Finite Group Databasebeta

← #150475

Subgroups — #150475

Order 512.

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

normal subgroups only

This group has 262 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#27674C211yesmaximal, characteristic
2#12623C211yesmaximal, characteristic
3#11537C211yesmaximal, characteristic
4#7411C212yesmaximal, complemented
5#7410C212yesmaximal, complemented
6#3093—21no—
7#2929C2211yescharacteristic
8C2×#427C411yescharacteristic, complemented
9#2166—21no—
10#2120—21no—
11#2090C412yescomplemented
12#2079C2212yes—
13#1952—21no—
14#1822—21no—
15#1816—21no—
16#1810C411yescharacteristic, complemented
17#1757—21no—
18#1392C2211yescharacteristic
19#1391C2211yescharacteristic
20#1321C2212yes—
21#1320—21no—
22C2×#131D411yescharacteristic
23C23×C8C2×C411yescharacteristic
24#525D411yescharacteristic
25C2×#117—21no—
26#485—42no—
27#482—42no—
28#435—41no—
29C8×D4—41no—
30C2×#108—21no—
31#428—22no—
32#427—41no—
33#427—41no—
34C2×#107—21no—
35C2×#106C2×C412yes—
36#424C2×C411yescharacteristic
37#417—22no—
38#413—21no—
39#410—21no—
40#408C2311yescharacteristic
41#407—21no—
42#407D411yescharacteristic
43#405—41no—
44#405D411yescharacteristic
45#364—41no—
46#352—22no—
47AΣL(1,𝔽16)—22no—
48#349—22no—
49#347—41no—
50#344—41no—
51#343—22no—
52#343—22no—
53#341C2×C412yes—
54C2×#41C2×D411yescharacteristic
55C23×C4C22⋊C411yescharacteristic
56#131—41no—
57#131—41no—
58C2×M16—21no—
59C2×M16—22no—
60C2×M16—41no—
61C2×M16C22⋊C411yescharacteristic
62C2×M16C2×D411yescharacteristic
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22×C411yescharacteristic
70#123—21no—
71C22⋊Q8—21no—
72#121—21no—
73#120—21no—
74C4×D4—41no—
75#117—41no—
76#117—42no—
77#117—42no—
78#117C22⋊C411yescharacteristic
79C2×C4⋊C4—22no—
80C2×C22⋊C4—41no—
81C2×C42—41no—
82M32—42no—
83#108—41no—
84#108—41no—
85#107—42no—
86#107—41no—
87#107—41no—
88#106—42no—
89#106—42no—
90C4⋊C8—41no—
91#103—42no—
92#102—42no—
93#101—41no—
94#100—22no—
95#99—22no—
96#99—41no—
97C22⋊C8—41no—
98C22⋊C8—41no—
99C22⋊C8—42no—
100#97—21no—
101#97—81no—
102C4×C8—21no—
103C4×C8—81no—
104C24#10411yescharacteristic
105#41—41no—
106#41—41no—
107C2×Q8AGL(1,ℤ/8)11yescharacteristic
108C2×D4—21no—
109C2×D4—41no—
110C2×D4AGL(1,ℤ/8)11yescharacteristic
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—42no—
117C22×C4#10411yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119M16—41no—
120M16—41no—
121M16—42no—
122M16—42no—
123M16—42no—
124M16—81no—
125M16—81no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—42no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—42no—
138C2×C8—81no—
139C2×C8#9911yescharacteristic
140C2×C8#9911yescharacteristic
141C4⋊C4—21no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—81no—
146C22⋊C4—21no—
147C22⋊C4—41no—
148C22⋊C4—41no—
149C22⋊C4—42no—
150C22⋊C4—82no—
151C42—21no—
152C42—82no—
153C42—81no—
154C16—82no—
155C23—21no—
156C23—21no—
157C23—41no—
158C23—81no—
159C23C2×#10411yescharacteristic
160Q8—41no—
161D4—41no—
162D4—81no—
163D4—82no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—162no—
178C2×C4C2×#9911yescharacteristic
179C2×C4#41811yescharacteristic
180C8—21no—
181C8—41no—
182C8—41no—
183C8—41no—
184C8—82no—
185C8—82no—
186C8#35211yescharacteristic
187C8#35211yescharacteristic
188C22—21no—
189C22—41no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—81no—
195C22—82no—
196C22#150711yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—81no—
201C4—162no—
202C4—162no—
203C4C2×#35211yescharacteristic
204C4#212911yescharacteristic
205C2—21no—
206C2—41no—
207C2—81no—
208C2—81no—
209C2#1177311yescharacteristic
subgroupG/H
center, socleC2#11773
commutatorC22×C8C22×C4
Frattini, cosocle#408C23
prime radical#525D4
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2929 ◁ G
LECSG ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7411C2semidirect
5. #7410C2semidirect
8. C2×#427C4semidirect
11. #2090C4semidirect
16. #1810C4semidirect

Maximal subgroups

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

subgroupindex
#276742
#126232
#115372
#74112
#74102