#339—22no—
47#338—41no—
48H(3,ℤ/4)C2×C412yes—
49C2×#41C22⋊C411yescharacteristic
50C2×#41C2×D411yescharacteristic
51#131—41no—
52#131—41no—
53#131—41no—
54#131—41no—
55#131—82no—
56C2×M16—21no—
57C2×M16—21no—
58C2×M16—22no—
59C2×M16—42no—
60C2×M16—42no—
61C2×M16C22⋊C411yescharacteristic
62C2×M16C22×C411yescharacteristic
63C22×C8—21no—
64C22×C8—21no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C4⋊Q8SD1611yescharacteristic
68#127Q1611yescharacteristic
69#124SD1611yescharacteristic
70#124Q1611yescharacteristic
71#121—41no—
72C4×D4—41no—
73#117—22no—
74C2×C42C22⋊C411yescharacteristic
75C2×C16—42no—
76#108—42no—
77#108—41no—
78#108—41no—
79#108—81no—
80#108—81no—
81#107—41no—
82#106—41no—
83C4⋊C8—41no—
84C4⋊C8—82no—
85#104—82no—
86#101—42no—
87#100—42no—
88#99—42no—
89C22⋊C8—41no—
90C22⋊C8—41no—
91C4×C8—21no—
92C4×C8—21no—
93#41—41no—
94#41—41no—
95#41—41no—
96#41—41no—
97C2×Q8#10311yescharacteristic
98C2×Q8C2×SD1611yescharacteristic
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4#10311yescharacteristic
103C2×D4C2×Q1611yescharacteristic
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4C2×C22⋊C411yescharacteristic
108M16—42no—
109M16—41no—
110M16—41no—
111M16—42no—
112M16—42no—
113M16—81no—
114M16—82no—
115M16—82no—
116M16—82no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—42no—
128C2×C8—82no—
129C2×C8—82no—
130C2×C8#9911yescharacteristic
131C2×C8#9911yescharacteristic
132C4⋊C4—41no—
133C4⋊C4—42no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—42no—
137C42—21no—
138C42—42no—
139C42#10311yescharacteristic
140C42#10311yescharacteristic
141C16—82no—
142C23—21no—
143C23—21no—
144C23#42111yescharacteristic
145Q8—41no—
146Q8—41no—
147D4—41no—
148D4—41no—
149D4—42no—
150D4—81no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4—82no—
164C2×C4C2×#9911yescharacteristic
165C2×C4C2×#10311yescharacteristic
166C8—21no—
167C8—41no—
168C8—41no—
169C8—41no—
170C8—82no—
171C8—162no—
172C8#35211yescharacteristic
173C8#35211yescharacteristic
174C22—21no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—81no—
179C22—81no—
180C22#152011yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—41no—
186C4—162no—
187C4C2×#35211yescharacteristic
188C4#212811yescharacteristic
189C2—21no—
190C2—41no—
191C2—81no—
192C2—81no—
193C2#1189911yescharacteristic
subgroupG/H
centerC4#2128
commutatorC2×M16C22×C4
Frattini, cosocle#408C23
socleC2#11899
prime radical#532D4
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G
LECSG ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7439C2semidirect

Maximal subgroups

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

subgroupindex
#276932
#128092
#114912
#74392