Finite Group Databasebeta

← #93023

Subgroups — #93023

Order 512.

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

normal subgroups only

This group has 241 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#13190C211yesmaximal, characteristic
2#8612C211yesmaximal, characteristic
3#8609C211yesmaximal, characteristic
4C2×C8×D4C2211yescharacteristic
5C2×C4⋊C16C411yescharacteristic, complemented
6C2×C22⋊C16—21no—
7C2×C22⋊C16—21no—
8C2×#346C411yescharacteristic, complemented
9#1810—21no—
10#1808—21no—
11#1799—21no—
12#1465—21no—
13#1293C412yescomplemented
14#1289C412yes—
15C23×C8D411yescharacteristic
16C23×C8D411yescharacteristic
17C2×C4×D4D411yescharacteristic
18C22×C16—21no—
19C22×C16—21no—
20C8×D4—41no—
21C8×D4—41no—
22C2×#108—41no—
23C2×#107—41no—
24C2×C4⋊C8Q811yescharacteristic
25C2×#99—41no—
26#407—22no—
27C2×C22⋊C8C2×C411yescharacteristic
28C2×C22⋊C8C2×C411yescharacteristic
29C2×C4×C8C2×C411yescharacteristic
30#380—22no—
31C4⋊C16—41no—
32C22⋊C16—42no—
33C22⋊C16—42no—
34#346—41no—
35#342—41no—
36#341—41no—
37#340—41no—
38#324—41no—
39#323—41no—
40C22×D4SD1611yescharacteristic
41C23×C4C22⋊C411yescharacteristic
42C23×C4C22⋊C411yescharacteristic
43C2×M16—42no—
44C22×C8—21no—
45C22×C8—21no—
46C22×C8—21no—
47C22×C8—21no—
48C22×C8—41no—
49C22×C8—41no—
50C22×C8—41no—
51C22×C8—41no—
52C22×C8C4211yescharacteristic
53C22×C8C22⋊C411yescharacteristic
54C22×C8C4⋊C411yescharacteristic
55C22×C8C4⋊C411yescharacteristic
56C4×D4—41no—
57C4×D4—41no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4SD1611yescharacteristic
60C2×C22⋊C4—42no—
61C2×C22⋊C4D811yescharacteristic
62C2×C22⋊C4Q1611yescharacteristic
63C2×C42C4⋊C411yescharacteristic
64C2×C16—22no—
65C2×C16—41no—
66C2×C16—41no—
67C2×C16—41no—
68C2×C16—41no—
69C2×C16—41no—
70#108—81no—
71#108—81no—
72#107—81no—
73#107—81no—
74C4⋊C8—41no—
75#99—82no—
76C22⋊C8—41no—
77C22⋊C8—41no—
78C22⋊C8—42no—
79C22⋊C8—42no—
80C4×C8—21no—
81C4×C8—21no—
82#95—42no—
83C24#10211yescharacteristic
84C24#10311yescharacteristic
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—41no—
88C2×D4—41no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—82no—
98C22×C4#9511yescharacteristic
99C22×C4#10211yescharacteristic
100C22×C4#10311yescharacteristic
101C22×C4#10611yescharacteristic
102C22×C4#10711yescharacteristic
103M16—82no—
104M16—82no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—81no—
118C2×C8—81no—
119C2×C8—82no—
120C2×C8#9911yescharacteristic
121C2×C8#9911yescharacteristic
122C4⋊C4—41no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—82no—
128C22⋊C4—82no—
129C42—21no—
130C42—21no—
131C16—42no—
132C16—81no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23#34011yescharacteristic
143C23#36111yescharacteristic
144C23#36211yescharacteristic
145D4—42no—
146D4—42no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—162no—
165C2×C4—162no—
166C2×C4—162no—
167C2×C4H(3,ℤ/4)11yescharacteristic
168C2×C4#34211yescharacteristic
169C8—21no—
170C8—41no—
171C8—41no—
172C8—41no—
173C8—162no—
174C8#35311yescharacteristic
175C8#35411yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22#128911yescharacteristic
191C22#138711yescharacteristic
192C22#138911yescharacteristic
193C4—21no—
194C4—41no—
195C4—41no—
196C4—41no—
197C4—41no—
198C4—322no—
199C4#139111yescharacteristic
200C4#139311yescharacteristic
201C2—21no—
202C2—21no—
203C2—41no—
204C2—41no—
205C2—81no—
206C2#722711yescharacteristic
207C2#723511yescharacteristic
208C2#723711yescharacteristic
subgroupG/H
center, socleC22#1289
commutatorC22×C8C42
Frattini, cosocleC2×C8×D4C22
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊C16C4semidirect
8. C2×#346C4semidirect
13. #1293C4semidirect

Maximal subgroups

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

subgroupindex
#131902
#86122
#86092