Finite Group Databasebeta

← #134814

Subgroups — #134814

Order 512.

345 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 76 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#22208C211yesmaximal, characteristic
2#11531C212yesmaximal
3#8612C212yesmaximal
4#7249C212yesmaximal
5C2×C8×Q8C411yescharacteristic
6C2×C8×D4C2211yescharacteristic
7C4×#124C2211yescharacteristic
8#1984C411yescharacteristic
9#1954C2211yescharacteristic
10#1923—22no—
11#1850—22no—
12#1842C411yescharacteristic
13#1810—22no—
14#1744C411yescharacteristic
15#1465C2214yes—
16#1324—22no—
17#1319—22no—
18#1293—22no—
19C23×C8D412yes—
20C2×#124Q811yescharacteristic
21C2×C4×Q8C2×C411yescharacteristic
22C2×C4×D4C2311yescharacteristic
23C8×Q8—42no—
24C8×D4—42no—
25C2×#108—42no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30#407—24no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8D412yes—
33C2×C4×C8—21no—
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8C2×C411yescharacteristic
36#391D411yescharacteristic
37#390Q811yescharacteristic
38#388—22no—
39#382D411yescharacteristic
40C4×C22⋊C4—22no—
41C43C2×C411yescharacteristic
42#341—42no—
43#336—21no—
44#336—22no—
45#336—21no—
46#324—24no—
47#323—24no—
48C22×Q8C4⋊C411yescharacteristic
49C22×D4C2×Q811yescharacteristic
50C23×C4C2×D412yes—
51C2×M16—44no—
52C22×C8—22no—
53C22×C8—22no—
54C22×C8—21no—
55C22×C8—21no—
56C22×C8—82no—
57C22×C8C22⋊C412yes—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60#124—22no—
61#124—22no—
62C4×Q8—41no—
63C4×Q8—41no—
64C4×D4—41no—
65C4×D4—41no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C4⋊C411yescharacteristic
68C2×C4⋊C4C2×D411yescharacteristic
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4#4112yes—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C4⋊C411yescharacteristic
76C2×C42C4⋊C411yescharacteristic
77C2×C42C22×C411yescharacteristic
78#108—84no—
79C4⋊C8—41no—
80C4⋊C8—41no—
81C4⋊C8—42no—
82C4⋊C8—82no—
83C22⋊C8—42no—
84C22⋊C8—44no—
85C22⋊C8—44no—
86C22⋊C8—82no—
87C4×C8—22no—
88C4×C8—41no—
89C4×C8—42no—
90#95—22no—
91#95—22no—
92C24C22⋊Q812yes—
93C2×Q8—21no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×Q8#10812yescomplemented
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—41no—
100C2×D4—41no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4C2×C22⋊C411yescharacteristic
110C22×C4C2×C4⋊C411yescharacteristic
111C22×C4#11711yescharacteristic
112C22×C4#12112yes—
113M16—84no—
114M16—84no—
115C2×C8—21no—
116C2×C8—22no—
117C2×C8—41no—
118C2×C8—44no—
119C2×C8—42no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—42no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—82no—
127C2×C8—84no—
128C2×C8#9912yes—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—42no—
132C22⋊C4—28no—
133C22⋊C4—42no—
134C22⋊C4—82no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—41no—
139C42—41no—
140C42—41no—
141C42—41no—
142C42—41no—
143C42—81no—
144C42#10812yescomplemented
145C23—22no—
146C23—22no—
147C23—22no—
148C23—42no—
149C23—42no—
150C23#38011yescharacteristic
151Q8—42no—
152Q8—42no—
153D4—42no—
154D4—42no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—82no—
177C2×C4C2×#9911yescharacteristic
178C2×C4#41011yescharacteristic
179C2×C4C2×#10811yescharacteristic
180C2×C4C2×#10811yescharacteristic
181C8—22no—
182C8—42no—
183C8—41no—
184C8—81no—
185C8—81no—
186C8—164no—
187C22—21no—
188C22—21no—
189C22—44no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—82no—
194C22#179911yescharacteristic
195C22#181011yescharacteristic
196C22#181311yescharacteristic
197C4—21no—
198C4—21no—
199C4—21no—
200C4—42no—
201C4—42no—
202C4—41no—
203C4—42no—
204C4—81no—
205C4—81no—
206C4#212711yescharacteristic
207C4#212911yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#861211yescharacteristic
212C2#1280011yescharacteristic
213C2#1281011yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
96. C2×Q8#108semidirect
144. C42#108semidirect

Maximal subgroups

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

subgroupindex
#222082
#115312
#86122
#72492