Finite Group Databasebeta

← #150530

Subgroups — #150530

Order 512.

264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 54 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#27691C211yesmaximal, characteristic
2#11537C211yesmaximal, characteristic
3#8238C211yesmaximal, characteristic
4#7433C212yesmaximal, complemented
5#7432C212yesmaximal, complemented
6#2929C2211yescharacteristic
7#2916C411yescharacteristic, complemented
8#2166—21no—
9#2103C412yescomplemented
10#2091C2212yes—
11#1953—21no—
12#1891—21no—
13#1858—21no—
14#1813C411yescharacteristic, complemented
15#1810—21no—
16C8×C22⋊C4—21no—
17#1501—21no—
18#1321—21no—
19#1320C2212yes—
20#1298C2211yescharacteristic
21#1297C2211yescharacteristic
22#568—21no—
23C2×#131C2×C411yescharacteristic
24C23×C8D411yescharacteristic
25#525D411yescharacteristic
26#435—41no—
27C8×D4—41no—
28#430—22no—
29#430—41no—
30#430C2×C412yes—
31C2×#108—21no—
32C2×#108—21no—
33C2×#108—41no—
34#424C2×C411yescharacteristic
35#413—21no—
36#408D411yescharacteristic
37#407—22no—
38#407C2311yescharacteristic
39C2×C22⋊C8—21no—
40#405D411yescharacteristic
41C2×C4×C8—41no—
42C4×C22⋊C4—21no—
43#364—41no—
44#349—22no—
45#347—41no—
46#344—22no—
47#341—22no—
48#338C2×C412yes—
49#336—41no—
50#328—22no—
51#327—22no—
52C2×#41C22⋊C411yescharacteristic
53C23×C4C2×D411yescharacteristic
54#131—42no—
55#131—82no—
56C2×M16—21no—
57C2×M16—22no—
58C2×M16—22no—
59C2×M16—42no—
60C2×M16C22×C411yescharacteristic
61C2×M16C2×D411yescharacteristic
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—42no—
66C22×C8—41no—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69#123—21no—
70C22⋊Q8—21no—
71#121—21no—
72#120—21no—
73C4×D4—41no—
74#117C22⋊C411yescharacteristic
75C2×C22⋊C4—41no—
76C2×C42—41no—
77M32—42no—
78#108—42no—
79#108—42no—
80#108—41no—
81#108—41no—
82#108—82no—
83C4⋊C8—41no—
84#101—41no—
85#100—41no—
86C22⋊C8—22no—
87C22⋊C8—41no—
88C22⋊C8—41no—
89C22⋊C8—41no—
90C22⋊C8—82no—
91#97—21no—
92C4×C8—21no—
93C4×C8—82no—
94#95—22no—
95C24#13711yescharacteristic
96#41—41no—
97#41—41no—
98C2×Q8#10411yescharacteristic
99C2×D4—21no—
100C2×D4—41no—
101C2×D4#10411yescharacteristic
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—42no—
108C22×C4C2×C22⋊C411yescharacteristic
109C22×C4#13711yescharacteristic
110M16—42no—
111M16—41no—
112M16—41no—
113M16—42no—
114M16—42no—
115M16—42no—
116M16—42no—
117M16—82no—
118C2×C8—22no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—41no—
125C2×C8—42no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—81no—
129C2×C8—81no—
130C2×C8—81no—
131C2×C8—82no—
132C2×C8#9911yescharacteristic
133C2×C8#9911yescharacteristic
134C4⋊C4—21no—
135C4⋊C4—41no—
136C22⋊C4—21no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—44no—
140C42—21no—
141C42—81no—
142C42—81no—
143C16—82no—
144C23—21no—
145C23—21no—
146C23—42no—
147C23—41no—
148C23#41911yescharacteristic
149Q8—41no—
150D4—41no—
151D4—42no—
152D4—82no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—81no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4C2×#9911yescharacteristic
168C2×C4C2×#10411yescharacteristic
169C8—21no—
170C8—41no—
171C8—41no—
172C8—41no—
173C8—82no—
174C8—82no—
175C8—82no—
176C8#35212yes—
177C22—22no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—81no—
185C22—81no—
186C22#150811yescharacteristic
187C4—21no—
188C4—41no—
189C4—41no—
190C4—81no—
191C4—84no—
192C4C2×#35211yescharacteristic
193C4#212811yescharacteristic
194C2—21no—
195C2—42no—
196C2—41no—
197C2—81no—
198C2#1178611yescharacteristic
subgroupG/H
centerC4#2128
commutatorC2×M16C22×C4
Frattini, cosocle#407C23
socleC2#11786
prime radical#525D4
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2929 ◁ G
LECSG ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7433C2semidirect
5. #7432C2semidirect
7. #2916C4semidirect
9. #2103C4semidirect
14. #1813C4semidirect

Maximal subgroups

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

subgroupindex
#276912
#115372
#82382
#74332
#74322