Finite Group Databasebeta

← #513275

Subgroups — #513275

Order 512.

299 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 114 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#33962C211yesmaximal, characteristic
2#33950C211yesmaximal, characteristic
3#13724C212yesmaximal
4#13720C211yesmaximal, characteristic
5#13708C212yesmaximal
6#13704C211yesmaximal, characteristic
7#13688C211yesmaximal, characteristic
8#13688C212yesmaximal
9#13660C211yesmaximal, characteristic
10#13653C212yesmaximal
11C4×Q64C211yesmaximal, characteristic
12#3395—21no—
13#3383—21no—
14C2×Q64C2211yescharacteristic
15C2×Q64C2212yes—
16#2253C2212yes—
17#2252C2211yescharacteristic
18#2247C2211yescharacteristic
19#2247C2212yes—
20#2229—22no—
21#2225—21no—
22#2209—21no—
23#2209—22no—
24Q8×C16C2211yescharacteristic
25C4×Q32C2211yescharacteristic
26C4×Q32C2211yescharacteristic
27C4×Q32C2212yes—
28C4×Q32C2212yes—
29#1424C2212yes—
30#1423C2211yescharacteristic
31C4⋊C32C2211yescharacteristic
32C4⋊C32C2212yes—
33#1416C2211yescharacteristic
34#1416C2211yescharacteristic
35#1416C2212yes—
36#1416C2212yes—
37#1416C2212yes—
38#1416C2212yes—
39#1397C2212yes—
40C4×C32C2211yescharacteristic
41#557—41no—
42#557—41no—
43C2×Q32—21no—
44C2×Q32—22no—
45C2×Q32C2311yescharacteristic
46C2×Q32C2312yes—
47C2×Q32C2311yescharacteristic
48#500—21no—
49#499—22no—
50#494—22no—
51#494—21no—
52#479—42no—
53#475—41no—
54#462—41no—
55#462—42no—
56C8×Q8D411yescharacteristic
57C4×Q16—21no—
58C4×Q16—22no—
59C4×Q16—22no—
60C4×Q16—21no—
61Q64—22no—
62Q64—24no—
63C2×C32C2311yescharacteristic
64C2×C32C2311yescharacteristic
65C2×C32C2312yes—
66#367—22no—
67#366—21no—
68#366C2311yescharacteristic
69#366C2311yescharacteristic
70#366C2312yes—
71C4⋊C16D411yescharacteristic
72C4⋊C16D412yes—
73#358—21no—
74#358—22no—
75#358—22no—
76#358—21no—
77C4×C16C2311yescharacteristic
78C4×C16C2312yes—
79C2×Q16—21no—
80C2×Q16—21no—
81C2×Q16—22no—
82C2×Q16—42no—
83C2×Q16—41no—
84C4⋊Q8—41no—
85C4⋊Q8—42no—
86C4⋊Q8—41no—
87C4⋊Q8—81no—
88#125—42no—
89#125—81no—
90#125—82no—
91#125—82no—
92C4×Q8—42no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×Q8—42no—
96C4×Q8—81no—
97C4×Q8—81no—
98C4×Q8D811yescharacteristic
99Q32—21no—
100Q32—22no—
101Q32—21no—
102Q32—21no—
103Q32—22no—
104Q32—81no—
105Q32—82no—
106Q32#4112yes—
107C2×C16C2×D411yescharacteristic
108C2×C16C2×D412yes—
109C2×C16C2411yescharacteristic
110#107—21no—
111#107—22no—
112#107—21no—
113#107—41no—
114#106—42no—
115C4⋊C8D811yescharacteristic
116C4⋊C8D812yes—
117#103—42no—
118#103—41no—
119#103—42no—
120#103—41no—
121C4×C8C2×D412yes—
122C4×C8C2×D411yescharacteristic
123C32—21no—
124C32—22no—
125C32—21no—
126C2×Q8—41no—
127C2×Q8—42no—
128C2×Q8—41no—
129C2×Q8—81no—
130C2×Q8D1611yescharacteristic
131Q16—22no—
132Q16—41no—
133Q16—42no—
134Q16—41no—
135Q16—42no—
136Q16—41no—
137Q16—162no—
138Q16—161no—
139C2×C8C2×D812yes—
140C2×C8C2×D811yescharacteristic
141C2×C8C22×D411yescharacteristic
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C4⋊C4—81no—
146C4⋊C4—82no—
147C4⋊C4—81no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C4⋊C4—81no—
152C4⋊C4—82no—
153C4⋊C4—81no—
154C4⋊C4—82no—
155C4⋊C4—81no—
156C4⋊C4D1611yescharacteristic
157C4⋊C4D1612yes—
158C42—82no—
159C42—82no—
160C42—81no—
161C42—81no—
162C42C2×D812yes—
163C42C2×D811yescharacteristic
164C16—21no—
165C16—22no—
166C16C2×#4111yescharacteristic
167C162−1+411yescharacteristic
168Q8—22no—
169Q8—42no—
170Q8—81no—
171Q8—82no—
172Q8—81no—
173Q8—81no—
174Q8—82no—
175Q8—162no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—81no—
182C2×C4C2×D1612yes—
183C2×C4C2×D1612yes—
184C2×C4C2×D1611yescharacteristic
185C2×C4C2×D1611yescharacteristic
186C2×C4C22×D811yescharacteristic
187C8—22no—
188C8—21no—
189C8#54811yescharacteristic
190C8#57711yescharacteristic
191C22C22×D1611yescharacteristic
192C4—22no—
193C4—22no—
194C4—21no—
195C4—82no—
196C4—162no—
197C4—161no—
198C4—162no—
199C4—161no—
200C4—161no—
201C4#226412yes—
202C4#335811yescharacteristic
203C4#341611yescharacteristic
204C2#3392511yescharacteristic
205C2C2×#226411yescharacteristic
206C2#3398411yescharacteristic
subgroupG/H
center, socle, prime radicalC22C22×D16
commutator, Frattini, cosocleC2×C16C24
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ Q8×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#339622
#339502
#137242
#137202
#137082
#137042
#136882
#136602
#136532
C4×Q642