Finite Group Databasebeta

← #134817

Subgroups — #134817

Order 512.

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

normal subgroups only

This group has 327 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#11530C212yesmaximal
3#8623C212yesmaximal, complemented
4#7248C212yesmaximal, complemented
5C2×C8×Q8C2211yescharacteristic
6C2×C8×D4C411yescharacteristic
7C4×#124C2211yescharacteristic
8#1984C2211yescharacteristic
9#1954C411yescharacteristic
10#1923—22no—
11#1850—22no—
12#1842C411yescharacteristic
13#1744C411yescharacteristic
14#1568—22no—
15#1467C2214yes—
16#1324—22no—
17#1318—22no—
18#1290—22no—
19C23×C8—21no—
20C2×#124Q811yescharacteristic
21C2×C4×Q8C2311yescharacteristic
22C2×C4×D4C2×C411yescharacteristic
23C8×Q8—42no—
24C8×D4—42no—
25#432—24no—
26C2×C4⋊C8—22no—
27C2×C4⋊C8C2×C411yescharacteristic
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8D412yes—
30C2×C22⋊C8—21no—
31C2×C22⋊C8—22no—
32C2×C4×C8C2×C411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D412yes—
35#391Q811yescharacteristic
36#390D411yescharacteristic
37#388—22no—
38#382D411yescharacteristic
39C4×C22⋊C4—22no—
40C43C2×C411yescharacteristic
41#336—21no—
42#336—21no—
43#336—22no—
44#335—42no—
45#334—42no—
46#324—24no—
47#324—24no—
48C22×Q8C2×Q811yescharacteristic
49C22×D4C4⋊C411yescharacteristic
50C23×C4—21no—
51C22×C8—21no—
52C22×C8—21no—
53C22×C8—42no—
54C22×C8—82no—
55C22×C8C22⋊C411yescharacteristic
56C22×C8C22⋊C412yes—
57C22×C8C22⋊C411yescharacteristic
58#124—22no—
59#124—22no—
60C4×Q8—41no—
61C4×Q8—41no—
62C4×D4—41no—
63C4×D4—41no—
64C2×C4⋊C4C4⋊C411yescharacteristic
65C2×C4⋊C4C2×D411yescharacteristic
66C2×C4⋊C4#4112yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42C4⋊C411yescharacteristic
73C2×C42C4⋊C411yescharacteristic
74C2×C42C22×C411yescharacteristic
75C2×C42C2×D412yes—
76C4⋊C8—41no—
77C4⋊C8—42no—
78C4⋊C8—41no—
79C4⋊C8—44no—
80C4⋊C8—82no—
81C22⋊C8—42no—
82C22⋊C8—44no—
83C22⋊C8—82no—
84#97—44no—
85C4×C8—22no—
86C4×C8—22no—
87C4×C8—22no—
88C4×C8—41no—
89#95—22no—
90#95—22no—
91C24—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—41no—
95C2×Q8—41no—
96C2×D4—21no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4#10812yes—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4C2×C22⋊C411yescharacteristic
108C22×C4C2×C4⋊C411yescharacteristic
109C22×C4#11711yescharacteristic
110C22×C4#12112yes—
111C22×C4C22⋊Q812yes—
112C2×C8—21no—
113C2×C8—22no—
114C2×C8—41no—
115C2×C8—42no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—42no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—84no—
123C2×C8—84no—
124C2×C8#9912yes—
125C4⋊C4—41no—
126C4⋊C4—42no—
127C4⋊C4—41no—
128C22⋊C4—28no—
129C22⋊C4—42no—
130C22⋊C4—82no—
131C42—22no—
132C42—21no—
133C42—21no—
134C42—22no—
135C42—21no—
136C42—41no—
137C42—41no—
138C42—41no—
139C42—81no—
140C42#10812yes—
141C23—22no—
142C23—42no—
143C23—42no—
144C23—41no—
145C23—41no—
146C23#38011yescharacteristic
147Q8—42no—
148Q8—42no—
149D4—42no—
150D4—42no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—82no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4C2×#9911yescharacteristic
174C2×C4#41011yescharacteristic
175C2×C4C2×#10811yescharacteristic
176C2×C4C2×#10811yescharacteristic
177C8—22no—
178C8—42no—
179C8—41no—
180C8—81no—
181C8—81no—
182C8—164no—
183C22—21no—
184C22—21no—
185C22—44no—
186C22—42no—
187C22—82no—
188C22—81no—
189C22—81no—
190C22#179911yescharacteristic
191C22#181011yescharacteristic
192C22#181311yescharacteristic
193C4—21no—
194C4—21no—
195C4—21no—
196C4—42no—
197C4—42no—
198C4—41no—
199C4—42no—
200C4—81no—
201C4—81no—
202C4#213311yescharacteristic
203C4#213411yescharacteristic
204C2—21no—
205C2—21no—
206C2—82no—
207C2#861211yescharacteristic
208C2#1280311yescharacteristic
209C2#1280811yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ 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. #8623C2semidirect
4. #7248C2semidirect

Maximal subgroups

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

subgroupindex
#222082
#115302
#86232
#72482