Finite Group Databasebeta

← #128982

Subgroups — #128982

Order 512.

337 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 90 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#22211C211yesmaximal, characteristic
2#11734C212yesmaximal
3#8165C212yesmaximal, complemented
4#7286C211yesmaximal, characteristic, complemented
5#7284C211yesmaximal, characteristic, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×#436C411yescharacteristic
8#2398C2211yescharacteristic
9#1985C2211yescharacteristic
10#1955C411yescharacteristic
11#1923—22no—
12#1851—22no—
13#1843C411yescharacteristic
14C4×#101C2212yes—
15#1746C411yescharacteristic
16#1461C2212yes—
17#1451—22no—
18#1353C414yes—
19#1345C414yes—
20#1283—22no—
21C22×M16—21no—
22C2×C4×Q8C2311yescharacteristic
23C2×C4×D4C2×C411yescharacteristic
24C8×Q8—42no—
25#436—42no—
26#431—22no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8D412yes—
31C2×#101—22no—
32C2×C22⋊C8—21no—
33C2×C22⋊C8—22no—
34C4×M16—22no—
35C2×#97C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8D412yes—
38#400C2×C412yes—
39#397—21no—
40#394C2×C412yes—
41#393—21no—
42C4×C22⋊C4—22no—
43#376C2×C411yescharacteristic
44#343C2×C414yes—
45#336—21no—
46#336—21no—
47#336—22no—
48#324—22no—
49#324C2×C414yes—
50#322—44no—
51C22×Q8#4111yescharacteristic
52C22×D4C4211yescharacteristic
53C23×C4—21no—
54C2×M16—24no—
55C2×M16—42no—
56C2×M16—82no—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8C22⋊C411yescharacteristic
60C22×C8C22⋊C412yes—
61C22×C8C22⋊C411yescharacteristic
62C4×Q8—41no—
63C4×Q8—41no—
64C4×D4—41no—
65C4×D4—41no—
66C2×C4⋊C4C4211yescharacteristic
67C2×C4⋊C4C22×C412yes—
68C2×C4⋊C4#4111yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42C22×C411yescharacteristic
74C2×C42C2×D412yes—
75C4⋊C8—41no—
76C4⋊C8—42no—
77C4⋊C8—41no—
78C4⋊C8—42no—
79C4⋊C8—82no—
80#101—44no—
81C22⋊C8—24no—
82C22⋊C8—42no—
83C22⋊C8—82no—
84#97—41no—
85#97—42no—
86C4×C8—24no—
87C4×C8—22no—
88C4×C8—42no—
89C4×C8—42no—
90#95—21no—
91#95—22no—
92#95—22no—
93#95C4212yes—
94C24—21no—
95C2×Q8—24no—
96C2×Q8—41no—
97C2×D4—41no—
98C2×D4—81no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4C2×C4211yescharacteristic
106C22×C4C2×C22⋊C411yescharacteristic
107C22×C4#11711yescharacteristic
108C22×C4C4×D412yes—
109C22×C4C4×D412yes—
110M16—44no—
111M16—44no—
112M16—84no—
113C2×C8—22no—
114C2×C8—21no—
115C2×C8—41no—
116C2×C8—42no—
117C2×C8—41no—
118C2×C8—44no—
119C2×C8—44no—
120C2×C8—41no—
121C2×C8—42no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8#9912yes—
126C4⋊C4—42no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C22⋊C4—42no—
130C22⋊C4—44no—
131C22⋊C4—82no—
132C42—22no—
133C42—21no—
134C42—22no—
135C42—21no—
136C42—41no—
137C42—41no—
138C23—41no—
139C23—42no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23C4×C22⋊C411yescharacteristic
144Q8—44no—
145D4—82no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—82no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4C2×#9911yescharacteristic
165C2×C4#41011yescharacteristic
166C2×C4#44414yes—
167C8—22no—
168C8—42no—
169C8—41no—
170C8—81no—
171C8—84no—
172C8—84no—
173C8—81no—
174C22—21no—
175C22—21no—
176C22—82no—
177C22—82no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22C4×#9911yescharacteristic
182C22#176512yes—
183C4—24no—
184C4—21no—
185C4—41no—
186C4—42no—
187C4—41no—
188C4—82no—
189C4—81no—
190C4#213311yescharacteristic
191C4#213411yescharacteristic
192C2—22no—
193C2—161no—
194C2#823911yescharacteristic
195C2#1262411yescharacteristic
196C2#1263211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C4×#99
prime radicalC22×D4C42
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8165C2semidirect
4. #7286C2semidirect
5. #7284C2semidirect

Maximal subgroups

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

subgroupindex
#222112
#117342
#81652
#72862
#72842